"""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 the core of the IAPWS-95 and IAPWS-97 standards.
The objective of this module is to contain extremely fast functions to
calculate several basic properties of water.
The simplest interfaces are :obj:`iapws95_rho` for density calculation only and
:obj:`iapws95_properties` for some basic properties.
For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/chemicals/>`_.
.. contents:: :local:
IAPWS-95 Basic Solvers
------------------------
.. autofunction:: chemicals.iapws.iapws95_rho
.. autofunction:: chemicals.iapws.iapws95_P
.. autofunction:: chemicals.iapws.iapws95_T
IAPWS-97 Basic Solvers
------------------------
.. autofunction:: chemicals.iapws.iapws97_rho
.. autofunction:: chemicals.iapws.iapws97_P
.. autofunction:: chemicals.iapws.iapws97_T
IAPWS-95 Properties
------------------------
.. autofunction:: chemicals.iapws.iapws95_properties
IAPWS Saturation Pressure/Temperature
----------------------------------------
.. autofunction:: chemicals.iapws.iapws95_Psat
.. autofunction:: chemicals.iapws.iapws95_dPsat_dT
.. autofunction:: chemicals.iapws.iapws92_Psat
.. autofunction:: chemicals.iapws.iapws92_dPsat_dT
.. autofunction:: chemicals.iapws.iapws95_Tsat
.. autofunction:: chemicals.iapws.iapws95_saturation
.. autofunction:: chemicals.iapws.iapws11_Psub
IAPWS Saturation Density
------------------------
.. autofunction:: chemicals.iapws.iapws95_rhol_sat
.. autofunction:: chemicals.iapws.iapws95_rhog_sat
.. autofunction:: chemicals.iapws.iapws95_drhol_sat_dT
.. autofunction:: chemicals.iapws.iapws92_rhol_sat
.. autofunction:: chemicals.iapws.iapws92_rhog_sat
IAPWS Constants
---------------
.. autodata:: chemicals.iapws.iapws95_Tc
.. autodata:: chemicals.iapws.iapws95_Pc
.. autodata:: chemicals.iapws.iapws95_rhoc
.. autodata:: chemicals.iapws.iapws95_MW
.. autodata:: chemicals.iapws.iapws95_R
.. autodata:: chemicals.iapws.iapws97_R
.. autodata:: chemicals.iapws.iapws95_Tt
IAPWS-97 Region 1
-----------------
.. autofunction:: chemicals.iapws.iapws97_G_region1
.. autofunction:: chemicals.iapws.iapws97_dG_dpi_region1
.. autofunction:: chemicals.iapws.iapws97_d2G_dpi2_region1
.. autofunction:: chemicals.iapws.iapws97_dG_dtau_region1
.. autofunction:: chemicals.iapws.iapws97_d2G_dtau2_region1
.. autofunction:: chemicals.iapws.iapws97_d2G_dpidtau_region1
IAPWS-97 Region 2
-----------------
.. autofunction:: chemicals.iapws.iapws97_G0_region2
.. autofunction:: chemicals.iapws.iapws97_dG0_dtau_region2
.. autofunction:: chemicals.iapws.iapws97_d2G0_dtau2_region2
.. autofunction:: chemicals.iapws.iapws97_Gr_region2
.. autofunction:: chemicals.iapws.iapws97_dGr_dpi_region2
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dpi2_region2
.. autofunction:: chemicals.iapws.iapws97_dGr_dtau_region2
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dtau2_region2
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dpidtau_region2
IAPWS-97 Region 3
-----------------
.. autofunction:: chemicals.iapws.iapws97_A_region3
.. autofunction:: chemicals.iapws.iapws97_dA_ddelta_region3
.. autofunction:: chemicals.iapws.iapws97_d2A_ddelta2_region3
.. autofunction:: chemicals.iapws.iapws97_dA_dtau_region3
.. autofunction:: chemicals.iapws.iapws97_d2A_dtau2_region3
.. autofunction:: chemicals.iapws.iapws97_d2A_ddeltadtau_region3
IAPWS-97 Region 3 PT Backwards Equation Boundaries
--------------------------------------------------
.. autofunction:: chemicals.iapws.iapws97_boundary_3uv
.. autofunction:: chemicals.iapws.iapws97_boundary_3ef
.. autofunction:: chemicals.iapws.iapws97_boundary_3cd
.. autofunction:: chemicals.iapws.iapws97_boundary_3gh
.. autofunction:: chemicals.iapws.iapws97_boundary_3ij
.. autofunction:: chemicals.iapws.iapws97_boundary_3jk
.. autofunction:: chemicals.iapws.iapws97_boundary_3mn
.. autofunction:: chemicals.iapws.iapws97_boundary_3qu
.. autofunction:: chemicals.iapws.iapws97_boundary_3rx
.. autofunction:: chemicals.iapws.iapws97_boundary_3wx
.. autofunction:: chemicals.iapws.iapws97_boundary_3ab
.. autofunction:: chemicals.iapws.iapws97_boundary_3op
IAPWS-97 Region 3 PT Backwards Equations
----------------------------------------
.. autofunction:: chemicals.iapws.iapws97_region3_a
.. autofunction:: chemicals.iapws.iapws97_region3_b
.. autofunction:: chemicals.iapws.iapws97_region3_c
.. autofunction:: chemicals.iapws.iapws97_region3_d
.. autofunction:: chemicals.iapws.iapws97_region3_e
.. autofunction:: chemicals.iapws.iapws97_region3_f
.. autofunction:: chemicals.iapws.iapws97_region3_g
.. autofunction:: chemicals.iapws.iapws97_region3_h
.. autofunction:: chemicals.iapws.iapws97_region3_i
.. autofunction:: chemicals.iapws.iapws97_region3_j
.. autofunction:: chemicals.iapws.iapws97_region3_k
.. autofunction:: chemicals.iapws.iapws97_region3_l
.. autofunction:: chemicals.iapws.iapws97_region3_m
.. autofunction:: chemicals.iapws.iapws97_region3_n
.. autofunction:: chemicals.iapws.iapws97_region3_o
.. autofunction:: chemicals.iapws.iapws97_region3_p
.. autofunction:: chemicals.iapws.iapws97_region3_q
.. autofunction:: chemicals.iapws.iapws97_region3_r
.. autofunction:: chemicals.iapws.iapws97_region3_s
.. autofunction:: chemicals.iapws.iapws97_region3_t
.. autofunction:: chemicals.iapws.iapws97_region3_u
.. autofunction:: chemicals.iapws.iapws97_region3_v
.. autofunction:: chemicals.iapws.iapws97_region3_w
.. autofunction:: chemicals.iapws.iapws97_region3_x
.. autofunction:: chemicals.iapws.iapws97_region3_y
.. autofunction:: chemicals.iapws.iapws97_region3_z
IAPWS-97 Region 5
-----------------
.. autofunction:: chemicals.iapws.iapws97_G0_region5
.. autofunction:: chemicals.iapws.iapws97_dG0_dtau_region5
.. autofunction:: chemicals.iapws.iapws97_d2G0_dtau2_region5
.. autofunction:: chemicals.iapws.iapws97_Gr_region5
.. autofunction:: chemicals.iapws.iapws97_dGr_dpi_region5
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dpi2_region5
.. autofunction:: chemicals.iapws.iapws97_dGr_dtau_region5
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dtau2_region5
.. autofunction:: chemicals.iapws.iapws97_d2Gr_dpidtau_region5
IAPWS-95 Ideal Gas Terms
------------------------
.. autofunction:: chemicals.iapws.iapws95_A0
.. autofunction:: chemicals.iapws.iapws95_dA0_dtau
.. autofunction:: chemicals.iapws.iapws95_d2A0_dtau2
.. autofunction:: chemicals.iapws.iapws95_d3A0_dtau3
.. autofunction:: chemicals.iapws.iapws95_A0_tau_derivatives
IAPWS-95 Residual Terms
-----------------------
.. autofunction:: chemicals.iapws.iapws95_Ar
.. autofunction:: chemicals.iapws.iapws95_dAr_ddelta
.. autofunction:: chemicals.iapws.iapws95_d2Ar_ddelta2
.. autofunction:: chemicals.iapws.iapws95_d3Ar_ddelta3
.. autofunction:: chemicals.iapws.iapws95_dAr_dtau
.. autofunction:: chemicals.iapws.iapws95_d2Ar_dtau2
.. autofunction:: chemicals.iapws.iapws95_d2Ar_ddeltadtau
.. autofunction:: chemicals.iapws.iapws95_d3Ar_ddeltadtau2
.. autofunction:: chemicals.iapws.iapws95_d3Ar_ddelta2dtau
.. autofunction:: chemicals.iapws.iapws95_d4Ar_ddelta2dtau2
"""
from fluids.numerics import broyden2, cbrt, exp, horner, horner_and_der, log, newton, newton_system, solve_2_direct, sqrt, translate_bound_f_jac, trunc_exp
from chemicals.utils import mark_numba_uncacheable
from chemicals.vapor_pressure import Psat_IAPWS, Tsat_IAPWS
__all__ = ['iapws97_boundary_2_3', 'iapws97_boundary_2_3_reverse',
'iapws97_identify_region_TP', 'iapws97_region_3', 'iapws97_region3_rho',
'iapws97_region1_rho', 'iapws97_region2_rho', 'iapws97_region5_rho',
'iapws95_rho', 'iapws95_P', 'iapws95_T', 'iapws97_rho_extrapolated',
'iapws97_rho', 'iapws97_P', 'iapws97_T', 'iapws95_Psat', 'iapws95_dPsat_dT',
'iapws95_Tsat', 'iapws92_rhol_sat', 'iapws92_rhog_sat', 'iapws95_rhol_sat',
'iapws95_rhog_sat', 'iapws95_saturation', 'iapws95_A0', 'iapws95_dA0_dtau',
'iapws95_d2A0_dtau2', 'iapws95_d3A0_dtau3', 'iapws95_A0_tau_derivatives',
'iapws95_Ar', 'iapws95_d3Ar_ddeltadtau2', 'iapws95_d3Ar_ddelta2dtau',
'iapws95_dAr_ddelta', 'iapws95_d2Ar_ddelta2', 'iapws95_d3Ar_ddelta3',
'iapws95_dAr_dtau', 'iapws95_d2Ar_dtau2', 'iapws95_d2Ar_ddeltadtau',
'iapws95_MW', 'iapws95_Pc', 'iapws95_Tc', 'iapws95_rhoc', 'iapws95_R',
'iapws97_R', 'iapws97_G_region1', 'iapws95_drhol_sat_dT',
'iapws97_dG_dpi_region1', 'iapws97_d2G_dpi2_region1',
'iapws97_dG_dtau_region1', 'iapws97_d2G_dtau2_region1',
'iapws97_d2G_dpidtau_region1', 'iapws97_Gr_region2',
'iapws97_dGr_dpi_region2', 'iapws97_d2Gr_dpi2_region2',
'iapws97_dGr_dtau_region2', 'iapws97_d2Gr_dtau2_region2',
'iapws97_d2Gr_dpidtau_region2', 'iapws97_G0_region2',
'iapws97_dG0_dtau_region2', 'iapws97_d2G0_dtau2_region2',
'iapws97_Gr_region5', 'iapws97_dGr_dpi_region5', 'iapws97_d2Gr_dpi2_region5',
'iapws95_d4Ar_ddelta2dtau2',
'iapws97_dGr_dtau_region5', 'iapws97_d2Gr_dtau2_region5',
'iapws97_d2Gr_dpidtau_region5', 'iapws97_G0_region5',
'iapws97_dG0_dtau_region5', 'iapws97_d2G0_dtau2_region5', 'iapws97_A_region3',
'iapws97_dA_ddelta_region3', 'iapws97_d2A_ddelta2_region3',
'iapws97_dA_dtau_region3', 'iapws97_d2A_dtau2_region3',
'iapws97_d2A_ddeltadtau_region3', 'iapws97_boundary_3uv',
'iapws97_boundary_3ef', 'iapws97_boundary_3ef', 'iapws97_boundary_3cd',
'iapws97_boundary_3gh', 'iapws97_boundary_3ij', 'iapws97_boundary_3jk',
'iapws97_boundary_3mn', 'iapws97_boundary_3qu', 'iapws97_boundary_3rx',
'iapws97_boundary_3wx', 'iapws97_boundary_3ab', 'iapws97_boundary_3op',
'iapws97_region3_a', 'iapws97_region3_b', 'iapws97_region3_c',
'iapws97_region3_d', 'iapws97_region3_e', 'iapws97_region3_f',
'iapws97_region3_g', 'iapws97_region3_h', 'iapws97_region3_i',
'iapws97_region3_j', 'iapws97_region3_k', 'iapws97_region3_l',
'iapws97_region3_m', 'iapws97_region3_n', 'iapws97_region3_o',
'iapws97_region3_p', 'iapws97_region3_q', 'iapws97_region3_r',
'iapws97_region3_s', 'iapws97_region3_t', 'iapws97_region3_u',
'iapws97_region3_v', 'iapws97_region3_w', 'iapws97_region3_x',
'iapws97_region3_y', 'iapws97_region3_z',
'iapws95_properties',
'iapws92_Psat', 'iapws92_dPsat_dT',
'iapws11_Psub',
]
iapws95_R = 461.51805
"""Specific gas constant in J/(kg*K) according to IAPWS-95"""
iapws97_R = 461.526
"""Specific gas constant in J/(kg*K) according to IAPWS-97"""
iapws95_MW = 18.015268
"""Molecular weight of water in g/mol according to IAPWS-95, also used in IAPWS-97"""
iapws95_Pc = 22064000.0
"""Critical pressure of water in Pa according to IAPWS-95, also used in IAPWS-97"""
iapws95_Tc = 647.096
"""Critical temperature of water in K according to IAPWS-95, also used in IAPWS-97"""
iapws95_Tc_inv = 1.0 / iapws95_Tc
iapws95_rhoc = 322.0
"""Critical density of water in kg/m^3 according to IAPWS-95, also used in IAPWS-97"""
iapws95_rhoc_inv = 1.0 / iapws95_rhoc
iapws95_R_rhoc_inv2 = iapws95_R*iapws95_rhoc_inv*iapws95_rhoc_inv
iapws95_Tt = 273.16
"""Triple temperature of water in K according to IAPWS"""
def use_mpmath_backend():
import mpmath as mp
globals()['exp'] = mp.exp
globals()['log'] = mp.log
globals()['sqrt'] = mp.sqrt
globals()['iapws95_R'] = mp.mpf("461.51805")
globals()['iapws97_R'] = mp.mpf("461.526")
globals()['iapws95_MW'] = mp.mpf("18.015268")
globals()['iapws95_Tc'] = mp.mpf("647.096")
globals()['Tc_inv'] = 1/mp.mpf("647.096")
globals()['rhoc'] = mp.mpf("322")
globals()['iapws95_rhoc_inv'] = 1/mp.mpf("322")
import fluids.numerics
fluids.numerics.exp = mp.exp
fluids.numerics.log = mp.log
fluids.numerics.trunc_exp = mp.exp
fluids.numerics.trunc_log = mp.log
return mp
def reset_backend():
import math
globals()['exp'] = math.exp
globals()['log'] = math.log
globals()['sqrt'] = math.sqrt
globals()['iapws95_R'] = 461.51805
globals()['iapws97_R'] = 461.526
globals()['iapws95_MW'] = 18.015268
globals()['iapws95_Tc'] = 647.096
globals()['Tc_inv'] = 1/647.096
globals()['rhoc'] = 322.0
globals()['iapws95_rhoc_inv'] = 1/322.0
import fluids.numerics
import fluids.numerics.special
fluids.numerics.exp = math.exp
fluids.numerics.log = math.log
fluids.numerics.trunc_exp = fluids.numerics.special.trunc_exp
fluids.numerics.trunc_log = fluids.numerics.special.trunc_log
def iapws97_boundary_2_3(T):
'''Above this pressure we are in region 3.
>>> iapws97_boundary_2_3(0.623150000E3)
16529164.2526216
'''
return (0.34805185628969E9 - T*(0.11671859879975E7 - 0.10192970039326E4*T))
### IAPWS 07 Region 3 P boundaries
def iapws97_boundary_2_3_reverse(P):
'''
>>> iapws97_boundary_2_3_reverse(16529164.2526216)
623.15000000000
'''
return (116.85603437027114637*sqrt(7.1845068690062562659e-8*P - 1.0)
+ 572.54459862744727161)
[docs]def iapws97_boundary_3uv(P):
'''
>>> iapws97_boundary_3uv(22.3E6)
647.7996121480069
'''
return (P*(P*(2.867916822636969863e-21*P - 2.228141349037550121e-13)
+ 8.905796021353068107e-6) + 528.1996462630620499)
# P = P/1E6
# T = sum([nis3uv[i]*P**Iis3uv[i] for i in range(4)])
# return T
[docs]def iapws97_boundary_3ef(P):
'''
>>> iapws97_boundary_3ef(40E6)
713.959399239744
'''
return 3.7278880039999996e-6*P + 564.843879079744056
# P = P/1E6
# T = 3.727888004*(P-22.064)+Tc
# return T
[docs]def iapws97_boundary_3cd(P):
'''
>>> iapws97_boundary_3cd(25E6)
649.3659208321279
'''
return (P*(P*(1.59090746562728991e-22*P - 1.2728354929587799e-14)
+ 2.78233532206914969e-6) + 585.27696669634895)
# P = P/1E6
# T = sum([nis3cd[i]*P**Iis3cd[i] for i in range(4)])
# return T
[docs]def iapws97_boundary_3gh(P):
'''
>>> iapws97_boundary_3gh(25E6)
656.69805722612
'''
return (P*(P*(P*(7.51608051114156857e-18 - 7.87105249910382769e-26*P)
- 2.69029173140129987e-10) + 0.0042814358479154593) - 24928.4240900418008)
# P = P/1E6
# T = sum([nis3gh[i]*P**Iis3gh[i] for i in range(5)])
# return T
[docs]def iapws97_boundary_3ij(P):
'''
>>> iapws97_boundary_3ij(25E6)
660.7865756716819
'''
return (P*(P*(P*(5.15308185433081825e-29*P - 5.87071076864458977e-21)
+ 2.60763050899561981e-13) - 6.16179320924617007e-7) + 584.814781649163024)
# P = P/1E6
# T = sum([nis3ij[i]*P**Iis3ij[i] for i in range(5)])
# return T
[docs]def iapws97_boundary_3jk(P):
'''
>>> iapws97_boundary_3jk(25E6)
668.1915358826951
'''
return (P*(P*(P*(1.37897492684193974e-28*P - 1.57391839848015003e-20)
+ 6.97072596851896056e-13) - 7.70600270141674947e-6) + 617.229772068439047)
# P = P/1E6
# T = sum([nis3jk[i]*P**Iis3jk[i] for i in range(5)])
# return T
[docs]def iapws97_boundary_3mn(P):
'''
>>> iapws97_boundary_3mn(22.8E6)
649.6054132953997
'''
return (P*(P*(1.92871054508107992e-21*P - 1.58365725441647998e-13)
+ 7.61978122720127966e-6) + 535.339483742384004)
# P = P/1E6
# T = sum([nis3mn[i]*P**Iis3mn[i] for i in range(4)])
# return T
[docs]def iapws97_boundary_3qu(P):
'''
>>> iapws97_boundary_3qu(22E6)
645.6355027340121
'''
return (P*(P*(1.22240301070144985e-21*P - 1.02020639611015996e-13)
+ 5.29062258221221963e-6) + 565.60364823912596)
# P = P/1E6
# T = sum([nis3qu[i]*P**Iis3qu[i] for i in range(4)])
# return T
[docs]def iapws97_boundary_3rx(P):
'''
>>> iapws97_boundary_3rx(22E6)
648.26227536701
'''
return (P*(P*(2.43293362700451993e-13 - 2.94905044740798962e-21*P)
- 1.02961025163668997e-6) + 584.561202520005963)
# P = P/1E6
# T = sum([nis3rx[i]*P**Iis3rx[i] for i in range(4)])
# return T
### IAPWS 07 Region 3 log P boundaries
[docs]def iapws97_boundary_3wx(logP_MPa, logP_MPa_inv):
'''
>>> iapws97_boundary_3wx(log(22.3), 1/log(22.3))
648.204947950734
'''
return (logP_MPa*(14.7370491183190993*logP_MPa + 97.3505869861951965)
+ 7.28052609145380014 + logP_MPa_inv*(329.196213998375015 + 873.371668682416953*logP_MPa_inv))
# P = P/1E6
# T = sum([nis3wx[i]*logP_MPa**Iis3wx[i] for i in range(5)])
# return T
[docs]def iapws97_boundary_3ab(logP_MPa, logP_MPa_inv):
'''
>>> iapws97_boundary_3ab(log(40), 1/log(40))
693.0341408296053
'''
return (logP_MPa*(21.3144632222113017*logP_MPa - 187.661219490113012)
+ 1547.93642129415002 - logP_MPa_inv*(1918.87498864292002 - 918.419702359447001*logP_MPa_inv))
# T = sum([nis3ab[i]*logP_MPa**Iis3ab[i] for i in range(5)])
# return T
[docs]def iapws97_boundary_3op(logP_MPa, logP_MPa_inv):
'''
>>> iapws97_boundary_3op(log(22.8), 1/log(22.8))
650.010694314133
'''
return (logP_MPa*(64.2859598466067013*logP_MPa - 332.500170441277987)
+ 969.461372400213008 + logP_MPa_inv*(773.845935768222034 - 1523.13732937084001*logP_MPa_inv))
# T = sum([nis3op[i]*logP_MPa**Iis3op[i] for i in range(5)])
# return T
region3_boundary_doc = """
Calculates the transition temperature for a region 3 PT backwards equation
transition.
Parameters
----------
P : float
Pressure [Pa]
Returns
-------
T_trans : float
Transition temperature [K]
Examples
--------
"""
region3_logboundary_doc = """
Calculates the transition temperature for a region 3 PT backwards equation
transition (for one of "wx", "ab", or "op"; the others do not use a log fit).
The parameters are provided in the specific units for speed savings only.
Parameters
----------
logP_MPa : float
Natural logarithm of pressure in units of MPa [log(MPa)]
logP_MPa_inv : float
Inverse of Natural logarithm of pressure in units of MPa [1/log(MPa)]
Returns
-------
T_trans : float
Transition temperature [K]
Examples
--------
"""
try:
for func in (iapws97_boundary_3uv, iapws97_boundary_3ef, iapws97_boundary_3cd, iapws97_boundary_3gh,
iapws97_boundary_3ij, iapws97_boundary_3jk, iapws97_boundary_3mn, iapws97_boundary_3qu,
iapws97_boundary_3rx):
func.__doc__ = region3_boundary_doc + func.__doc__
for func in (iapws97_boundary_3wx, iapws97_boundary_3ab, iapws97_boundary_3op):
func.__doc__ = region3_logboundary_doc + func.__doc__
except: # except is needed for running Python under -OO flag
pass
### Region 1
[docs]def iapws97_G_region1(tau, pi):
r'''Calculates the dimensionless Gibbs free energy for water according to
the IAPWS-97 standard (for region 1).
.. math::
\gamma = \sum_{i=1}^{34} I_i(7.1-\pi)^{I_i}(\tau - 1.222)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
G : float
Dimensionless Gibbs energy G/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_G_region1(1386/277.15, 101325/16.53E6)
-0.00016341033954414
'''
pit = 7.1 - pi
taut = tau - 1.222
taut_inv = 1.0/taut
pit2 = pit*pit
pit4 = pit2*pit2
taut_inv2 = pit4*pit4 # abuse taut_inv2 variable as a temporary
pit21 = taut_inv2*taut_inv2*pit4*pit
pit29 = pit21*taut_inv2
taut_inv2 = taut*taut # abuse taut_inv2 variable as a temporary
taut3 = taut_inv2*taut
taut5 = taut_inv2*taut3
taut_inv2 = taut_inv*taut_inv
taut_inv3 = taut_inv2*taut_inv
taut_inv9 = taut_inv3*taut_inv3*taut_inv3
taut_inv29 = taut_inv9*taut_inv9*taut_inv9*taut_inv2
return (pit*(-0.02184171717541399937*taut - 0.01899006821841900047*taut_inv
- 0.0325297487705049973 - 0.00005283835796993000233*taut3
- 0.000607063015658739955*taut_inv3*taut_inv3*taut_inv
+ 0.0002831908012380400042*taut_inv9)
+ taut*(3.385516916838500201
+ 0.00004766139390698700138*pit2
- 0.9579196338787200338*taut)
- 0.8454818716911399745*taut_inv
- 3.756360367204000017
+ 0.1463297121316700089*taut_inv2
+ pit2*(-0.0003000178079302599906
- 4.414184533084599669e-6*taut3 - 0.0004718432107326699771*taut_inv3
- 7.269499629759400146e-16*taut5*taut5*taut5*taut*taut
+ pit*(-2.827079798531199973e-6 - 0.00003167964484505400157*taut_inv3*taut_inv
- 8.520512812010300437e-10*taut5*taut))
+ taut3*(0.157720385132280011 - 0.01661641719950100043*taut)
+ 0.0008121462998356799657*taut5
+ pit4*(taut_inv2*(- 6.517122289560100218e-7 - taut_inv3*(2.242528190799999857e-6
+ 4.051699686011699983e-7*pit*taut_inv3)
- 1.273430174164099942e-9*pit4*taut_inv9)
- 1.434172993792399922e-13*taut5*taut5
- 1.742487123063400057e-10*pit4*taut_inv3*taut_inv3)
+ taut_inv29*(1.44783078285210013e-20*pit21*pit2*taut_inv2
- 6.876213129553099646e-19*pit21
+ pit29*taut_inv9*(2.633578166279499979e-23
- 1.194762264007099993e-23*pit*taut_inv
+ 1.822809458140400033e-24*pit2*taut_inv2
- 9.353708729245799802e-26*pit2*pit*taut_inv3)))
[docs]def iapws97_dG_dpi_region1(tau, pi):
r'''Calculates the derivative of dimensionless Gibbs free energy
with respect to `pi` for water according to the IAPWS-97 standard
(for region 1).
.. math::
\frac{\partial \gamma}{\partial \pi} = \sum_{i=1}^{34}
-n_i I_i(7.1-\pi)^{I_i-1}(\tau - 1.222)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
dG_dpi : float
Derivative of dimensionless Gibbs energy G/(RT) with respect to `pi`,
[-]
Notes
-----
Used in density solution.
This contains a hand-optimized implementation with a single division,
no power operations, 65 multiplications, 16 local variables, and a
minimum number of additions.
Examples
--------
>>> iapws97_dG_dpi_region1(1386/277.15, 101325/16.53E6)
0.1292327182544
'''
pit = 7.1 - pi
taut = tau - 1.222
pit2 = pit*pit
pit3 = pit2*pit
pit7 = pit3*pit3*pit
pit14 = pit7*pit7
taut3 = taut*taut*taut
taut6 = taut3*taut3
taut10 = taut6*taut3*taut
taut_inv = 1.0/taut
taut_inv2 = taut_inv*taut_inv
taut_inv4 = taut_inv2*taut_inv2
taut_inv5 = taut_inv4*taut_inv
taut_inv7 = taut_inv5*taut_inv2
taut_inv17 = taut_inv5*taut_inv5*taut_inv7
pit14taut_inv17 = taut_inv17*pit14
return (pit*(-0.0000953227878139740028*taut + 0.000600035615860519981 + 8.82836906616919934e-6*(taut3)
+ 0.000943686421465339954*(taut_inv2*taut_inv) + 1.45389992595188003e-15*(taut10*taut6*taut))
+ 0.0218417171754139994*taut + 0.0189900682184190005*taut_inv + 0.0325297487705049973
+ pit2*(8.48123939559359992e-6 + 0.0000950389345351620047*(taut_inv4) +
2.55615384360309023e-9*(taut6)) + 2.60684891582404009e-6*(pit3)*(taut_inv2)
+ 8.97011276319999943e-6*(pit3)*(taut_inv5) + 5.73669197516959969e-13*(pit3)*(taut10)
+ 0.0000528383579699300023*(taut3)
+ 1.39398969845072005e-9*(pit7)*(taut_inv5*taut_inv)
+ taut_inv7*(2.02584984300584983e-6*(pit3*pit)*taut_inv
+ 1.01874413933127995e-8*(pit7)*(taut_inv4)
+ 0.000607063015658739955 - 0.000283190801238040004*taut_inv2)
+ pit14taut_inv17*(1.44400475720615078e-17*(pit3*pit3)*(taut_inv7*taut_inv5)
- 3.33001080055983015e-19*pit7*pit*taut_inv7*taut_inv7
+ pit14taut_inv17*(3.58428679202129995e-22*(pit)*(taut_inv5)
- 7.6373766822105502e-22*taut_inv4
- 5.65070932023524029e-23*pit2*taut_inv5*taut_inv
+ 2.99318679335865594e-24*(pit3)*(taut_inv7))))
[docs]def iapws97_d2G_dpi2_region1(tau, pi):
r'''Calculates the second derivative of dimensionless Gibbs free energy
with respect to `pi` for water according to the IAPWS-97 standard
(for region 1).
.. math::
\frac{\partial^2 \gamma}{\partial \pi^2} = \sum_{i=1}^{34}
n_i I_i(I_i-1)(7.1-\pi)^{I_i-2}(\tau - 1.222)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
d2G_dpi2 : float
Second Derivative of dimensionless Gibbs energy G/(RT) with respect to
`pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2G_dpi2_region1(1386/277.15, 101325/16.53E6)
-0.0010570100274769
'''
pit = 7.1 - pi
taut = tau - 1.222
taut_inv = 1.0/taut
pit2 = pit*pit
pit3 = pit2*pit
tmp = pit3*pit3*pit3
pit18 = tmp*tmp
pit27 = pit18*tmp
taut3 = taut*taut*taut
taut7 = taut3*taut3*taut
taut_inv2 = taut_inv*taut_inv
taut_inv3 = taut_inv*taut_inv2
tmp = taut_inv3*taut_inv3 # 6
tmp *= tmp # 12; also temporary variable
taut_inv29 = tmp*tmp*taut_inv2 # 26
taut_inv38 = taut_inv29*tmp # 26 + 12 = 38 end
taut_inv29 *= taut_inv3 # 29 end
# some savings remain below via horner with appropriate testing
return (-0.00001696247879118719984*pit - 0.0001900778690703240094*pit*taut_inv3*taut_inv
- 5.112307687206180469e-9*pit*taut3*taut3 + 0.00009532278781397400275*taut
- 0.0006000356158605199813 - 7.820546747472120262e-6*pit2*taut_inv2
- 0.00002691033828959999829*pit2*taut_inv3*taut_inv2 - 1.721007592550879906e-12*pit2*taut7*taut3
- 8.103399372023399331e-6*pit3*taut_inv3*taut_inv3*taut_inv2 - 8.828369066169199338e-6*taut3
- 0.0009436864214653399542*taut_inv3 - 9.757927889155040732e-9*pit3*pit3*taut_inv3*taut_inv3
- 7.131208975318960007e-8*pit3*pit3*taut_inv3*taut_inv3*taut_inv3*taut_inv2
- 1.453899925951880029e-15*taut7*taut7*taut3 - 2.888009514412301439e-16*pit18*pit*taut_inv29
+ 7.326023761231625652e-18*pit18*pit3*taut_inv29*taut_inv2 + 2.138465471018954057e-20*pit27*taut_inv38
- 1.039443169686176943e-20*pit27*pit*taut_inv38*taut_inv
+ 1.695212796070572203e-21*pit27*pit2*taut_inv38*taut_inv2
- 9.278879059411833807e-23*pit27*pit3*taut_inv38*taut_inv3)
[docs]def iapws97_dG_dtau_region1(tau, pi):
r'''Calculates the derivative of dimensionless Gibbs free energy
with respect to `tau` for water according to the IAPWS-97 standard
(for region 1).
.. math::
\frac{\partial \gamma}{\partial \tau} = \sum_{i=1}^{34}
n_i(7.1-\pi)^{I_i}J_i(\tau - 1.222)^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
dG_dtau : float
Derivative of dimensionless Gibbs energy G/(RT) with respect to `tau`,
[-]
Notes
-----
Examples
--------
>>> iapws97_dG_dtau_region1(1386/277.15, 101325/16.53E6)
0.026440334282967
'''
pit = 7.1 - pi
taut = tau - 1.222
taut_inv = 1.0/taut
pit2 = pit*pit
pit4 = pit2*pit2
tmp = pit4*pit4 # 8
pit21 = tmp*tmp*pit4*pit # 20
pit29 = pit21*tmp
taut2 = taut*taut
taut4 = taut2*taut2
taut_inv2 = taut_inv*taut_inv
taut_inv3 = taut_inv2*taut_inv
tmp = taut_inv3*taut_inv3 # 6
taut_inv9 = tmp*taut_inv3
taut_inv30 = taut_inv9*tmp # 15
taut_inv30 *= taut_inv30
return (-0.02184171717541399937*pit - 0.0001585150739097900138*pit*taut2 +
0.01899006821841900047*pit*taut_inv2 + 0.004249441109611180011*pit*taut_inv3*taut_inv3*taut_inv2
- 0.002548717211142359929*pit*taut_inv9*taut_inv - 1.915839267757440068*taut
+ 3.385516916838500201 + 0.00004766139390698700138*pit2 - 0.00001324255359925379985*pit2*taut2
+ 0.001415529632198009877*pit2*taut_inv3*taut_inv
- 1.235814937059098064e-14*pit2*taut4*taut4*taut4*taut4 + 0.4731611553968400052*taut2
+ 0.8454818716911399745*taut_inv2 - 5.112307687206180469e-9*pit2*pit*taut4*taut
+ 0.0001267185793802160063*pit2*pit*taut_inv3*taut_inv2 - 0.06646566879800400174*taut2*taut
- 0.2926594242633400178*taut_inv3 + 1.303424457912020044e-6*pit4*taut_inv3
+ 0.00001121264095399999929*pit4*taut_inv3*taut_inv3 - 1.434172993792399922e-12*pit4*taut4*taut4*taut
+ 0.00406073149917839972*taut4 + 3.241359748809359987e-6*pit4*pit*taut_inv9
+ 1.045492273838040137e-9*pit4*pit4*taut_inv3*taut_inv3*taut_inv
+ 1.400773191580509978e-8*pit4*pit4*taut_inv9*taut_inv3 + 1.99410180757039883e-17*pit21*taut_inv30
- 4.488275426841510012e-19*pit21*pit2*taut_inv30*taut_inv2
- 1.000759703186210033e-21*pit29*taut_inv30*taut_inv9
+ 4.659572829627690075e-22*pit29*pit*taut_inv30*taut_inv9*taut_inv
- 7.291237832561599838e-23*pit29*pit2*taut_inv30*taut_inv9*taut_inv2
+ 3.835020578990777884e-24*pit29*pit2*pit*taut_inv30*taut_inv9*taut_inv3)
[docs]def iapws97_d2G_dtau2_region1(tau, pi):
r'''Calculates the second derivative of dimensionless Gibbs free energy
with respect to `tau` for water according to the IAPWS-97 standard
(for region 1).
.. math::
\frac{\partial^2 \gamma}{\partial \tau^2} = \sum_{i=1}^{34}
n_i(7.1-\pi)^{I_i}J_i(J_i-1)(\tau - 1.222)^{J_i-2}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
d2G_dtau2 : float
Second Derivative of dimensionless Gibbs energy G/(RT) with respect to
`tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2G_dtau2_region1(1386/277.15, 101325/16.53E6)
-0.3645169808573
'''
pit = 7.1 - pi
taut = tau - 1.222
taut_inv = 1.0/taut
pit2 = pit*pit
pit4 = pit2*pit2
taut_inv2 = pit4*pit4 # abuse taut_inv2 variable as a temporary
pit21 = taut_inv2*taut_inv2*pit4*pit
pit29 = pit21*taut_inv2
taut2 = taut*taut
taut4 = taut2*taut2
taut_inv2 = taut_inv*taut_inv
taut_inv4 = taut_inv2*taut_inv2
taut_inv9 = taut_inv4*taut_inv4*taut_inv
taut_inv31 = taut_inv9*taut_inv9*taut_inv9*taut_inv4
return (-0.0003170301478195800275*pit*taut - 0.03798013643683800095*pit*taut_inv2*taut_inv
- 0.03399552887688944008*pit*taut_inv9 + 0.02548717211142359843*pit*taut_inv9*taut_inv2
+ 0.9463223107936800105*taut - 0.00002648510719850759971*taut*pit2 - 1.915839267757440068
- 0.005662118528792039508*pit2*taut_inv4*taut_inv
- 1.977303899294556903e-13*pit2*taut4*taut4*taut4*taut2*taut - 0.1993970063940120052*taut2
- 2.556153843603090152e-8*pit2*pit*taut4 - 0.0006335928969010799772*pit2*pit*taut_inv4*taut_inv2
+ 0.01624292599671359888*taut2*taut - 1.690963743382279949*taut_inv2*taut_inv
- 3.910273373736060131e-6*pit4*taut_inv4 - 0.00006727584572399999572*pit4*taut_inv4*taut_inv2*taut_inv
- 1.29075569441316001e-11*pit4*taut4*taut4 + 0.8779782727900200534*taut_inv4
- 0.0000291722377392842403*pit4*pit*taut_inv9*taut_inv - 7.318445916866280962e-9*pit4*pit4*taut_inv4*taut_inv4
- 1.680927829896611973e-7*pit4*pit4*taut_inv9*taut_inv4 - 5.982305422711196613e-16*pit21*taut_inv31
+ 1.436248136589283204e-17*pit21*pit2*taut_inv31*taut_inv2 + 3.902962842426219073e-20*pit29*taut_inv31*taut_inv9
- 1.863829131851076105e-20*pit29*pit*taut_inv31*taut_inv9*taut_inv
+ 2.989407511350256051e-21*pit29*pit2*taut_inv31*taut_inv9*taut_inv2
- 1.610708643176126667e-22*pit29*pit2*pit*taut_inv31*taut_inv9*taut_inv2*taut_inv)
[docs]def iapws97_d2G_dpidtau_region1(tau, pi):
r'''Calculates the second derivative of dimensionless Gibbs free energy
with respect to `tau` and `pi` for water according to the IAPWS-97 standard
(for region 1).
.. math::
\frac{\partial^2 \gamma}{\partial \tau \partial \pi} = \sum_{i=1}^{34}
-n_iI_i(7.1-\pi)^{I_i}J_i(\tau - 1.222)^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (1386 K)/T [-]
pi : float
Dimensionless pressure, P/(16.53 MPa), [-]
Returns
-------
d2G_dpidtau : float
Second Derivative of dimensionless Gibbs energy G/(RT) with respect to
`tau` and `pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2G_dpidtau_region1(1386/277.15, 101325/16.53E6)
0.025837659858819
'''
pit = 7.1 - pi
taut = tau - 1.222
taut_inv = 1.0/taut
pit2 = pit*pit
pit3 = pit2*pit
pit7 = pit3*pit3*pit
pit28 = pit7*pit7
pit28 *= pit28
taut2 = taut*taut
taut8 = taut2*taut2
taut8 *= taut8
taut_inv2 = taut_inv*taut_inv
taut_inv3 = taut_inv2*taut_inv
taut_inv30 = taut_inv3*taut_inv3 # 6 temporarily
taut_inv9 = taut_inv30*taut_inv3
taut_inv30 = taut_inv9*taut_inv30
taut_inv30 *= taut_inv30 # 30 finally
return (-0.00009532278781397400275*pit + 0.00002648510719850759971*pit*taut2
- 0.002831059264396019754*pit*taut_inv3*taut_inv + 2.471629874118196128e-14*pit*taut8*taut8
+ 0.02184171717541399937 + 1.533692306161854223e-8*pit2*taut2*taut2*taut
- 0.0003801557381406480188*pit2*taut_inv3*taut_inv2 + 0.0001585150739097900138*taut2
- 0.01899006821841900047*taut_inv2 - 5.213697831648080174e-6*pit3*taut_inv3
- 0.00004485056381599999715*pit3*taut_inv3*taut_inv3 + 5.736691975169599686e-12*pit3*taut8*taut
- 0.00001620679874404679866*pit3*pit*taut_inv9 - 8.3639381907043211e-9*pit7*taut_inv3*taut_inv3*taut_inv
- 1.120618553264407982e-7*pit7*taut_inv9*taut_inv3 - 0.004249441109611180011*taut_inv3*taut_inv3*taut_inv2
+ 0.002548717211142359929*taut_inv9*taut_inv - 4.187613795897837728e-16*pit7*pit7*pit3*pit3*taut_inv30
+ 1.03230334817354737e-17*pit7*pit7*pit7*pit*taut_inv30*taut_inv2
+ 2.902203139240009378e-20*pit28*taut_inv30*taut_inv9
- 1.397871848888307079e-20*pit28*pit*taut_inv30*taut_inv9*taut_inv
+ 2.26028372809409602e-21*pit28*pit2*taut_inv30*taut_inv9*taut_inv2
- 1.227206585277048923e-22*pit28*pit3*taut_inv30*taut_inv9*taut_inv3)
### Region 2
[docs]def iapws97_G0_region2(tau, pi):
r'''Calculates the dimensionless ideal gas Gibbs free energy for water
according to the IAPWS-97 standard (for region 2).
.. math::
\gamma^\circ = \ln \pi + \sum_{i=1}^9 n_i^\circ \tau^{J_i^\circ}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
G0 : float
Dimensionless ideal gas Gibbs energy G0/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_G0_region2(540/300.0, 101325/1e6)
3.3180953922351
'''
tau_inv = 1.0/tau
return (tau*(tau*(0.0212684637533070015*tau - 0.284086324607719987)
+ 10.0866559680180004) + tau_inv*(tau_inv*(tau_inv*(tau_inv*(0.0714527380814549973
- 0.00560879112830200022*tau_inv) - 0.407104982239279989) + 1.42408191714439991)
- 4.38395113194500041) + log(pi) - 9.69276865002169963)
[docs]def iapws97_dG0_dtau_region2(tau, pi):
r'''Calculates the first derivative of dimensionless ideal gas Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial \gamma^\circ}{\partial \tau} =\sum_{i=1}^9 n_i^\circ
J_i^\circ\tau^{J_i^\circ-1}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dG0_dtau : float
First derivative of dimensionless ideal gas Gibbs energy G0/(RT)
with respect to `tau`, [-]
Notes
-----
This function does not depend on `pi` but it is accepted for consistency.
Examples
--------
>>> iapws97_dG0_dtau_region2(540/300.0, 101325/1e6)
10.2374188173906
'''
tau_inv = 1.0/tau
return (tau*(0.0638053912599210044*tau - 0.568172649215439973)
+ tau_inv*tau_inv*(tau_inv*(tau_inv*(tau_inv*(0.0280439556415100003*tau_inv
- 0.285810952325819989) + 1.22131494671783991) - 2.84816383428879982)
+ 4.38395113194500041) + 10.0866559680180004)
[docs]def iapws97_d2G0_dtau2_region2(tau, pi):
r'''Calculates the second derivative of dimensionless ideal gas Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial^2 \gamma^\circ}{\partial \tau^2} =\sum_{i=1}^9 n_i^\circ
J_i^\circ( J_i^\circ-1)\tau^{J_i^\circ-2}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2G0_dtau2 : float
Second derivative of dimensionless ideal gas Gibbs energy G0/(RT)
with respect to `tau`, [-]
Notes
-----
This function does not depend on `pi` but it is accepted for consistency.
Examples
--------
>>> iapws97_d2G0_dtau2_region2(540/300.0, 101325/1e6)
-1.2472096479372
'''
tau_inv = 1.0/tau
return (0.127610782519842009*tau + tau_inv*tau_inv*tau_inv*(tau_inv*(tau_inv*(
tau_inv*(1.42905476162910006 - 0.168263733849060015*tau_inv)
- 4.88525978687135964) + 8.54449150286639991)
- 8.76790226389000082) - 0.568172649215439973)
[docs]def iapws97_Gr_region2(tau, pi):
r'''Calculates the dimensionless residual Gibbs free energy for water
according to the IAPWS-97 standard (for region 2).
.. math::
\gamma^r = \sum_{i=1}^{43} n_i \pi^{I_i} (\tau - 0.5)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
Gr : float
Dimensionless residual Gibbs energy Gr/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_Gr_region2(540/300.0, 101325/1e6)
-0.71851548053980
'''
taut = tau - 0.5
pi2 = pi*pi
pi3 = pi2*pi
pi4 = pi2*pi2
pi8 = pi4*pi4
pi20 = pi8*pi8*pi4
taut2 = taut*taut
taut3 = taut2*taut
taut4 = taut2*taut2
taut10 = taut4*taut4*taut2
taut25 = taut10*taut10*taut2*taut3
return (-0.01783486229235799886*pi*taut - 0.001773174247321299916*pi
- 0.04599601369636500264*pi*taut2 - 0.05758125908343200011*pi*taut3
- 0.05032527872793000207*pi*taut4*taut2 - 0.00003303264167020299999*taut*pi2
+ 4.387066728443500103e-7*taut*pi3 - 7.884730955936700091e-10*taut*pi4
- 0.000189489875163149999*pi2*taut2 - 0.003939277724335500143*pi2*taut4
- 0.04379729565057299823*pi2*taut4*taut3 - 0.00002667454791408700141*pi2*taut25*taut10*taut
+ 2.048173769230899887e-8*pi3 - 0.0000322776772385700023*pi3*taut3
- 0.001503392454214799983*pi3*taut4*taut2 - 0.04066825356264899827*pi3*taut25*taut10
+ 1.279071785228500082e-8*pi4*taut2 + 4.82253727185070016e-7*pi4*taut3
+ 2.292207633766100113e-6*pi4*pi*taut4*taut3 - 1.671476645106100115e-11*pi4*pi2*taut3
- 0.002117147232135499837*pi4*pi2*taut10*taut4*taut2 - 23.89574193410399872*pi4*pi2*taut25*taut10
- 5.905956432427000368e-18*pi4*pi3 - 1.262180889910100042e-6*pi4*pi3*taut10*taut
- 0.03894684243573900279*pi4*pi3*taut25 + 1.125621136045899983e-11*pi8*taut4*taut4
- 8.231134089799800435*pi8*taut25*taut10*taut + 1.980971280208800021e-8*pi8*pi*taut10*taut3
+ 1.040696521017399955e-19*pi8*pi2*taut4 - 1.023474709592900015e-13*pi8*pi2*taut10
- 1.001817937951099974e-9*pi8*pi2*taut10*taut4 - 8.088290864698499771e-11*pi8*pi8*taut25*taut4
+ 0.1069303187940899985*pi8*pi8*taut25*taut25
- 0.3366225057417099875*pi8*pi8*pi2*taut25*taut25*taut4*taut3
+ 8.918584535542099871e-25*pi20*taut10*taut10 + 3.062931687623199748e-13*pi20*taut25*taut10
- 4.200246769820800092e-6*pi20*taut25*taut10*taut10*taut3
- 5.905602968563900274e-26*pi20*pi*taut10*taut10*taut + 3.782694761345700151e-6*pi20*pi2*taut25*taut25*taut3
- 1.276860893468100005e-15*pi20*pi3*taut25*taut10*taut4 + 7.308761059506100019e-29*pi20*pi4*taut25*taut
+ 5.541471535077800115e-17*pi20*pi4*taut25*taut10*taut4*taut
- 9.436970724120999844e-7*pi20*pi4*taut25*taut25*taut4*taut4)
[docs]def iapws97_dGr_dpi_region2(tau, pi):
r'''Calculates the first derivative of dimensionless residual Gibbs free
energy with respect to `pi` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial \gamma^r}{\partial \pi} = \sum_{i=1}^{43} n_i I_i
\pi^{I_i-1} (\tau - 0.5)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dGr_dpi : float
Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect
to `pi`, [-]
Notes
-----
Used in density solution.
Examples
--------
>>> iapws97_dGr_dpi_region2(540/300.0, 101325/1e6)
-27.7714056629532
'''
taut = tau - 0.5
pi2 = pi*pi
pi3 = pi*pi2
pi4 = pi*pi3
pi5 = pi2*pi3
pi6 = pi3*pi3
pi7 = pi6*pi
pi9 = pi7*pi2
pi15 = pi9*pi6
pi19 = pi9*pi9*pi
pi23 = pi9*pi9*pi5
taut2 = taut*taut
taut3 = taut*taut2
taut4 = taut2*taut2
taut6 = taut3*taut3
taut7 = taut*taut6
taut10 = taut4*taut6
taut20 = taut10*taut10
taut35 = taut20*taut10*taut3*taut2
taut36 = taut*taut35
taut50 = taut35*taut10*taut2*taut3
return (pi19*(-2.93678005497663003e-14*pi3*taut35*taut4 + 0.0000832192847496054092*pi2*taut36*taut10*taut7
- 1.24017662339841913e-24*pi*taut10*taut10*taut) - 6.05920510335077989*pi15*pi2*taut50*taut7
+ pi4*taut7*(1.78287415218792009e-7*pi4*taut6 + 0.0000114610381688305001)
- 0.000066065283340406*pi*taut - 0.000378979750326299998*pi*taut2
- 0.00787855544867100029*pi*taut4 - 0.0875945913011459965*pi*taut7
- 0.0000533490958281740028*pi*taut36
+ taut10*(taut20*taut10*(taut10*(1.71088510070543998*pi15
- 0.0000226487297378903988*taut4*taut4*pi23)
- 0.0000840049353964159951*taut4*taut4*pi19)
# Cannot factor out taut here due to numerical issues
+ taut10*(1.32995316841867198e-15*taut20*pi23 - 1.29412653835175996e-9*taut7*taut2*pi15
+ 1.78371690710842005e-23*pi19 + 1.75410265428146418e-27*taut6*pi23
- 0.27262789705017304*taut4*taut*pi6)
- 0.012702883392812999*taut6*pi5
- 1.00181793795109993e-8*taut4*pi9 - 8.83526622937069987e-6*taut*pi6
- 1.02347470959289996e-12*pi9) + 9.00496908836719986e-11*taut4*taut4*pi7
+ 1.31612001853305008e-6*taut*pi2 - 3.15389238237468004e-9*taut*pi3
- 0.0178348622923579989*taut - 0.0000968330317157100001*pi2*taut3
- 0.00451017736264439952*pi2*taut6 - 0.122004760687946995*pi2*taut35
+ 6.14452130769269999e-8*pi2 - 4.13416950269890026e-17*pi6
- 1.00288598706366e-10*pi5*taut3 - 143.374451604623999*pi5*taut35
- 65.8490727183984035*pi7*taut36 + 1.04069652101739995e-18*pi9*taut4
+ 6.1258633752463995e-12*pi19*taut35 + 5.11628714091400033e-8*taut2*pi3
- 0.0459960136963650026*taut2 + 1.92901490874028006e-6*taut3*pi3
- 0.0575812590834320001*taut3 - 0.0503252787279300021*taut6 - 0.00177317424732129992)
[docs]def iapws97_d2Gr_dpi2_region2(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `pi` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial^2 \gamma^r}{\partial \pi^2} = \sum_{i=1}^{43} n_i I_i
(I_i-1)\pi^{I_i-2} (\tau - 0.5)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dpi2 : float
Second Derivative of dimensionless residual Gibbs energy Gr/(RT) with
respect to `pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dpi2_region2(540/300.0, 101325/1e6)
-983.15187604898
'''
taut = tau - 0.5
pi2 = pi*pi
pi4 = pi2*pi2
pi8 = pi4*pi4
pi18 = pi2*pi8*pi8
taut2 = taut*taut
taut3 = taut2*taut
taut4 = taut2*taut2
taut10 = taut4*taut4*taut2
taut25 = taut10*taut10*taut4*taut
return (2.632240037066100168e-6*pi*taut + 1.228904261538539999e-7*pi - 0.0001936660634314200003*pi*taut3
- 0.00902035472528879903*pi*taut4*taut2 - 0.2440095213758939896*pi*taut25*taut10
- 0.00006606528334040599997*taut - 9.461677147124039696e-9*taut*pi2 + 1.534886142274200165e-7*pi2*taut2
+ 5.787044726220840615e-6*pi2*taut3 - 0.0003789797503262999981*taut2
+ 0.00004584415267532200057*pi2*pi*taut4*taut3 - 5.014429935318299763e-10*pi4*taut3
- 0.06351441696406499859*pi4*taut10*taut4*taut2 - 716.8722580231200254*pi4*taut25*taut10
- 0.007878555448671000286*taut4 - 2.480501701619340401e-16*pi4*pi
- 0.00005301159737622419582*pi4*pi*taut10*taut - 1.635767382301038353*pi4*pi*taut25
+ 6.303478361857040293e-10*pi4*pi2*taut4*taut4 - 460.9435090287888102*pi4*pi2*taut25*taut10*taut
+ 1.426299321750336068e-6*pi4*pi2*pi*taut10*taut3 - 0.08759459130114599645*taut4*taut3
+ 9.366268689156599206e-18*pi8*taut4 - 9.211272386336099881e-12*pi8*taut10
- 9.016361441559899391e-8*pi8*taut10*taut4 - 1.941189807527639966e-8*pi8*pi4*pi2*taut25*taut4
+ 25.66327651058159987*pi8*pi4*pi2*taut25*taut25 - 103.0064867569632554*pi8*pi8*taut25*taut25*taut4*taut3
+ 3.389062123505998002e-22*pi18*taut10*taut10 + 1.16391404129681584e-10*pi18*taut25*taut10
- 0.001596093772531903933*pi18*taut25*taut10*taut10*taut3 - 2.4803532467968384e-23*pi18*pi*taut10*taut10*taut
+ 0.001747604979741713581*pi18*pi2*taut25*taut25*taut3
- 6.460916120948586575e-13*pi18*pi2*pi*taut25*taut10*taut4 + 4.034436104847367623e-26*pi18*pi4*taut25*taut
+ 3.058892287362945313e-14*pi18*pi4*taut25*taut10*taut4*taut
- 0.0005209207839714791177*pi18*pi4*taut25*taut25*taut4*taut4 - 0.00005334909582817400282*taut25*taut10*taut)
[docs]def iapws97_dGr_dtau_region2(tau, pi):
r'''Calculates the first derivative of dimensionless residual Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial \gamma^r}{\partial \tau} = \sum_{i=1}^{43} n_i
\pi^{I_i} J_i (\tau - 0.5)^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dGr_dtau : float
Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect
to `tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_dGr_dtau_region2(540/300.0, 101325/1e6)
-18.1535856049444
'''
taut = tau - 0.5
pi2 = pi*pi
pi3 = pi2*pi
pi4 = pi2*pi2
pi8 = pi4*pi4
pi20 = pi8*pi8*pi4
taut2 = taut*taut
taut4 = taut2*taut2
taut5 = taut4*taut
taut13 = taut4*taut4*taut5 # 13
taut34 = taut13*taut4 # 9 + 8 = 17
taut34 *= taut34 # 34 end
return (-0.09199202739273000529*pi*taut - 0.01783486229235799886*pi - 0.1727437772502959934*pi*taut2 - 0.3019516723675800263*pi*taut5 - 0.0003789797503262999981*taut*pi2 + 2.558143570457000165e-8*taut*pi4 - 0.00003303264167020299999*pi2 - 0.01575711089734200057*pi2*taut2*taut - 0.3065810695540109876*pi2*taut5*taut - 0.0009602837249071320778*pi2*taut34*taut + 4.387066728443500103e-7*pi3 - 0.00009683303171571000013*pi3*taut2 - 0.00902035472528879903*pi3*taut5 - 1.423388874692715023*pi3*taut34 - 7.884730955936700091e-10*pi4 + 1.446761181555210154e-6*pi4*taut2 + 0.00001604545343636269952*pi4*pi*taut5*taut - 5.014429935318300022e-11*pi4*pi2*taut2 - 0.0338743557141679974*pi4*pi2*taut13*taut2 - 836.3509676936399728*pi4*pi2*taut34 - 0.00001388398978901110004*pi4*pi3*taut5*taut5 - 0.9736710608934751043*pi4*pi3*taut13*taut5*taut5*taut + 9.004969088367199864e-11*pi8*taut5*taut2 - 296.3208272327927943*pi8*taut34*taut + 2.575262664271440094e-7*pi8*pi*taut5*taut5*taut2 + 4.162786084069599818e-19*pi8*pi2*taut2*taut - 1.023474709592899964e-12*pi8*pi2*taut5*taut2*taut2 - 1.402545113131540004e-8*pi8*pi2*taut13 - 2.345604350762564972e-9*pi8*pi8*taut13*taut13*taut2 + 5.34651593970450012*pi8*pi8*taut34*taut13*taut2 - 19.18748282727747068*pi8*pi8*pi2*taut34*taut13*taut5*taut2*taut2 + 1.783716907108420048e-23*pi20*taut13*taut5*taut + 1.072026090668119912e-11*pi20*taut34 - 0.0002016118449513984044*pi20*taut34*taut13 - 1.240176623398419126e-24*pi20*pi*taut13*taut5*taut2 + 0.0002004828223513221118*pi20*pi2*taut34*taut13*taut5 - 4.979757484525589725e-14*pi20*pi3*taut34*taut2*taut2 + 1.90027787547158596e-27*pi20*pi4*taut13*taut5*taut5*taut2 + 2.216588614031120095e-15*pi20*pi4*taut34*taut5 - 0.00005473443019990179931*pi20*pi4*taut34*taut13*taut5*taut5)
[docs]def iapws97_d2Gr_dtau2_region2(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial^2 \gamma^r}{\partial \tau^2} = \sum_{i=1}^{43} n_i
\pi^{I_i} J_i (J_i-1) (\tau - 0.5)^{J_i-2}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dtau2 : float
Second derivative of dimensionless residual Gibbs energy Gr/(RT) with
respect to `tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dtau2_region2(540/300.0, 101325/1e6)
-470.9302933324787
'''
taut = tau - 0.5
pi2 = pi*pi
pi3 = pi2*pi
pi4 = pi2*pi2
pi8 = pi4*pi4
pi20 = pi8*pi8*pi4
taut2 = taut*taut
taut4 = taut2*taut2
taut5 = taut4*taut
taut18 = taut4*taut4
taut18 *= taut18 # 16
taut33 = taut18*taut18*taut #32
taut18 *= taut2 # 18
return (-0.3454875545005919868*pi*taut - 0.09199202739273000529*pi - 1.509758361837900242*pi*taut4 - 0.0001936660634314200003*taut*pi3 + 2.893522363110420308e-6*taut*pi4 - 1.002885987063660004e-10*taut*pi4*pi2 - 0.0003789797503262999981*pi2 - 0.04727133269202600518*pi2*taut2 - 1.839486417324065926*pi2*taut5 - 0.03360993037174962034*pi2*taut33*taut - 0.04510177362644399168*pi3*taut4 - 48.3952217395523121*pi3*taut33 + 2.558143570457000165e-8*pi4 + 0.00009627272061817619036*pi4*pi*taut5 - 0.5081153357125199888*pi4*pi2*taut5*taut5*taut4 - 28435.93290158375748*pi4*pi2*taut33 - 0.0001388398978901110071*pi4*pi3*taut5*taut4 - 23.3681054614434025*pi4*pi3*taut18*taut5 + 6.303478361857040293e-10*pi8*taut5*taut - 10371.22895314774723*pi8*taut33*taut + 3.090315197125728325e-6*pi8*pi*taut5*taut5*taut + 1.248835825220879946e-18*pi8*pi2*taut2 - 9.211272386336099881e-12*pi8*pi2*taut5*taut2*taut - 1.823308647071001956e-7*pi8*pi2*taut5*taut5*taut2 - 6.567692182135182584e-8*pi8*pi8*taut18*taut5*taut4 + 261.9792810455205085*pi8*pi8*taut33*taut5*taut5*taut5 - 1074.499038327538301*pi8*pi8*pi2*taut33*taut18*taut4 + 3.389062123505998002e-22*pi20*taut18 + 3.644888708271607926e-10*pi20*taut33 - 0.009475756712715725089*pi20*taut33*taut5*taut5*taut2*taut - 2.4803532467968384e-23*pi20*pi*taut18*taut + 0.01042510676226874981*pi20*pi2*taut33*taut18 - 1.892307844119724095e-12*pi20*pi3*taut33*taut4 + 4.750694688678964685e-26*pi20*pi4*taut18*taut5*taut + 8.644695594721368726e-14*pi20*pi4*taut33*taut5 - 0.003119862521394402635*pi20*pi4*taut33*taut18*taut5)
[docs]def iapws97_d2Gr_dpidtau_region2(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `tau` and `pi` for water
according to the IAPWS-97 standard (for region 2).
.. math::
\frac{\partial^2 \gamma^r}{\partial \tau \partial \pi} = \sum_{i=1}^{43} n_i
I_i \pi^{I_i-1} J_i (\tau - 0.5)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dpidtau_ : float
Second derivative of dimensionless residual Gibbs energy Gr/(RT) with
respect to `tau` and `pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dpidtau_region2(540/300.0, 101325/1e6)
-735.391845360247
'''
taut = tau - 0.5
pi2 = pi*pi
pi3 = pi2*pi
pi6 = pi3*pi3
pi17 = pi6*pi6*pi3*pi2
taut2 = taut*taut
taut4 = taut2*taut2
taut5 = taut4*taut
taut13 = taut4*taut4*taut5 # 13
taut34 = taut13*taut4 # 9 + 8 = 17
taut34 *= taut34 # 34 end
return (-0.0007579595006525999962*pi*taut - 0.00006606528334040599997*pi - 0.03151422179468400114*pi*taut2*taut - 0.6131621391080219752*pi*taut5*taut - 0.001920567449814264156*pi*taut34*taut - 0.09199202739273000529*taut + 1.023257428182800066e-7*taut*pi3 - 0.01783486229235799886 + 1.316120018533050084e-6*pi2 - 0.0002904990951471300275*pi2*taut2 - 0.02706106417586639709*pi2*taut5 - 4.270166624078144402*pi2*taut34 - 0.1727437772502959934*taut2 - 3.153892382374680036e-9*pi3 + 5.787044726220840615e-6*pi3*taut2 + 0.000080227267181813501*pi3*pi*taut5*taut - 3.008657961190980272e-10*pi3*pi2*taut2 - 0.2032461342850079844*pi3*pi2*taut13*taut2 - 5018.10580616183961*pi3*pi2*taut34 - 0.3019516723675800263*taut5 - 0.00009718792852307769686*pi6*taut5*taut5 - 6.815697426254326174*pi6*taut13*taut5*taut5*taut + 7.203975270693759891e-10*pi6*pi*taut5*taut2 - 2370.566617862342355*pi6*pi*taut34*taut + 2.317736397844296243e-6*pi6*pi2*taut5*taut5*taut2 + 4.162786084069599818e-18*pi6*pi3*taut2*taut - 1.023474709592900005e-11*pi6*pi3*taut5*taut2*taut2 - 1.402545113131539905e-7*pi6*pi3*taut13 - 3.752966961220103956e-8*pi6*pi6*pi3*taut13*taut13*taut2 + 85.54425503527200192*pi6*pi6*pi3*taut34*taut13*taut2 - 345.3746908909944295*pi17*taut34*taut13*taut5*taut2*taut2 + 3.567433814216839978e-22*pi17*pi2*taut13*taut5*taut + 2.144052181336239759e-10*pi17*pi2*taut34 - 0.004032236899027968197*pi17*pi2*taut34*taut13 - 2.604370909136680202e-23*pi17*pi3*taut13*taut5*taut2 + 0.004410622091729086459*pi17*pi3*pi*taut34*taut13*taut5 - 1.145344221440885637e-12*pi17*pi3*pi2*taut34*taut2*taut2 + 4.560666901131806878e-26*pi17*pi6*taut13*taut5*taut5*taut2 + 5.319812673674687913e-14*pi17*pi6*taut34*taut5 - 0.001313626324797643238*pi17*pi6*taut34*taut13*taut5*taut5)
### Region 3 A formulations
[docs]def iapws97_A_region3(tau, delta):
r'''Calculates the dimensionless Helmholtz free energy for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{f(\rho, T)}{RT} = \phi(\delta, \tau) = n_1\ln\delta
+ \sum_{i=2}^{40} n_i \delta^{I_i}\tau^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
A : float
Helmholtz free energy A/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_A_region3(647.096/500.0, 400.0/322.0)
-3.0336402168865
'''
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
delta5 = delta4*delta
tau2 = tau*tau
tau4 = tau2*tau2
tau6 = tau4*tau2
tau13 = tau6*tau6*tau
return (-1.265431547771399989*delta*tau2 - 1.152440780668100073*delta*tau6
+ 0.8852104398431800414*delta*tau13*tau2 - 0.6420776518160700164*delta*tau13*tau4
+ 20.94439697430700065*tau + 0.1077051262633200029*tau*delta5
- 0.0001655767979503699929*tau*delta5*delta5 + 1.065807002851300034*log(delta)
- 15.73284529023900014 + 0.3849346018667100244*delta2 - 0.8521470882420599802*delta2*tau2
+ 4.897228154187700078*delta2*tau6 - 3.050261725696500115*delta2*tau6*tau
+ 0.03942053687915399868*delta2*tau13*tau6*tau2*tau + 0.1255840842430800131*delta2*tau13*tau13
- 7.686770787871600064*tau2 - 0.2799932969871000155*delta3 + 1.389979956946000073*delta3*tau2
- 2.018991502357000201*delta3*tau4 - 0.008214763717396300277*delta3*tau13*tau2*tau
- 0.4759603573492299788*delta3*tau13*tau13 + 0.0439840744735000011*delta4
- 0.4447643542873899736*delta4*tau2 + 0.9057207071973299994*delta4*tau4
+ 0.7052245008796700354*delta4*tau13*tau13 - 0.3291362325895400009*delta5*tau2*tau
- 0.5087106204115799946*delta5*tau13*tau13 - 0.02217540087309599964*delta5*delta
+ 0.0942607516650919991*delta5*delta*tau2 + 0.1643627844796100024*delta5*delta*tau13*tau13
- 0.01350337224134800021*delta5*delta2*tau2 + 2.618594778795400035*tau6*tau
- 0.01483434535247200002*delta5*delta3*tau13*tau13 + 0.0005792295362808399465*delta5*delta4*tau2
+ 0.00323089047037109986*delta5*delta4*tau13*tau13 + 0.00008096480299621500545*delta5*delta5
- 2.808078114861999985*tau6*tau4 - 0.00004492389906181499664*delta5*delta5*delta*tau13*tau13
+ 1.205336969651700008*tau6*tau6 - 0.008456681281250200827*tau13*tau6*tau4)
[docs]def iapws97_dA_ddelta_region3(tau, delta):
r'''Calculates the derivative of dimensionless Helmholtz free energy
with respect to `delta` for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{\partial \phi(\delta, \tau)}{\partial \delta} = \frac{n_1}{\delta}
+ \sum_{i=2}^{40} n_i I_i \delta^{I_i-1}\tau^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
dA_ddelta : float
Derivative of dimensionless Helmholtz free energy with respect to
`delta`, [-]
Notes
-----
Examples
--------
>>> iapws97_dA_ddelta_region3(647.096/500.0, 400.0/322.0)
7.35562435092
'''
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
delta5 = delta4*delta
tau2 = tau*tau
tau3 = tau2*tau
tau6 = tau3*tau3
tau13 = tau6*tau6*tau
return (0.7698692037334200489*delta - 1.70429417648411996*delta*tau2
+ 9.794456308375400155*delta*tau6 - 6.100523451393000229*delta*tau6*tau
+ 0.07884107375830799735*delta*tau13*tau6*tau3
+ 0.2511681684861600261*delta*tau13*tau13 + 0.5385256313166000286*tau*delta4
- 0.001655767979503699984*tau*delta5*delta4 + 1.065807002851300034/delta
- 0.8399798909613001019*delta2 + 4.169939870838000218*delta2*tau2
- 6.056974507071000602*delta2*tau3*tau - 0.0246442911521888991*delta2*tau13*tau3
- 1.427881072047689992*delta2*tau13*tau13 - 1.265431547771399989*tau2
+ 0.1759362978940000044*delta3 - 1.779057417149559894*delta3*tau2
+ 3.622882828789319998*delta3*tau3*tau + 2.820898003518680142*delta3*tau13*tau13
- 1.645681162947699949*delta4*tau3 - 2.543553102057900084*delta4*tau13*tau13
- 0.1330524052385760048*delta5 + 0.5655645099905519668*delta5*tau2
+ 0.9861767068776600142*delta5*tau13*tau13 - 0.09452360568943600494*delta5*delta*tau2
- 1.152440780668100073*tau6 - 0.1186747628197760002*delta5*delta2*tau13*tau13
+ 0.005213065826527559302*delta5*delta3*tau2 + 0.02907801423333989874*delta5*delta3*tau13*tau13
+ 0.0008096480299621500003*delta5*delta4 - 0.0004941628896799649291*delta5*delta5*tau13*tau13
+ 0.8852104398431800414*tau13*tau2 - 0.6420776518160700164*tau13*tau3*tau)
[docs]def iapws97_d2A_ddelta2_region3(tau, delta):
r'''Calculates the second derivative of dimensionless Helmholtz free energy
with respect to `delta` for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{\partial^2 \phi(\delta, \tau)}{\partial \delta^2} = \frac{-n_1}{\delta^2}
+ \sum_{i=2}^{40} n_i I_i (I_i-1)\delta^{I_i-2}\tau^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2A_ddelta2 : float
Second derivative of dimensionless Helmholtz free energy with respect to
`delta`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2A_ddelta2_region3(647.096/500.0, 400.0/322.0)
-2.2858869882497
'''
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
tau2 = tau*tau
tau4 = tau2*tau2
tau6 = tau4*tau2
tau20 = tau6*tau6*tau6*tau2
return ( - 1.065807002851300034/delta2 -1.679959781922600204*delta
+ 8.339879741676000435*delta*tau2 - 12.1139490141420012*delta*tau4
- 0.04928858230437779819*delta*tau6*tau6*tau4 - 2.855762144095379984*delta*tau20*tau6
+ 2.154102525266400114*tau*delta3 - 0.0149019118155332992*tau*delta4*delta4
+ 0.7698692037334200489 + 0.5278088936820000132*delta2 - 5.337172251448679461*delta2*tau2
+ 10.86864848636795955*delta2*tau4 + 8.462694010556040425*delta2*tau20*tau6
- 1.70429417648411996*tau2 - 6.582724651790799797*delta3*tau2*tau
- 10.17421240823160034*delta3*tau20*tau6 - 0.6652620261928799961*delta4
+ 2.827822549952760056*delta4*tau2 + 4.930883534388300404*delta4*tau20*tau6
- 0.5671416341366160019*delta4*delta*tau2 - 0.8307233397384320428*delta4*delta2*tau20*tau6
+ 9.794456308375400155*tau6 + 0.04170452661222047441*delta4*delta3*tau2
+ 0.2326241138667191899*delta4*delta3*tau20*tau6 - 6.100523451393000229*tau6*tau
+ 0.007286832269659350436*delta4*delta4 - 0.004941628896799649291*delta4*delta4*delta*tau20*tau6
+ 0.07884107375830799735*tau20*tau2 + 0.2511681684861600261*tau20*tau6)
[docs]def iapws97_dA_dtau_region3(tau, delta):
r'''Calculates the derivative of dimensionless Helmholtz free energy
with respect to `tau` for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{\partial \phi(\delta, \tau)}{\partial \tau} =
+ \sum_{i=2}^{40} n_i J_i \delta^{I_i}\tau^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
dA_dtau : float
Derivative of dimensionless Helmholtz free energy with respect to
`tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_dA_dtau_region3(647.096/500.0, 400.0/322.0)
-24.9687028688
'''
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
delta5 = delta4*delta
tau11 = tau*tau # tau2
tau3 = tau11*tau
tau5 = tau3*tau11
tau11 = tau5*tau5 # 10
tau25 = tau11*tau11*tau5 # 25
tau11 *= tau
return (-2.530863095542799979*delta*tau - 6.914644684008599995*delta*tau5
+ 13.27815659764770118*delta*tau11*tau3 - 10.91532008087319028*delta*tau11*tau5
- 15.37354157574320013*tau - 1.70429417648411996*tau*delta2
+ 2.779959913892000145*tau*delta3 - 0.8895287085747799471*tau*delta4
+ 0.1885215033301839982*tau*delta5*delta - 0.02700674448269600042*tau*delta5*delta2
+ 0.001158459072561679893*tau*delta5*delta4 + 20.94439697430700065
+ 29.38336892512619869*delta2*tau5 - 21.35183207987549991*delta2*tau5*tau
+ 0.867251811341387957*delta2*tau11*tau5*tau5 + 3.265186190320080506*delta2*tau25
- 8.075966009428000802*delta3*tau3 - 0.1314362194783408044*delta3*tau11*tau3*tau
- 12.37496929107997978*delta3*tau25 + 3.622882828789319998*delta4*tau3
+ 18.33583702287142003*delta4*tau25 + 0.1077051262633200029*delta5
- 0.9874086977686200584*delta5*tau*tau - 13.22647613070108008*delta5*tau25
+ 4.273432396469860173*delta5*delta*tau25 + 18.33016345156779892*tau5*tau
- 0.3856929791642719763*delta5*delta3*tau25 + 0.08400315222964860329*delta5*delta4*tau25
- 28.08078114861999808*tau5*tau3*tau - 0.0001655767979503699929*delta5*delta5
- 0.001168021375607189872*delta5*delta5*delta*tau25 + 14.4640436358203992*tau11
- 0.1945036694687546086*tau11*tau11)
[docs]def iapws97_d2A_dtau2_region3(tau, delta):
r'''Calculates the second derivative of dimensionless Helmholtz free energy
with respect to `tau` for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{\partial^2 \phi(\delta, \tau)}{\partial \tau^2} =
+ \sum_{i=2}^{40} n_i J_i (J_i-1)\delta^{I_i}\tau^{J_i-2}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2A_dtau2 : float
Second derivative of dimensionless Helmholtz free energy with respect
to `tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2A_dtau2_region3(647.096/500.0, 400.0/322.0)
-373.6565823701
'''
delta2 = delta*delta
delta3 = delta2*delta
delta6 = delta3*delta3
tau2 = tau*tau
tau4 = tau2*tau2
tau10 = tau4*tau4*tau2
tau24 = tau4*tau10*tau10
return (-2.530863095542799979*delta - 34.57322342004299998*delta*tau4
+ 185.8941923670678307*delta*tau10*tau2*tau - 174.6451212939710445*delta*tau10*tau4*tau
- 1.974817395537240117*tau*delta3*delta2 - 15.37354157574320013 - 1.70429417648411996*delta2
+ 146.9168446256310006*delta2*tau4 - 128.1109924792530137*delta2*tau4*tau
+ 18.21228803816914876*delta2*tau10*tau10 + 81.62965475800201887*delta2*tau24
+ 2.779959913892000145*delta3 - 24.22789802828400241*delta3*tau2
- 1.97154329217511215*delta3*tau10*tau4 - 309.3742322769995212*delta3*tau24
- 0.8895287085747799471*delta3*delta + 10.86864848636795955*delta3*delta*tau2
+ 458.395925571785483*delta3*delta*tau24 - 330.6619032675270091*delta3*delta2*tau24
+ 109.9809807094067935*tau4*tau + 0.1885215033301839982*delta6
+ 106.8358099117465088*delta6*tau24 - 0.02700674448269600042*delta6*delta
- 9.642324479106799018*delta6*delta2*tau24 - 252.7270303375799756*tau4*tau4
+ 0.001158459072561679893*delta6*delta3 + 2.100078805741214971*delta6*delta3*tau24
+ 159.1044799940243877*tau10 - 0.02920053439017974636*delta6*delta3*delta2*tau24
- 4.279080728312601778*tau10*tau10*tau)
[docs]def iapws97_d2A_ddeltadtau_region3(tau, delta):
r'''Calculates the second derivative of dimensionless Helmholtz free energy
with respect to `tau` and `delta` for water
according to the IAPWS-97 standard (for region 3).
.. math::
\frac{\partial^2 \phi(\delta, \tau)}{\partial \tau \partial \delta} =
+ \sum_{i=2}^{40} n_i J_i \delta^{I_i-1}\tau^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2A_ddeltadtau : float
Second derivative of dimensionless Helmholtz free energy with respect
to `tau` and `delta`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2A_ddeltadtau_region3(647.096/500.0, 400.0/322.0)
145.85190014717
'''
tau2 = tau*tau
delta2 = tau2*tau2 # abuse - tau4
tau5 = delta2*tau
tau14 = tau5*tau5 # 10
tau25 = tau14*tau14*tau5
tau14 *= delta2
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
delta5 = delta4*delta
return (-3.408588352968239921*delta*tau + 58.76673785025239738*delta*tau5
- 42.70366415975099983*delta*tau5*tau + 1.734503622682775914*delta*tau14*tau5*tau2
+ 6.530372380640161012*delta*tau25 - 2.530863095542799979*tau
+ 8.339879741676000435*tau*delta2 - 3.558114834299119789*tau*delta3
+ 1.131129019981103934*tau*delta5 - 0.1890472113788720099*tau*delta5*delta
+ 0.0104261316530551186*tau*delta5*delta3 - 24.22789802828400241*delta2*tau2*tau
- 0.3943086584350223855*delta2*tau14*tau - 37.12490787323994113*delta2*tau25
+ 14.49153131515727999*delta3*tau2*tau + 73.34334809148568013*delta3*tau25
+ 0.5385256313166000286*delta4 - 4.937043488843100292*delta4*tau2
- 66.13238065350540751*delta4*tau25 + 25.64059437881915926*delta5*tau25
- 6.914644684008599995*tau5 - 3.08554383331417581*delta5*delta2*tau25
+ 0.7560283700668373186*delta5*delta3*tau25 - 0.001655767979503699984*delta5*delta4
- 0.01284823513167908729*delta5*delta5*tau25 + 13.27815659764770118*tau14
- 10.91532008087319028*tau14*tau2)
### Region 5
[docs]def iapws97_G0_region5(tau, pi):
r'''Calculates the dimensionless ideal gas Gibbs free energy for water
according to the IAPWS-97 standard (for region 5).
.. math::
\gamma^\circ = \ln \pi + \sum_{i=1}^6 n_i^\circ \tau^{J_i^\circ}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
G0 : float
Dimensionless ideal gas Gibbs energy G/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_G0_region5(1000.0/1500, 101325/1e6)
-14.9741430290056
'''
tau_inv = 1.0/tau
return (tau*(6.85408416344340043 - 0.329616265389169993*tau)
+ tau_inv*(tau_inv*(0.369015349803330006 - 0.0248051489334660015*tau_inv)
- 3.1161318213925) + log(pi) - 13.1799836742010008)
[docs]def iapws97_dG0_dtau_region5(tau, pi):
r'''Calculates the first derivative of dimensionless ideal gas Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial \gamma^\circ}{\partial \tau} =\sum_{i=1}^6 n_i^\circ
J_i^\circ\tau^{J_i^\circ-1}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dG0_dtau : float
First derivative of dimensionless ideal gas Gibbs energy G/(RT)
with respect to `tau`, [-]
Notes
-----
This function does not depend on `pi` but it is accepted for consistency.
Examples
--------
>>> iapws97_dG0_dtau_region5(1000.0/1500, 101325/1e6)
11.311766995978
'''
tau_inv = 1.0/tau
return (-0.659232530778339987*tau + tau_inv*tau_inv*(tau_inv*(0.074415446800398008*tau_inv
- 0.738030699606660012) + 3.1161318213925) + 6.85408416344340043)
[docs]def iapws97_d2G0_dtau2_region5(tau, pi):
r'''Calculates the second derivative of dimensionless ideal gas Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial^2 \gamma^\circ}{\partial \tau^2} =\sum_{i=1}^6 n_i^\circ
J_i^\circ( J_i^\circ-1)\tau^{J_i^\circ-2}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2G0_dtau2 : float
Second derivative of dimensionless ideal gas Gibbs energy G/(RT)
with respect to `tau`, [-]
Notes
-----
This function does not depend on `pi` but it is accepted for consistency.
Examples
--------
>>> iapws97_d2G0_dtau2_region5(1000.0/1500, 101325/1e6)
-12.744650271463655
'''
tau_inv = 1.0/tau
return (tau_inv*tau_inv*tau_inv*(tau_inv*(2.21409209881998015
- 0.297661787201592032*tau_inv)
- 6.232263642785) - 0.659232530778339987)
[docs]def iapws97_Gr_region5(tau, pi):
r'''Calculates the dimensionless residual Gibbs free energy for water
according to the IAPWS-97 standard (for region 5).
.. math::
\gamma^r = \sum_{i=1}^{6} n_i \pi^{I_i} (\tau)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
Gr : float
Dimensionless residual Gibbs energy Gr/(RT), [-]
Notes
-----
Examples
--------
>>> iapws97_Gr_region5(1000/300.0, 101325/1e6)
-0.0194648291645718
'''
tau3 = tau*tau*tau
return (pi*(pi*tau3*(3.79194548229549995e-8*pi*tau*tau3 + (2.24400374094849992e-6
- 4.11632754534709986e-6*tau3*tau3)) + tau*(tau*(0.000901537616739440007
- 0.00502700776776479966*tau) + 0.00157364048552589993)))
[docs]def iapws97_dGr_dpi_region5(tau, pi):
r'''Calculates the first derivative of dimensionless residual Gibbs free
energy with respect to `pi` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial \gamma^r}{\partial \pi} = \sum_{i=1}^{6} n_i I_i
\pi^{I_i-1} (\tau)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dGr_dpi : float
Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect
to `pi`, [-]
Notes
-----
Used in density solution.
Examples
--------
>>> iapws97_dGr_dpi_region5(1000/300.0, 101325/1e6)
-0.213281155629998
'''
tau3 = tau*tau
tau3 *= tau
return (pi*tau3*(4.48800748189699983e-6 + tau3*(1.13758364468865005e-7*pi*tau - 8.23265509069419973e-6*tau3))
+ tau*(tau*(0.000901537616739440007 - 0.00502700776776479966*tau) + 0.00157364048552589993))
[docs]def iapws97_d2Gr_dpi2_region5(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `pi` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial^2 \gamma^r}{\partial \pi^2} = \sum_{i=1}^{6} n_i I_i (I_i-1)
\pi^{I_i-2} (\tau)^{J_i}
Parameters
----------
tau : float
Dimensionless temperature, (540 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dpi2 : float
Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect
to `pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dpi2_region5(1000/300.0, 101325/1e6)
-0.4179905782304291
'''
tau2 = tau*tau
return (tau2*tau*(tau2*tau2*(2.2751672893773001e-7*pi - 8.23265509069419973e-6*tau2)
+ 4.48800748189699983e-6))
[docs]def iapws97_dGr_dtau_region5(tau, pi):
r'''Calculates the first derivative of dimensionless residual Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial \gamma^r}{\partial \tau} = \sum_{i=1}^{6} n_i
\pi^{I_i} J_i(\tau)^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
dGr_dtau : float
Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect
to `tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_dGr_dtau_region5(1000/300.0, 101325/1e6)
-0.02200629869194
'''
tau2 = tau*tau
return pi*(0.001803075233478880013*tau + 0.001573640485525899932
+ tau2*(-0.01508102330329439897 + pi*(6.732011222845499322e-6
+ tau2*tau2*(-0.00003704694790812389877*tau2
+ 2.654361837606849767e-7*pi))))
[docs]def iapws97_d2Gr_dtau2_region5(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `tau` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial^2 \gamma^r}{\partial \tau^2} = \sum_{i=1}^{6} n_i
\pi^{I_i} J_i(J_i-1)(\tau)^{J_i-2}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dtau2 : float
Second derivative of dimensionless residual Gibbs energy Gr/(RT) with
respect to `tau`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dtau2_region5(1000/300.0, 101325/1e6)
-0.0239165867999155
'''
tau2 = tau*tau
# # 10 before
# return (0.00180307523347888001*pi + tau*(0.0000134640224456909986*pi2
# - 0.0301620466065887979*pi + tau**4*pi2*(1.59261710256410975e-6*pi*
# - 0.00029637558326499119*tau2)))
return pi*(0.001803075233478880013 + tau*(-0.03016204660658879794
+ pi*(0.00001346402244569099864 + tau2*tau2*(-0.0002963755832649911902*tau2
+ 1.592617102564109754e-6*pi))))
[docs]def iapws97_d2Gr_dpidtau_region5(tau, pi):
r'''Calculates the second derivative of dimensionless residual Gibbs free
energy with respect to `tau` and `pi` for water
according to the IAPWS-97 standard (for region 5).
.. math::
\frac{\partial^2 \gamma^r}{\partial \tau \partial \pi} = \sum_{i=1}^{6}
n_i I_i \pi^{I_i-1} J_i(\tau)^{J_i-1}
Parameters
----------
tau : float
Dimensionless temperature, (1000 K)/T [-]
pi : float
Dimensionless pressure, P/(1 MPa), [-]
Returns
-------
d2Gr_dpidtau : float
Second derivative of dimensionless residual Gibbs energy Gr/(RT) with
respect to `tau` and `pi`, [-]
Notes
-----
Examples
--------
>>> iapws97_d2Gr_dpidtau_region5(1000/300.0, 101325/1e6)
-0.27438379131103097
'''
tau2 = tau*tau
return (0.001573640485525899932 + tau*(0.001803075233478880013
+ tau*(-0.01508102330329439897 + pi*(0.00001346402244569099864
+ tau2*tau2*(-0.00007409389581624779755*tau2 + 7.963085512820550889e-7*pi)))))
### Region 3 subregion density calls
REGION_3A = 1
REGION_3B = 2
REGION_3C = 3
REGION_3D = 4
REGION_3E = 5
REGION_3F = 6
REGION_3G = 7
REGION_3H = 8
REGION_3I = 9
REGION_3J = 10
REGION_3K = 11
REGION_3L = 12
REGION_3M = 13
REGION_3N = 14
REGION_3O = 15
REGION_3P = 16
REGION_3Q = 17
REGION_3R = 18
REGION_3S = 19
REGION_3T = 20
REGION_3U = 21
REGION_3V = 22
REGION_3W = 23
REGION_3X = 24
REGION_3Y = 25
REGION_3Z = 26
REGION_3_AS_LETTERS = {1: 'A', 2: 'B', 3: 'C', 4: 'D', 5: 'E', 6: 'F', 7: 'G',
8: 'H', 9: 'I', 10: 'J', 11: 'K', 12: 'L', 13: 'M',
14: 'N', 15: 'O', 16: 'P', 17: 'Q', 18: 'R', 19: 'S',
20: 'T', 21: 'U', 22: 'V', 23: 'W', 24: 'X', 25: 'Y', 26: 'Z'}
def iapws97_region_3(T, P):
r'''Identify the subregion in the IAPWS-97 region 3 given a temperature and
pressure point. No cheking that the original point is in region 3 is
performed.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
subregion : int
One of 1 to 26 inclusive, [-]
Examples
--------
>>> iapws97_identify_region_TP(648.6, 22.5e6)
3
'''
logP_MPa = log(P*1e-6)
logP_MPa_inv = 1.0/logP_MPa
region = 0
if P > 100e6:
return region
Tab = iapws97_boundary_3ab(logP_MPa, logP_MPa_inv)
if P > 40e6: # and P <= 100e6
if T <= Tab:
return REGION_3A
return REGION_3B
Tcd = iapws97_boundary_3cd(P)
Tef = iapws97_boundary_3ef(P)
if P > 25e6 and P <= 40e6:
if T <= Tcd:
return REGION_3C
elif Tcd < T <= Tab:
return REGION_3D
elif Tab < T <= Tef:
return REGION_3E
else:
return REGION_3F
Tgh = iapws97_boundary_3gh(P)
Tij = iapws97_boundary_3ij(P)
Tjk = iapws97_boundary_3jk(P)
if P > 23.5E6 and P <= 25e6:
if T <= Tcd:
return REGION_3C
elif Tcd < T and T <= Tgh:
return REGION_3G
elif Tgh < T and T <= Tef:
return REGION_3H
elif Tef < T and T <= Tij:
return REGION_3I
elif Tij < T and T<= Tjk:
return REGION_3J
return REGION_3K
if 23e6 < P <= 23.5e6:
if T <= Tcd:
return REGION_3C
elif Tcd < T <= Tgh:
return REGION_3L
elif Tgh < T <= Tef:
return REGION_3H
elif Tef < T <= Tij:
return REGION_3I
elif Tij < T <= Tjk:
return REGION_3J
return REGION_3K
Tmn = iapws97_boundary_3mn(P)
Top = iapws97_boundary_3op(logP_MPa, logP_MPa_inv)
if 22.5e6 < P <= 23e6:
if T <= Tcd:
return REGION_3C
elif Tcd < T <= Tgh:
return REGION_3L
elif Tgh < T <= Tmn:
return REGION_3M
elif Tmn < T <= Tef:
return REGION_3N
elif Tef < T <= Top:
return REGION_3O
elif Top < T <= Tij:
return REGION_3P
elif Tij < T <= Tjk:
return REGION_3J
else:
return REGION_3K
Trx = iapws97_boundary_3rx(P)
Tqu = iapws97_boundary_3qu(P)
Psat_643 = 21043367.318975247 # Psat_IAPWS(643.15)
Tsat = Tsat_IAPWS(P)
if Psat_643 < P <= 22.5e6:
if T <= Tcd:
return REGION_3C
elif Tcd < T <= Tqu:
return REGION_3Q
elif Trx < T <= Tjk:
return REGION_3R
elif T > Tjk:
return REGION_3K
else:
Tuv = iapws97_boundary_3uv(P)
Twx = iapws97_boundary_3wx(logP_MPa, logP_MPa_inv)
if 22.11e6 < P <= 22.5e6:
if Tqu < T <= Tuv:
return REGION_3U
elif Tuv < T <= Tef:
return REGION_3V
elif Tef < T <= Twx:
return REGION_3W
elif Twx < T <= Trx:
return REGION_3X
if 22.064e6 < P <= 22.11e6:
if Tqu < T <= Tuv:
return REGION_3U
elif Tuv < T <= Tef:
return REGION_3Y
elif Tef < T <= Twx:
return REGION_3Z
elif Twx < T <= Trx:
return REGION_3X
if T > Tsat:
if Psat_643 < P <= 2.190096265E7:
#if T < Trx:
return REGION_3X
if 2.190096265E7 < P <= 22.064e6:
if T <= Twx:
return REGION_3Z
elif Twx < T <= Trx:
return REGION_3X
else:
if Psat_643 < P <= 2.193161551E7:
return REGION_3U
if 2.193161551E7 < P <= 22.064E6:
if Tqu < T <= Tuv:
return REGION_3U
return REGION_3Y
if 20.5e6 < P <= Psat_643:
if T <= Tcd:
return REGION_3C
elif Tcd < T <= Tsat:
return REGION_3S
elif Tsat <= T <= Tjk:
return REGION_3R
else:
return REGION_3K
P3cd = 19.00881189E6
if P3cd < P <= 20.5e6:
if T < Tcd:
return REGION_3C
elif Tcd < T < Tsat:
return REGION_3S
else:
return REGION_3T
Psat_623 = 16529164.25260448# Psat_IAPWS(623.15)
if Psat_623 < P <= P3cd:
if T <= Tsat:
return REGION_3C
return REGION_3T
[docs]def iapws97_region3_a(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*1e-08 - 0.085
thetat = T*0.0013157894736842105 - 0.817
pit_inv = 1.0/pit
pit2 = pit*pit
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.050226879086966297*pit + 0.040874841585674497*pit_inv + thetat*(0.47468639786331202*pit_inv + thetat*(-0.36964530819337699*pit + 1.1864681499791501*pit_inv + thetat*(1.3582570312914*(pit_inv2) + (thetat2)*(thetat*((thetat2)*((thetat2)*((thetat2)*(234105.65413187601*(pit_inv10) - 76705.194838085197*pit_inv10*pit_inv2) + 28277.661724328598*(pit_inv5) - 26989.395617661299*pit_inv5*pit_inv3 + 6280.0804934568896*(pit_inv10) + 572.61674081061597*(pit_inv10*pit_inv2)) - 2424.3152002952302*pit_inv2*pit_inv2 - 156.237904341963*pit_inv5*pit_inv3) + 44.272952105831401*(pit_inv3)) + 26.698704085604*(pit_inv5) + 0.216867826045856*(pit_inv5*pit_inv3) - 0.0253321069529674*pit_inv10 + 0.00110879558823853*(pit_inv10*pit_inv2)) + 1.7935760401998899*(pit_inv3)) + 0.079744179390101699*(pit2) + 0.45318626168577397*(pit_inv2)) + 0.19526677045264301 - 0.0122494831387441*pit_inv3 + 0.0011673222766826099*(pit_inv5) - 0.00018040710008550501*pit_inv5*pit_inv) + 0.54698726572754897 + 0.0063382803752842004*(pit2) - 0.0059322348901834198*pit_inv2 + 0.00043521732302273302*(pit_inv3))
V = main*0.0024
return 1.0/V
[docs]def iapws97_region3_b(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*1e-08 - 0.28
thetat = T*0.0011627906976744186 - 0.779
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.102845919373532*pit + 0.0122261479925384*pit_inv + thetat*(thetat*(-0.49267663758928398*pit + 2.1635605769293802*pit_inv + thetat*(thetat*(thetat*(thetat*((thetat2)*((thetat2)*((thetat2)*(-29103.2084950276*pit_inv10*thetat2 + 41.688712601056501*(pit_inv10*pit_inv2)) - 1406.99677420738*pit_inv5 - 0.082767047000362096*pit_inv10*pit_inv2) + 361.18245261214901*(pit_inv5) + 140.24499760965799*(pit_inv5*pit_inv) - 111.422582236948*pit_inv5*pit_inv3 + 0.048365198219705897*(pit_inv10)) + 294.00250933851498*(pit_inv5*pit_inv)) + 23.960066025616101*(pit_inv2) - 50.667329572163702*pit_inv3 - 4.2559780405863199*pit_inv2*pit_inv2 - 344.38415881145897*pit_inv5 - 0.020230008390401399*pit_inv5*pit_inv) + 171.346792457471*(pit_inv2*pit_inv2)) - 41.375495701104199*pit_inv3) + 6.0881736840178498*(pit_inv2) - 0.24046253507852999*pit3 - 0.041637529016623598*pit_inv3 - 0.00202023902676481*pit_inv2*pit_inv2) - 0.116892827834085 + 0.00151140509678925*(pit_inv3) + 0.128369435967012*(pit2*pit2)) + 0.398198903368642 + 0.065554045640679001*(pit2) - 0.00057221296556902296*pit_inv2 + 6.9134608500033399e-6*(pit_inv3) - 0.026979818031007501*pit2*pit2)
V = main*0.0041
return 1.0/V
[docs]def iapws97_region3_c(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*2.5e-08 - 0.259
thetat = T*0.0014492753623188406 - 0.903
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.156531975531713*pit + 0.0158646812591361*pit_inv + thetat*(0.70790633624184296*pit_inv + thetat*(11.770743004815801*pit + 12.601622514657*pit_inv + thetat*(thetat*(thetat*(thetat*(thetat*(thetat*((thetat2)*(-79389204.982125103*pit_inv10 + 32258310.340326902*(pit_inv10*pit_inv2)) - 3820310.2057081298*pit_inv2*pit_inv2 + 7912143.6522279195*(pit_inv5*pit_inv) - 899732.52990737697*pit_inv10 + 27671.3458847564*(pit_inv10*pit_inv2)) + 1232904.23502494*(pit2) - 1070777.1666086901*pit3 - 833426.56321285095*pit_inv5 + 175336.67532249901*(pit_inv5*pit_inv3)) + 2297.8474234507198*(pit_inv5*pit_inv3) - 342.41606509536302*pit_inv10 + 3.1196778876303002*(pit_inv10*pit_inv2)) + 3775.1566896695099*(pit_inv2) + 95.319300321738794*(pit_inv5*pit_inv3)) + 234.604891591616*(pit_inv2) - 65.950886355576699*pit_inv5) - 44.017020394964497*pit2 + 31.032749849200801*(pit_inv3)) - 17.810058818913699 + 0.064573468058329198*(pit_inv2*pit_inv2)) + 0.67654426899910103 - 0.18644246747194901*pit2 + 3.1993334584420903e-5*(pit_inv5) + 0.0438319858566475*(pit5*pit3)) + 0.73614365577215202 + 0.084014365386044704*(pit2) - 0.00089299671848372397*pit_inv2 - 0.0240650039730845*pit3 + 4.0639884847007902e-5*(pit_inv3))
V = main*0.0022
return 1.0/V
[docs]def iapws97_region3_d(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*2.5e-08 - 0.559
thetat = T*0.0014492753623188406 - 0.939
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.032665012142638297*pit + 0.0050083391537212099*pit_inv + thetat*(0.38784248299841101*pit_inv + thetat*(thetat*(thetat*(thetat*(-1385.35367777182*pit_inv + thetat*(4980.4417172787698*pit + thetat*(thetat*((thetat2)*((thetat2)*((thetat2)*(-41125421794.6539*pit_inv10 + 1153711331204.97*(pit_inv10*pit_inv2)*(thetat2)) - 12712303.684593201*pit_inv10*pit_inv2) - 19541952.506071299*pit_inv5*pit_inv3 + 1443694.8990905299*(pit_inv10) + 508.05887480834502*(pit_inv10*pit_inv2)) + 2240407.5442698798*(pit_inv5*pit_inv) - 68931.508793315807*pit_inv5*pit_inv3 - 20.3578994462286*pit_inv10) + 125031.83535173599*(pit_inv2*pit_inv2) - 385294.21355528902*pit_inv5 - 22.177428114603799*pit_inv5*pit_inv3 - 0.0021499135204754499*pit_inv10*pit_inv2) + 3163.7351056401499*(pit_inv5*pit_inv) + 0.00277211346836625*(pit_inv10) + 3.1521038953880098e-5*(pit_inv10*pit_inv2)) - 348.15320341466298*pit_inv5) + 225.66051751243799*(pit_inv3) - 1.56481703640525e-6*pit_inv10 - 4.5248484717164498e-10*pit_inv10*pit_inv2) + 6.23449786243773e-6*(pit_inv5*pit_inv3)) + 1.7194625206874199 + 0.096812367845584099*(pit_inv3) - 0.00040421385283399602*pit_inv5 + 2.4155480603397201e-11*(pit_inv10)) - 0.029962841081922899*pit_inv2 + 0.00013464838327108901*(pit_inv2*pit_inv2) - 4.3670134792235601e-6*pit_inv5) + 0.87074524597177305 - 0.000190102435341872*pit_inv2 + 0.0055147802276508699*(pit3) + 1.3520370009940299e-7*(pit_inv2*pit_inv2) - 1.97805728776273e-16*pit_inv10)
V = main*main
V *= V*0.0029
return 1.0/V
[docs]def iapws97_region3_e(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*2.5e-08 - 0.587
thetat = T*0.0014084507042253522 - 0.918
pit_inv = 1.0/pit
pit2 = pit*pit
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.54467412487890998*pit - 0.015749651617430801*pit_inv + thetat*(thetat*(thetat*(thetat*(2045.29931318843*pit - 1043.9079421301101*pit_inv2 + (thetat2)*(-22834.235932875199*pit + thetat*(thetat*((thetat2)*((thetat2*thetat2)*((thetat2)*(79497740233.560303*(pit_inv10) - 114328360753.44901*pit_inv10*pit_inv2) + 5353641749.6012697*(pit_inv10) + 715815808.40472102*(pit_inv10*pit_inv2)) - 1117963.81424162*pit_inv5*pit_inv3 + 665695.90883625206*(pit_inv10)) - 142586.073991215*pit_inv5*pit_inv3) - 266627.75039034098*pit_inv3 + 92.223056342143707*(pit_inv5*pit_inv3)) + 47599.266771712399*(pit_inv3) - 6699.8923907049102*pit_inv5 + 8961.2162964075997*(pit_inv5*pit_inv) - 9.0398366869115705e-5*pit_inv10) - 33.973132597771297*pit_inv2*pit_inv2) + 123.654999499486*(pit_inv2) + 3.7653100201571999e-12*(pit_inv10)) - 34.193183591040501*pit2 - 1.2052311155227799*pit_inv3 + 0.0045124253848683399*(pit_inv2*pit_inv2)) + 1.7837346287390301 + 0.30563840482826499*(pit_inv2)) + 0.685331118940253 + 0.41319748151589902*(pit2) - 0.00015331495438652401*pit_inv2)
V = main*0.0032
return 1.0/V
[docs]def iapws97_region3_f(T, P):
# This function was automatically generated. Do not edit it directly!
pit = sqrt(P*2.5e-08 - 0.587)
thetat = T*0.0013698630136986301 - 0.891
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-0.00141987303638727*pit*thetat_inv + thetat*(2.69251915156554*pit + thetat*(-30.020869577178299*pit*thetat + 34.974181585872202*pit - 16.517557195908601) + 2.1410775923648599 - 8.3909127728616895*pit2 - 1.4977453386065001*pit5 + 1.5120553127513301*(pit5*pit2)) - 0.00100615977450049*thetat_inv + 0.99996914025219197 - 1.31546288252539*pit2 + 6.0130719366876297e-6*(thetat_inv2) + 1.52115067087106*(pit3) - 0.00059109920647890904*pit3*thetat_inv2 + 1.81545608337015e-10*(pit3)*(thetat_inv5) - 2.5175654779232502e-8*thetat_inv3 + 2.5295647066322501e-5*(pit2*pit2)*(thetat_inv3) + 1.0072626520378601e-15*(pit5)*(thetat_inv5*thetat_inv3) - 7.9394097056296902e-10*pit5*pit*thetat_inv5*thetat_inv - 0.000150290891264717*pit5*pit2*thetat_inv2*thetat_inv2 + 4.7094260622165203e-6*(pit10)*(thetat_inv5*thetat_inv) + 0.00060437464020126502*(pit10*pit2)*(thetat_inv2*thetat_inv2) - 9.1162788626607693e-9*pit10*pit2*thetat_inv5*thetat_inv3 + 1.95049710391712e-13*(pit10*pit2)*(thetat_inv10) - 0.00091929673666610605*pit10*pit2*pit2*thetat_inv2*thetat_inv2 - 1.3779607079840901e-5*pit10*pit2*pit2*thetat_inv5*thetat_inv - 3.0306390804340398e-7*pit10*pit2*pit2*thetat_inv5*thetat_inv3 + 6.1091697358298098e-12*(pit10*pit2*pit2)*(thetat_inv10) - 2.2513293390013599e-16*pit10*pit2*pit2*thetat_inv10*thetat_inv2 + 7.5325947989869902e-7*(pit10*pit3*pit3)*(thetat_inv5*thetat_inv3) + 6.3928822313254498e-10*(pit10*pit3*pit3)*(thetat_inv10) + 7.5614029435161407e-9*(pit10*pit5*pit3)*(thetat_inv10) - 4.0032147868292898e-13*pit10*pit5*pit3*thetat_inv10*thetat_inv2 + 2.6958601059187399e-5*(pit10*pit5*pit3*pit2)*(thetat_inv5*thetat_inv) - 2.3761238114053899e-8*pit10*pit5*pit3*pit2*thetat_inv10 - 9.1208205403489092e-12*pit10*pit5*pit3*pit2*thetat_inv10*thetat_inv2 - 7.3282813515783902e-11*pit10*pit10*pit2*thetat_inv10*thetat_inv2 - 0.00040573553273032199*pit10*pit10*pit2*pit2*thetat_inv2*thetat_inv2 + 2.4199557830666001e-10*(pit10*pit10*pit2*pit2)*(thetat_inv10*thetat_inv2) + 1.8942414349801099e-10*(pit10*pit10*pit5*pit3)*(thetat_inv10*thetat_inv2) - 4.8663296507456297e-10*pit10*pit10*pit10*pit2*thetat_inv10*thetat_inv2)
V = main*main
V *= V*0.0064
return 1.0/V
[docs]def iapws97_region3_g(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4e-08 - 0.872
thetat = T*0.0015151515151515152 - 0.971
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.11048023927260101*pit + thetat*(5.3651603187505899*pit + thetat*(30.433158444409301*pit_inv + thetat*(-2914.41872156205*pit - 91.078254013468097*pit_inv2 + (thetat2)*(thetat*(thetat*(thetat*(5932507979.59445*pit_inv + (thetat2)*((thetat2)*((thetat2)*(-7.1294938340821105e+18*pit_inv2 + (thetat2*thetat2)*(-3.6417406211079798e+27*pit_inv + (thetat2)*((thetat2)*((thetat2)*(-1.04578785289542e+36*pit_inv2 + 6.16338176535305e+39*(pit3) + 7.2537907205934803e+29*(pit_inv10) - 1.05549884548496e+28*pit_inv10*pit_inv2) + 8.0205671552837802e+31*(pit_inv2*pit_inv2) - 1.2088917586118e+38*pit5 - 2.8021431005410101e+30*pit_inv5*pit_inv + 4.962507048713e+24*(pit_inv10*pit_inv2)) + 6.1375422916861901e+27*(pit_inv5) - 9.2217276959610093e+22*pit_inv10) - 1.9578886571897101e+17*pit_inv10*pit_inv2) - 758642165988.27795*pit_inv10 + 9481808850.3208008*(pit_inv10*pit_inv2)) + 8.1839602452461196e+22*(pit5*pit) + 228646846221.83099*(pit_inv5*pit_inv3) - 1149872.3828058699*pit_inv10*pit_inv2) - 37954580.233648703*pit_inv5*pit_inv3) - 4997410.93010619*pit_inv5*pit_inv + 10755.5033344858*(pit_inv5*pit_inv3)) - 29861781.9828065*pit_inv3 + 1049154.0676958601*(pit_inv5) - 61.7718249205859*pit_inv5*pit_inv3 + 4.1220902065299602e-5*(pit_inv10*pit_inv2)) - 8375139317986550.0*pit10) + 135033.22728156499*(pit_inv2)) + 940781944.83582902*(pit5*pit3)) - 72.464414375850794) - 0.33769360965747097) + 0.92179140353246103 - 36727.966954544798*pit10)
V = main*main
V *= V*0.0027
return 1.0/V
[docs]def iapws97_region3_h(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4e-08 - 0.898
thetat = T*0.0015151515151515152 - 0.983
pit_inv = 1.0/pit
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.21134640224085799*pit + 0.0014719965861807599*pit_inv + thetat*(thetat*(24.997175295749098*pit + 20.261848702557799*pit_inv + thetat*(thetat*((thetat2)*(thetat*(thetat*((thetat2)*((thetat2)*((thetat2)*(17195156812433.699*(pit_inv10) - 18546115498514500.0*pit_inv10*thetat2) + 7741354215.8708296*(pit_inv10*pit_inv2)) - 605971823.58500504*pit_inv10) - 6561744.2199959401*pit_inv5*pit_inv + 17768333.734819099*(pit_inv5*pit_inv3) + 1936.9655876492*(pit_inv10) + 0.056137967888757703*(pit_inv10*pit_inv2)) - 2120.1062070121998*pit_inv5*pit_inv3) - 234396.09169331301*pit_inv5*pit_inv - 170.875935679023*pit_inv5*pit_inv3 - 0.00143987128208183*pit_inv10) + 1394.99167345464*(pit_inv2*pit_inv2) + 13.5249306374858*(pit_inv5) + 11.017744362957499*(pit_inv5*pit_inv) + 1.11482975877938e-9*(pit_inv10)) - 2.1294625702140002*pit_inv5) - 0.15201104438964799*pit_inv3 + 0.17718916414581301*(pit_inv2*pit_inv2) + 1.5636221297739601e-5*(pit_inv5)) - 0.0070367093203638799*pit_inv3 - 3.9546432784610502e-14*pit_inv5*pit_inv3) + 0.89934551894423997 + 9.8191692299111303e-5*(pit_inv2) + 3.8785116807801003e-17*(pit_inv5*pit_inv3))
V = main*main
V *= V*0.0032
return 1.0/V
[docs]def iapws97_region3_i(T, P):
# This function was automatically generated. Do not edit it directly!
pit = sqrt(P*4e-08 - 0.91)
thetat = T*0.0015151515151515152 - 0.984
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat5 = thetat3*thetat2
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-0.0023530288573684901*pit*thetat_inv - 0.269226321968839*pit - 5.6662075717003204e-7*pit*thetat_inv2 - 4.4635205567874902e-12*pit*thetat_inv2*thetat_inv2 + thetat*(thetat*((thetat3)*((thetat5)*(259862256980408.0 - 2.2328627042235599e+21*pit5*thetat2 + 4.2273953705724101e+19*(pit5*pit3)) - 99226310037675.0*pit10*pit2*pit2) + 42227580030.4086*(pit10*pit5*pit3)) - 1.4862085792233299) - 2.3177966967562398*thetat_inv*pit2*pit2 + 4.8133713145289097*thetat_inv*(pit5) + 1.0690568435913601 + 9.22024992944392*(pit2) - 17.394256556222199*pit3 + 3.5763350550377198e-12*(pit3)*(thetat_inv5) - 0.00026705035107576803*pit2*pit2*thetat_inv2 + 7.0068178555622898e-6*(pit2*pit2)*(thetat_inv3) - 7.5353304697975201e-13*pit5*thetat_inv5*thetat_inv + 0.0064641293413649596*(pit5*pit2)*(thetat_inv3) - 1.18746004987383e-5*pit5*pit2*thetat_inv2*thetat_inv2 - 4.1058853633093698e-10*pit5*pit3*thetat_inv5*thetat_inv + 3.1369818047381199e-13*(pit10)*(thetat_inv5*thetat_inv3) - 0.013526863990502101*pit10*pit2*thetat_inv2*thetat_inv2 - 3.39823323754373e-6*pit10*pit2*thetat_inv5*thetat_inv + 1.6439533434503999e-24*(pit10*pit2)*(thetat_inv10*thetat_inv2) - 0.046395953375238497*pit10*pit2*pit2*thetat_inv2*thetat_inv2 + 1.84386437538366e-9*(pit10*pit2*pit2)*(thetat_inv5*thetat_inv3) - 7.2325251421162496e-15*pit10*pit2*pit2*thetat_inv10 + 0.00345570606200257*(pit10*pit5*pit3)*(thetat_inv5*thetat_inv) - 5.4084301862408299e-8*pit10*pit5*pit3*thetat_inv5*thetat_inv3 - 2.2262099845219699e-11*pit10*pit5*pit3*thetat_inv10 + 6.8816915443933499e-17*(pit10*pit5*pit3)*(thetat_inv10*thetat_inv2) + 9.2723798515367901e-10*(pit10*pit5*pit3*pit2)*(thetat_inv10) - 1.2697447877048701e-15*pit10*pit5*pit3*pit2*thetat_inv10*thetat_inv2 + 6.1267081201648904e-14*(pit10*pit10*pit2)*(thetat_inv10*thetat_inv2) - 0.00038366950263682198*pit10*pit10*pit2*pit2*thetat_inv5*thetat_inv3 - 7.22693924063497e-12*pit10*pit10*pit2*pit2*thetat_inv10*thetat_inv2 - 93197.689751108599*pit10*pit10*pit10*pit2*thetat_inv5 + 0.000374684572410204*(pit10*pit10*pit10*pit2)*(thetat_inv10) + 65.811054675947403*(pit10*pit10*pit10*pit3*pit3)*(thetat_inv5*thetat_inv3) - 0.0247690616026922*pit10*pit10*pit10*pit3*pit3*thetat_inv10)
V = main*main
V *= V*0.0041
return 1.0/V
[docs]def iapws97_region3_j(T, P):
# This function was automatically generated. Do not edit it directly!
pit = sqrt(P*4e-08 - 0.875)
thetat = T*0.0014925373134328358 - 0.964
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-0.000728541958464774*pit*thetat_inv + 1.79058760078792e-6*pit*(thetat_inv2) + thetat*(-18.757613337170401*pit + thetat*(6223.2297178647304*thetat*(pit5*pit) - 198.704578406823*pit2*pit2) + 5.3061558192897902 + 24.357475537729002*(pit2)) - 0.00011137131739554*thetat_inv + 0.0019906087407184901*thetat_inv*(pit2) + 1.0034289242368499 - 0.000177040785499444*pit3*thetat_inv2 - 0.0025968038522712999*pit2*pit2*thetat_inv2 - 1.6102312131433301*pit5 - 0.0023626469284413801*pit5*thetat_inv2 + 7.3862779022428697e-5*(pit5)*(thetat_inv3) - 9.6075411670166893e-9*pit10*thetat_inv5*thetat_inv + 0.0076737378140421097*(pit10*pit2)*(thetat_inv3) - 5.1057226972048803e-11*pit10*pit2*thetat_inv5*thetat_inv3 + 1.46564542926508e-5*(pit10*pit2*pit2)*(thetat_inv5) - 7.1759073552674496e-10*pit10*pit2*pit2*thetat_inv5*thetat_inv3 + 6.6385546948525403e-15*(pit10*pit2*pit2)*(thetat_inv10) + 3.0902947427701301e-12*(pit10*pit3*pit3)*(thetat_inv10) - 4.6421630097170797e-16*pit10*pit5*pit3*thetat_inv10*thetat_inv2 - 2.36716126781431e-10*pit10*pit5*pit3*pit2*thetat_inv10 - 3.9049963796116099e-14*pit10*pit5*pit3*pit2*thetat_inv10*thetat_inv2 - 0.0042227178748249702*pit10*pit10*pit2*pit2*thetat_inv5*thetat_inv + 4.54652854268717e-12*(pit10*pit10*pit2*pit2)*(thetat_inv10*thetat_inv2) + 2.7092900272022802*(pit10*pit10*pit5*pit3)*(thetat_inv5) + 2.8391174235470601e-11*(pit10*pit10*pit5*pit3)*(thetat_inv10*thetat_inv2))
V = main*main
V *= V*0.0054
return 1.0/V
[docs]def iapws97_region3_k(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4e-08 - 0.802
thetat = T*0.0014705882352941176 - 0.935
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-3.4470960548668601*pit - 0.00027761760697574801*pit*thetat_inv2 + 1.7034307284185001e-6*pit*(thetat_inv3) + 3.9472147136367802e-15*pit_inv*(thetat_inv5) + thetat*(22.133386244709499*pit + thetat*(-194.64611003707901*pit + thetat*(thetat*(3289.13873658481*(pit2) + (thetat2)*(37262.996737414702*pit_inv + (thetat2*thetat2)*(-401215699.57609898*pit_inv2 + (thetat2)*(48450147831.840599*(pit_inv2) - 29102685116444.398*thetat2)))) + 589.70277127742895) - 104.52963483027899) + 12.243316265660001) - 0.000879148916140706*thetat_inv + 0.84431786384433105 + 2.5583029857902702*(pit2) - 0.00181057560300994*pit2*thetat_inv2 - 6.96664158132412e-6*pit2*thetat_inv3 - 1.8084520914547001e-11*pit2*thetat_inv5*thetat_inv + 8.0835463977282498e-16*(pit2)*(thetat_inv5*thetat_inv3) + 4.7536162997023301e-7*(thetat_inv2) - 0.0039568892342125*pit5*thetat_inv3 - 6.6187679255803405e-7*pit5*thetat_inv5*thetat_inv - 1.73270241249904e-19*pit5*thetat_inv10*thetat_inv2 + 3.8371940902555598e-5*(pit5*pit)*(thetat_inv5) + 1.6075110746495799e-9*(pit5*pit)*(thetat_inv5*thetat_inv3) - 4.00879935920517e-14*pit5*pit*thetat_inv10 + 6.0420329981913199e-18*(pit5*pit)*(thetat_inv10*thetat_inv2) - 3.8043640701245203e-15*thetat_inv5*thetat_inv - 6.4956544670245702e-15*pit5*pit3*thetat_inv10*thetat_inv2 - 1.4909532850600001e-12*pit10*thetat_inv10*thetat_inv2 + 5.4144937732958098e-9*(pit10*pit2)*(thetat_inv10) - 3.6979437416866603e-30*thetat_inv10*thetat_inv2)
V = main*0.0077
return 1.0/V
[docs]def iapws97_region3_l(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.166666666666667e-08 - 0.908
thetat = T*0.0015384615384615385 - 0.989
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat5 = thetat3*thetat2
thetat10 = thetat5*thetat5
main = (-0.198068404154428*pit + 2.6583961888553e-5*pit_inv + thetat*(0.025339239288975399*pit_inv + thetat*(thetat*(-214.443041836579*pit_inv + thetat*(thetat*(thetat*(thetat*(thetat*((thetat2)*(-306367307532219.0*pit_inv2 + (thetat2)*(4.9423723717971798e+20 + (thetat2)*((thetat2)*(-1.4141534988114001e+30*pit + (thetat2)*((thetat2)*((thetat2)*((thetat2)*(5.21635864527315e+34*(pit_inv5*pit_inv3) - 4.87095672740742e+54*pit_inv5*pit_inv3*thetat10*thetat2 - 3.8145826048995503e+32*pit_inv10) + 4.1386518684890801e+26*(pit_inv10*pit_inv2)) - 6.9595362234882901e+32*pit_inv3 - 7.5896694638775806e+22*pit_inv10*pit_inv2) + 2.7070611108523801e+29*(pit_inv3) - 4.4435947874629502e+22*pit_inv5*pit_inv3 + 5.5492387028966697e+18*(pit_inv10*pit_inv2)) - 1.0810548079647099e+24*pit_inv2*pit_inv2 - 188277213604704.0*pit_inv10*pit_inv2) + 1.1666212121932199e+32*(pit5*pit) + 5294829964228630.0*(pit_inv5*pit_inv3) - 815038000738.06006*pit_inv10 + 2607020586.4753699*(pit_inv10*pit_inv2)) - 495017809506.71997*pit_inv5*pit_inv3 - 4.4570336919694501e+32*pit10) + 22609563.143717401*(pit_inv5*pit_inv3) + 6.4279493237369401e+32*(pit10*pit2*pit2)) - 714430.20993754698*pit_inv5*pit_inv) + 7774514.3796098996*(pit_inv2*pit_inv2)) - 0.0123239564600519*pit_inv5*pit_inv3) - 10.0752127917598*pit_inv5) + 0.127868634615495*(pit_inv5) - 3158749762715330.0*pit10) + 72.155916336135405*(pit_inv2) - 2.1285716942348398*pit_inv3) + 33.840122250919102 + 0.11060902747228001*(pit_inv2)) + 2.2318404310169999 - 99.386242161365104*pit2 - 3.5757858116965902e-6*pit_inv3 + 47313.790987276501*(pit5)) + 0.93784660148966703 + 125.07053414273101*(pit2*pit2) - 996.473529004439*pit5)
V = main*main
V *= V*0.0026
return 1.0/V
[docs]def iapws97_region3_m(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 1.0
thetat = sqrt(sqrt(T*0.0015384615384615385 - 0.997))
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
main = (0.81138436348184695 + (thetat2)*((thetat3)*(-81456.820934687203*pit + thetat*(458384.82859394897*pit + thetat*(thetat*((thetat2)*(-5475783138.9909697*pit2 + (thetat2)*((thetat2)*(-170451090076.38501*pit + 185135446.82833701 + (thetat2*thetat2)*(157890366037614.0*pit + (thetat2)*(-2025305097487740.0*pit + (thetat2)*(1.70215539458936e+17*(pit2) + (thetat2)*(-821698160721956.0 + (thetat2*thetat2)*(2.3341586947851002e+17 - 6.0007993458680299e+22*pit3 + 5.9458438227338403e+24*(pit2*pit2) + (thetat2*thetat2)*(1.88813911076809e+21*pit + (thetat2*thetat2)*(-3.2942192395146001e+21 - 1.37570282536696e+25*pit2 + 1.8150899630390199e+27*(pit3) - 3.4686512276835299e+29*pit2*pit2 - 2.1196114877426e+37*pit5*pit3 - 1.2861789988767499e+48*pit10*pit2*pit2 + 4.79817895699239e+64*(pit10*pit10*pit2*pit2)) + 1.11052244098768e+35*(pit5*pit3) + 2.9113395860250301e+45*(pit10*pit2*pit2)) + 1.8946127934949199e+39*(pit10*pit2) - 8.1009342884264494e+45*pit10*pit3*pit3) - 7.9526024187230606e+23*pit5) + 6.3923490991874096e+41*(pit10*pit3*pit3)) + 3.6819392618356999e+59*(pit10*pit10*pit5*pit3)))) + 1850072455632.3899*(pit3)) + 200725701112386.0*(pit5)) + 939454935735.56299*(pit2*pit2) + 2.6657285643293799e+27*(pit10*pit2*pit2)) + 45373580.000427298*(pit2)) - 38575400038384.797*pit5*pit) - 65977456.760287397*pit3 - 15286114865.930201*pit2*pit2 - 560165667510.44604*pit5) + 7.9553765761342698e+31*(pit10*pit5*pit3*pit2)) - 5681.99310990094*pit3 - 17865719817.2556*pit5*pit3)
V = main*0.0028
return 1.0/V
[docs]def iapws97_region3_n(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.976
thetat = T*0.0015384615384615385 - 0.997
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-302.807107747776*pit + thetat*(232534.27270987601*pit + thetat*((-86987136466.276901*pit + thetat*(400849240129329.0*pit*thetat + 354542769185.67102))*(thetat2) - 792681.20713260002) + 1591.5874831459901) - 0.00089076330670130501*thetat_inv - 4402.0959940771399*thetat_inv*pit3 - 4.9311136203016203e-11*pit2*thetat_inv5 + 5.4127691156417598e-14*(pit2)*(thetat_inv5*thetat_inv) + 2.4056080832171301e-7*(thetat_inv2) - 0.0064306413263692502*pit3*thetat_inv3 + 7.0541210077369902e-12*(pit3)*(thetat_inv5*thetat_inv) - 3.34952758812999e-19*pit3*thetat_inv5*thetat_inv3 - 6.0724664397089301e-24*pit3*thetat_inv10 + 6.14869006573609e-31*(pit3)*(thetat_inv10*thetat_inv2) + 0.0022001990172961501*(pit2*pit2)*(thetat_inv2*thetat_inv2) - 1.5864969989454301e-6*pit2*pit2*thetat_inv5 + 2.5858588789748602e-9*(pit2*pit2)*(thetat_inv5*thetat_inv) + 5.8223866704894202e-28*(pit2*pit2)*(thetat_inv10*thetat_inv2) + 62.915414901504803*(pit5)*(thetat_inv3) - 4.0235211523449402e-19*pit5*thetat_inv10 + 135.14731861706099*(pit5*pit)*(thetat_inv3) - 7.44938506925544e-17*pit5*pit*thetat_inv10 + 3.9062836923846201e-23*(pit5*pit)*(thetat_inv10*thetat_inv2) - 0.52503742788609997*pit5*pit2*thetat_inv5 - 4.2153772609838902e-9*pit5*pit2*thetat_inv5*thetat_inv3 + 8.2144575825511904e-21*(pit5*pit2)*(thetat_inv10*thetat_inv2) + 1.8991720652623699e-13*(pit5*pit3)*(thetat_inv10) + 1.7727487236194601e-26*(thetat_inv5*thetat_inv3) + 4.0213796184277599e-15*(pit10)*(thetat_inv10*thetat_inv2) - 1.35031446451331e-32*thetat_inv10 - 0.039104816792964903*pit10*pit2*thetat_inv5*thetat_inv3 + 3.64975183508473e-6*(pit10*pit2)*(thetat_inv10) + 6.5171817187830098e-13*(pit10*pit2)*(thetat_inv10*thetat_inv2) + 2.80967799943151e-39*(thetat_inv10*thetat_inv2) - 2.1177335580305798e-8*pit10*pit2*pit2*thetat_inv10*thetat_inv2 + 0.0026495335438007201*(pit10*pit5*pit3)*(thetat_inv10*thetat_inv2))
V = exp(main)*0.0031
return 1.0/V
[docs]def iapws97_region3_o(T, P):
# This function was automatically generated. Do not edit it directly!
pit = sqrt(P*4.347826086956522e-08 - 0.974)
thetat = T*0.0015384615384615385 - 0.996
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (0.0028907869214915001*thetat_inv + 0.244482731907223*thetat_inv*(pit2) + 1.38647388209306*thetat_inv*(pit2*pit2) + 1.4173349203098499e-24*(pit3)*(thetat_inv10) + 2.0137732541180301e-6*(pit2*pit2)*(thetat_inv2*thetat_inv2) - 5.8518840178277898e-9*pit2*pit2*thetat_inv5 - 5.9453920290143103e-18*pit2*pit2*thetat_inv5*thetat_inv3 - 3.5453385305947602e-29*pit2*pit2*thetat_inv10*thetat_inv2 - 7.35234770382342e-12*thetat_inv2*thetat_inv2 + 0.00137680878349369*(pit5)*(thetat_inv3) - 1.7395936508477201e-5*pit5*thetat_inv2*thetat_inv2 + 8.1489760580551298e-15*(pit5*pit)*(thetat_inv5*thetat_inv3) + 4.2559663135183898e-26*(pit5*pit2)*(thetat_inv10*thetat_inv2) - 0.0017184963895152099*pit5*pit3*thetat_inv2*thetat_inv2 + 1.3981474793024e-13*(pit5*pit3)*(thetat_inv5*thetat_inv3) - 3.8744911378775499e-18*pit5*pit3*thetat_inv10 + 1.1896057807201801e-11*(pit10)*(thetat_inv5*thetat_inv3) + 6.4189052951329603e-22*(pit10)*(thetat_inv10*thetat_inv2) + 1.2874602397971801e-35*(thetat_inv10*thetat_inv2) + 2.33907907347507e-8*(pit10*pit2*pit2)*(thetat_inv5*thetat_inv3) - 1.55282762571611e-18*pit10*pit2*pit2*thetat_inv10*thetat_inv2 + 3.7768264908914897e-9*(pit10*pit5*pit3*pit2)*(thetat_inv10) - 1.7409324776621299e-13*pit10*pit5*pit3*pit2*thetat_inv10*thetat_inv2 - 5.1672023657530198e-11*pit10*pit10*pit2*pit2*thetat_inv10*thetat_inv2)
V = main*0.0034
return 1.0/V
[docs]def iapws97_region3_p(T, P):
# This function was automatically generated. Do not edit it directly!
pit = sqrt(P*4.347826086956522e-08 - 0.972)
thetat = T*0.0015384615384615385 - 0.997
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (thetat*(-1235.9234861013699*pit + 3246.64750281543*thetat + 116.033094095084) - 9.8282534201036601e-5*thetat_inv - 0.056140345001349498*thetat_inv*pit2 + 1.0514570085061199 + 236.31342539392401*(pit3) + 8.5667740164086898e-8*(pit3)*(thetat_inv3) + 0.0097250329235010896*(pit2*pit2)*(thetat_inv2) - 1.03001994531927*pit5*pit*thetat_inv2 - 2.15743778861592e-5*pit5*pit2*thetat_inv2*thetat_inv2 - 1.4965370619916199e-9*pit5*pit2*thetat_inv5 - 8.3445219829144506*pit5*pit3*thetat_inv2 + 0.586602660564988*(pit10)*(thetat_inv3) + 0.00294985697916798*(pit10*pit2)*(thetat_inv5) + 8.1625609594702108e-6*(pit10*pit2)*(thetat_inv5*thetat_inv) + 3.4348002210496797e-26*(pit10*pit2)*(thetat_inv10*thetat_inv2) + 10.776602703285301*(pit10*pit2*pit2)*(thetat_inv3) + 4.0095476380694099e-10*(pit10*pit2*pit2)*(thetat_inv5*thetat_inv3) + 7.1173046627658394e-17*(pit10*pit2*pit2)*(thetat_inv10) - 4.0944959913818202e-7*pit10*pit3*pit3*thetat_inv5*thetat_inv3 - 7.2912130775890196e-6*pit10*pit5*pit3*thetat_inv5*thetat_inv3 + 6.7710797093890902e-9*(pit10*pit5*pit3*pit2)*(thetat_inv10) + 6.0274597302297505e-8*(pit10*pit10*pit2)*(thetat_inv10) + 0.00179946628317437*(pit10*pit10*pit2*pit2)*(thetat_inv5*thetat_inv3) - 3.82323011855257e-11*pit10*pit10*pit2*pit2*thetat_inv10*thetat_inv2 - 0.000345042834640005*pit10*pit10*pit10*pit3*pit3*thetat_inv10*thetat_inv2)
V = main*0.0041
return 1.0/V
[docs]def iapws97_region3_q(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.848
thetat = T*0.0015384615384615385 - 0.983
pit_inv = 1.0/pit
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
main = (-0.0838165632204598*pit + 0.0010035565172151*pit_inv + thetat*(2.4779590841149202*pit + 0.33349145514351602*pit_inv + thetat*(1.0969757688887301*pit_inv + thetat*(-3191.1496900653301*pit + thetat*(thetat*(thetat*(thetat*(thetat*((thetat2)*(-1729857814.3333499*pit_inv10 - 82043.384325994994*pit_inv10*pit_inv2 + 47327151846.1586*(pit_inv10*pit_inv2)*(thetat2)) + 35176923.2729192*(pit_inv5*pit_inv3) - 3566.1702998249002*pit_inv10) + 32.860002543598*(pit_inv10)) - 775489.25998514402*pit_inv5*pit_inv - 0.080595002100541296*pit_inv10) + 99349.988382027397*(pit_inv5)) + 2256.8993916191798*(pit_inv2) - 6128.4281682008304*pit_inv2*pit_inv2) + 232.808472983776*(pit_inv3) - 0.64209417190456997*pit_inv2*pit_inv2) - 4.2857722747561402*pit_inv2 + 7.1034669196601803e-5*(pit_inv5)) - 0.0064359606067845602*pit_inv2) + 0.96191737937645205 - 1.42808220416837e-5*pit_inv2)
V = main*main
V *= V*0.0022
return 1.0/V
[docs]def iapws97_region3_r(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.874
thetat = T*0.0015384615384615385 - 0.982
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat5 = thetat3*thetat2
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (thetat*(3.0530889006508901 + (thetat2)*(thetat*(thetat*(thetat*((thetat2)*(490112654.15421098*(pit_inv3) - 7014385996282.5801*pit_inv5*pit_inv3*thetat5*thetat) + 0.0014416595566086299*(pit_inv5*pit_inv3)) - 3997452.76971264 - 10433.403065402101*pit_inv3) + 393.09721470624498*(pit_inv3)) + 0.26197513536810901*(pit_inv3))) - 0.000147104222772069*thetat_inv + 1.0360274804340801 - 0.046492350440777798*pit3*thetat_inv2 + 5.6923371959374999e-12*(pit3)*(thetat_inv5*thetat_inv) - 8.30946716459219e-17*pit_inv3*thetat_inv3 + 0.015953672241120199*(pit5*pit3)*(thetat_inv5) - 5.3647956020181104e-7*pit5*pit3*thetat_inv5*thetat_inv3 + 3.9998879569316202e-13*(pit5*pit3)*(thetat_inv10) - 5.3540039651290598e-18*pit5*pit3*thetat_inv10*thetat_inv2 + 150764.97412551101*(pit10)*(thetat_inv2) - 14336.5406393758*pit10*thetat_inv3 + 546.49132352849097*(pit10)*(thetat_inv2*thetat_inv2) - 9.9345695784500592*pit10*thetat_inv5 + 0.066351314422445407*(pit10)*(thetat_inv5*thetat_inv) - 9.8343063671645403e-6*pit10*thetat_inv5*thetat_inv3 + 2.4424745385850599e-8*(pit10)*(thetat_inv10) + 2.70303248860217e-15*(pit10)*(thetat_inv10*thetat_inv2) - 3.37209709340105e-10*pit10*pit2*thetat_inv10*thetat_inv2 + 3.7750198002546901e-9*(pit10*pit2*pit2)*(thetat_inv10*thetat_inv2))
V = main*0.0054
return 1.0/V
[docs]def iapws97_region3_s(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.761904761904762e-08 - 0.886
thetat = T*0.0015625 - 0.99
pit_inv = 1.0/pit
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat5 = thetat3*thetat2
main = (-0.198339358557937*pit + thetat*(0.0012139997999321701*pit_inv + thetat*(1.88317043049455*pit_inv + thetat*(-1670.7350396206*pit_inv + thetat*(-65391.5627346115 + 18751.4491833092*(pit_inv2) + (thetat2)*((thetat2)*(-5041607241.3259001*pit_inv3 + 88458547.259613395*(pit_inv5) + (thetat5*thetat)*((thetat2)*((thetat2)*((thetat2)*((thetat2)*((thetat2)*(-6.1655261113579205e+45*pit2*pit2 + (thetat2*thetat2)*(6.0401220016344396e+49 + (thetat2*thetat2)*(-1.7598409016350101e+57*pit - 1.8566232754532401e+53*pit_inv2*pit_inv2 + 2.02281884477061e+58*(pit_inv5*pit_inv)*(thetat2*thetat2))) + 1.00415480000824e+31*(pit_inv10*pit_inv2) + 9.5089817042504195e+53*(pit10*pit2*pit2)) - 1.9154000182136699e+29*pit_inv10) - 5.3246661214025397e+22*pit_inv10*pit_inv2) + 4.37796099975134e+33*(pit2*pit2)) + 1.66540181638363e+22*(pit_inv5)) + 10561837780884700.0*(pit_inv5*pit_inv3))) - 313563.19766911102*pit_inv2*pit_inv2) + 1935687689.1779699*(pit5)) - 0.0624942093918942*pit_inv3 - 10917404.498782899*pit2*pit2) + 45621.341533807099*(pit3)) + 2.94885696802488 - 575.99125514438401*pit3) + 0.96596165059977501 + 3.5631488140398702*(pit3))
V = main*main
V *= V*0.0022
return 1.0/V
[docs]def iapws97_region3_t(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*5e-08 - 0.803
thetat = T*0.0015384615384615385 - 1.02
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat2 = thetat*thetat
main = (-2.9100291578376098*pit + thetat*(6.6423511500903096 + (thetat2)*(thetat*(-2893.6623672720998 + (thetat2)*(thetat*(thetat*((thetat2)*(-829088246858.08301*pit + (thetat2)*(-3859232023098.48 + (thetat2)*(1.6046460868783398e+17*(pit2) + (thetat2)*((thetat2)*((thetat2)*((thetat2)*((thetat2)*((thetat2*thetat2)*((thetat2*thetat2)*((thetat2*thetat2)*(-3.4155204086064402e+50*pit5*pit2 - 7.0577262332637399e+64*pit10*pit10*pit2 - 4.4422736775830395e+71*pit10*pit10*pit10*pit2 - 2.81396013562745e+76*pit10*pit10*pit10*pit3*pit3) + 5.6902145441327e+57*(pit10*pit5*pit3*pit2) + 4.2843233862067799e+68*(pit10*pit10*pit10*pit2)) - 3.00475129680486e+60*pit10*pit10*pit5*pit3) + 1.6686117620014801e+52*(pit10*pit10*pit2*pit2)) - 6.5647528033941103e+35*pit10 - 7.0058454643311301e+47*pit10*pit10*pit2 - 6.6848129519680801e+50*pit10*pit10*pit10*pit2) - 3.2791059208652301e+30*pit5*pit2) + 3.5528604551230099e+38*(pit10*pit5*pit3)) + 3.58958955867578e+28*(pit10)) - 1.68776617209269e+26*pit10) + 2.4537564093705501e+23*(pit10)) - 2297462376236920.0*pit2*pit2 - 5.2725133970904698e+20*pit10) + 1566374275417.29*(pit3)) - 67707383068734.898*pit5*pit2) - 534686695.71346903*pit2) + 1105544467.9054301*(pit5*pit2)) + 196435.366560186*(pit3) + 38565900.164800599*(pit5*pit2))) + 1.5528724958626801 + 1.76814899675218*(pit2) - 1.78154560260006*pit2*pit2)
V = main*0.0088
return 1.0/V
[docs]def iapws97_region3_u(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.902
thetat = T*0.0015384615384615385 - 0.988
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (0.00105581745346187*pit*(thetat_inv2) - 0.00012270822923564101*pit_inv*thetat_inv + thetat*(-106.201671767107*pit_inv + thetat*(thetat*(thetat*(thetat*(thetat*((thetat2)*((thetat2)*(-1.6011681327467599e+24*pit2 + (thetat2)*(9.03443213959313e+24*pit_inv + (thetat2)*(-6.9399627037085199e+27*pit_inv - 3.1443257755155201e+21*pit_inv5*pit_inv3 + 2.5992951084949901e+19*(pit_inv10) + 1.2208834925835501e+17*(pit_inv10*pit_inv2)) - 8.4340592684641799e+20*pit_inv5 + 1.5907964819684901e+20*(pit_inv5*pit_inv) - 8.7847358505008499e+17*pit_inv5*pit_inv3 - 8826669315646520.0*pit_inv10) + 222612779142211.0*(pit_inv5*pit_inv3) + 1042164686.08488*(pit_inv10)) + 8843876513378.3594*(pit_inv5) - 2169349169962.8501*pit_inv5*pit_inv + 1.04674840020929e+26*(pit5*pit3)) - 7.8175450769884603e+27*pit10*pit2*pit2) - 651903203602581.0*pit2) - 339.56761730342299*pit_inv5 + 2.2614596374788099e+21*(pit10*pit2)) + 276378438378930.0*(pit5)) + 11.4178193518022*(pit_inv3) + 677143292290.14404*(pit5) - 30142694798017.102*pit5*pit) + 7189.5756712785096) - 5.1025429423783704e-9*pit3*thetat_inv5 + 6.4891671896557497e-9*(thetat_inv3) - 0.152355388953402*pit5*thetat_inv2*thetat_inv2 + 0.0116862983141686*(pit5*pit)*(thetat_inv5) + 1.6971981388484e-8*(pit5*pit3)*(thetat_inv5*thetat_inv3) - 10801.690456013999*pit10*thetat_inv2*thetat_inv2 + 5361164.8360273801*(pit10*pit2)*(thetat_inv2*thetat_inv2) - 9.9062360193429495e-13*pit10*pit2*thetat_inv10*thetat_inv2 - 22770.046464391999*pit10*pit2*pit2*thetat_inv5*thetat_inv + 1.5100154888067e-5*(pit10*pit2*pit2)*(thetat_inv10) - 4.8873156577621002e-10*pit10*pit2*pit2*thetat_inv10*thetat_inv2)
V = main*0.0026
return 1.0/V
[docs]def iapws97_region3_v(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.96
thetat = T*0.0015384615384615385 - 0.995
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (2.7603260114515101e-29*pit*(thetat_inv10) - 1.11526741826431e-35*pit*thetat_inv10*thetat_inv2 + 0.018458726111483699*pit_inv - 1.8821488234144798e-9*pit_inv*thetat_inv2 + thetat*(thetat*(thetat*(thetat*(thetat*(thetat*(-72368188562634800.0 + (thetat2)*((thetat2)*(-2.2344919405412401e+26 + (thetat2)*(7.4270572330273797e+26*(pit_inv3) - 1.03977184454767e+28*pit_inv5*thetat2) - 1.92359972440634e+22*pit_inv3 - 4.6813835890873197e+31*pit2*pit2 + 5.8779310562074801e+20*(pit_inv2*pit_inv2) - 6.9759575034739098e+18*pit_inv5 + 31308029991594400.0*(pit_inv5*pit_inv)) + 5131174628650.4404*(pit_inv5) - 62418400710.315804*pit_inv5*pit_inv) + 654144.37374993705*(pit_inv5) + 59461.976619346002*(pit_inv5*pit_inv)) - 25.912373638026899*pit_inv5*pit_inv) + 8206120.4864546899*(pit_inv2)) + 134856491567853.0*(pit3) + 51065511977436000.0*(pit2*pit2)) - 51.742968245060503*pit_inv2 - 7606674911832790.0*pit5 - 1.92824336984852e-6*pit_inv5) + 1.05006446192036e-9*(pit_inv5) + 2.4776139232905799e+26*(pit10*pit2*pit2)) - 1.3583040778266301e-6*thetat_inv2 + 2.80375725094731e-18*(pit_inv3)*(thetat_inv3) + 6.5244029334585999e-10*(pit2*pit2)*(thetat_inv5*thetat_inv) + 9.2699003653063902e-30*(pit_inv2*pit_inv2)*(thetat_inv5*thetat_inv) - 4.3667703405165502e-42*pit_inv2*pit_inv2*thetat_inv10 + 1.1956313554066601e-48*(pit_inv2*pit_inv2)*(thetat_inv10*thetat_inv2) + 3.5925221360411398e-26*(pit_inv5*pit_inv)*(thetat_inv3) - 3.5707866820337699e-55*pit_inv5*pit_inv*thetat_inv10*thetat_inv2 - 4.1724798698682099e-19*pit5*pit3*thetat_inv10*thetat_inv2 + 1.7744174292404301e-61*(pit_inv5*pit_inv3)*(thetat_inv10*thetat_inv2) + 31254567775610.398*(pit10)*(thetat_inv2) - 4.15652812061591e-55*pit_inv10*thetat_inv5*thetat_inv3 - 100375333864186.0*pit10*pit2*thetat_inv3)
V = main*0.0031
return 1.0/V
[docs]def iapws97_region3_w(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.959
thetat = T*0.0015384615384615385 - 0.995
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
pit_inv10 = pit_inv5*pit_inv5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat5 = thetat3*thetat2
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (0.92326135790147001*pit*thetat_inv + 2.7170023573989301e-15*pit_inv*(thetat_inv2*thetat_inv2) + 2.3741673261664401e-27*pit_inv*(thetat_inv5*thetat_inv3) + thetat*(-90.788621348359996*pit_inv + thetat*(thetat*(-1033.08436323771*pit_inv3 + (thetat3)*((thetat2)*(-96110924.098574698*pit_inv5*pit_inv + (thetat5*thetat)*(-1.5854860965500201e+18*pit_inv5*pit_inv3 - 89446035500.552597*pit_inv10*pit_inv2) + 22827.6853990249*(pit_inv5*pit_inv3) + 0.109892402329239*(pit_inv10) - 5.8621913381701603e-8*pit_inv10*pit_inv2) - 0.0575368389425212*pit_inv5*pit_inv3) + 0.725937724828145*(pit_inv2*pit_inv2)) + 3219887.6763638901*(pit2) + 579.51404176570998*(pit_inv2) + 6.15762068640611e-9*(pit_inv5*pit_inv)) + 156.792067854621 - 0.066255281634216803*pit_inv2) - 5.9786598842257703*thetat_inv*pit2 - 4.7110372549807704e-13*thetat_inv*pit_inv2*pit_inv2 + 5.3116803751977397e-31*thetat_inv*(pit_inv10) + 4.9342908604698102e-8*(pit3)*(thetat_inv5) - 3.9944139004220299e-30*pit3*thetat_inv10*thetat_inv2 + 1.8776852576368201e-39*(pit_inv3)*(thetat_inv10) - 3.4082129141971899e-7*pit5*thetat_inv5*thetat_inv - 2.0761028465413701e-12*pit5*thetat_inv5*thetat_inv3 + 8.1203698337056496e-20*(pit5)*(thetat_inv10) - 4.0627428665262499e-45*pit_inv5*thetat_inv10 - 6.3498798119066896e-25*pit_inv5*pit_inv*thetat_inv3 + 3.29865748576503e-28*(pit_inv5*pit_inv)*(thetat_inv2*thetat_inv2) - 8.5671158651021397e-13*pit5*pit3*thetat_inv10 + 5.4200057337223301e-18*(pit5*pit3)*(thetat_inv10*thetat_inv2) + 8.58133791857099e-6*(pit10)*(thetat_inv5*thetat_inv3) + 2.6617045440598101e-14*(pit10)*(thetat_inv10*thetat_inv2) - 1.71242509570207e-37*thetat_inv10*thetat_inv2)
V = main*main
V *= V*0.0039
return 1.0/V
[docs]def iapws97_region3_x(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.347826086956522e-08 - 0.91
thetat = T*0.0015384615384615385 - 0.988
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
thetat_inv10 = thetat_inv5*thetat_inv5
main = (-0.32085055136733398*pit*thetat_inv + 2.1578022250902002e-27*pit*(thetat_inv10) + thetat*(thetat*(thetat*(thetat*(thetat*(64866249228.068199*pit_inv + thetat*(-38264244845861000.0*pit2 + (thetat2)*((thetat2)*(1.6989448143359199e+21 + (thetat2)*((thetat2)*(-4.2599956229273801e+23*pit_inv2*pit_inv2 + 3.7737374129815101e+18*(pit_inv5*pit_inv3)) + 1.0731906585576701e+21*(pit_inv3)) - 1033632255988600.0*pit_inv5 - 5071008837229.1299*pit_inv5*pit_inv) - 3.2606864627931401e+20*pit3 + 2.6241320970635799e+24*(pit5*pit3)))) - 8515357334.8425798) + 39794900155318.398*(pit2*pit2)) - 0.00092472937839094499*pit_inv2*pit_inv2) + 1.8479081432077301e-6*(pit_inv2*pit_inv2) - 43235522531.974503*pit5 - 592874245598.60999*pit5*pit + 25818961427085.301*(pit5*pit3)) + 2.44200600688281 - 563199.25339166599*pit3 - 2.7538607767442098e-29*pit3*thetat_inv10*thetat_inv2 - 4.6230777187397299e-13*pit_inv3*thetat_inv2 + 16223.4569738433*(pit5)*(thetat_inv2) + 1.00824008584757e-7*(pit5)*(thetat_inv5*thetat_inv) + 1573381.97797544*(pit5*pit3)*(thetat_inv3) + 1.3306164728110601*(pit5*pit3)*(thetat_inv5*thetat_inv) - 0.092001193743114204*pit10*thetat_inv5*thetat_inv3 - 592910695.76253605*pit10*pit2*thetat_inv2*thetat_inv2 + 8470048.7061208691*(pit10*pit2)*(thetat_inv5) - 11.043375910954699*pit10*pit2*thetat_inv5*thetat_inv3 + 0.0022021376590542598*(pit10*pit2)*(thetat_inv10) + 4308676.5806146804*(pit10*pit2*pit2)*(thetat_inv5*thetat_inv) - 1192.28759669889*pit10*pit2*pit2*thetat_inv5*thetat_inv3 + 0.18133960351630199*(pit10*pit2*pit2)*(thetat_inv10) - 1.8302717326966e-5*pit10*pit2*pit2*thetat_inv10*thetat_inv2)
V = main*0.0049
return 1.0/V
[docs]def iapws97_region3_y(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.545454545454546e-08 - 0.996
thetat = T*0.0015384615384615385 - 0.994
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit10 = pit5*pit5
thetat2 = thetat*thetat
thetat3 = thetat*thetat2
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
main = (thetat*(5834.4130522840696 + (thetat3)*(thetat*(thetat*((thetat2)*(-1.5909649090470799e+26*pit + 1.18973500934212e+25 - 2.66713136106469e+30*pit3) + 1.4933391705313e+27*(pit2*pit2)) - 13477896845792500.0 - 5.2711465785069603e+21*pit2) + 3.27777227273171e+18*(pit2) + 7.0510622439983402e+20*(pit3)) - 3818819062711000.0*pit5) + 496.21219715823901*thetat_inv*(pit2) - 3.15839902302021e-7*pit2*thetat_inv2*thetat_inv2 + 2.10017506281863e-17*(pit3)*(thetat_inv5*thetat_inv3) - 5.2559799502463301e-10*thetat_inv3 - 1.4537051255456199e-8*pit2*pit2*thetat_inv5*thetat_inv - 14979562.028764101*pit5*thetat_inv2 - 93780816955019.297*pit5*pit3*thetat_inv2 + 7.2466016558579696e-5*(pit5*pit3)*(thetat_inv5*thetat_inv3) + 5144114683.7638302*(pit10)*(thetat_inv5) - 82819.859404014103*pit10*pit2*thetat_inv5*thetat_inv3)
V = main*main
V *= V*0.0031
return 1.0/V
[docs]def iapws97_region3_z(T, P):
# This function was automatically generated. Do not edit it directly!
pit = P*4.545454545454546e-08 - 0.993
thetat = T*0.0015384615384615385 - 0.994
pit_inv = 1.0/pit
thetat_inv = 1.0/thetat
pit2 = pit*pit
pit3 = pit*pit2
pit5 = pit3*pit2
pit_inv2 = pit_inv*pit_inv
pit_inv3 = pit_inv*pit_inv2
pit_inv5 = pit_inv3*pit_inv2
thetat2 = thetat*thetat
thetat_inv2 = thetat_inv*thetat_inv
thetat_inv3 = thetat_inv*thetat_inv2
thetat_inv5 = thetat_inv3*thetat_inv2
main = (-8.1973155961052001e-21*pit_inv*thetat_inv5*thetat_inv + thetat*(-62500479.117154002*pit + thetat*(thetat*(328380587890.65997 + (thetat2)*(thetat*(8.0319795746202006e+20*(pit2) + (thetat2)*(9238140070232500.0*(pit_inv2*pit_inv2) + 3277763028588600.0*(pit_inv5)) - 167170186672140.0*pit_inv3 - 3238999157299.6001*pit_inv2*pit_inv2 + 7288032747.7770996*(pit_inv5) - 4630574.3033124004*pit_inv5*pit_inv) + 663221436245.51001*(pit_inv3) - 1105981701.1840999*pit_inv2*pit_inv2) + 2.4400789229064998e-11*(pit_inv5*pit_inv3)) + 2537.4935870139002*(pit_inv2))) - 68285901137.457001*thetat_inv*pit5*pit - 3783.9104705594*pit3*thetat_inv2 - 2.0439701133835e-11*pit3*thetat_inv5*thetat_inv + 8.4225008041371002e-13*(pit_inv3)*(thetat_inv2) - 3739.6286292864002*pit5*pit*thetat_inv2*thetat_inv2 + 15.435572168146001*(pit5*pit)*(thetat_inv5) + 0.0097287654593861995*(pit5*pit)*(thetat_inv5*thetat_inv) + 3945360.4949706998*(pit5*pit3)*(thetat_inv2*thetat_inv2) - 0.00024848801561453999*pit5*pit3*thetat_inv5*thetat_inv3)
V = main*main
V *= V*0.0038
return 1.0/V
region3_backwards_PT_doc = """
Calculate the mass density water in one of the 26 region 3
backwards regions of the IAPWS-97 standard.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
rho : float
Mass density of water in region 3, [kg/m^3]
Notes
-----
Significant discontinuities exist between each region.
These functions are automatically generated and are not to be edited directly.
"""
try:
for _ in (iapws97_region3_a, iapws97_region3_b, iapws97_region3_c, iapws97_region3_d,
iapws97_region3_e, iapws97_region3_f, iapws97_region3_g, iapws97_region3_h,
iapws97_region3_i, iapws97_region3_j, iapws97_region3_k, iapws97_region3_l,
iapws97_region3_m, iapws97_region3_n, iapws97_region3_o, iapws97_region3_p,
iapws97_region3_q, iapws97_region3_r, iapws97_region3_s, iapws97_region3_t,
iapws97_region3_u, iapws97_region3_v, iapws97_region3_w, iapws97_region3_x,
iapws97_region3_y, iapws97_region3_z):
_.__doc__ = region3_backwards_PT_doc
except: # except is needed for running Python under -OO flag
pass
def iapws97_region3_rho(T, P):
r'''Calculate the mass density of water in region 3 of the IAPWS-97 standard.
No cheking that the original point is in region 3 is performed.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
rho : float
Mass density of water in region 3, [kg/m^3]
Notes
-----
Significant discontinuities exist between each region.
Examples
--------
>>> iapws97_region3_rho(648.6, 22.5e6)
353.06081088726
'''
region = iapws97_region_3(T, P)
if region == REGION_3A:
rho = iapws97_region3_a(T, P)
elif region == REGION_3B:
rho = iapws97_region3_b(T, P)
elif region == REGION_3C:
rho = iapws97_region3_c(T, P)
elif region == REGION_3D:
rho = iapws97_region3_d(T, P)
elif region == REGION_3E:
rho = iapws97_region3_e(T, P)
elif region == REGION_3F:
rho = iapws97_region3_f(T, P)
elif region == REGION_3G:
rho = iapws97_region3_g(T, P)
elif region == REGION_3H:
rho = iapws97_region3_h(T, P)
elif region == REGION_3I:
rho = iapws97_region3_i(T, P)
elif region == REGION_3J:
rho = iapws97_region3_j(T, P)
elif region == REGION_3K:
rho = iapws97_region3_k(T, P)
elif region == REGION_3L:
rho = iapws97_region3_l(T, P)
elif region == REGION_3M:
rho = iapws97_region3_m(T, P)
elif region == REGION_3N:
rho = iapws97_region3_n(T, P)
elif region == REGION_3O:
rho = iapws97_region3_o(T, P)
elif region == REGION_3P:
rho = iapws97_region3_p(T, P)
elif region == REGION_3Q:
rho = iapws97_region3_q(T, P)
elif region == REGION_3R:
rho = iapws97_region3_r(T, P)
elif region == REGION_3S:
rho = iapws97_region3_s(T, P)
elif region == REGION_3T:
rho = iapws97_region3_t(T, P)
elif region == REGION_3U:
rho = iapws97_region3_u(T, P)
elif region == REGION_3V:
rho = iapws97_region3_v(T, P)
elif region == REGION_3W:
rho = iapws97_region3_w(T, P)
elif region == REGION_3X:
rho = iapws97_region3_x(T, P)
elif region == REGION_3Y:
rho = iapws97_region3_y(T, P)
elif region == REGION_3Z:
rho = iapws97_region3_z(T, P)
else:
raise ValueError("Could not identify region 3 subregion")
return rho
def iapws97_identify_region_TP(T, P, use_95_boundary=False):
r'''Identify the main region given a temperature and pressure point
according to the IAPWS-97 standard.
Raises a ValueError if the input point is out of bounds.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
use_95_boundary : bool, optional
If True, the SF-95 formulation vapor pressure fit will be used instead
of the simpler vapor pressure curve associated with IF-97, [-]
Returns
-------
region : int
One of 1, 2, 3, or 5; region 4 is the two-phase region which cannot
be reached with a TP call, [-]
Examples
--------
>>> iapws97_identify_region_TP(400, 1e6)
1
'''
under_623 = T <= 623.15
if under_623:
if use_95_boundary:
Psat = iapws95_Psat(T)
else:
Psat = Psat_IAPWS(T)
two_to_three = iapws97_boundary_2_3(T)
if 273.15 <= T <= 623.15 and Psat < P <= 100E6:
return 1
elif 273.15 <= T <= 623.15 and P <= Psat:
return 2
elif 623.15 <= T <= 1073.15 and P <= 100E6 and P <= two_to_three:
return 2
elif 623.15 <= T <= 1073.15 and P <= 100E6 and P > two_to_three:
return 3
elif 1073.15 <= T <= 2273.15 and P <= 50E6:
return 5
else:
raise ValueError("For box (1,2,3,4) 273.15 K <= T <= 1073.15 K and P <= 100 MPa; "
"for box 5, 1073.15 K <= T <= 2273.15 K and P <= 50 MPa.")
def iapws97_region1_rho(T, P):
# Useful to separate this out for confirming derivatives numerically
pi = P*6.049606775559589e-08 #1/16.53E6
tau = 1386.0/T
dG_dpi = iapws97_dG_dpi_region1(tau, pi)
return P/(iapws97_R*T*pi*dG_dpi)
def iapws97_region2_rho(T, P):
pi = P*1e-6
tau = 540.0/T
dG_dpi = 1.0/pi + iapws97_dGr_dpi_region2(tau, pi)
return P/(iapws97_R*T*pi*dG_dpi)
def iapws97_region5_rho(T, P):
pi = P*1e-6
tau = 1000.0/T
dG_dpi = 1.0/pi + iapws97_dGr_dpi_region5(tau, pi)
return P/(iapws97_R*T*pi*dG_dpi)
[docs]def iapws97_rho(T, P, use_95_boundary=False):
r'''Calculate the density of water in kg/m^3 according to the IAPWS-97
standard.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
use_95_boundary : bool, optional
If True, respect the IAPWS-95 vapor pressure curve instead of the IF-97
one, [-]
Returns
-------
rho : float
Mass density of water, [kg/m^3]
Notes
-----
The range of validity of this formulation is as follows:
For :math:`P \le 100 \text{ MPa}`:
.. math::
273.15 \text{ K} \le T \le 1073.15 \text{ K}
For :math:`P \le 50 \text{ MPa}`:
.. math::
1073.15 \text{ K} \le T \le 2273.15 \text{ K}
A ValueError is raised if the temperature or the pressure is out of bounds.
IAPWS is implemented in four regions in the `T`-`P` domain:
Region 1 (liquid), region 2 (gas and supercritical gas), region 5
(high temperature gas), and region 3 (near-critical).
Significant discontinuities exist between the transitions of each regions.
In region 3, there are 26 sub-regions and the correlation has the least
accuracy.
For many applications, the discontinuities in IF-97 can be problematic and
the slower IAPWS-95 must be used. IAPWS-95 also has a wider range of
applicability.
Examples
--------
>>> iapws97_rho(648.6, 22.5e6)
353.06081088726
>>> iapws97_rho(330.0, 8e5)
985.10498080770
>>> iapws97_rho(823.0, 14e6)
40.39293607288123
>>> iapws97_rho(2000.0, 3e7)
32.11456228328856
References
----------
.. [1] Cooper, JR, and RB Dooley. "Revised Release on the IAPWS Industrial
Formulation 1997 for the Thermodynamic Properties of Water and Steam."
The International Association for the Properties of Water and Steam 1
(2007): 48.
'''
region = iapws97_identify_region_TP(T, P, use_95_boundary)
if region == 1:
return iapws97_region1_rho(T, P)
elif region == 2:
return iapws97_region2_rho(T, P)
elif region == 3:
return iapws97_region3_rho(T, P)
elif region == 5:
return iapws97_region5_rho(T, P)
else:
raise ValueError("Out of bounds")
def iapws97_rho_extrapolated(T, P, use_95_boundary=False):
# Intended to extend the range using first derivatives
# for use in iapws-95 solver.
out_of_range = (T < 273.15
or T > 2273.15
or P > 100e6
or (T > 1073.15 and 50e6 <= P <= 100e6))
if not out_of_range:
return iapws97_rho(T, P, use_95_boundary)
if T > 2273.15 and P < 50E6:
T_border = 2273.15
Pref = 1e6
Tref = 1e3
P_inv = 1.0/P
pi = P*1e-6
tau = Tref/T_border
# region 5 upwards T extrapolate drho_dT
dGr_dpi = iapws97_dGr_dpi_region5(tau, pi)
dG_dpi = 1e6*P_inv + dGr_dpi
rho = P/(iapws97_R*T_border*pi*dG_dpi)
d2Gr_dpidtau = iapws97_d2Gr_dpidtau_region5(tau, pi)
x0 = (dGr_dpi + Pref*P_inv)
x1 = iapws97_R*T_border*T_border
drho_dT = -Pref/(x1*x0) + Pref*Tref*d2Gr_dpidtau/(x1*T_border*x0*x0)
rho = rho + drho_dT*(T - T_border)
elif T > 1073.15 and 50e6 <= P <= 100e6:
T_border = 1073.15
Pref = 1e6
Tref = 540.0
P_inv = 1.0/P
pi = P*1e-6
tau = Tref/T_border
# region 5 upwards T extrapolate drho_dT
dGr_dpi = iapws97_dGr_dpi_region2(tau, pi)
dG_dpi = 1e6*P_inv + dGr_dpi
rho = P/(iapws97_R*T_border*pi*dG_dpi)
d2Gr_dpidtau = iapws97_d2Gr_dpidtau_region2(tau, pi)
x0 = (dGr_dpi + Pref*P_inv)
x1 = iapws97_R*T_border*T_border
drho_dT = -Pref/(x1*x0) + Pref*Tref*d2Gr_dpidtau/(x1*T_border*x0*x0)
drho = drho_dT*(T - T_border)
if (rho + drho) > .1*rho:
# Do not take the extrapolation if density has decreased too much
rho = rho + drho
elif T < 273.15:
# region 1 - fairly incompressible, don't bother extrapolating
if P > 100e6: P = 100e6
rho = iapws97_region1_rho(273.15, P)
else:
# Don't bother extrapolating to higher P at this point, the density
# change is small there
if P > 100e6:
P = 100e6
if T > 1073.15:
T = 1073.15
rho = iapws97_rho(T, P, use_95_boundary)
return rho
def iapws_97_Trho_err_region1(P, T, rho):
pi_region1 = P*6.049606775559589e-08 #1/16.53E6
tau_region1 = 1386.0/T
dG_dpi_region1 = iapws97_dG_dpi_region1(tau_region1, pi_region1)
rhol = P/(iapws97_R*T*pi_region1*dG_dpi_region1)
err = rhol - rho
d2G_dpi2_region1 = iapws97_d2G_dpi2_region1(tau_region1, pi_region1)
derr = -d2G_dpi2_region1/(iapws97_R*T*dG_dpi_region1*dG_dpi_region1)
# print(P, err, derr)
return err, derr
def iapws_97_Trho_err_region2(P, T, rho):
pi_region2 = P*1e-6
tau_region2 = 540.0/T
dG_dpi_region2 = 1/pi_region2 + iapws97_dGr_dpi_region2(tau_region2, pi_region2)
rhog = P/(iapws97_R*T*pi_region2*dG_dpi_region2)
err = rhog - rho
d2G_dpi2_region2 = iapws97_d2Gr_dpi2_region2(tau_region2, pi_region2)
d2G_dpi2_region2 -= 1e12/(P*P) # ideal part
# checked numerically
derr = -d2G_dpi2_region2/(iapws97_R*T*dG_dpi_region2*dG_dpi_region2)
# print(P, T, rho, err, derr)
return err, derr
def iapws_97_Trho_err_region5(P, T, rho):
pi_region5 = P*1e-6
tau_region5 = 1000.0/T
dG_dpi_region5 = 1/pi_region5 + iapws97_dGr_dpi_region5(tau_region5, pi_region5)
rhog = P/(iapws97_R*T*pi_region5*dG_dpi_region5)
err = rhog - rho
d2G_dpi2_region5 = iapws97_d2Gr_dpi2_region5(tau_region5, pi_region5)
d2G_dpi2_region5 -= 1e12/(P*P) # ideal part
derr = -d2G_dpi2_region5/(iapws97_R*T*dG_dpi_region5*dG_dpi_region5)
return err, derr
[docs]def iapws97_P(T, rho):
r'''Calculate the pressure of water according to the IAPWS-97
standard given a temperature `T` and mass density `rho`.
Parameters
----------
T : float
Temperature, [K]
rho : float
Mass density of water, [kg/m^3]
Returns
-------
P : float
Pressure, [Pa]
Notes
-----
The range of validity of this formulation is as follows:
For :math:`P \le 100 \text{ MPa}`:
.. math::
273.15 \text{ K} \le T \le 1073.15 \text{ K}
For :math:`P \le 50 \text{ MPa}`:
.. math::
1073.15 \text{ K} \le T \le 2273.15 \text{ K}
A ValueError is raised if the temperature or density is out of bounds.
Newton's method with analytical derivatives is used here to solve these
equations. The solver tolerance is as tight as it can be without causing
wasted iterations that do not improve the result at all. Pressure changes
quickly with density however, and some discrepancy between solvers is to be
expected.
For region 3, there are really two formulations present in IAPWS-97. There
is a Helmholtz energy equation (Temperature and density dependent), and
also 26 separate backwards equations for `rho` which depend on `T` and `P`.
The Helmholtz energy equation is much more accurate and does not have
discontinuities. The two sets of equations agree closely not not perfectly.
By design, :obj:`iapws97_rho` implements the 26 T-P equations and this
implements the Helmholtz energy equation. This means that in region 3
solutions will not be consistent. For consistency requirements, IAPWS-95
is recommended.
This solver does not have any issues with multiple solutions. The solvers
have been checked to achieve a relative solution tolerance of 5e-9 on
100 million points.
Examples
--------
>>> iapws97_P(330.0, iapws97_rho(T=330.0, P=8e5))
8e5
>>> iapws97_P(823.0, 40.39293607288123)
14e6
>>> iapws97_P(T=2000.0, rho=32.11456228328856)
3e7
Region 3 point - does not implement the same equations as
:obj:`iapws97_rho`!
>>> iapws97_P(648.6, iapws97_rho(T=648.6, P=22.5e6))
22499974.093936257
References
----------
.. [1] Cooper, JR, and RB Dooley. "Revised Release on the IAPWS Industrial
Formulation 1997 for the Thermodynamic Properties of Water and Steam."
The International Association for the Properties of Water and Steam 1
(2007): 48.
'''
if T < 273.15:
raise ValueError("T is under minimum value of 273.15 K")
elif T <= 1073.15:
if T <= 623.15:
# region 2 to region 1 only - easy solver under 623K
# Compute the density borders at the saturation pressure, and then decide which to pursue
Psat = Psat_IAPWS(T)
pi_region2 = Psat*1e-6
tau_region2 = 540.0/T
dG_dpi_region2 = 1.0/pi_region2 + iapws97_dGr_dpi_region2(tau_region2, pi_region2)
rhog_sat = Psat/(iapws97_R*T*pi_region2*dG_dpi_region2)
pi_region1 = Psat*6.049606775559589e-08 #1/16.53E6
tau_region1 = 1386.0/T
dG_dpi_region1 = iapws97_dG_dpi_region1(tau_region1, pi_region1)
rhol_sat = Psat/(iapws97_R*T*pi_region1*dG_dpi_region1)
if rhog_sat < rho < rhol_sat:
raise ValueError("Specified density is not a stable state at T")
elif rho > rhol_sat:
return newton(iapws_97_Trho_err_region1, Psat*10.0, fprime=True, bisection=True,
low=Psat, high=100e6, args=(T, rho), xtol=3e-12)
else:
return newton(iapws_97_Trho_err_region2, Psat*.1, fprime=True, bisection=True,
low=Psat*1e-20, high=Psat, args=(T, rho), xtol=3e-12)
P_region2_border = iapws97_boundary_2_3(T)
pi_region2 = P_region2_border*1e-6
tau_region2 = 540.0/T
dG_dpi_region2 = 1.0/pi_region2 + iapws97_dGr_dpi_region2(tau_region2, pi_region2)
rhog_region2_border = P_region2_border/(iapws97_R*T*pi_region2*dG_dpi_region2)
if rho < rhog_region2_border or P_region2_border > 100e6:
return newton(iapws_97_Trho_err_region2, P_region2_border*.1, fprime=True, bisection=True,
low=P_region2_border*1e-20, high=P_region2_border, args=(T, rho), xtol=3e-12)
else:
# region 3
tau = iapws95_Tc / T
delta = rho * iapws95_rhoc_inv
dA_ddelta = iapws97_dA_ddelta_region3(tau, delta)
return dA_ddelta*delta*rho*iapws97_R*T
elif T <= 2273.15:
return newton(iapws_97_Trho_err_region5, 1e6, fprime=True, bisection=True,
low=1e-10, high=50e6, args=(T, rho), xtol=1e-12)
else:
raise ValueError("T is above maximum value of 2273.15 K")
def iapws_97_Prho_err_region1(T, P, rho):
pi_region1 = P*6.049606775559589e-08 #1/16.53E6
tau_region1 = 1386.0/T
dG_dpi_region1 = iapws97_dG_dpi_region1(tau_region1, pi_region1)
rhol = P/(iapws97_R*T*pi_region1*dG_dpi_region1)
err = rhol - rho
# what it is supposed to be
drhol = (-16.53E6/(iapws97_R*T*T*dG_dpi_region1)
+ 16.53E6*1386.0*iapws97_d2G_dpidtau_region1(tau_region1, pi_region1)/(iapws97_R*T*T*T*dG_dpi_region1*dG_dpi_region1))
return err, drhol
def iapws_97_Prho_err_region2(T, P, rho):
pi_region2 = P*1e-6
tau_region2 = 540.0/T
dG_dpi_region2 = 1.0/pi_region2 + iapws97_dGr_dpi_region2(tau_region2, pi_region2)
rhol = P/(iapws97_R*T*pi_region2*dG_dpi_region2)
err = rhol - rho
drhol = (-1e6/(iapws97_R*T*T*dG_dpi_region2)
+ 1E6*540.0*iapws97_d2Gr_dpidtau_region2(tau_region2, pi_region2)/(iapws97_R*T*T*T*dG_dpi_region2*dG_dpi_region2))
return err, drhol
def iapws_97_Prho_err_region5(T, P, rho):
pi_region5 = P*1e-6
tau_region5 = 1000/T
dG_dpi_region5 = 1.0/pi_region5 + iapws97_dGr_dpi_region5(tau_region5, pi_region5)
rhol = P/(iapws97_R*T*pi_region5*dG_dpi_region5)
err = rhol - rho
drhol = (-1e6/(iapws97_R*T*T*dG_dpi_region5)
+ 1E6*1000*iapws97_d2Gr_dpidtau_region5(tau_region5, pi_region5)/(iapws97_R*T*T*T*dG_dpi_region5*dG_dpi_region5))
return err, drhol
def iapws_97_Prho_err_region3(T, P, rho):
tau = iapws95_Tc / T
delta = rho * iapws95_rhoc_inv
dA_ddelta = iapws97_dA_ddelta_region3(tau, delta)
P_calc = dA_ddelta*delta*rho*iapws97_R*T
err = P_calc - P
d2A_ddeltadtau = iapws97_d2A_ddeltadtau_region3(tau, delta)
derr = iapws97_R*rho**2*dA_ddelta/iapws95_rhoc - iapws97_R*iapws95_Tc*rho**2*d2A_ddeltadtau/(T*iapws95_rhoc)
return err, derr
[docs]@mark_numba_uncacheable
def iapws97_T(P, rho):
r'''Calculate the temperature of water according to the IAPWS-97
standard given a pressure `P` and mass density `rho`.
Parameters
----------
P : float
Pressure, [Pa]
rho : float
Mass density of water, [kg/m^3]
Returns
-------
T : float
Temperature, [K]
Notes
-----
The range of validity of this formulation is as follows:
For :math:`P \le 100 \text{ MPa}`:
.. math::
273.15 \text{ K} \le T \le 1073.15 \text{ K}
For :math:`P \le 50 \text{ MPa}`:
.. math::
1073.15 \text{ K} \le T \le 2273.15 \text{ K}
A ValueError is raised if the pressure or density is out of bounds.
Newton's method with analytical derivatives is used here to solve these
equations. The solver tolerance is as tight as it can be without causing
wasted iterations that do not improve the result at all.
Due to water's unique density curve, there is a temperature region
spanning 273.15 K to 280.005 K where there are two solutions. No guarantee
is made as to which solution will be returned.
Examples
--------
>>> iapws97_T(8e5, iapws97_rho(T=330.0, P=8e5))
330.0
>>> iapws97_T(14e6, 40.39293607288123)
823.0
>>> iapws97_T(P=3e7, rho=32.11456228328856)
2000.0
References
----------
.. [1] Cooper, JR, and RB Dooley. "Revised Release on the IAPWS Industrial
Formulation 1997 for the Thermodynamic Properties of Water and Steam."
The International Association for the Properties of Water and Steam 1
(2007): 48.
'''
solve_region = 0
if P > 100e6:
raise ValueError("P is above maximum value of 100 MPa")
if P < 50E6:
# Calculate the 2-5 border using region 5's equations
pi_5_border = P*1e-6
tau_5_border = 1000.0/1073.15
dG_dpi_5_border = 1.0/pi_5_border + iapws97_dGr_dpi_region5(tau_5_border, pi_5_border)
rho_25_border = P/(iapws97_R*1073.15*pi_5_border*dG_dpi_5_border)
# get rho minimum at 1073.15
if rho <= rho_25_border:
rho_5end_border = iapws97_rho(2273.15, P)
if rho < rho_5end_border:
raise ValueError("Density is lower then region 5 limit")
else:
solve_region = 5
if solve_region == 0:
no_region_3 = False
if P > 13918839.778885445:
T_23 = iapws97_boundary_2_3_reverse(P)
if T_23 < 623.15:
no_region_3 = True
else:
no_region_3 = True
if no_region_3:
if P < 0.005706860511498897: # where the vapor pressure equation dies
solve_region = 2 # always gas
T_23 = 273.15
else:
Tsat = Tsat_IAPWS(P)
if Tsat < 273.15:
solve_region = 2 # always gas
T_23 = 273.15
else:
pi_region1_sat = P*6.049606775559589e-08 #1/16.53E6
tau_region1_sat = 1386.0/Tsat
dG_dpi_region1_sat = iapws97_dG_dpi_region1(tau_region1_sat, pi_region1_sat)
rho1_sat = P/(iapws97_R*Tsat*pi_region1_sat*dG_dpi_region1_sat)
pi_region2_sat = P*1e-6
tau_region2_sat = 540.0/Tsat
dG_dpi_region2_sat = 1.0/pi_region2_sat + iapws97_dGr_dpi_region2(tau_region2_sat, pi_region2_sat)
rho2_sat = P/(iapws97_R*Tsat*pi_region2_sat*dG_dpi_region2_sat)
# if rho2_sat < rho < rho1_sat:
# raise ValueError("At specified pressure, density is not a stable solution")
if rho > rho2_sat:
solve_region = 1
else:
solve_region = 2
T_23 = Tsat # for initial guess bondary only
else:
rho_23_side3 = iapws97_rho(T_23 * (1 + 1e-12), P)
# Calculate the 2-5 border using region 2's equations
pi_region2_25_on2 = P*1e-6
tau_region2_25_on2 = 540.0/1073.15
dG_dpi_region2_25_on2 = 1.0/pi_region2_25_on2 + iapws97_dGr_dpi_region2(tau_region2_25_on2, pi_region2_25_on2)
rho2_25_on2 = P/(iapws97_R*1073.15*pi_region2_25_on2*dG_dpi_region2_25_on2)
if rho2_25_on2 <= rho <= rho_23_side3:
solve_region = 2
if solve_region == 0:
rho_13 = iapws97_rho(623.15-1e-12, P)
if rho_23_side3 <= rho <= rho_13:
solve_region = 3
if solve_region == 0:
rho_Tmin_border = iapws97_rho(273.15, P)
if rho_13 <= rho <= rho_Tmin_border:
solve_region = 1
else:
# Sometimes liquid water has a higher density at a higher temperature. Weird!
solve_region = 1
if solve_region == 5:
return newton(iapws_97_Prho_err_region5, 1673.15, fprime=True, bisection=True,
low=1073.15, high=2273.15, args=(P, rho), xtol=1e-12)
elif solve_region == 2:
return newton(iapws_97_Prho_err_region2, 0.5*(T_23 + 1073.15), fprime=True, bisection=True,
low=T_23 , high=1073.15, args=(P, rho), xtol=1e-12)
elif solve_region == 3:
return newton(iapws_97_Prho_err_region3, 0.5*(T_23 + 623.15), fprime=True, bisection=True,
low=623.15 , high=T_23, args=(P, rho), xtol=1e-12)
elif solve_region == 1:
try:
Tsat = Tsat_IAPWS(P)
except:
Tsat = 623.15
return newton(iapws_97_Prho_err_region1, 0.5*(Tsat + 273.15), fprime=True, bisection=True,
low=273.15 , high=Tsat, args=(P, rho), xtol=1e-12)
else:
raise ValueError("Could not detect region")
### IAPWS95
### IAPWS 95 Initial Guesses
[docs]def iapws92_rhol_sat(T):
r'''Calculates saturation liquid mass density of water using the IAPWS
SR1-86(1992) [1]_ [2]_ explicit equation.
.. math::
\frac{\rho^{sat}_l}{\rho_c} = 1 + b_1\tau^{1/3} + b_2\tau^{2/3}
+ b_3 \tau^{5/3} + b_4\tau^{16/3} + b_5\tau^{43/3} + b_6\tau^{110/3}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
rhol_sat : float
Saturation liquid mass density of water [kg/m^3]
Notes
-----
This equation is fit to experimental data to within its accuracy. It does
not satisfy the equilibrium conditions for the IAPWS-95 or IAPWS-97
formulations.
The values of the constants are as follows:
b1 = 1.99274064;
b2 = 1.09965342;
b3 = -0.510839303;
b4 = -1.75493479;
b5 = -45.5170352;
b6 = -6.74694450e5
See Also
--------
iapws95_rhol_sat
Examples
--------
>>> iapws92_rhol_sat(300.)
996.5089712803
References
----------
.. [1] IAPWS, Secretariat, B Dooley, and EPRI. "Revised Supplementary
Release on Saturation Properties of Ordinary Water Substance", 1992.
.. [2] Wagner, Wolfgang, and A. Pruss. "International Equations for the
Saturation Properties of Ordinary Water Substance. Revised According to
the International Temperature Scale of 1990. Addendum to J. Phys. Chem.
Ref. Data 16, 893 (1987)." Journal of Physical and Chemical Reference
Data 22, no. 3 (May 1, 1993): 783-87. https://doi.org/10.1063/1.555926.
'''
tau = 1.0 - T * iapws95_Tc_inv
tau_cbrt = tau**(1.0/3.0)
ratio = 1.0 + 1.99274064*tau_cbrt
tau_cbrt4 = tau_cbrt*tau_cbrt # still 2 for first term
ratio += 1.09965342*tau_cbrt4 # still b2*tau^(2/3)
tau_cbrt4 = tau_cbrt4*tau_cbrt4
ratio += -0.510839303*tau_cbrt4*tau_cbrt
tau_cbrt8 = tau_cbrt4*tau_cbrt4
ratio += -1.75493479*tau_cbrt8*tau_cbrt8
tau_cbrt4 = tau_cbrt8*tau_cbrt # repurpse 4 as 9
tau_cbrt = tau_cbrt4*tau_cbrt8 # 17 - repurpose variable as tau_cbrt34
tau_cbrt *= tau_cbrt # 17+17 = 34
ratio += tau_cbrt*(-45.5170352*tau_cbrt4 -6.74694450e5*tau_cbrt*tau_cbrt*tau_cbrt8)
# ratio += -6.74694450e5*tau_cbrt*tau_cbrt*tau_cbrt*tau_cbrt8
return ratio * iapws95_rhoc
[docs]def iapws92_rhog_sat(T):
r'''Calculates saturation vapor mass density of water using the IAPWS
SR1-86(1992) [1]_ [2]_ explicit equation.
.. math::
\ln \left(\frac{\rho^{sat}_g}{\rho_c}\right) = 1 + c_1\tau^{2/6} + c_2\tau^{4/6}
+ c_3 \tau^{8/6} + c_4\tau^{18/6} + c_5\tau^{37/6} + c_6\tau^{71/6}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
rhog_sat : float
Saturation vapor mass density of water [kg/m^3]
Notes
-----
This equation is fit to experimental data to within its accuracy. It does
not satisfy the equilibrium conditions for the IAPWS-95 or IAPWS-97
formulations.
The values of the constants are as follows:
c1 = -2.03150240;
c2 = -2.68302940;
c3 = -5.38626492;
c4 = -17.2991605;
c5 = -44.7586581;
c6 = -63.9201063
See Also
--------
iapws95_rhog_sat
Examples
--------
>>> iapws92_rhog_sat(300.)
0.0255887212886
References
----------
.. [1] IAPWS, Secretariat, B Dooley, and EPRI. "Revised Supplementary
Release on Saturation Properties of Ordinary Water Substance", 1992.
.. [2] Wagner, Wolfgang, and A. Pruss. "International Equations for the
Saturation Properties of Ordinary Water Substance. Revised According to
the International Temperature Scale of 1990. Addendum to J. Phys. Chem.
Ref. Data 16, 893 (1987)." Journal of Physical and Chemical Reference
Data 22, no. 3 (May 1, 1993): 783-87. https://doi.org/10.1063/1.555926.
'''
tau = 1.0 - T * iapws95_Tc_inv
tau_6rt = tau**(1.0/6.0)
tau_6rt2 = tau_6rt*tau_6rt
ratio = -2.03150240*tau_6rt2
tau_6rt8 = tau_6rt2*tau_6rt2 # start it off as 4
ratio += -2.68302940*tau_6rt8
tau_6rt8 *= tau_6rt8
ratio += -5.38626492*tau_6rt8
tau_6rt16 = tau_6rt8*tau_6rt8
tau_6rt18 = tau_6rt16*tau_6rt2
ratio += -17.2991605*tau_6rt18
tau_6rt2 = tau_6rt18*tau_6rt18*tau_6rt # 37 - reuse tau_6rt2
ratio += tau_6rt2*(-44.7586581 - 63.9201063*tau_6rt16*tau_6rt18) # 71
return exp(ratio) * iapws95_rhoc
### IAPWS 95 fundamental derivatives
[docs]def iapws95_A0(tau, delta):
r'''Calculates the ideal gas Helmholtz energy of water according to the
IAPWS-95 standard.
.. math::
\phi^\circ = \ln \delta + n_1 + n_2\tau + n_3\ln \tau
+ \sum_{i=4}^8 n_i \ln \left[1 - \exp(-\gamma_i \tau) \right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
A0 : float
Ideal gas dimensionless Helmholtz energy A/(RT) [-]
Notes
-----
This implementation is checked to have a relative error always under 1e-15.
Examples
--------
>>> iapws95_A0(647.096/300.0, 999.0/322)
9.537075529761053
'''
# only way to optimize is likely to save transcendentals between calls
# It should also be possible to replace the tau bit with polynomial fits
# which are nearly bit for bit identical, including the derivatives.
return (6.68321052759320011*tau + log(delta) + 3.00632*log(tau)
+ 0.24873*log(1. - exp(-27.5075105*tau))
+ 0.96956*log(1. - exp(-9.24437796*tau))
+ 1.2795*log(1. - exp(-7.74073708*tau))
+ 0.97315*log(1. - exp(-3.53734222*tau))
+ 0.012436*log(1. - exp(-1.28728967*tau))
- 8.32044648374970031)
[docs]def iapws95_dA0_dtau(tau, delta):
r'''Calculates the first derivative of ideal gas Helmholtz energy of water
with respect to `tau` according to the IAPWS-95 standard.
.. math::
\frac{\partial \phi^\circ}{\partial \tau} = n_2 + \frac{n_3}{\tau}
+ \sum_{i=4}^8 n_i\gamma_i \left[\left(1-\exp(-\gamma_i \tau)
\right)^{-1} - 1 \right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
dA0_dtau : float
First derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
Notes
-----
This implementation is checked to have a relative error always under 1e-15.
Examples
--------
>>> iapws95_dA0_dtau(647.096/300.0, 999.0/322)
8.079705548882
'''
return (-22.4843580635585205
+ 0.01600873433612/(1.0 - exp(-1.28728967*tau))
+ 3.44236458139299994/(1.0 - exp(-3.53734222*tau))
+ 9.90427309386000054/(1.0 - exp(-7.74073708*tau))
+ 8.96297909489759981/(1.0 - exp(-9.24437796*tau))
+ 6.84194308666500017/(1.0 - exp(-27.5075105*tau))
+ 3.00632/tau)
[docs]def iapws95_d2A0_dtau2(tau, delta):
r'''Calculates the second derivative of ideal gas Helmholtz energy of water
with respect to `tau` according to the IAPWS-95 standard.
.. math::
\frac{\partial^2 \phi^\circ}{\partial \tau^2} = \frac{n_3}{\tau^2}
+ \sum_{i=4}^8 n_i\gamma_i ^2 \exp(-\gamma_i \tau)
\left[\left(1-\exp(-\gamma_i \tau)
\right)^{-2}\right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2A0_dtau2 : float
Second derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
Notes
-----
This implementation is checked to have a relative error always under 1e-15.
Examples
--------
>>> iapws95_d2A0_dtau2(647.096/300.0, 999.0/322)
-0.653543047751809
'''
x0 = exp(-27.5075105*tau)
x1 = exp(-9.24437796*tau)
x2 = exp(-7.74073708*tau)
x3 = exp(-3.53734222*tau)
x4 = exp(-1.28728967*tau)
return (-188.204821296839867*x0/((1.0 - x0)*(1.0 - x0))
- 82.8571664008121047*x1/((1.0 - x1)*(1.0 - x1))
- 76.6663739880884236*x2/((1.0 - x2)*(1.0 - x2))
- 12.1768215703940861*x3/((1.0 - x3)*(1.0 - x3))
- 0.0206078783406615819*x4/((1.0 - x4)*(1.0 - x4))
- 3.00632/(tau*tau))
[docs]def iapws95_d3A0_dtau3(tau, delta):
r'''Calculates the third derivative of ideal gas Helmholtz energy of water
with respect to `tau` according to the IAPWS-95 standard.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d3A0_dtau3 : float
Third derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
Notes
-----
This implementation is checked to have a relative error always under 1e-15.
This equation is not explicitly in IAPWS-95, but is needed to compute some
second derivatives.
Examples
--------
>>> iapws95_d3A0_dtau3(647.096/300.0, 999.0/322)
0.6222542507278
'''
tot = 0.0
x0 = exp(-27.5075105*tau)
x1 = 1.0/(1.0 - x0)
x2 = exp(-9.24437796*tau)
x3 = 1.0/(1.0 - x2)
x4 = exp(-7.74073708*tau)
x5 = 1.0/(1.0 - x4)
x6 = exp(-3.53734222*tau)
x7 = 1.0/(1.0 - x6)
x8 = exp(-1.28728967*tau)
x9 = 1.0/(1.0 - x8)
return (x1*x1*x0*(5177.04609797344619
+ 10354.0921959468924*x0*x1)
+ (x3*x3*x2)*(765.962962903719927
+ 1531.92592580743985*x2*(x3))
+ x5*x5*x4*(593.454243918743487
+ 1186.90848783748697*x4*(x5))
+ x7*x7*x6*(86.1471700927234139*x6*(x7)
+ 43.0735850463617069)
+ x9*x9*x8*(0.0265283089085503951
+ 0.0530566178171007902*x8*(x9))
+ 6.01264/(tau*tau*tau))
[docs]def iapws95_A0_tau_derivatives(tau, delta):
r'''Calculates the ideal gas Helmholtz energy of water
and its first three derivatives with respect to `tau` according to the
IAPWS-95 standard. As each of those calls spends most of their time
computing exponentials which are the same for each function, function
offers a time saving.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
A0 : float
Ideal gas dimensionless Helmholtz energy A/(RT) [-]
dA0_dtau : float
First derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
d2A0_dtau2 : float
Second derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
d3A0_dtau3 : float
Third derivative of ideal gas dimensionless Helmholtz energy A/(RT)
with respect to `tau` [-]
Notes
-----
The extra cost of calling this function vs :obj:`iapws95_A0` alone is
~15% with numba, ~40% with PyPy, and 120% with CPython.
Examples
--------
>>> iapws95_A0_tau_derivatives(647.096/300.0, 999.0/322)
(9.53707552976, 8.0797055488, -0.65354304775, 0.62225425072)
'''
_exp, _log = exp, log
# 6 divisions, 5 exp, 7 log
x0 = _exp(-27.5075105*tau)
x1 = 1.0/(1.0 - x0)
x2 = _exp(-9.24437796*tau)
x3 = 1.0/(1.0 - x2)
x4 = _exp(-7.74073708*tau)
x5 = 1.0/(1.0 - x4)
x6 = _exp(-3.53734222*tau)
x7 = 1.0/(1.0 - x6)
x8 = _exp(-1.28728967*tau)
x9 = 1.0/(1.0 - x8)
tau_inv = 1.0/tau
A0 = (6.68321052759320011*tau + _log(delta) + 3.00632*_log(tau)
+ 0.24873*_log(1. - x0)
+ 0.96956*_log(1. - x2)
+ 1.2795*_log(1. - x4)
+ 0.97315*_log(1. - x6)
+ 0.012436*_log(1. - x8)
- 8.32044648374970031)
dA0_dtau = (-22.4843580635585205
+ 0.01600873433612*x9
+ 3.44236458139299994*x7
+ 9.90427309386000054*x5
+ 8.96297909489759981*x3
+ 6.84194308666500017*x1
+ 3.00632*tau_inv)
x0 *= x1
x2 *= x3
x4 *= x5
x6 *= x7
x8 *= x9
d2A0_dtau2 = (-188.204821296839867*x0*x1
- 82.8571664008121047*x2*x3
- 76.6663739880884236*x4*x5
- 12.1768215703940861*x6*x7
- 0.0206078783406615819*x8*x9
- 3.00632*tau_inv*tau_inv)
d3A0_dtau3 = (x1*x0*(5177.04609797344619
+ 10354.0921959468924*x0)
+ (x3*x2)*(765.962962903719927
+ 1531.92592580743985*x2)
+ x5*x4*(593.454243918743487
+ 1186.90848783748697*x4)
+ x7*x6*(86.1471700927234139*x6
+ 43.0735850463617069)
+ x9*x8*(0.0265283089085503951
+ 0.0530566178171007902*x8)
+ 6.01264*tau_inv*tau_inv*tau_inv)
return (A0, dA0_dtau, d2A0_dtau2, d3A0_dtau3)
[docs]def iapws95_Ar(tau, delta):
r'''Calculates the residual Helmholtz energy of water according to the
IAPWS-95 standard.
.. math::
\phi^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} \delta^{d_{i}}
\tau^{t_{i}}+\sum_{i=8}^{51} n_{i} \delta^{d_{i}} \tau^{t_{i}}
\mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}}
\tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}
\right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}+\sum_{i=55}^{56}
n_{i} \Delta^{b_{i}} \delta \psi
.. math::
\Delta=\theta^{2}+B_{i}\left[(\delta-1)^{2}\right]^{a_{i}}
.. math::
\theta=(1-\tau)+A_{i}\left[(\delta-1)^{2}\right]^{\frac{1}{2 \beta_{i}}}
.. math::
\psi=e^{-C_{i}(\delta-1)^{2}-D_{i}(\tau-1)^{2}}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
Ar : float
Residual Helmholtz energy A/(RT) [-]
Notes
-----
This is an optimized implementatation taking 9 exp calls, 4 sqrts,
and 3 powers. It was generated using SymPy's CSE functionality, with
select polynomial optimizations by hand as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative
error was 5.0416E-15, with a maximum relative error of 1.118E-9 and a
standard deviation of 5.773e-13.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 90000 points, the mean relative
error was 3.14E-15, with a maximum relative error of 3.54e-12 and a
standard deviation of 3.017E-14.
This comparison indicates that this implementation is more accurate than
the straightforward implementation.
Examples
--------
>>> iapws95_Ar(647.096/300.0, 999.0/322)
-9.57577716026768
'''
_sqrt, _exp = sqrt, exp
taurt = _sqrt(tau)
tau_quarter = _sqrt(taurt)
tau_eighth = _sqrt(tau_quarter) # tau checked, is not causing the small discrepancies.
deltam1sqr = (delta - 1.0)
deltam1sqr *= deltam1sqr
taum1sqr = (tau - 1.0)
taum1sqr *= taum1sqr
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau*tau3
tau6 = tau4*tau2
x29 = tau4*tau4
tau9 = tau6*tau3
tau10 = tau*tau9
tau11 = tau10*tau
tau13 = tau10*tau3
tau50 = tau10*tau10
tau50 *= tau50*tau10
delta2 = delta*delta
delta4 = delta2*delta2
delta5 = delta*delta4
delta6 = delta*delta5
delta8 = delta6*delta2
delta9 = delta*delta8
delta3 = delta*delta2
expnd = _exp(-delta)
expnd2 = _exp(-delta2)
expnd3 = _exp(-delta3)
expnd6 = _exp(-delta6)
tauexpnd = tau*expnd
tau4expnd = expnd*tau4
deltaexpnd2 = delta*expnd2
tau7expnd2 = tau6*tau*expnd2
delta3expnd = delta3*expnd
delta8expnd2 = delta8*expnd2
tau10expnd2 = tau10*expnd2
tau23expnd3 = tau10*tau13*expnd3
delta5expnd6 = expnd6*delta5
delta9expnd2 = delta9*expnd2
x32 = -20.0*deltam1sqr
x33 = (0.826446280991736*tau - 1.0)
x33 = delta2*_exp(x32 - 219.615*x33*x33)
x50 = (-tau + 0.32*(deltam1sqr)**1.66666666666666674 + 1.0)
x51 = 0.8*tau - 1.0
x35 = 0.2*_sqrt(deltam1sqr)*deltam1sqr*deltam1sqr*deltam1sqr + x50*x50
x35_n05 = x35**-0.05
return (delta*(delta*(delta9*(delta*(delta*(delta*(-6.26395869124539993e-10*delta*tauexpnd
- 0.000062689710414685001*tau10*_exp(-delta4))
- 1.32511800746680002e-12*tau13*expnd) - 0.0000163885683425300002*x29*expnd2)
+ 3.65821651442040008e-7*tau4expnd)
+ taurt*(-0.26145533859358*tau_quarter + 0.318025093454180008)
-tauexpnd*(0.25709043003437998*tau4 + 0.192327211560020001)
- 0.00781997516879810034*tau_quarter*tau_eighth*delta
)
+ 7.89576347228280007*tau**0.875
+ tau11*(tau11*tau11*tau11*delta5expnd6*(0.317774973307380026*tau2
- 0.199057183544080002)
+ expnd*(1.15379964229510002e-9*delta9
- 0.0000662126050396869941*tau))
# These two terms catastrophic cancel somewhat; cannot change them any at all
+ 0.0349940054637650003*tau11*tau11*delta3*expnd3
+ 0.0436136157238109987*tau4*tau*tau11*delta2*expnd3
+ tau50*(expnd6*delta2*(-0.118411824259810006*delta3
- 5.57111185656449973e-10))
+ tau*(0.00880894931021340005*delta3
+ 0.00833265048807130086*delta8expnd2
+ 0.0176114910087519991*deltaexpnd2
+ 31.5461402377809996*x33
- 8.78032033035609949)
+delta3*(expnd2*tau3*(tau4*(-0.0313587007125490022
- 0.743159297103409999*tau3)
+ 0.0049969146990805997)
# tau23expnd3 needs to do in both
- 0.0767881978446210006*tau23expnd3
+delta*(0.0224462773320059997*tau23expnd3
- 7.59413770881439999e-6*tau9*expnd
+ 0.478073299154800013*tau10expnd2))
- 0.107936009089319995*tau7expnd2
+ 0.000158703083241569997*tau9*delta9expnd2
+ 0.221322951675460011*tau9*deltaexpnd2
- 0.40247669763527999*tau10*deltaexpnd2
- 0.0400928289258069975*tau2*delta3expnd
- 0.029052336009585001*tau2*delta8expnd2
+ 3.93434226032540015e-7*delta3expnd*tau13
+ 0.000562509793518880044*delta6*tau3*expnd
+ 0.0141806344006170006*delta6*tau10expnd2
+ 0.0386150855742060026*tau3*delta8expnd2
- 0.0000156086522571349985*delta8*tau4expnd
+ 0.580833999857589989*delta2*tau10expnd2
- 2521.31543416949989*delta2*tau4*_exp(x32 - 390.625*x51*x51)
+ 0.160748684862510011*delta2*tau4expnd
- 0.0203934865137039983*delta8expnd2*tau4
- 0.136364351103430009*tau10expnd2*delta5
+ 0.0205279408959480013*delta5*tau6*expnd2
+ 0.204338109509650007*expnd*tau6
+ 0.00199555719795409996*delta9expnd2*tau6
- 0.00165540500637340006*delta8expnd2*x29
- 31.3062603234350014*x33
- 0.148746408567240002*x35*x35_n05*x35_n05*x35_n05*_exp(-28.0*deltam1sqr - 700.0*taum1sqr)
+ 0.318061108784439994*x35*x35_n05*_exp(-32.0*deltam1sqr - 800.0*taum1sqr)
- 0.668565723079650009*tau4expnd + 0.0125335479355230001/taurt))
[docs]def iapws95_dAr_ddelta(tau, delta):
r'''Calculates the first derivative of residual Helmholtz energy of water
with respect to `delta` according to the IAPWS-95 standard.
.. math::
\phi_{\delta}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} d_{i} \delta^{d_{i}-1}
\tau^{t_{i}}+\sum_{i=8}^{51} n_{i} \mathrm{e}^{-\delta^{c_{i}}}\left[
\delta^{d_{i}-1} \tau^{t_{i}}\left(d_{i}-c_{i} \delta^{c_{i}}\right)
\right]+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}}
\mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}\right)^{2}-\beta_{i}
\left(\tau-\gamma_{i}\right)^{2}}\left[\frac{d_{i}}{\delta}-2 \alpha_{i}
\left(\delta-\varepsilon_{i}\right)\right]+\sum_{i=55}^{56} n_{i}
\left[\Delta^{b_{i}}\left(\psi+\delta \frac{\partial \psi}{\partial
\delta}\right)+\frac{\partial \Delta^{b_{i}}}{\partial \delta} \delta
\psi\right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
dAr_ddelta : float
First derivative of residual Helmholtz energy A/(RT) with respect to
`delta`, [-]
Notes
-----
This is an optimized implementatation taking 8 exp calls, 4 sqrts,
and 2 powers. It was generated using SymPy's CSE functionality, with
select polynomial optimizations by hand as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative
error was 4.033E-15, with a maximum relative error of 3.8765e-10 and a
standard deviation of 3.189e-13.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 90000 points, the mean relative
error was 6.046E-15, with a maximum relative error of 3.39E-10 and a
standard deviation of 7.056E-13.
There was a singularity at `tau` = `delta` = 1, but the limit is correctly
returned.
Examples
--------
>>> iapws95_dAr_ddelta(647.096/300.0, 999.0/322)
-0.3093321202374
'''
if tau == 1.0 and delta == 1.0:
# Evaluated with sympy's limit command, otherwise divide by zero
return -0.7705590295466400609
_sqrt, _exp = sqrt, exp
# Variables which do not depend on delta
taurt = _sqrt(tau)
tau4rt = _sqrt(taurt)
tau8rt = _sqrt(tau4rt) # tau checked, is not causing the small discrepancies.
tau875 = tau**0.875#tau8rt*tau4rt*taurt#tau**0.875
c54 = (tau - 1.21)
taum1sqr = (tau - 1.0)
taum1sqr *= taum1sqr
x64 = 700.0*taum1sqr
x100 = (tau - 1.25)
x59 = 800.0*taum1sqr
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau*tau3
tau6 = tau3*tau3
tau7 = tau6*tau
tau9 = tau6*tau3
tau10 = tau*tau9
tau12 = tau10*tau2
tau13 = tau10*tau3
tau23 = tau10*tau13
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta*delta3
delta5 = delta*delta4
delta6 = delta*delta5
delta8 = delta6*delta2
delta9 = delta*delta8
# 10, 11, 12, 13, 14 also used for delta but not needed
x6 = delta - 1.0
expnd = _exp(-delta)
expnd2 = _exp(-delta2)
expnd6 = _exp(-delta6)
expnd3 = _exp(-delta3)
expnd4 = _exp(-delta4)
x8 = x6*expnd
x10 = tau*expnd
x11 = delta*(delta - 2.0)
x12 = tau4*expnd
x14 = delta3*expnd*(delta - 4.0)
x21 = delta8*tau4
x24 = delta2 + delta2
x26 = tau*tau6*expnd2
x27 = delta*(delta2 - 1.0)
x28 = expnd2*x27
x30 = expnd2*tau10
x33 = delta3*(delta2 - 2.0)
x35 = delta5*(delta2 - 3.0)
x38 = delta5*(delta6 - 1.0)
x39 = expnd6*x38
x40 = tau23*tau23*tau4*expnd6
x41 = delta9*expnd2*(delta2 - 5.0)
x42 = tau4*tau4*expnd2
x43 = 3.0*delta3
x44 = delta3*(x43 - 4.0)
x45 = tau23*expnd3
x46 = x24 - 9.0
x47 = expnd2*x46
x48 = delta8*x47
x50 = delta - 1.0
x51 = x50*x50
dm1rt23 = cbrt(x51)#**(1.0/3.0) # numba will make this a cbrt, PyPy does not care, CPython slows a by a multiply and assign
dm1rt23 *= dm1rt23
x65 = x51*dm1rt23
x52 = -20.0*x51
x53 = (-40.0*delta4 + 40.0*delta3 + 3.0*delta2)
x51_2_x51sqrt = x51*x51*_sqrt(x51)
y50 = 0.32*x65 + 1.0 - tau
x56 = 0.2*x51*x51_2_x51sqrt + y50*y50
x57 = 32.0*x51
x62 = delta*x6
x63 = 28.0*x51
exp100 = _exp(0.125*(-x57 - x59))
exp200 = exp100*exp100
exp400 = exp200*exp200
exp800 = exp400*exp400
exp700 = exp400*exp200*exp100
pow005 = x56**(-0.05)
exp250 = _exp(x52 - 250.0*x100*x100)
exp150 = _exp(x52 - 150.0*c54*c54)
x54 = x53*exp150
x67 = x62*(1.4*x51_2_x51sqrt + (-2.13333333333333333*tau + 0.68266666666666666*x65 + 2.13333333333333333)*dm1rt23)
return (-3.6582165144204e-7*delta9*delta*x12*(delta - 11.0) + 3.277713668506e-5*delta9*delta2*x42*(delta2 - 6.0)
+ 1.3251180074668e-12*delta9*delta3*tau13*expnd*(delta - 13.0)
+ 0.00012537942082937*delta9*delta4*tau10*(delta4 + delta4 - 7.0)*expnd4
+ 6.2639586912454e-10*delta9*delta5*x10*(delta - 15.0) - 0.023459925506394301*tau4rt*tau8rt*delta2
- 0.52291067718716*tau4rt*taurt*delta
+ 7.89576347228280007*tau875#tau_quarter*taurt*tau_eighth # improves average but increases max.
+ 0.25709043003438*tau4*tau*x11*expnd
- 1.1537996422951e-9*tau*tau10*delta9*expnd*(delta - 10.0) + 6.6212605039687e-5*tau12*x8
- 0.130840847171432989*tau9*tau7*delta2*expnd3*(delta3 - 1.0) - 0.0349940054637650003*tau10*tau12*expnd3*x44
+ 1.19434310126448007*tau23*tau12*tau9*x39 - 1.90664983984428016*tau23*tau23*x39 + 0.636050186908360016*taurt*delta
- 0.0352229820175039982*tau*x28 + 0.0352357972408536002*tau*delta3 - 0.00833265048807130086*tau*x48
+ 31.546140237781*tau*x54 - 8.7803203303561*tau + 0.19232721156002*x10*x11
- 0.16074868486251*x12*delta2*(delta - 3.0) + 0.0400928289258069975*tau2*x14 + 0.029052336009585001*tau2*x48
- 3.9343422603254e-7*x14*tau13 + 7.5941377088144e-6*delta4*tau9*expnd*(delta - 5.0)
- 0.4780732991548*delta4*x30*(x24 - 5.0) - 0.0224462773320059997*delta4*x45*(x43 - 5.0)
- 0.442645903350920022*tau9*x28 - 0.000317406166483139994*tau9*x41 - 0.000562509793518880044*delta6*tau3*expnd*(delta - 7.0)
- 0.0141806344006170006*delta6*x30*(x24 - 7.0) - 0.0099938293981611994*tau3*expnd2*x33
- 0.0386150855742060026*tau3*x48 + 0.00165540500637340006*delta8*x42*x46 + 0.0203934865137039983*x21*x47
+ 1.5608652257135e-5*x21*expnd*(delta - 9.0) - 0.0410558817918960026*expnd2*x35*tau6 + 0.0627174014250980044*x26*x33
+ 0.10793600908932*x26*(x24 - 1.0) + 0.804953395270559979*x27*x30 - 0.58083399985759*delta2*x30*(x24 - 3.0)
+ 1.67133355696935002e-9*delta2*x40*(delta6 + delta6 - 1.0) + 1.48631859420682*x30*x33 + 0.272728702206860019*x30*x35
+ 0.710470945558860034*x38*x40 - 0.00399111439590819992*x41*tau6 + 0.0767881978446210006*x44*x45
- 2521.3154341695*tau4*x53*exp250 + 0.668565723079650009*tau4*x8
- 31.3062603234350014*x54 - 0.14874640856724*x56*pow005*pow005*pow005*(-56.0*x62*exp700
+ exp700) + 0.31806110878444*x56*pow005*(-64.0*x62*exp800
+ exp800) - 0.126434447282154*pow005*pow005*pow005*x67*exp700
+ 0.302158053345218003*pow005*x67*exp800 - 0.204338109509650007*x8*tau6 + 0.0125335479355230001/taurt)
def iapws95_d2Ar_ddelta2_delta_1(tau):
# Derived with sympy
exp = trunc_exp
return (-0.046919851012788602*tau**0.375 + 0.636050186908360016*tau**0.5
- 0.52291067718716*tau**0.75 + 1.56820593144323182*tau**50
- 4.20850366554920272*tau**46 + 2.63624563596086548*tau**44
+ 0.196436526075828838*tau**23 - 0.115862176569242416*tau**22
- 0.144400973219474584*tau**16 + 7.23617955691807259e-7*tau**13
+ 0.0000243582561405054769*tau**12 + 3.01366009018245322e-8*tau**11
+ 0.31651327600812403*tau**10 - 0.322675438839726093*tau**9
- 0.0236600953022247575*tau**8 + 0.102487320000099597*tau**7
+ 0.023417142139243132*tau**6 + 0.0945782837315734979*tau**5
+ 1.93563785043781879e-165*tau**4*exp(625.0*tau)*exp(-250.0*tau**2)
+ 0.0196947947463537576*tau**4 + 0.542141065361544732*tau**3
- 0.479881408666422377*tau**2 - 4.49616772203662689e-93*tau*exp(363.0*tau)*exp(-150.0*tau**2)
+ 0.267030565833120499*tau
+ 1.48037819526523354e-303*(0.5*tau**2 - tau + 0.5)**0.849999999999999978
*exp(1400*tau)*exp(-700*tau**2) - 1.44239270039235679e-346*(0.5*tau**2 - tau
+ 0.5)**0.949999999999999956*exp(1600*tau)*exp(-800*tau**2)
+ 4.46197842597955621e-93*exp(363.0*tau)*exp(-150.0*tau**2))
[docs]def iapws95_d2Ar_ddelta2(tau, delta):
r'''Calculates the second derivative of residual Helmholtz energy of water
with respect to `delta` according to the IAPWS-95 standard.
.. math::
\begin{aligned} \phi_{\delta \delta}^{\mathrm{r}}=& \sum_{i=1}^{7}
n_{i} d_{i}\left(d_{i}-1\right) \delta^{d_{i}-2} \tau^{t_{i}}
+\sum_{i=8}^{51} n_{i} \mathrm{e}^{-\delta^{6}}\left[\delta^{d_{i}-2}
\tau^{t_{i}}\left(\left(d_{i}-c_{i} \delta^{c_{i}}\right)\left(d_{i}
-1-c_{i} \delta^{c_{i}}\right)-c_{i}^{2} \delta^{c_{i}}\right)\right]
+\sum_{i=52}^{54} n_{i} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}
\left(\delta-\varepsilon_{i}\right)^{2}-\beta_{i}\left(\tau-\gamma_{i}
\right)^{2}} \\ & \cdot\left[-2 \alpha_{i} \delta^{d_{i}}+4
\alpha_{i}^{2} \delta^{d_{i}}\left(\delta-\varepsilon_{i}\right)^{2}
-4 d_{i} \alpha_{i} \delta^{d_{i}-1}\left(\delta-\varepsilon_{i}\right)
+d_{i}\left(d_{i}-1\right) \delta^{d_{i}-2}\right]+\sum_{i=55}^{56}
n_{i}\left[\Delta^{b_{i}}\left(2 \frac{\partial \psi}{\partial \delta}
+\delta \frac{\partial^{2} \psi}{\partial \delta^{2}}\right)+2
\frac{\partial \Delta^{b_{i}}}{\partial \delta}\left(\psi+\delta
\frac{\partial \psi}{\partial \delta}\right)+\frac{\partial^{2}
\Delta^{b_{i}}}{\partial \delta^{2}} \delta \psi\right]\end{aligned}
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2Ar_ddelta2 : float
Second derivative of residual Helmholtz energy A/(RT) with respect to
`delta`, [-]
Notes
-----
This is an optimized implementatation taking 4 exp calls, 4 sqrts,
and 2 powers. It was generated using SymPy's CSE functionality, with
select polynomial optimizations by hand as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative
error was 9.566e-16, with a maximum relative error of 1.0518E-10 and a
standard deviation of 6.20265E-14.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 250000 points, the mean relative
error was 1.039E-15, with a maximum relative error of 2.431E-11 and a
standard deviation of 5.31708E-14.
Examples
--------
>>> iapws95_d2Ar_ddelta2(647.096/300.0, 999.0/322)
1.7862535141735987
'''
if delta == 1.0:
return iapws95_d2Ar_ddelta2_delta_1(tau)
# 4 sqrt; 4 exp; 2 power; 3 div; loads of multiplies and adds.
# What depends only on tau - keep it separated out
_sqrt, _exp = sqrt, exp
taurt = _sqrt(tau)
tau4rt = _sqrt(taurt)
tau8rt = _sqrt(tau4rt)
tau3 = tau*tau*tau
tau4 = tau3*tau
tau6 = tau3*tau3
taum1sqr = (tau - 1.0)*(tau - 1.0)
y63 = (0.8*tau - 1.0)
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta2*delta2
y1 = delta - 1.0
y2 = delta*y1
y3 = y1*y1
y4 = 800.0*delta2*y3 - 20.0*delta2 - 120.0*y2 + 3.0
y5 = -20.0*y3
y6 = (0.826446280991736*tau - 1.0)
exp150 = _exp(y5 - 219.615*y6*y6)
exp100 = _exp(-100.0*taum1sqr - 4.0*y3)
exp200 = exp100*exp100
exp400 = exp200*exp200
exp700 = exp400*exp200*exp100
exp800 = exp400*exp400
exp250 = _exp(y5 - 390.625*y63*y63)
y6 = y4*exp150
y7 = delta*tau
y9 = 2.0 - 2.0*delta
dm1rt23 = cbrt(y3)
y12 = y3*dm1rt23*dm1rt23
y3rt = _sqrt(y3)
y16 = y3*y3*y3rt
y13 = 0.32*y12 + 1.0 - tau
y13 = 0.2*y3*y16 + y13*y13
pow005 = y13**(-0.05)
y15 = pow005*pow005*pow005
y17 = dm1rt23*dm1rt23*(-2.1333333333333333*tau + 0.682666666666666755*y12 + 2.1333333333333333)
y18 = 1.4*y16 + y17
y19 = y1*y18
y21 = (y16 + 0.714285714285714191*y17)
y21 *= y3*y21
y22 = (y18 + y3*((-2.84444444444444455*tau + 0.910222222222222488*y12
+ 2.84444444444444455)/dm1rt23 + 2.275555555555556*y3*dm1rt23 + 7.*y3*y3rt))
delta5 = delta4*delta
delta6 = delta4*delta2
delta7 = delta*delta6
delta8 = delta*delta7
delta9 = delta*delta8
delta12 = delta6*delta6
delta15 = delta8*delta7
expnd = _exp(-delta)
expnd2 = _exp(-delta2)
expnd4 = _exp(-delta4)
expnd3 = _exp(-delta3)
expnd6 = _exp(-delta6)
y24 = delta*expnd
y26 = delta2*expnd
y29 = delta2*expnd2
y31 = expnd2*delta4
y33 = expnd2*delta7
y35 = expnd2*delta9
y37 = expnd2*delta5*delta6
y39 = expnd*delta3
y40 = expnd*delta4
y42 = expnd*delta5
y44 = expnd*delta7
y45 = expnd2*delta6
y47 = expnd*delta8
y48 = expnd*delta9
y50 = expnd*delta5*delta5
y51 = expnd*delta5*delta6
y52 = expnd2*delta8
y53 = expnd2*delta5*delta5
y55 = expnd2*delta12
y56 = delta*expnd2
y57 = expnd2*delta3
y60 = delta*expnd3
y61 = delta4*expnd3
y62 = delta7*expnd3
return (-0.046919851012788602*delta*tau8rt*tau4rt + 0.148746408567240002*delta*exp700*(
0.2499*y15/y13*y21 - 0.85*y15*y22)
- 0.318061108784439994*delta*exp800*(0.0931*pow005/y13*y21
- 0.949999999999999956*pow005*y22) - 5042.63086833899979*delta*y4*tau4*exp250
- 62.6125206468700028*delta*y6 + taurt*(0.636050186908360016 - 0.52291067718716*tau4rt)
+ 0.105707391722560801*tau*delta2 - tau*(1.31543132516153401e-7*delta7*delta6*expnd
+ tau*(-tau*(tau*(tau*(tau*(tau*(tau*(tau*(tau*(-0.00100303536663496002*delta15*delta5*expnd4
+ 0.0077735240914209398*delta8*delta8*expnd4 + tau*(tau*(0.000132425210079373988*expnd
- 0.0000662126050396869941*y24 + y7*(tau3*(tau6*(tau*(tau3*tau6*tau6*tau6*(tau*tau*(11.439899039065681*delta15
- 32.41304727735276*delta9 + 9.53324919922140168*delta3 + tau4*(-4.26282567335316021*delta15
+ 12.078006074500621*delta9 - 3.55235472779430017*delta3 + 3.67693382533256956e-8*delta6
- 2.00560026836322003e-8*delta12 - 3.34266711393870005e-9)) - 7.16605860758687996*delta15
+ 20.3038327214961605*delta9 - 5.97171550632239967*delta3)*expnd6 + 0.448925546640119966*delta2*expnd3
- 0.80806598395221596*delta5*expnd3 + 0.20201649598805399*delta8*expnd3 - 0.921458374135452063*y60
+ 2.30364593533863005*y61 - 0.691093780601588992*y62) + 0.419928065565180031*y60
- 1.04982016391295008*y61 + 0.314946049173885023*y62) - 1.04672677737146391*delta3*expnd3
+ 0.392522541514298995*delta6*expnd3 + 0.261681694342865978*expnd3) - 1.32511800746680002e-12*expnd*delta12
+ 4.72121071239048018e-6*y24 - 3.14747380826032012e-6*y26 + 3.93434226032540015e-7*y39
- 2.06718409164820794e-10*y50 + 3.44530681941368034e-11*y51)) + 1.03841967806558997e-7*y47
- 2.30759928459020012e-8*y48 + 1.15379964229510002e-9*y50) - 7.59868993714932728*expnd2*delta5
- 0.804953395270559979*expnd2 - 4.89314458888812087*y29 + 7.67603002421735958*y31 + 1.48687416460069*y33
+ 0.0567225376024680025*y35 + 0.572835940275540079*y45 - 0.545457404413720037*y52
- 0.0114095272954726698*delta12*expnd4 + 3.48500399914553993*y56 + 1.42978998508973909*y57)
+ 0.442645903350920022*expnd2 - 2.21322951675460011*y29 + 0.885291806701840045*y31
- 0.00015188275417628801*y39 + 0.000075941377088144005*y40 - 7.59413770881439999e-6*y42
+ 0.0142832774917412992*y52 - 0.0066655294961459402*y53 + 0.000634812332966279988*y55)
- 0.0000655542733701200008*expnd2*delta7*delta7 - 0.119189160458884807*y33 + 0.0629053902421892047*y35
- 0.00662162002549360022*y37 - 0.00216329102121396019*y53 + 0.000819428417126499982*y55)
- 0.376304408550587999*y29 + 0.564456612825881998*y31 - 0.125434802850196009*y45
+ 0.647616054535919972*y56 - 0.431744036357279981*y57) - 0.408676219019300013*expnd
+ 0.204338109509650007*y24 + 0.615838226878440032*y31 - 0.533726463294648013*y45
+ 0.26171191139966099*y52 - 0.083813402314072194*y53 + 0.00798222879181639984*y55)
- 0.51418086006875996*expnd + 1.02836172013751992*y24 - 0.25709043003437998*y26)
+ 1.33713144615930002*expnd + 0.295926386095409999*y24 - 0.964492109175060008*y26
- 1.46833102898668777*y33 + 0.774952487520751965*y35 - 0.0815739460548159934*y37
+ 0.160748684862510011*y39 - 0.00112382296251371978*y44 + 0.000280955740628429946*y47
+ 0.0000246317294014894049*y48 - 8.04807633172487966e-6*y50 + 3.65821651442040008e-7*y51)
- 0.00787513710926432062*expnd*delta6 + 0.059962976388967193*y29 - 0.0899444645834507894*y31
+ 2.78028616134283224*y33 - 1.46737325181982814*y35 + 0.15446034229682401*y37
+ 0.0236254113277929619*y42 + 0.000562509793518880044*y44 + 0.0199876587963223988*y45)
+ 0.48111394710968397*y26 + 2.09176819269011993*y33 - 1.10398876836422999*y35
+ 0.116209344038340004*y37 - 0.32074263140645598*y39 + 0.0400928289258069975*y40)
- 1.87918760737362006e-8*expnd*delta7*delta7 + 6.26395869124539993e-10*expnd*delta15 + 0.384654423120040001*expnd
- 0.769308846240080002*y24 - 0.0352229820175039982*expnd2 + 0.192327211560020001*y26
+ 0.176114910087520005*y29 - 0.0704459640350079963*y31 - 0.599950835141133676*y33
+ 0.316640718546709443*y35 - 0.0333306019522852034*y37)
- 8.32979887976543942*exp700*y13*y15*(delta*(56.0*y3 - 1.0) + y9)
+ 0.252868894564308*exp700*y15*y19*(56.0*y2 - 1.0)
+ 20.3559109622041596*y13*pow005*exp800*(delta*(64.0*y3 - 1.0) + y9)
- 0.604316106690436006*exp800*y19*pow005*(64.0*y2 - 1.0) + 63.0922804755619993*y6*y7)
[docs]def iapws95_d3Ar_ddelta3(tau, delta):
r'''Calculates the third derivative of residual Helmholtz energy of water
with respect to `delta` according to the IAPWS-95 standard.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d3Ar_ddelta3 : float
Third derivative of residual Helmholtz energy A/(RT) with respect to
`delta`, [-]
Notes
-----
No equation is given for this in IAPWS-95, and the derivative was
symbolically computed with SymPy.
This is an optimized implementatation.
It was generated using SymPy's CSE functionality.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-4
kg/m^3 to 5000 kg/m^3, 90000 points were evaluated. The mean
relative error was 5.41E-13, with a maximum relative error of 6.3957e-11 and a
standard deviation of 3.346e-12.
90000 points were also evaluated with mpmath. The mean
relative error was 1.41959E-14, with a maximum relative error of 5.8878E-10
and a standard deviation of 1.978E-12.
Evaluating 10000 points in the 1e-10 to 1e-4 range, the
mean relative error was 1.2E-16, maximum relative error 1.2e-16, and
standard deviation 6.66e-16.
Examples
--------
>>> iapws95_d3Ar_ddelta3(647.096/300.0, 999.0/322)
0.33621190578
'''
# Gonna have to reduce the number of constants quite a bit for pypy
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta*delta3
delta5 = delta*delta4
delta6 = delta*delta5
delta7 = delta*delta6
delta8 = delta*delta7
delta9 = delta*delta8
delta10 = delta*delta9
delta11 = delta*delta10
delta12 = delta*delta11
delta13 = delta*delta12
delta15 = delta2*delta13
delta21 = delta15*delta6
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau*tau3
tau6 = tau2*tau4
tau8 = tau4*tau4
tau9 = tau*tau8
tau10 = tau*tau9
tau40 = tau10*tau10
tau40 *= tau40
x0 = delta*tau
x1 = exp(-delta)
x2 = tau*x1
x3 = 1.15396326936012006*x2
x5 = x1*tau4
x6 = tau*tau4*x1
x7 = 1.54254258020627999*x6
x9 = x1*tau6
x10 = tau6*tau6*x1
x12 = exp(-delta2)
x14 = x1*tau2
x15 = tau10*tau3*x1
x19 = tau6*tau*x12
x21 = x12*tau10
x23 = exp(-delta3)
x24 = tau10*tau6*x23
x26 = exp(-delta6)
x27 = tau40*tau10*x26
x29 = x12*tau3
x31 = x12*tau9
x32 = tau10*tau10*tau2*x23
x33 = tau10*tau10*tau3*x23
x34 = 0.000455648262528864003*x1
x35 = tau*x12
x36 = delta3*tau9
x38 = x1*tau3
x39 = x1*tau9
x42 = tau10*tau*x1
x44 = x12*delta8
x48 = x12*delta12
x49 = x12*tau6
x50 = tau40*tau4*x26
x51 = tau40*tau6*x26
x52 = x12*tau2
x53 = x12*tau4
x55 = x12*tau8
x56 = tau10*exp(-delta4)
x58 = delta - 1.0
x59 = x58*x58
x60 = -20.0*x59
x61 = (0.826446280991736*tau - 1.0)
x61 = exp(x60 - 219.615*x61*x61)
x62 = tau*x61
x63 = delta2*x61
x64 = (0.8*tau - 1.0)
x64 = tau4*exp(x60 - 390.625*x64*x64)
x65 = x58*x61
x66 = tau*x63
x67 = delta2*x64
x68 = delta*x58
x69 = delta3*x58
x70 = x58*x58*x58
x71 = delta3*x70
x72 = sqrt(x59)*x59*x59*x59
x73 = x59**1.66666666666666674
x74 = -tau + 0.320000000000000007*x73 + 1.0
x75 = 0.2*x72 + x74*x74
pow005 = x75**(-0.05)
x76 = (tau - 1.0)
x76 *= x76
x77 = exp(-28.0*x59 - 700.0*x76)
x78 = x75*pow005*pow005*pow005*x77
x79 = exp(-32.0*x59 - 800.0*x76)
x80 = x75*pow005*x79
x81 = 3908.33490474319842*x80
x82 = delta*x70
x83 = x73*x74
x84 = 1.19000000000000017*x72 + 1.81333333333333346*x83
x85 = pow005*pow005*pow005*x77
x86 = x84*x85
x87 = 1.33*x72 + 2.02666666666666684*x83
x88 = pow005*x79
x89 = x87*x88
x90 = delta/x58
x91 = 61.0677328866124753*x90
x92 = 1.0/x59
x93 = 0.446239225701720033*x92
x94 = x73*x73
x95 = 7.13999999999999879*x72 + 4.2311111111111126*x83 + 1.93422222222222251*x94
x96 = x85*x95
x97 = 0.954183326353319927*x92
x98 = 7.98000000000000043*x72 + 4.72888888888889003*x83 + 2.16177777777777802*x94
x99 = x88*x98
x100 = delta/x70
x101 = 0.148746408567240002*x100
x102 = 0.318061108784439994*x100
x103 = 24.9893966392963236*x90
x104 = 0.0700000000000000622*x72 + 0.106666666666666771*x83
x105 = pow005/x75
x106 = x105*x79*x87
x107 = x104*x106
x108 = 0.210000000000000048*x72 + 0.320000000000000062*x83
x109 = x105*pow005*pow005
x110 = x109*x77*x84
x111 = x108*x110
x112 = x104*x79
x113 = x108*x77
x114 = sqrt(sqrt(tau))
return (0.00401214146653984007*delta9*delta9*delta5*x56 - 0.0511548036983829613*delta9*delta10*x56
+ 1.20336016101793195e-7*delta9*delta9*x27 - 2.81878141106042993e-8*delta9*delta5*x2
+ 0.0000662126050396869941*delta*x10 - 0.962227894219367941*delta*x14
+ 9.44242142478096036e-6*delta*x15 - 0.752608817101175998*delta*x19
- 8.1763823872351189*delta*x21 + 0.119925952777934386*delta*x29 - delta*x3
- 5.31175084021104027*delta*x31 + 0.839856131130360062*delta*x32
- 1.84291674827090413*delta*x33 - 2.22491060444553002*delta*x5
+ 22540.5074328732007*delta*x65 - delta*x7 - 0.204338109509650007*delta*x9
- 0.046919851012788602*sqrt(x114)*x114 - 4.0496681372026524*tau*x44
- 0.0666612039045704069*tau*x48 - 0.422675784210048033*x0*x12
- 22713.2209712023177*x0*x65 + 0.211414783445121601*x0 - 0.000198637815119060996*x10
- 0.636122217568879988*x100*x105*x112*x98 + 0.297492817134480003*x100*x109*x113*x95
+ x101*x110*(1.26000000000000001*x72 + 0.746666666666666812*x83 + 0.341333333333333433*x94)
- x101*x113*x105*x105*pow005*x84*(1.60999999999999988*x72 + 2.45333333333333359*x83)
- x101*x85*(35.6999999999999886*x72 + 5.6414814814814811*x83 + 13.5395555555555589*x94)
- x102*x106*(0.420000000000000373*x72 + 0.248888888888889104*x83 + 0.113777777777777894*x94)
+ x102*x112*x105/x75*x87*(1.4700000000000002*x72 + 2.24000000000000021*x83)
+ x102*x88*(39.9000000000000057*x72 + 6.30518518518518789*x83 + 15.132444444444447*x94)
- x103*x111 + x103*x96 + x107*x91 - x107*x97 + 1.44334184132905197*delta2*x14
- 0.0000141636321371714414*delta2*x15 - 2.59046421814367989*delta2*x19 + 0.192327211560020001*delta2*x2
- 2.68063804302186082*delta2*x21 - delta2*tau9*x34 + 1.34677663992036001*delta2*x33
+ 1.44673816376259001*delta2*x5 + 0.25709043003437998*delta2*x6 + x111*x93 + 7.9676262603165604*x12*x36
+ 14.11943530065831*tau2*x44 + 0.232418688076680008*tau2*x48 - 0.48111394710968397*x14*delta3
+ 0.0400928289258069975*x14*delta4 - 5.1679602291205205e-11*x15*delta12 + 1.32511800746680002e-12*x15*delta13
+ 4.72121071239048018e-6*x15*delta3 - 3.93434226032540015e-7*x15*delta4 - 2.27390250081302894e-9*x15*delta10
+ 6.20155227494462486e-10*x15*delta11 - 1.71006072270999421e-6*delta12*x2 - 4.81344064407172886e-7*delta12*x27
- 0.308920684593648021*delta12*x29 + 0.0132432400509872004*delta12*x55 + 3.94629397548460203e-7*delta13*x2
- 0.00126962466593255998*delta13*x31 - 0.0159644575836327997*delta13*x49 - 0.00255661666143467987*delta13*x55
+ 6.26395869124539993e-10*delta15*x2 - 140.673247220654275*delta15*x27 - 236.479934050367035*delta15*x50
+ 377.516668289167455*delta15*x51 + 0.000131108546740240002*delta15*x55 + 0.17001449464462573*delta15*x56
+ 3.01043526840470443*x19*delta3 + 0.863488072714559962*x19*delta4 - 1.88152204275294022*x19*delta5
+ 0.250869605700392018*x19*delta7 + 0.647616054535919972*x19 + 40.4904092746456783*x21*delta3
+ 25.6054990265034803*x21*delta6 - 40.8530296559261075*x21*delta4 - 11.9150444067814796*x21*delta5
- 5.50933111586084046*x21*delta7 - 2.4632454907791681*x21*delta8 + 1.09091480882744007*x21*delta9
- 0.113445075204936005*x21*delta10 + 3.48500399914553993*x21 - 4.97195219251445408*delta3*x24
- 14.2094189111772007*delta3*x27 - 0.479703811111737544*delta3*x29 + 0.634013676315071995*delta3*x35
+ 2.46335290751376013*delta3*x49 - 0.160748684862510011*delta3*x5 - 23.8868620252896022*delta3*x50
+ 38.1329967968856067*delta3*x51 - 150270.049552488024*delta3*x65 + 5.88783812271448515*x24*delta6
- 1.17756762454289698*x24*delta9 + 0.261681694342865978*x24 + 2.77441370456912092e-7*delta6*x27
+ 19.4620031293998252*delta6*x29 + 4.19965584598793562*delta6*x35 + 0.0118127056638964809*delta6*x38
- 0.00786676073759603849*delta6*x5 - 14.6423773488308395*delta6*x52 - 10.2783172029068144*delta6*x53
- 0.834324123212193625*delta6*x55 + 142.09418911177201*x27*delta9 + 25.576954040118963*x27*delta21
- 3.34266711393870005e-9*x27 - 18.766931589064118*tau3*x44 + 0.299814881944835965*x29*delta5
- 0.0399753175926447976*x29*delta7 + 4.63381026890472025*x29*delta10 + x3 - 1.77058361340368009*x31*delta5
+ 0.114266219933930394*x31*delta7 - 0.0952218499449419969*x31*delta9 + 0.0209488069878872411*x31*delta11
- 6.50888501626029026*x32*delta4 + 5.6690288851299302*x32*delta7 - 0.94483814752165507*x32*delta10
+ 14.2826047990995058*x33*delta4 - 6.19517254363365666*x33*delta5 - 12.4396880508286038*x33*delta7
+ 4.24234641574913418*x33*delta8 + 2.07328134180476731*x33*delta10 - 0.606049487964162026*x33*delta11
+ x34*x36 - 0.140891928070015993*x35*delta5 + 0.99991805856855609*x35*delta10
+ 0.118127056638964806*delta4*x38 - 0.000113912065632216001*delta4*x39 - 0.070876233983378889*x38*delta5
- 0.000562509793518880044*x38*delta7 + 7.59413770881439999e-6*x39*delta5 + 9.911234445660142*tau4*x44
+ 0.163147892109631987*tau4*x48 - 4.43403523352476814*delta5*x49 + 8.30735742452472082e-7*delta7*x42
+ 3.16114821778658417*delta7*x49 + 0.00337146888754115935*delta7*x5 - 3.11525903419677031e-7*x42*delta8
+ 3.46139892688530034e-8*x42*delta9 - 1.15379964229510002e-9*x42*delta10 - 0.0000592701760150253563*delta8*x5
+ 0.804526833097472416*x44*tau8 - 1.36155784594004392*delta9*x49 - 0.000105112492718738184*delta9*x5
+ 238.868620252896022*delta9*x50 - 381.329967968855954*delta9*x51 - 0.0216329102121396027*delta9*x55
+ 0.0000120721144975873203*delta10*x5 - 3.48628032115020003*delta10*x52 - 2.44721838164447991*delta10*x53
- 0.198648600764808003*delta10*x55 + 0.263413550129941165*delta11*x49 - 3.65821651442040008e-7*delta11*x5
+ 0.0141597230479459206*delta11*x55 - 0.136914327545672065*delta11*x56 - 1.04120506006389002*x5
+ 42.9963516455212797*x50*delta21 - 68.6393942343940893*x51*delta21 - 450810.148657464015*x59*x63
+ 454264.419424046355*x59*x66 - 36306942.2520408034*x59*x67 - 1399.40621180059406*x59*x78 + x59*x81
+ 2003600.66069984017*x61*x71 - 187.837561940610016*x61 + 151421.473141348804*x62*x69
- 2018952.97521798406*x62*x71 + 189.276841426685991*x62 + 11270.2537164366004*x63 + 1815347.11260203994*x64*x68
- 12102314.0840135999*x64*x69 + 161364187.786847979*x64*x71 - 15127.8926050169994*x64 - 11356.6104856011589*x66
+ 907673.556301020086*x67 - 1399.40621180059384*x68*x78 + x68*x81 - 1399.40621180059406*x68*x86
+ 3908.33490474319842*x68*x89 + x7 + 26122.2492869444213*x78*x82 + 24.9893966392963165*x78
- 83377.8113011882378*x80*x82 - 61.0677328866124753*x80 + 24.9893966392963165*x86*x90 + 49.9787932785926472*x86
- x89*x91 - 122.135465773224951*x89 + 0.613014328528949992*x9 - x91*x99 - x93*x96 + x97*x99)
[docs]def iapws95_d3Ar_ddelta2dtau(tau, delta):
r'''Calculates the third derivative of residual Helmholtz energy of water
with respect to `delta` twice and `tau` one according to the IAPWS-95
standard.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d3Ar_ddeltadtau2 : float
Third derivative of residual Helmholtz energy A/(RT) with respect to
`delta` twice and `tau` once, [-]
Notes
-----
This is an optimized implementatation.
It was generated using SymPy's CSE functionality.
No equation is given for this in IAPWS-95, and the derivative was
symbolically computed with SymPy.
Like many higher-order derivatives of functions with exponentials, this one
balloons to use many, many terms.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean
relative error was 3.629e-15, with a maximum relative error of 8.38E-11 and a
standard deviation of 2.1214E-13.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 10000 points, the mean relative
error was 2.4e-15, with a maximum relative error of 7.62E-12 and a
standard deviation of 7.818E-14.
Examples
--------
>>> iapws95_d3Ar_ddelta2dtau(647.096/300.0, 999.0/322)
0.015646982949077
'''
# simplest when derivaed from diff((calcA_res(tau, delta)), delta, delta, tau)
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau*tau3
tau5 = tau*tau4
tau6 = tau*tau5
tau7 = tau*tau6
tau8 = tau*tau7
tau9 = tau*tau8
tau12 = tau9*tau3
tau40 = tau12*tau12*tau12*tau4
tau_inv = 1.0/tau
taurtinv = sqrt(tau_inv)
tau4rtinv = sqrt(taurtinv)
tau8rtinv = sqrt(tau4rtinv)
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta*delta3
delta5 = delta*delta4
delta6 = delta*delta5
delta7 = delta2*delta5
delta8 = delta*delta7
delta9 = delta*delta8
delta10 = delta*delta9
delta11 = delta*delta10
delta12 = delta*delta11
delta13 = delta*delta12
delta14 = delta*delta13
delta16 = delta2*delta14
x1 = exp(-delta)
x3 = exp(-delta2)
x2 = delta*x1
x4 = delta2*x1
x6 = x1*delta13
x9 = x1*tau3
x11 = x1*tau4
x13 = tau9*tau2*x1
x15 = tau*x1
x17 = delta2*x3
x18 = delta4*x3
x20 = delta7*x3
x22 = delta9*x3
x24 = delta11*x3
x26 = tau8*x3
x28 = tau9*x3
x30 = tau6*x3
x31 = exp(-delta3)
x32 = tau9*tau6*x31
x34 = exp(-delta6)
x35 = tau9*tau40*x34
x37 = x1*tau8
x38 = x1*tau12
x39 = x1*delta4
x42 = x1*tau2
x44 = tau*tau9*x1
x47 = tau12*tau9*x31
x48 = tau9*tau9*tau4*x31
x49 = tau3*tau40*x34
x50 = tau5*tau40*x34
x51 = x3*delta6
x53 = tau5*x3
x54 = x3*tau7
x55 = tau9*exp(-delta4)
x57 = delta - 1.0
x58 = x57*x57
x59 = -20.0*x58
x60 = (0.826446280991736*tau - 1.0)
x60 = exp(x59 - 219.615*x60*x60)
x61 = delta*x60
x62 = 189.276841426685991*x61
x63 = delta3*x60
x64 = 1261.8456095112399*x63
x65 = (0.8*tau - 1.0)
x65 = exp(x59 - 390.625*x65*x65)
x66 = x65*tau3
x67 = 300.0*tau - 363.0
x68 = delta2*x57
x69 = x60*x68
x70 = 7571.07365706743985*x69
x71 = delta3*x66
x72 = tau*x67
x73 = x58*x63
x74 = 50473.8243804496014*x73
x75 = tau4*x65*(500.*tau - 625.0)
x76 = delta3*x75
x77 = tau - 1.0
x78 = delta*x77
x79 = x77*x77
x80 = exp(-28.0*x58 - 700.0*x79)
x81 = sqrt(x58)*x58*x58*x58
x82 = x58**1.66666666666666666
x83 = -tau + 0.32*x82 + 1.0
x84 = 0.2*x81 + x83*x83
x85 = x80*x84**0.85
x86 = x78*x85
x87 = exp(-32.0*x58 - 800.0*x79)
x88 = x84**0.949999999999999956*x87
x89 = x78*x88
x90 = x57*x77
x84_inv = 1.0/x84
pow005 = x84**(-0.05)
x91 = x80*pow005*pow005*pow005
x92 = delta*x82
x93 = x91*x92
x94 = pow005*x87
x95 = x92*x94
x96 = 1.0/x57
x97 = x82*x96
x98 = 1.0/x58
x99 = x91*(-1.7*tau + 0.544*x82 + 1.7)
x100 = delta*x99
x101 = x94*(-1.9*tau + 0.608*x82 + 1.9)
x102 = delta*x101
x103 = x82*x83
x104 = 1.81333333333333346*x103 + 1.19*x81
x105 = x104*x91
x106 = 2.02666666666666684*x103 + 1.33*x81
x107 = x106*x94
x108 = x77*x96
x109 = 0.106666666666666771*x103 + 0.07*x81
x110 = x84_inv*pow005*x87
x111 = x92*x98
x112 = 0.32*x103 + 0.21*x81
x113 = x80*x84_inv*pow005*pow005*pow005
x114 = x104*x113
x115 = x106*x110
x116 = -0.3*tau + 0.096*x82 + 0.3
x117 = x114*x116
x118 = -0.1*tau + 0.032*x82 + 0.1
x119 = x115*x118
x120 = x58**3.333333333333333333
x121 = x98*(4.2311111111111126*x103 + 1.93422222222222251*x120 + 7.13999999999999879*x81)
x122 = 208.244971994135994*x78
x123 = x98*(4.72888888888889003*x103 + 2.16177777777777802*x120 + 7.98000000000000043*x81)
x124 = 508.89777405510398*x78
x125 = 0.148746408567240002*delta
x126 = 0.318061108784439994*delta
x127 = x109*x98
x128 = x112*x98
return (-0.0100303536663495993*delta10*delta10*x55 - 6.26395869124539993e-10*delta5*delta10*x1
- 0.017594944129795724*delta*taurtinv*tau8rtinv + 16.6595977595308788*delta*x117
- 40.7118219244083193*delta*x119 - 0.000794551260476243984*delta*x13 + 34.8500399914553967*delta*x28
+ 4.5333123817514398*delta*x30 + 4.18690710948585565*delta*x32 - 1.67133355696935012e-7*delta*x35
- 60511.5704200679975*delta*x66 + 15127.8926050169994*delta*x75 + 1.18370554438164*delta*x9
+ 0.318025093454180008*taurtinv - 0.392183007890369972*tau4rtinv - 4.18353638538023986*tau*x20
+ 2.20797753672845998*tau*x22 - 0.232418688076680008*tau*x24 - 0.962227894219367941*tau*x4
- 1.28545215017189984*delta2*x11 - 19.9190656507914028*delta2*x26 + 9.23841744243395979*delta2*x47
- 21.1935426051153968*delta2*x48 - 3.85796843670024003*delta2*x9 + 0.105707391722560801*delta2
- 2.45205731411579997*x1*tau5 + 1.87918760737362006e-8*x1*delta14 - 0.384654423120040001*x1
+ 5.14180860068759937*tau4*x2 + 466.46873726686465*x100*x58 - 8.32979887976543942*x100
+ 40.7118219244083193*x101*x57 - 1302.77830158106622*x102*x58 + 20.3559109622041596*x102
+ x104*x125*x128*x80*x84_inv*x84_inv*pow005*pow005*pow005*(-2.29999999999999982*tau + 0.736*x82
+ 2.3) + 416.489943988271989*x105*x108 - 23323.4368633432314*x105*x78
- x106*x126*x127*x84_inv*x84_inv*pow005*x87*(-2.1*tau + 0.672*x82
+ 2.1) - 1017.79554811020796*x107*x108 + 65138.9150790533095*x107*x78
+ 0.644603847136465014*x109*x110*x111 - 2.57090430034379969*x11 + x110*x118*x123*x126
- 0.26972682086859523*x111*x112*x113 - 0.0475988507415168113*x111*x114 + 0.0339265182703403015*x111*x115
- x113*x116*x121*x125 - x114*x122*x128 + x115*x124*x127 - 0.297492817134480003*x117*x96
+ 0.636122217568879988*x119*x96 + 3.69502936127063997*tau5*x18 + 1.22602865705789998*tau5*x2
- 3.20235877976788785*tau5*x51 + x121*x122*x91 - x123*x124*x94 + 0.00158910252095248797*x13
+ 0.64148526281291196*delta3*x15 + 14.2978998508973945*delta3*x28 - 3.02220825450095987*delta3*x30
- 0.00136694478758659201*delta3*x37 - 0.0000409171595073841633*delta3*x38 + 10.3252875727227593*delta3*x48
+ 0.642994739450040043*delta3*x9 - 0.0801856578516139951*x15*delta4 + 7.9676262603165604*delta4*x26
- 16.7476284379434226*delta4*x32 - 177.61773638971502*delta4*x35 - 262.755482278185639*delta4*x49
+ 438.529463164184449*delta4*x50 - 48.9314458888811927*x17*tau9 - 2.63413085985411577*x17*tau6
+ 0.179888929166901579*x17*tau2 - 0.176114910087520005*x17 + 76.7603002421735994*x18*tau9
+ 3.95119628978117454*x18*tau6 - 0.269833393750352368*x18*tau2 + 0.0704459640350079963*x18
+ 6.28036066422878392*delta7*x32 + 1.83846691266628511e-6*delta7*x35 + 0.00168752938055664013*delta7*x42
- 0.00449529185005487913*delta7*x9 + 0.769308846240080002*x2 + 14.8687416460068995*x20*tau9
+ 8.34085848402849628*x20*tau2 - 0.95351328367107846*x20*tau7 - 5.87332411594675108*x20*tau3
+ 0.599950835141133676*x20 + 0.567225376024680039*delta9*x28 - 2.5383592130492199e-7*delta9*x44
+ 4.64637940772524249*delta9*x48 + 0.0000985269176059575924*delta9*x9 - 4.4021197554594842*x22*tau2
+ 0.503243121937513638*x22*tau7 + 3.09980995008300741*x22*tau3 - 0.316640718546709443*x22
- 2.68733931914267053e-9*delta11*x38 + 1.46328660576816003e-6*delta11*x9 + 0.463381026890472003*x24*tau2
- 0.0529729602039488018*x24*tau7 - 0.326295784219263973*x24*tau3 + 0.0333306019522852034*x24
+ 0.000683472393793296005*tau8*x39 + 0.128549497425671705*x26*delta8 - 0.0599897654653134618*x26*delta10
+ 0.00571331099669651968*x26*delta12 + 3.9838131301582802*x26 + 5.72835940275540079*x28*delta6
- 75.9868993714932515*x28*delta5 - 5.45457404413720059*x28*delta8 - 8.04953395270560002*x28
- 0.878043619951372034*tau6*x51 + 0.0352229820175039982*x3 - 0.0236254113277929619*delta6*x42
- 18.58551763090097*delta6*x48 + 603.900303725031108*x35*delta10 - 1.00280013418161002e-6*x35*delta13
- 213.141283667658001*x35*delta16 + 5.11464493842302041e-6*tau12*x39 + 0.0000613757392610762381*tau12*x4
- 1.72265340970684017e-11*tau12*x6 - 0.0000683472393793296059*x37*delta5 + 4.47889886523778457e-10*x38*delta12
- 0.192327211560020001*x4 + 0.070876233983378889*delta5*x42 - 23.0960436060849013*delta5*x47
+ 52.983856512788492*delta5*x48 + 0.059962976388967193*tau2*x51 + 1.14226164587214911e-6*delta8*x44
+ 6.92881308182547073*delta8*x47 - 15.8951569538365494*delta8*x48 + 1.57027146839796594*delta8*x53
+ 0.00112382296251371978*delta8*x9 + 1.26917960652461008e-8*x44*delta10 + 893.368639745831047*delta10*x49
- 1491.00017475822688*delta10*x50 - 0.502880413884433164*delta10*x53 - 0.0173063281697116815*delta10*x54
- 0.0000321923053268995186*delta10*x9 + 0.0478933727508984025*delta12*x53 + 0.00655542733701199986*delta12*x54
- 0.114095272954726712*delta12*x55 - 315.306578733822732*x49*delta16 + 526.235355797021271*x50*delta16
- 0.000524434186960960006*x54*delta14 + 0.0777352409142094136*x55*delta16 - 16.6595977595308788*x57*x99
- 16136418.7786847986*x58*x71 + 4034104.69467119966*x58*x76 + 653056.232173610479*x58*x86 - 2084445.2825297059*x58*x89
- 1.31543132516153401e-7*x6 + 187.837561940610016*x61*x67 - x62*x72 + x62 - 1252.25041293740014*x63*x67 + x64*x72
- x64 + 2420462.81680272007*x66*x68 - 7513.50247762440085*x67*x69 + 50090.0165174960057*x67*x73
- 605115.704200680018*x68*x75 + x70*x72 - x70 + 403410.469467120012*x71 - x72*x74 + x74 - 100852.617366780003*x76
- 23323.4368633432314*x85*x90 - 11661.7184316716157*x86 + 65138.9150790533095*x88*x90 + 32569.4575395266547*x89
+ 5.34852578463720008*x9 + 0.539453641737190459*x91*x97 + 0.62936258202672235*x93*x98 - 30.2094039372826693*x93
- 1.28920769427293003*x94*x97 - 1.50407564331841881*x95*x98 + 82.5092924334675217*x95)
[docs]def iapws95_d3Ar_ddeltadtau2(tau, delta):
r'''Calculates the third derivative of residual Helmholtz energy of water
with respect to `delta` once and `tau` twice according to the IAPWS-95
standard.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d3Ar_ddeltadtau2 : float
Third derivative of residual Helmholtz energy A/(RT) with respect to
`delta` once and `tau` twice, [-]
Notes
-----
This is an optimized implementatation.
It was generated using SymPy's CSE functionality.
No equation is given for this in IAPWS-95, and the derivative was
symbolically computed with SymPy.
Like many higher-order derivatives of functions with exponentials, this one
balloons to use many, many terms.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean
relative error was 7.936e-16, with a maximum relative error of 1.965E-11 and a
standard deviation of 4.7938E-14.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 90000 points, the mean relative
error was 6.08E-16, with a maximum relative error of 3.537E-12 and a
standard deviation of 1.85197E-14.
Examples
--------
>>> iapws95_d3Ar_ddeltadtau2(647.096/300.0, 999.0/322)
1.081479970332
'''
tau_inv = 1.0/tau
taurtinv = sqrt(tau_inv)
tau4rtinv = sqrt(taurtinv)
tau8rtinv = sqrt(tau4rtinv)
tau2 = tau*tau
tau4 = tau2*tau2
tau6 = tau4*tau2
tau7 = tau6*tau
tau8 = tau4*tau4
tau11 = tau8*tau2*tau
tau44 = tau11*tau11
tau44 *= tau44
tau48 = tau44*tau4 # 11*4+4
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta2*delta2
delta5 = delta*delta4
delta6 = delta4*delta2
delta7 = delta*delta6
delta8 = delta*delta7
delta9 = delta*delta8
delta10 = delta*delta9
delta11 = delta*delta10
delta13 = delta2*delta11
x1 = exp(-delta)
x16 = exp(-delta2)
x35 = exp(-delta3)
x37 = exp(-delta6)
x3 = x1*delta3
x5 = x1*delta4
x7 = x1*tau2
x8 = 8.02278867695580011*x7
x10 = 6.13014328528949992*x1*tau4
x11 = 0.00874006386523868382*tau8*tau2*x1
x12 = tau2*tau*x1
x14 = tau*x1
x18 = x16*delta8
x20 = x16*delta10
x21 = tau*tau4*x16
x23 = x16*tau7
x24 = 31.8705050412662416*x23
x26 = x16*tau8
x27 = tau*x16
x31 = tau*tau8*x1
x33 = x1*tau11
x36 = 31.4018033211439231*tau7*tau7*x35
x39 = x37*tau48
x40 = tau8*tau8*tau4*x35
x41 = tau7*tau7*tau7*x35
x42 = x16*tau4
x43 = delta5*x37
x44 = 2259.69714759239605*tau11*tau11*tau11*tau8*tau
x45 = 3946.76516847766015*tau44
x47 = x16*tau6
x48 = delta11*x37
x49 = tau8*exp(-delta4)
x50 = 0.82644628099173556*tau
x51 = (x50 - 1.0)
x51 *= x51
x52 = delta - 1.0
x53 = x52*x52
x54 = -20.0*x53
x55 = exp(-219.615*x51 + x54)
x56 = x55*(131769.0*x51 - 300.0)
x57 = delta3*x52
x58 = tau_inv*tau_inv
x60 = (-x50 + 0.00275482093663911853*tau_inv + 1.0)
x60 = tau*x55*(x58 - 131769.0*x60*x60 + 300.0)
x61 = 0.800000000000000044*tau
x62 = (0.00640000000000000031*tau_inv - x61 + 1.0)
x62 = tau4*(4.0*x58 - 390625.0*x62*x62 + 500.0)*exp(x54 - 390.625*(x61 - 1.0)*(x61 - 1.0))
x63 = tau - 1.0
x64 = x63*x63
x65 = exp(-28.0*x53 - 700.0*x64)
x66 = sqrt(x53)*x53*x53*x53
x67 = x53**1.66666666666666674
x68 = -tau + 0.320000000000000007*x67 + 1.0
x69 = x68*x68
x70 = 0.2*x66 + x69
x71 = 1400.0*x64 - 1.0
pow005 = x70**(-0.05)
x72 = x70*pow005*pow005*pow005*x71
x73 = exp(-32.0*x53 - 800.0*x64)
x74 = 1600.0*x64 - 1.0
x75 = x70*pow005*x74
x76 = -1.69999999999999996*tau + 0.544000000000000039*x67 + 1.69999999999999996
x77 = pow005*pow005*pow005
x78 = x63*x77
x79 = x76*x78
x80 = -1.89999999999999991*tau + 0.607999999999999985*x67 + 1.89999999999999991
x82 = x63*pow005
x83 = x80*x82
x70_inv = 1.0/x70
x88 = pow005*x70_inv
x84 = x88*pow005*pow005
x85 = 0.510000000000000009*x69
x86 = -1.69999999999999996*x77 + x84*x85
x87 = 0.148746408567240002*x65
x89 = 0.190000000000000169*x69
x90 = -1.89999999999999991*pow005 + x88*x89
x91 = 0.318061108784439994*x73
x92 = 1.0/x52
x93 = x67*x92
x94 = x67*x68
x95 = x84*(0.210000000000000048*x66 + 0.320000000000000062*x94)
x96 = x63*x92
x97 = x88*(0.0700000000000000622*x66 + 0.106666666666666771*x94)
return (0.0225682957492866001*delta13*delta4*x49 - 2.68733931914267053e-9*delta10*delta2*x33
- 0.159012546727090004*delta*taurtinv*tau_inv + 0.0980457519725924931*delta*tau_inv*tau4rtinv
- delta*x10 + delta*x11 - 10.2836172013751987*delta*x12 + delta*x24
- 72.4458055743504019*delta*x26 + delta*x8 - delta*x87*(-78400.0*x52*x72
- 156800.0*x52*x79 + 56.0*x52*x86 + 1400.0*x71*x77*x92*(1.19000000000000017*x66
+ 1.81333333333333346*x94) - 2800.0*x76*x95*x96 + 5077.33333333333394*x78*x93
- x92*(-x70_inv*x70_inv*x77*x85*(1.60999999999999988*x66 + 2.45333333333333359*x94)
+ 1.08800000000000008*x84*x94 + 1.69999999999999996*x95)) + delta*x91*(-102400.0*x52*x75
- 204800.0*x52*x83 + 64.0*x52*x90 + 1600.0*x74*pow005*x92*(1.33000000000000007*x66
+ 2.02666666666666684*x94) - 3200.0*x80*x96*x97 + 6485.33333333333303*x82*x93
- x92*(-pow005*x70_inv*x70_inv*x89*(1.4700000000000002*x66 + 2.24000000000000021*x94)
+ 0.405333333333333712*x88*x94 + 1.89999999999999991*x97)) + 0.00940016095164225053*taurtinv*tau_inv*tau_inv
+ 0.00549842004056116419*taurtinv*tau_inv*tau8rtinv*delta2 - 0.863599129780931229*tau_inv*tau8rtinv
+ 2.08521462100712407*tau*x18 - 0.463381026890472003*tau*x20 + 5.14180860068759937*delta2*x12
+ 9.0666247635028796*delta2*x21 + 156.825179961549281*delta2*x26 + delta2*x36 - 4.0947672145749067e-6*delta2*x39
- 93.9187809703050078*delta2*x56 - 94.6384207133429953*delta2*x60 + 7563.94630250849968*delta2*x62
+ 5.78695265505036005*delta2*x7 + 0.000546777915034636847*x1*tau7*delta5 + x10 - x11
+ 0.0236254113277929619*delta6*x14 - 77.1193941754752785*delta6*x26 - 48.5016915727782916*delta6*x40
+ 116.564484328134682*delta6*x41 - 0.00337505876111328026*x14*delta7 + 24.545583198617404*delta7*x26
- 34.0734489899851098*delta7*x41 - 1.23167645375688006*delta7*x42 + 8.18953442914981341e-6*delta8*x39
- 0.00168573444377057989*delta8*x7 - 2.55251419211106034*x18*tau8 - 0.834324123212193625*x18*tau6
- 2.20249654348003165*x18*tau2 - 0.522942048172529983*x18 - 1.26917960652460995e-7*delta10*x31
+ 0.0000482884579903492813*delta10*x7 - 5.26826171970823243*delta3*x21 - delta3*x24 - 195.091541382877153*delta3*x26
+ 0.119925952777934386*delta3*x27 + 64.6689220970377221*delta3*x40 - 155.4193124375129*delta3*x41
+ 0.185405360713820799*x20*tau6 + 0.48944367632889596*x20*tau2 + 0.116209344038340004*x20
+ 2.63413085985411621*x21*delta5 - 4.5333123817514398*x21 - 0.00273388957517318402*tau7*x5
+ 0.114266219933930394*x23*delta9 - 0.0228532439867860787*x23*delta11 + 60.1319238827615692*x26*delta5
+ 110.582864645293796*x26*delta4 - 0.059962976388967193*x27*delta5 + 0.000245502957044305007*tau11*x3
- 0.0000613757392610762517*tau11*x5 - delta5*x36 + 3.69502936127063997*delta5*x42 - 1.92898421835012002*x3*tau2
- 0.32074263140645598*x3 + 1.26917960652460984e-6*delta9*x31 + 0.598667159386230052*delta9*x42
+ 0.000187303827085619982*delta9*x7 + 1740.65381661920696*delta11*x39 - 0.119733431877245999*delta11*x42
- 0.0110131179261801597*delta11*x47 - 4.38985981730447989e-6*delta11*x7 + 2.0671840916482082e-10*x33*delta13
+ 0.00183551965436336002*delta13*x47 - 0.078989035122503104*delta13*x49 - 1740.65381661920696*tau48*x43
+ 56.7890816499751736*delta4*x41 - x43*x44 + x43*x45 + x44*x48 - x45*x48 + 0.0801856578516139951*x5
+ 1252.25041293740014*x56*x57 + 1261.8456095112399*x57*x60 - 100852.617366780003*x57*x62
- 208.244971994135994*x65*x72 - 416.489943988271989*x65*x79 + 508.89777405510398*x73*x75
+ 1017.79554811020796*x73*x83 - x8 + x86*x87 - x90*x91)
[docs]def iapws95_dAr_dtau(tau, delta):
r'''Calculates the first derivative of residual Helmholtz energy of water
with respect to `tau` according to the IAPWS-95 standard.
.. math::
\phi_{\tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} t_{i} \delta^{d_{i}}
\tau^{t_{i}-1}+\sum_{i=8}^{51} n_{i} t_{i} \delta^{d_{i}} \tau^{t_{i}-1}
\mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}}
\tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}
\right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}
\left[\frac{t_{i}}{\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right)
\right]+\sum_{i=55}^{56} n_{i} \delta\left[\frac{\partial
\Delta^{b_{i}}}{\partial \tau} \psi+\Delta^{b_{i}} \frac{\partial
\psi}{\partial \tau}\right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
dAr_dtau : float
Derivative of residual Helmholtz energy A/(RT) with respect to `tau`,
[-]
Notes
-----
This is an optimized implementatation taking 9 exp calls, 4 sqrts,
and 2 powers. It was generated using SymPy's CSE functionality, with a
limited amount of `horner` polynomial optimizations as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean relative
error was 5.68E-14, with a maximum relative error of 6.73E-9 and a standard
deviation of 1.35E-11.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 90000 points, the mean relative
error was 4.66E-14, with a maximum relative error of 4.25E-10 and a standard
deviation of 1.77E-12.
The maximum error ocurs in the extremely low density regime,
:math:`\rho < 1e-6`.
Examples
--------
>>> iapws95_dAr_dtau(647.096/300.0, 999.0/322)
-7.7043336309570
'''
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta2*delta2
delta8 = delta4*delta4
x1 = exp(-delta)
x4 = exp(-delta2)
x26 = exp(-delta3)
x2 = delta*x1
x5 = delta*x4
x7 = x4*delta8
x8 = 300.0*tau
x9 = (delta - 1.0)
x9 *= x9
x10 = -20.0*x9
x11 = (0.826446280991736*tau - 1.0)
x11 = exp(x10 - 219.615*x11*x11)
tau2 = tau*tau
tau4 = tau2*tau2
taurtinv = 1.0/sqrt(tau)
tau_inv = taurtinv*taurtinv
tau4inv = sqrt(taurtinv)
tau8inv = sqrt(tau4inv)
x15 = tau - 1.0
x13 = tau*delta2
x16 = x15*x15
x17 = x9**1.66666666666666674
x18 = (-tau + 0.32*x17 + 1.0)
x18 = 0.2*x9*x9*x9*sqrt(x9) + x18*x18
x18_05 = x18**-0.05
x19 = delta4*delta2
x20 = delta3*x4
x21 = delta8*delta2*x1
x22 = delta4*delta*x4
x24 = delta8*delta*x4
x27 = delta*x26
x50 = (0.8*tau - 1.0)
return (delta*(-6.26395869124539993e-10*delta8*delta4*delta2*x1 + 0.159012546727090004*delta*taurtinv
- 0.00626677396776150007*tau_inv*taurtinv - tau8inv*(0.196091503945184986*delta*tau8inv
+ 0.00293249068829928763*taurtinv*delta2 - 6.90879303824744984)
- tau*(-tau*(tau*(tau*(tau*(tau*(tau*(-0.000131108546740240002*delta8*delta3*x4
+ tau*(tau*(-0.000626897104146849956*delta8*delta4*delta*exp(-delta4) + tau*(tau*(-0.000794551260476243984*x1
+ x13*(tau2*tau*(tau4*tau2*(tau*(tau4*tau4*tau4*tau4*tau4*tau*(tau2*(14.6176487721394821*delta3 - tau4*(5.92059121299049984*delta3
+ 2.78555592828224976e-8)) - 8.75851607593951975*delta3)*exp(-x19) + 0.516264378636137944*x26*delta2
- 1.76612855042628292*x27) + 0.76986812020283002*x27) + 0.697817851580975979*x26)
+ 5.11464493842302041e-6*x2 - 1.72265340970684017e-11*x21)) + 1.26917960652461008e-8*x1*delta8*delta)
+ 0.14180634400617001*x19*x4 - 7.43159297103409955*x20 - 1.36364351103430015*x22
+ 4.78073299154799969*delta4*x4 + 5.80833999857589944*delta2*x4 - 4.02476697635280001*x5)
- 0.0000683472393793296059*x1*delta4 + 0.00142832774917412992*x24 + 1.9919065650791401*x5)
- 0.0132432400509872004*x7) - 0.219510904987843009*x20 - 0.755552063625239967*x4)
+ 1.22602865705789998*x1 + 0.123167645375688001*x22 + 0.0119733431877246006*x24)
- 1.28545215017189984*x2) + 0.642994739450040043*x1*delta2 - 0.000062434609028539994*x1*delta8
- 2.67426289231860004*x1 + 1.46328660576816003e-6*x21 - 0.0815739460548159934*x7)
+ 0.00168752938055664013*x1*x19 + 0.0149907440972417982*x20 + 0.115845256722618001*x7)
+ 0.0801856578516139951*delta3*x1 + 0.058104672019170002*x7) + 0.00880894931021340005*delta3
+ 31.5461402377809996*x11*x13*(tau_inv - x8 + 363.0) + 31.3062603234350014*x11*delta2*(x8 - 363.0)
- 2521.31543416949989*tau4*delta2*(-500.0*tau + 4.0*tau_inv + 625.0)*exp(x10 - 390.625*x50*x50)
- 0.192327211560020001*x2 + 0.0176114910087519991*x5 + 0.00833265048807130086*x7
+ 0.148746408567240002*(1400.0*x15*x18*x18_05*x18_05*x18_05
+ x18_05*x18_05*x18_05*(-1.7*tau + 0.544*x17
+ 1.7))*exp(-700.0*x16 - 28.0*x9) - 0.318061108784439994*(1600.0*x15*x18*x18_05
+ x18_05*(-1.9*tau + 0.608*x17 + 1.9))
*exp(-800.0*x16 - 32.0*x9) - 8.78032033035609949))
[docs]def iapws95_d2Ar_dtau2(tau, delta):
r'''Calculates the second derivative of residual Helmholtz energy of water
with respect to `tau` according to the IAPWS-95 standard.
.. math::
\phi_{\tau \tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} t_{i}\left(t_{i}
-1\right) \delta^{d_{i}} \tau^{t_{i}-2}+\sum_{i=8}^{51} n_{i} t_{i}
\left(t_{i}-1\right) \delta^{d_{i}} \tau^{t_{i}-2} \mathrm{e}^{
-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}}
\mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}\right)^{2}
-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}\left[\left(\frac{t_{i}}
{\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right)\right)^{2}-\frac{t_{i}}
{\tau^{2}}-2 \beta_{i}\right]
+\sum_{i=55}^{56} n_{i} \delta\left[\frac{\partial^{2} \Delta^{b_{i}}}
{\partial \tau^{2}} \psi+2 \frac{\partial \Delta^{b_{i}}}{\partial \tau}
\frac{\partial \psi}{\partial \tau}+\Delta^{b_{i}} \frac{\partial^{2}
\psi}{\partial \tau^{2}}\right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2Ar_dtau2 : float
Second derivative of residual Helmholtz energy A/(RT) with respect to
`tau`, [-]
Notes
-----
This is an optimized implementatation taking 9 exp calls, 4 sqrts,
and 2 powers. It was generated using SymPy's CSE functionality, with a
limited amount of `horner` polynomial optimizations as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative
error was 4.595E-16, with a maximum relative error of 1.835e-10 and a standard
deviation of 1.209E-13.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 250000 points, the mean relative
error was 2.6026E-16, with a maximum relative error of 2.36E-12 and a standard
deviation of 8.055E-15.
This comparison indicates this implementation is more accurate than the
straightforward implementation.
Examples
--------
>>> iapws95_d2Ar_dtau2(647.096/300.0, 999.0/322)
-1.2616419775539
'''
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta*delta3
delta5 = delta*delta4
delta6 = delta*delta5
delta8 = delta2*delta6
delta9 = delta8*delta
x1 = exp(-delta)
x3 = x1*delta3
tau2 = tau*tau
tau4 = tau2*tau2
tau6 = tau4*tau2
tau7 = tau*tau6
tau8 = tau*tau7
tau11 = tau8*tau2*tau
tau21 = tau11*tau8*tau2
tau48 = tau21*tau21*tau6
x5 = x1*tau2
x8 = exp(-delta2)
taurtinv = 1.0/sqrt(tau)
tau_inv = taurtinv*taurtinv
tau4inv = sqrt(taurtinv)
tau8inv = sqrt(tau4inv)
x10 = x8*delta8
x11 = tau*tau4*x8
x13 = delta*x8
x15 = delta3*x8
x19 = tau8*x8
x20 = exp(-delta3)
x22 = exp(-delta6)
x23 = delta3*x20
x26 = x22*delta5
x28 = delta9*x8
x29 = 0.82644628099173556*tau
x30 = (x29 - 1.0)
x30 *= x30
x31 = (delta - 1.0)
x31 *= x31
x32 = -20.0*x31
x33 = delta2*exp(-219.615*x30 + x32)
x34 = 1.0/tau2
x35 = 1.0/tau
x36 = 0.8*tau
x37 = tau - 1.0
x38 = x37*x37
x39 = x31**1.66666666666666674
x40 = (-tau + 0.32*x39 + 1.0)
x40 *= x40
x41 = 0.2*x31*x31*x31*sqrt(x31) + x40
x43 = x41**(-0.05)
x42 = x43*x43*x43
x50 = 1.0 - x29 + 0.00275482093663911853*x35
x51 = (0.0064*x35 - x36 + 1.0)
return (delta*(delta9*delta*(-0.00564207393732165004*delta3*tau8*exp(-delta4)
- 2.0671840916482082e-10*delta2*x1*tau11 - 0.000917759827181680011*delta*tau6*x8
+ 4.38985981730447989e-6*x5 )
- 5.14180860068759937*delta*tau2*tau*x1
+tau_inv*(
+ 0.0490228759862962465*delta*tau4inv
+taurtinv*(- 0.079506273363545002*delta
+ 0.00940016095164225053*tau_inv
+ 0.00183280668018705488*tau8inv*delta2)
- 0.863599129780931229*tau8inv)
+ tau8*tau*(tau11*(tau11*tau11*x26*(657.79419474627673*tau2
- 376.616191265399323) + 16.1672305242594305*x23)
+ tau*(10.4672677737146405*tau4*delta2*x20 - 0.00874006386523868382*x1)
+ 1.26917960652460995e-7*x1*delta9)
+ tau*(0.00337505876111328026*x1*delta6
+ 0.231690513445236002*x10 + 0.0299814881944835965*x15
- 31.5461402377809996*x33*(x34 - 131769.0*x50*x50
+ 300.0)) + 52.2750599871830985*delta2*x19 - 1.3649224048583023e-6*delta2*tau48*x22
+ 1.92898421835012002*delta2*x5 + 2521.31543416949989*delta2*tau4*(4.0*x34 - 390625.0*x51*x51
+ 500.0)*exp(x32 - 390.625*(x36 - 1.0)*(x36 - 1.0)) - 0.000546777915034636847*x1*tau7*delta4
+ 6.13014328528949992*x1*tau4 - 0.0927026803569103997*x10*tau6 - 0.24472183816444798*x10*tau2
- 0.058104672019170002*x10 - 1.31706542992705811*x11*delta3 - 4.5333123817514398*x11
+ 15.9352525206331208*tau7*x13 + 0.0114266219933930394*tau7*x28 - 36.222902787175201*x13*tau8
- 66.8843367393068888*tau8*x15 + 0.0000613757392610762517*tau11*x3 + 43.0265969239319972*delta4*x19
+ 11.3578163299950354*delta4*x20*tau21 - 12.272791599308702*x19*delta5 + 1.27625709605553017*x19*delta6
- 290.108969436534494*tau48*x26 - 38.8548281093782251*x23*tau21 + 0.615838226878440032*delta5*tau4*x8
+ 0.0598667159386229997*x28*tau4 - 0.0801856578516139951*x3 - 31.3062603234350014*x33*(131769.0*x30 - 300.0)
- 0.000187303827085619982*x5*delta8 - 8.02278867695580011*x5
- 0.148746408567240002*(2800.0*x37*x42*(-1.7*tau
+ 0.544*x39 + 1.7) - 0.51*x40*x42/(x41)
+ 1400.0*x41*x42*(1400.0*x38 - 1.0) + 1.7*x42)*exp(-28.0*x31 - 700.0*x38)
+ 0.318061108784439994*(3200.0*x37*x43*(-1.9*tau + 0.608*x39
+ 1.9) - 0.19*x40*x43/(x41)
+ 1600.0*x41*x43*(1600.0*x38 - 1.0) + 1.9*x43)*exp(-32.0*x31 - 800.0*x38)))
[docs]def iapws95_d2Ar_ddeltadtau(tau, delta):
r'''Calculates the second derivative of residual Helmholtz energy of water
with respect to `tau` and also `delta` according to the IAPWS-95 standard.
.. math::
\phi_{\delta \tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} d_{i} t_{i}
\delta^{d_{i}-1} \tau^{t_{i}-1}+\sum_{i=8}^{51} n_{i} t_{i}
\delta^{d_{i}-1} \tau^{t_{i}-1}\left(d_{i}-c_{i} \delta^{c_{i}}\right)
\mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}}
\tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}
\right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}\left[\frac{d_{i}}
{\delta}-2 \alpha_{i}\left(\delta-\varepsilon_{i}\right)\right]\left[
\frac{t_{i}}{\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right)\right]
\sum_{i=55}^{56} n_{i}\left[\Delta^{b_{i}}\left(\frac{\partial \psi}
{\partial \tau}+\delta \frac{\partial^{2} \psi}{\partial \delta
\partial \tau}\right)+\delta \frac{\partial \Delta^{b_{i}}}{\partial
\delta} \frac{\partial \psi}{\partial \tau}+\frac{\partial
\Delta^{b_{i}}}{\partial \tau}\left(\psi+\delta \frac{\partial \psi}
{\partial \delta}\right)+\frac{\partial^{2} \Delta^{b_{i}}}{\partial
\delta \partial \tau} \delta \psi\right]
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d2Ar_ddeltadtau : float
Second derivative of residual Helmholtz energy A/(RT) with respect to
`tau` and `delta`, [-]
Notes
-----
This is an optimized implementatation taking 11 exp calls, 4 sqrts,
and 3 powers. It was generated using SymPy's CSE functionality, with
select polynomial optimizations by hand as well. It is over
10x faster than a naive implementation.
This implementation has been tested against a straightforward
implementation with the equations given in IAPWS-95.
Over a linear temperature range of 200 K to 5000 K and a logarithmic
density range of 1E-10
kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative
error was 2.82E-14, with a maximum relative error of 8.404E-9 and a standard
deviation of 5.166e-12.
Over the same range, the model was evaluated to a precision of 50
decimal places with `mpmath`, and on 90000 points, the mean relative
error was 6.974E-14, with a maximum relative error of 4.286E-9 and a standard
deviation of 4.286E-12.
Examples
--------
>>> iapws95_d2Ar_ddeltadtau(647.096/300.0, 999.0/322)
-0.198403562385
'''
delta2 = delta*delta
delta3 = delta2*delta
delta4 = delta3*delta
delta5 = delta4*delta
delta6 = delta5*delta
delta8 = delta6*delta2
delta9 = delta8*delta
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau*tau3
tau5 = tau*tau4
tau7 = tau2*tau5
tau8 = tau*tau7
tau9 = tau*tau8
tau16 = tau8*tau8
tau32 = tau16*tau16
tau49 = tau32*tau*tau16
taurtinv = 1.0/sqrt(tau)
tau_inv = taurtinv*taurtinv
tau4inv = sqrt(taurtinv)
tau8inv = sqrt(tau4inv)
x2 = exp(-delta)
x10 = exp(-delta2)
x24 = exp(-delta3)
x29 = exp(-delta6)
x3 = delta*x2*(delta - 2.0)
x5 = delta - 1.0
x6 = x2*x5
x9 = delta3*(delta - 4.0)
x11 = delta*x10*(delta2 - 1.0)
x12 = x2*tau3
x13 = tau7*tau5*x2
x21 = 2.0*delta2
x22 = x10*delta8*(x21 - 9.0)
x23 = tau3*tau3*x10
x25 = delta3*(delta2 - 2.0)
x26 = x10*tau9
x28 = delta5*(delta2 - 3.0)
x30 = delta5*x29*(delta6 - 1.0)
x32 = x10*delta9*(delta2 - 5.0)
x34 = 3.0*delta3
x35 = delta3*(x34 - 4.0)
x36 = tau9*tau9*tau4*x24
x37 = 300.0*tau
x38 = x5*x5
x38rt = sqrt(x38)
x39 = -20.0*x38
x40 = delta3*(-40.0*delta + 40.0 + 3.0/delta)
x41 = (0.826446280991736*tau - 1.0)
x41 = x40*exp(x39 - 219.615*x41*x41)
x42 = 1.0/tau
x57 = x38**0.666666666666666666
x43 = x38*x57
x44 = 1.0 -tau + 0.32*x43
x44 = 0.2*x38*x38*x38*x38rt + x44*x44
x45 = delta*x5
x46 = 56.0*x45
x47 = tau - 1.0
x48 = x47*x47
x49 = exp(-28.0*x38 - 700.0*x48)
x50 = x47*x49
x51 = 64.0*x45
x52 = exp(-32.0*x38 - 800.0*x48)
x53 = x44**(-0.05)
x54 = x38 + 25.0*x48
x55 = x53*x53*x53
x56 = 1.4*x38*x38*x38rt
x58 = -2.1333333333333333*tau + 0.682666666666666755*x43 + 2.1333333333333333
x59 = x56 + x57*x58
x60 = x45*x52
x61 = x56 + x58*(x5*x5)**0.666666666666666741
x100 = 0.8*tau - 1.0
return (delta9*delta*(6.26395869124539993e-10*delta4*x2*(delta - 15.0)
+ 0.00125379420829369991*delta3*tau9*(2.0*delta4 - 7.0)*exp(-delta4)
+ 1.72265340970684017e-11*delta2*x13*(delta - 13.0)
+ 0.000262217093480480003*delta*x10*tau7*(delta2 - 6.0)
- 1.46328660576816003e-6*x12*(delta - 11.0) )
- 0.392183007890369972*delta*tau4inv
+ taurtinv*(0.318025093454180008*delta
- 0.00626677396776150007*tau_inv
- 0.00879747206489786202*tau8inv*delta2)
+ 6.90879303824744984*tau8inv
+ tau9*tau*(# These two terms do nto want to have tau32 factored out
-87.7058926328368926*tau3*tau32*x30
+ 52.5510964556371221*tau*tau32*x30
- 0.76986812020283002*tau9*tau2*x24*x35
- 2.09345355474292782*tau5*delta2*x24*(delta3 - 1.0)
+ 0.000794551260476243984*tau*x6
- 1.26917960652461008e-8*delta9*x2*(delta - 10.0))
+ 0.0801856578516139951*tau*x2*x9
+ 0.058104672019170002*tau*x22 + 31.5461402377809996*tau*x41*(-x37 + x42 + 363.0)
+ 0.0352357972408536002*delta3 - 0.642994739450040043*delta2*x12*(delta - 3.0)
- 5.80833999857589944*delta2*x26*(x21 - 3.0)
+ 8.35666778484674929e-8*delta2*x29*tau49*(2.0*delta6 - 1.0) - 0.0299814881944835965*x10*tau2*x25
- 0.246335290751376002*x10*x28*tau5 - 3.9838131301582802*x11*tau8 + 8.04953395270560002*x11*tau9
- 0.0352229820175039982*x11 + 0.000062434609028539994*x12*delta8*(delta - 9.0)
- 5.11464493842302041e-6*x13*x9 + 0.0000683472393793296059*delta4*tau8*x2*(delta - 5.0)
- 4.78073299154799969*delta4*x26*(x21 - 5.0) - 0.516264378636137944*delta4*x36*(x34 - 5.0)
- 0.00285665549834825984*tau8*x32 - 0.00168752938055664013*delta6*tau2*x2*(delta - 7.0)
- 0.14180634400617001*delta6*x26*(x21 - 7.0) - 0.115845256722618001*tau2*x22
+ 0.0132432400509872004*x22*tau7 + 0.0815739460548159934*x22*tau3 - 0.00833265048807130086*x22
+ 0.439021809975686017*x23*x25 + 0.755552063625239967*x23*(x21 - 1.0) + 14.8631859420681991*x25*x26
+ 2.7272870220686003*x26*x28 + 1.28545215017189984*x3*tau4 + 0.192327211560020001*x3
+ 35.5235472779429955*x30*tau49 - 0.0239466863754492013*x32*tau5 + 1.76612855042628292*x35*x36
+ 2.67426289231860004*tau3*x6 - 2521.31543416949989*x40*tau4*(-500.0*tau + 4.0*x42 + 625.0)*exp(
x39 - 390.625*x100*x100) + 31.3062603234350014*x41*(x37 - 363.0)
- 208.244971994135994*x44*x53*x53*x53*x50*(x46 - 1.0)
+ 508.89777405510398*x44*x53*x47*x52*(x51 - 1.0)
- 0.148746408567240002*x45*x49*(x53*x53*x53/x44*x61*(
-0.255*tau + 0.0816*x43 + 0.255)
- 1.81333333333333346*x55*x57) + 177.008226195015595*x45*x50*x55*x59
- 483.452885352348801*x47*x53*x59*x60 - 0.318061108784439994*x53*(-x51*x52
+ exp(-32.0*x54))*(-1.9*tau + 0.608*x43 + 1.9)
+ 0.148746408567240002*x55*(-x46*x49 + exp(-28.0*x54))*(-1.7*tau
+ 0.544*x43 + 1.7) - 1.22602865705789998*x6*tau5
+ 0.318061108784439994*x60*(x53/x44*x61*(-0.095*tau
+ 0.0304*x43 + 0.095) - 2.02666666666666666*x53*x57)
- 8.78032033035609949)
def iapws95_d4Ar_ddelta2dtau2_full(tau, delta):
x0 = exp(-delta)
x1 = delta**2
x2 = x0*x1
x3 = delta**3
x4 = x0*x3
x5 = delta**4
x6 = x0*x5
x7 = tau**2
x8 = x0*x7
x9 = tau**3
x10 = x0*x9
x11 = tau**4
x12 = x0*x11
x13 = tau**10*x0
x14 = delta**5
x15 = tau*x0
x16 = delta**6
x17 = delta**7
x18 = exp(-x1)
x19 = x17*x18
x20 = delta**9
x21 = x18*x20
x22 = delta**11
x23 = x18*x22
x24 = tau**7
x25 = x18*x24
x26 = tau**8
x27 = x18*x26
x28 = tau**5
x29 = x18*x28
x30 = exp(-x3)
x31 = tau**14*x30
x32 = exp(-x16)
x33 = tau**48
x34 = x32*x33
x35 = tau*x18
x36 = tau**11
x37 = delta**8
x38 = tau**9*x0
x39 = delta**10
x40 = x0*x36
x41 = delta**12
x42 = delta**13
x43 = tau**20*x30
x44 = tau**21*x30
x45 = x18*x5
x46 = tau**42
x47 = x32*x46
x48 = tau**44
x49 = x32*x48
x50 = x11*x18
x51 = tau**6
x52 = x18*x51
x53 = x32*x39
x54 = x18*x41
x55 = x26*exp(-x5)
x56 = delta**16
x57 = delta - 1.0
x58 = x57**2
x59 = -20.0*x58
x60 = (0.82644628099173556*tau - 1.0)**2
x61 = exp(x59 - 219.615*x60)
x62 = delta*x61
x63 = x3*x61
x64 = exp(x59 - 390.625*(0.8*tau - 1)**2)
x65 = delta*x64
x66 = x11*x65
x67 = tau*x63
x68 = 300.0*tau - 363.0
x69 = x1*x57
x70 = x61*x69
x71 = x3*x64
x72 = x7*x71
x73 = x11*x71
x74 = x60*x62
x75 = x58*x63
x76 = x9*(500.000000000000114*tau - 625.0)
x77 = x64*x69
x78 = x11*x77
x79 = x60*x63
x80 = (0.800000000000000155*tau - 1.0)**2
x81 = x71*x76
x82 = x60*x70
x83 = x73*x80
x84 = x58*x79
x85 = x58**3.5
x86 = x58**1.66666666666666674
x87 = -tau + 0.320000000000000007*x86 + 1.0
x88 = 0.200000000000000011*x85 + x87**2
x89 = tau - 1.0
x90 = x89**2
x91 = exp(-28*x58 - 700*x90)
x92 = x88**(-0.150000000000000022)*x91
x93 = delta*x92
x94 = exp(-32*x58 - 800*x90)
x95 = x88**(-0.0500000000000000444)*x94
x96 = delta*x95
x97 = x88**0.849999999999999978*x91
x98 = delta*x97
x99 = x88**0.949999999999999956*x94
x100 = delta*x99
x101 = x57*x92
x102 = x57*x95
x103 = x57*x97
x104 = x57*x99
x105 = x58*x93
x106 = x58*x96
x107 = x58*x98
x108 = x100*x90
x109 = 32652811.6086805239*x90
x110 = 104222264.126485303*x90
x111 = x86*x89
x112 = x111*x93
x113 = x111*x96
x114 = x88**(-1.14999999999999991)*x91
x115 = delta*x114
x116 = x58**3.33333333333333348
x117 = 1/x58
x118 = x116*x117
x119 = x88**(-1.05000000000000004)*x94
x120 = delta*x119
x121 = 1/x57
x122 = x121*x86
x123 = x121*x95
x124 = 23323.4368633432314*x93
x125 = -1.69999999999999996*tau + 0.544000000000000039*x86 + 1.69999999999999996
x126 = x125*x89
x127 = -1.89999999999999991*tau + 0.607999999999999985*x86 + 1.89999999999999991
x128 = x127*x89
x129 = -0.300000000000000044*tau + 0.0960000000000000159*x86 + 0.300000000000000044
x130 = x115*x129
x131 = x130*x86
x132 = -0.100000000000000089*tau + 0.0320000000000000284*x86 + 0.100000000000000089
x133 = x119*x132
x134 = delta*x86
x135 = x133*x134
x136 = x114*x129
x137 = x125*x130
x138 = x127*x133
x139 = delta*x138
x140 = x86*x87
x141 = 1.81333333333333346*x140 + 1.19000000000000017*x85
x142 = x115*x141
x143 = 2.02666666666666684*x140 + 1.33000000000000007*x85
x144 = x119*x143
x145 = delta*x144
x146 = x143*x96
x147 = x121*x141
x148 = x114*x147
x149 = x147*x92
x150 = x121*x144
x151 = 1017.79554811020796*x143
x152 = x117*(1.93422222222222251*x116 + 4.2311111111111126*x140 + 7.13999999999999879*x85)
x153 = x115*x152
x154 = x117*(2.16177777777777802*x116 + 4.72888888888889003*x140 + 7.98000000000000043*x85)
x155 = x152*x93
x156 = x154*x96
x157 = x117*(0.106666666666666771*x140 + 0.0700000000000000622*x85)
x158 = x134*x157
x159 = x117*(0.320000000000000062*x140 + 0.210000000000000048*x85)
x160 = x111*x117
x161 = x129*x89
x162 = x132*x89
x163 = 291542.960791790392*x90
x164 = 814236.43848816643*x90
x165 = x88**(-2.04999999999999982)
x166 = -2.10000000000000009*tau + 0.672000000000000042*x86 + 2.10000000000000009
x167 = x166*x94
x168 = x165*x167
x169 = x88**(-2.14999999999999991)
x170 = -2.29999999999999982*tau + 0.735999999999999988*x86 + 2.29999999999999982
x171 = x169*x170*x91
x172 = x117*x134
x173 = x143*x165*x94
x174 = x166*x173
x175 = delta*x141
x176 = x129*x171
x177 = x132*x174
x178 = delta*x154
x179 = delta*x157
x180 = x145*x157
x181 = x159*x175
x182 = x181*x91
x183 = x142*x159
return (-0.0902731829971464006*delta**20*x55 - 0.00367103930872672004*delta**14*x52 + 0.0109968400811223284*delta*tau**(-1.625) + 20.5672344027503975*delta*x10 + 6.13014328528949992*delta*x12 - 0.00874006386523868382*delta*x13 - 0.148746408567240002*delta*x152*x176 - 40.7118219244083193*delta*x177 + 313.650359923098563*delta*x27 + 27.1998742905086388*delta*x29 + 62.8036066422878321*delta*x31 - 8.1895344291498151e-6*delta*x34 + 3.55111663314491999*delta*x8 - 0.159012546727090004*tau**(-1.5) + 0.0980457519725924931*tau**(-1.25) + 16.6817169680569926*tau*x19 - 8.80423951091896839*tau*x21 + 0.926762053780944006*tau*x23 - 56783.0524280057944*tau*x62 + 2271322.09712023195*tau*x70 + 24940820.1179529876*tau*x74 - 166272134.119686574*tau*x79 - 997632804.718119502*tau*x82 + 6650885364.78746319*tau*x84 - 0.000546777915034636847*x0*x14*x24 - 5.14180860068759937*x1*x10 - 159.352525206331222*x1*x25 - 440.383012999930713*x1*x27 - 15.8047851591246946*x1*x29 + 0.359777858333803158*x1*x35 + 194.006766291113166*x1*x43 - 466.257937312538729*x1*x44 - 11.5739053101007201*x1*x8 - 10.2836172013751987*x10 - 2084445.2825297059*x100*x58 + 32569.4575395266547*x100 + 46646.8737266864628*x101*x126 + 28.3213161912024951*x101 - 130277.830158106619*x102*x128 - 77.3524616563758087*x102 + x103*x109 - 23323.4368633432314*x103 - x104*x110 + 65138.9150790533095*x104 - 1306112.46434722096*x105*x126 - 792.996853353669849*x105 + 4168890.56505941181*x106*x128 + 2475.27877300402588*x106 - 914278725.0430547*x107*x90 + 653056.232173610479*x107 + 3335112452.0475297*x108*x58 - 52111132.0632426515*x108 + x109*x141*x93 + 18.4751468063532016*x11*x45 + 0.239466863754491999*x11*x54 - x110*x146 + 755.235098432066707*x111*x115*x159 + 4125.46462167337631*x111*x123 - 1762.21522967482247*x112*x117 + 84586.3310243914602*x112 + 4813.04205861894025*x113*x117 - 264029.735787096084*x113 + 0.172625165355901*x115*x118 + 1.2587251640534447*x117*x131 - 3.00815128663683762*x117*x135 - 0.137515487389112678*x118*x120 - 2062.73231083668816*x119*x158*x89 - 12.2602865705789998*x12 - 0.0318061108784440244*x120*x154 + 0.636122217568879988*x121*x177 - 2.57841538854586005*x122*x133 + 1.07890728347438092*x122*x136 - 1510.47019686413341*x122*x89*x92 + 1628472.87697633286*x123*x143*x90 - x123*x151 + x124*x126 - x124*x141 - 16.6595977595308788*x125*x136*x57 - 65138.9150790533095*x128*x96 + 0.0174801277304773676*x13 - 60.4188078745653385*x131 + 0.318061108784439994*x132*x168*x178 - 1017.79554811020796*x133*x178*x89 - 0.539453641737190459*x134*x159*x171 + 165.018584866935043*x135 + 466.46873726686465*x137*x58 - 8.32979887976543942*x137 + 40.7118219244083193*x138*x57 - 1302.77830158106622*x139*x58 + 20.3559109622041596*x139 + 0.141752467966757778*x14*x15 - 683.882094343439235*x14*x27 - 485.01691572778293*x14*x43 + 1165.64484328134677*x14*x44 - 0.0951977014830336227*x141*x171*x172 + 133.276782076247088*x142*x160 - 46646.8737266864628*x142*x161 - 4.99787932785926525*x142 - 0.318061108784439994*x143*x167*x179*x88**(-3.04999999999999982)*(-4.09999999999999964*tau + 1.31199999999999983*x86 + 4.09999999999999964) - 108.56485846508896*x145*x160 + 130277.830158106619*x145*x162 + 4.07118219244083512*x145 + 65138.9150790533095*x146 - 0.297492817134480003*x147*x176 + 832.979887976543978*x148*x161 + 0.0892478451403440204*x148 - 583085.921583580784*x149*x90 + 416.489943988271989*x149 - 0.0472508226555859237*x15*x16 + 0.00337505876111328026*x15*x17 - 2035.59109622041592*x150*x162 - 0.0636122217568880488*x150 + x151*x168*x179*x89 + 416.489943988271989*x153*x161 + 0.0446239225701720102*x153 - x155*x163 + 208.244971994135994*x155 + x156*x164 - 508.89777405510398*x156 + 1.28920769427293003*x158*x168 + 51.555234624798608*x16*x27 - 5.26826171970823243*x16*x29 + 0.119925952777934386*x16*x35 - 408.881387879821318*x16*x44 - 16.0117938988394393*x16*x50 + x163*x183 - x164*x180 - 0.342116739704651951*x169*x182 + 133.818674814062092*x17*x27 + 94.2054099634317623*x17*x31 + 0.0000900848787206479763*x17*x34 - 0.0134858755501646374*x17*x8 + 0.148746408567240002*x170*x182*x88**(-3.14999999999999991)*(-4.29999999999999982*tau + 1.37599999999999989*x86 + 4.29999999999999982) - 416.489943988271989*x171*x181*x89 + 0.067853036540680603*x172*x174 + 0.667928328447323971*x173*x179 + 16.6595977595308788*x175*x176 + 508.89777405510398*x180 - 208.244971994135994*x183 - 6.674592985697549*x19*x51 - 17.6199723478402532*x19*x7 - 4.18353638538023986*x19 + 0.000736508871132914803*x2*x36 - 0.962227894219367941*x2 - 2.53835921304921969e-6*x20*x38 + 102.22034696995533*x20*x44 + 0.00029558075281787275*x20*x8 + 5.10502838422212069*x21*x26 + 3.52270185356259535*x21*x51 + 9.29942985024902313*x21*x7 + 2.20797753672845998*x21 - 3.22480718297120481e-8*x22*x40 + 4.38985981730447989e-6*x22*x8 - 0.370810721427641599*x23*x51 - 0.97888735265779192*x23*x7 - 0.232418688076680008*x23 - 0.0109355583006927361*x24*x4 + 0.00546777915034636804*x24*x6 + 1.02839597940537364*x25*x37 - 0.479918123722507695*x25*x39 + 0.0457064879735721574*x25*x41 + 63.7410100825324832*x25*x5 + 31.8705050412662416*x25 + 128.681098658076564*x27*x3 - 49.091166397234808*x27*x37 + 690.842702179562366*x27*x5 - 72.4458055743504019*x27 + 23.7071777386870473*x28*x45 - 18.1332495270057592*x29*x3 + 227.156326599900694*x3*x44 - 251.214426569151328*x31*x5 + 29591.1148825265227*x33*x53 - 0.0000491372065748988906*x34*x42 - 8703.26908309603641*x34*x5 - 10443.9228997152422*x34*x56 - 0.539666787500704737*x35*x5 - 0.000491005914088610013*x36*x4 + 0.0000613757392610762517*x36*x6 + 0.0000114226164587214907*x37*x38 + 145.505074718334896*x37*x43 - 349.693452984404075*x37*x44 + 7.85135734198982949*x37*x50 + 0.00337146888754115935*x37*x8 + 1.26917960652460995e-7*x38*x39 - 2.51440206942216582*x39*x50 - 0.121144297187981767*x39*x52 - 0.0000965769159806985627*x39*x8 + 1.92898421835012002*x4*x7 + 0.64148526281291196*x4 + 5.37467863828534189e-9*x40*x41 - 2.0671840916482082e-10*x40*x42 - 1.02685745659254035*x41*x55 + 38414.8515090707369*x46*x53 - 11298.4857379619825*x47*x5 - 13558.1828855543772*x47*x56 - 67095.0078641202126*x48*x53 + 19733.8258423883017*x49*x5 + 23680.5910108659555*x49*x56 + 0.0458879913590840016*x51*x54 + 0.699617168227884667*x55*x56 - 15142147.3141348809*x58*x67 - 48409256.3360543996*x58*x72 + 2017052347.33560038*x58*x73 + 32272837.5573695973*x58*x81 - 1575822146355.9375*x58*x83 - 0.0801856578516139951*x6 - 378.553682853371981*x62*x68 + 56351.2685821830019*x62 + 2523.6912190224798*x63*x68 - 375675.123881220061*x63 - 181534.711260204*x65*x7 + 121023.140840135995*x65*x76 - 5909333048.83476543*x66*x80 + 7563946.30250850134*x66 + 378553.682853371953*x67 + 15142.1473141348797*x68*x70 - 100947.648760899203*x68*x75 + 7261388.45040815976*x7*x77 - 2254050.74328732025*x70 + 1210231.40840136004*x72 - 50426308.6833900139*x73 - 24751167.6993522421*x74 + 15027004.9552487992*x75 - 4840925.63360544015*x76*x77 + 236373321953.390625*x78*x80 - 302557852.100340068*x78 + 165007784.66234827*x79 + 16.0455773539116002*x8 - 806820.938934240025*x81 + 990046707.974089622*x82 + 39395553658.8984375*x83 - 6600311386.49393082*x84 + 16326405.804340262*x90*x98 + 14.1606580956012458*x93 - 38.6762308281879044*x96 - 11661.7184316716157*x98)
[docs]def iapws95_d4Ar_ddelta2dtau2(tau, delta):
r'''Calculates the fourth derivative of residual Helmholtz energy of water
with respect to `tau` twice and `delta` twice according to the IAPWS-95
standard.
Parameters
----------
tau : float
Dimensionless temperature, (647.096 K)/T [-]
delta : float
Dimensionless density, rho/(322 kg/m^3), [-]
Returns
-------
d4Ar_ddelta2dtau2 : float
Fourth derivative of residual Helmholtz energy A/(RT) with respect to
`tau` and `delta`, [-]
Notes
-----
Examples
--------
>>> iapws95_d4Ar_ddelta2dtau2(647.096/300.0, 999.0/322)
-2.656422915480
'''
if 0.6 < tau < 1.6 and delta < 7.0:
return iapws95_d4Ar_ddelta2dtau2_full(tau, delta)
# Otherwise drop terms 51-54 and the last two
tau_inv = 1.0/tau
tau_invrt2 = sqrt(tau_inv)
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau2*tau2
tau5 = tau*tau4
tau6 = tau2*tau4
tau7 = tau*tau6
tau8 = tau4*tau4
tau9 = tau*tau8
tau10 = tau2*tau8
tau11 = tau*tau10
tau14 = tau4*tau10
tau20 = tau10*tau10
tau21 = tau*tau20
tau42 = tau21*tau21
tau24 = tau4*tau20
tau44 = tau20*tau24
tau16 = tau8*tau8
tau48 = tau24*tau24
delta2 = delta*delta
delta3 = delta*delta2
delta4 = delta2*delta2
delta5 = delta*delta4
delta6 = delta2*delta4
delta7 = delta*delta6
delta8 = delta4*delta4
delta9 = delta*delta8
delta10 = delta2*delta8
delta11 = delta*delta10
delta12 = delta4*delta8
delta13 = delta*delta12
delta16 = delta8*delta8
x0 = exp(-delta2)
x1 = delta11*x0
x2 = exp(-delta)
x3 = delta2*x2
x4 = delta3*x2
x5 = delta4*x2
x6 = delta7*x0
x7 = delta9*x0
x8 = tau10*x2
x9 = tau2*x2
x10 = tau3*x2
x11 = tau4*x2
x12 = tau7*x0
x13 = tau8*x0
x14 = exp(-delta3)
x15 = tau14*x14
x16 = exp(-delta6)
x17 = tau48*x16
x18 = tau5*x0
x19 = tau4*x0
x20 = delta10*x16
x21 = tau6*x0
x22 = tau9*x2
x23 = tau11*x2
x24 = delta12*x0
x25 = tau8*exp(-delta4)
x26 = tau42*x16
x27 = tau44*x16
x28 = tau*x0
x29 = tau20*x14
x30 = tau21*x14
x31 = delta4*x0
x32 = tau*x2
x33 = delta12
return (0.0109968400811223284*delta*tau_inv**1.625 + 20.5672344027503975*delta*x10 + 6.13014328528949992*delta*x11 + 313.650359923098563*delta*x13 + 62.8036066422878321*delta*x15 - 8.1895344291498151e-6*delta*x17 + 27.1998742905086388*delta*x18 - 0.00874006386523868382*delta*x8 + 3.55111663314491999*delta*x9 - 0.479918123722507695*delta10*x12 - 2.51440206942216582*delta10*x19 - 0.121144297187981767*delta10*x21 + 1.26917960652460995e-7*delta10*x22 - 0.0000965769159806985627*delta10*x9 - 3.22480718297120481e-8*delta11*x23 + 4.38985981730447989e-6*delta11*x9 + 0.0457064879735721574*delta12*x12 + 5.37467863828534189e-9*delta12*x23 - 1.02685745659254035*delta12*x25 - 0.0000491372065748988906*delta13*x17 - 2.0671840916482082e-10*delta13*x23 - 10443.9228997152422*delta16*x17 + 0.699617168227884667*delta16*x25 - 13558.1828855543772*delta16*x26 + 23680.5910108659555*delta16*x27 - 5.14180860068759937*delta2*x10 - 159.352525206331222*delta2*x12 - 440.383012999930713*delta2*x13 - 15.8047851591246946*delta2*x18 - 0.00367103930872672004*delta2*x21*x33 + 0.359777858333803158*delta2*x28 + 194.006766291113166*delta2*x29 - 466.257937312538729*delta2*x30 - 11.5739053101007201*delta2*x9 + 128.681098658076564*delta3*x13 - 18.1332495270057592*delta3*x18 + 227.156326599900694*delta3*x30 + 63.7410100825324832*delta4*x12 + 690.842702179562366*delta4*x13 - 251.214426569151328*delta4*x15 - 8703.26908309603641*delta4*x17 - 11298.4857379619825*delta4*x26 + 19733.8258423883017*delta4*x27 - 0.539666787500704737*delta4*x28 - 0.000546777915034636847*delta5*tau7*x2 - 683.882094343439235*delta5*x13 - 485.01691572778293*delta5*x29 + 1165.64484328134677*delta5*x30 + 0.141752467966757778*delta5*x32 + 51.555234624798608*delta6*x13 - 5.26826171970823243*delta6*x18 - 16.0117938988394393*delta6*x19 + 0.119925952777934386*delta6*x28 - 408.881387879821318*delta6*x30 - 0.0472508226555859237*delta6*x32 + 133.818674814062092*delta7*x13 + 94.2054099634317623*delta7*x15 + 0.0000900848787206479763*delta7*x17 + 0.00337505876111328026*delta7*x32 - 0.0134858755501646374*delta7*x9 + 1.02839597940537364*delta8*x12 - 49.091166397234808*delta8*x13 + 7.85135734198982949*delta8*x19 + 0.0000114226164587214907*delta8*x22 - 0.0902731829971464006*delta8*x25*x33 + 145.505074718334896*delta8*x29 - 349.693452984404075*delta8*x30 + 0.00337146888754115935*delta8*x9 - 2.53835921304921969e-6*delta9*x22 + 102.22034696995533*delta9*x30 + 0.00029558075281787275*delta9*x9 + 0.926762053780944006*tau*x1 + 16.6817169680569926*tau*x6 - 8.80423951091896839*tau*x7 + 0.000736508871132914803*tau11*x3 - 0.000491005914088610013*tau11*x4 + 0.0000613757392610762517*tau11*x5 - 0.97888735265779192*tau2*x1 + 1.92898421835012002*tau2*x4 - 17.6199723478402532*tau2*x6 + 9.29942985024902313*tau2*x7 + 0.239466863754491999*tau4*x24 + 18.4751468063532016*tau4*x31 + 38414.8515090707369*tau42*x20 - 67095.0078641202126*tau44*x20 + 29591.1148825265227*tau48*x20 + 23.7071777386870473*tau5*x31 - 0.370810721427641599*tau6*x1 + 0.0458879913590840016*tau6*x24 - 6.674592985697549*tau6*x6 + 3.52270185356259535*tau6*x7 - 0.0109355583006927361*tau7*x4 + 0.00546777915034636804*tau7*x5 + 5.10502838422212069*tau8*x7 + 0.0980457519725924931*tau_inv*tau_invrt2**0.5 - 0.159012546727090004*tau_inv*tau_invrt2 - 0.232418688076680008*x1 - 10.2836172013751987*x10 - 12.2602865705789998*x11 + 31.8705050412662416*x12 - 72.4458055743504019*x13 - 0.962227894219367941*x3 + 0.64148526281291196*x4 - 0.0801856578516139951*x5 - 4.18353638538023986*x6 + 2.20797753672845998*x7 + 0.0174801277304773676*x8 + 16.0455773539116002*x9)
### Vapor pressure solution
def _P_G_dG_dV_T_dG_dV_T(T, V):
'''For calculating vapor pressure'''
_MW_kg = iapws95_MW / 1000
R_MW_0_001 = _MW_kg*iapws95_R
rho = rho_mass = iapws95_MW / V / 1000
tau = iapws95_Tc / T
delta = rho_mass / iapws95_rhoc
tau = iapws95_Tc / T
delta = rho / iapws95_rhoc
# print(tau, delta, 'tau, delta')
A = iapws95_A0(tau, delta) + iapws95_Ar(tau, delta)
dA_dtau = iapws95_dAr_dtau(tau, delta) + iapws95_dA0_dtau(tau, delta)
d2A_dtau2 = iapws95_d2Ar_dtau2(tau, delta) + iapws95_d2A0_dtau2(tau, delta)
dA_ddelta = iapws95_dAr_ddelta(tau, delta) + 1/delta
d2A_ddelta2 = iapws95_d2Ar_ddelta2(tau, delta) -1/(delta*delta)
d2A_ddeltadtau = iapws95_d2Ar_ddeltadtau(tau, delta)
P = (dA_ddelta*delta)*rho*iapws95_R*T
S = (tau*dA_dtau - A)*R_MW_0_001
H = (tau*dA_dtau + dA_ddelta*delta)*T*R_MW_0_001
G = H - T*S
dP_dV = (-iapws95_MW*iapws95_MW*iapws95_R*T*(iapws95_MW*d2A_ddelta2/10**9 + V*iapws95_rhoc*dA_ddelta/500000)/(V*V*V*V*iapws95_rhoc*iapws95_rhoc))
dS_dV_T = ((dA_ddelta - iapws95_Tc * d2A_ddeltadtau / T)
* R_MW_0_001 * iapws95_MW / 1000 / (V * V * iapws95_rhoc))
dS_dP_T = dS_dV_T/dP_dV
dH_dV_T = (T * iapws95_MW * (-iapws95_MW * d2A_ddelta2 / (V * iapws95_rhoc * 10 ** 6) - dA_ddelta / 1000
- iapws95_Tc * d2A_ddeltadtau / T / 1000) * R_MW_0_001 / (iapws95_rhoc * V * V))
dH_dP_T = dH_dV_T/dP_dV
dG_dP_T = -T*dS_dP_T + dH_dP_T
dG_dV_T = dG_dP_T*dP_dV
return G, P, dG_dV_T, dP_dV
def iapws95_sat_err_and_jac(Vs, T):
#print(Vs, 'Vs')
V_l, V_g = Vs
G_l, P_l, dG_dV_l, dP_dV_l = _P_G_dG_dV_T_dG_dV_T(T, V_l)
G_g, P_g, dG_dV_g, dP_dV_g = _P_G_dG_dV_T_dG_dV_T(T, V_g)
err = [0.0]*2
err[0] = G_l - G_g
err[1] = P_l - P_g
jac = [[dG_dV_l, -dG_dV_g],
[dP_dV_l, -dP_dV_g]]
#print(err, 'err')
# print([float(i) for i in err], float(P_g), float(P_l), [float(i) for i in Vs])
return err, jac
[docs]def iapws95_saturation(T, xtol=1e-5, rhol_guess=None, rhog_guess=None):
r'''Solve the vapor-liquid saturation equations of IAPWS-95 given a
specified temperature. With floating point numbers, the achievable
tolerance is somewhat low so `xtol` is exposed as a setting - it can be
adjusted somewhat. Density guesses may be provided, otherwise they will
be estimated.
.. math::
G_{liq}(T, \rho_l) = G_{vap}(T, \rho_g)
.. math::
P_{liq}(T, \rho_l) = P_{vap}(T, \rho_g)
Parameters
----------
T : float
Temperature at which to solve for saturation condition, [K]
xtol : float
Tolerance for solver, [-]
rhol_guess : float, optional
Liquid density of water at saturation (guess), [kg/m^3]
rhog_guess : float, optional
Vapor density of water at saturation (guess), [kg/m^3]
Returns
-------
Psat : float
Saturation vapor pressure, 3[Pa]
rhol : float
Saturation liquid water density, [kg/m^3]
rhog : float
Saturation vapor water density, [kg/m^3]
Notes
-----
This is not a perfect function.
With `mpmath` multiple precision, the equation can be solved down to 233.6 K
and up to 647.095999995 K - within 10 parts in a billion of the critical
point exactly.
Reasons for non-convergence include floating point issues as delta
becomes 1, and zero division errors in the matrix inverse.
Examples
--------
>>> iapws95_saturation(400.0, xtol=1e-6)
(245769.345, 937.4860, 1.3694075)
>>> iapws95_saturation(647.0955, xtol=1e-7)
(22063866.35, 325.70, 318.277)
'''
if rhog_guess is None:
rhog = iapws92_rhog_sat(T)
else:
rhog = rhog_guess
if rhol_guess is None:
if T > 647.09:
# For unknown reasons, the worse estimates converge better near
# the critical point
rhol = iapws92_rhol_sat(T)
else:
rhol = iapws95_rhol_sat(max(T, 235))
else:
rhol = rhol_guess
Vg = iapws95_MW / (rhog * 1000)
Vl = iapws95_MW / (rhol * 1000)
V_crit_liq = iapws95_MW / (322.000000000001 * 1000) # Do not allow it to jump to delta=1
V_crit_gas = iapws95_MW / (321.999999999999 * 1000)
V_min = iapws95_MW / (1e-5 * 1000) # assume a minimum density of 1e-5 kg/m^3 for gas; stops converging at 18 times that
V_min = max(V_crit_gas*(1.0+1e-10), Vg*2.0) # Assume the vapor guess is within 100% of the actual value
V_max = iapws95_MW / (2000.0 * 1000) # use 2000 kg/m^3 as an upper bound
# Translate the function variables so that vapor density cannot go above
# 322 kg/m^3 and liquid cannot go under it
new_f_j, translate_into, translate_outof = translate_bound_f_jac(iapws95_sat_err_and_jac,
jac=True,
bounds=[(V_crit_gas, V_min), (V_max, V_crit_liq)])
guess = translate_into([Vg, Vl])
if T > 647.09599999:
# Avoid the jacobian - very badly behaved
ans, iters = broyden2(guess, fun=lambda x, T:new_f_j(x, T)[0], jac=new_f_j, xtol=xtol, maxiter=1000, jac_has_fun=True, skip_J=False, args=(T,))
else:
ans, iters = newton_system(new_f_j, guess, args=(T,),
jac=True, xtol=xtol, solve_func=solve_2_direct)
Vg, Vl = translate_outof(ans)
# Compute the densities and vapor pressure
rhol = iapws95_MW / (Vl * 1000)
rhog = iapws95_MW / (Vg * 1000)
Psat = iapws95_P(T, rhol)
return Psat, rhol, rhog
[docs]def iapws92_Psat(T):
r'''Compute the saturation pressure of the IAPWS-92 equation.
.. math::
P_{sat} = P_c \exp\left(\frac{T_c}{T}[a_1\tau + a_2\tau^{1.5} + a_3\tau^3 + a_4\tau^{3.5}
a_5\tau^4 + a_6\tau^{7.5}]\right)
Parameters
----------
T : float
Temperature at which to calculate the saturation condition and
its temperature derivative, [K]
Returns
-------
Psat : float
Saturation vapor pressure, [Pa]
Notes
-----
The coefficients are [-7.85951783, 1.84408259, -11.7866497, 22.6807411,
-15.9618719, 1.80122502]
Examples
--------
>>> iapws92_Psat(400.0)
245765.2635418
'''
Tr = T*iapws95_Tc_inv
tau = 1.0 - Tr
taurt2 = sqrt(tau)
tau2 = tau*tau
tau3 = tau*tau2
tau4 = tau2*tau2
return iapws95_Pc*exp((1.84408259*tau*taurt2 - 7.85951783*tau + 22.6807411*tau3*taurt2
+ 1.80122502*tau3*taurt2*tau4 - 11.7866497*tau3 - 15.9618719*tau4)/Tr)
[docs]def iapws92_dPsat_dT(T):
r'''Compute the temperature derivative of saturation pressure of the
IAPWS-92 equation.
.. math::
P_{sat} = P_c \exp\left(\frac{T_c}{T}[a_1\tau + a_2\tau^{1.5} + a_3\tau^3 + a_4\tau^{3.5}
a_5\tau^4 + a_6\tau^{7.5}]\right)
Parameters
----------
T : float
Temperature at which to calculate the saturation condition and
its temperature derivative, [K]
Returns
-------
dPsat_dT : float
First temperature derivative of saturation vapor pressure, [Pa/K]
Psat : float
Saturation vapor pressure, [Pa]
Notes
-----
The coefficients are [-7.85951783, 1.84408259, -11.7866497, 22.6807411,
-15.9618719, 1.80122502]
Examples
--------
>>> iapws92_dPsat_dT(400.0)
(7483.47094105, 245765.263541)
'''
Tr = T*iapws95_Tc_inv
T_inv = 1.0/T
Tr_inv = iapws95_Tc*T_inv
tau = 1.0 - Tr
taurt2 = sqrt(tau)
tau2 = tau*tau
tau4 = tau2*tau2
Psat = iapws95_Pc*exp(tau*(taurt2*(1.84408259 + (1.80122502*tau4 + 22.6807411)*tau2)
- tau2*(11.7866497 + 15.9618719*tau) - 7.85951783)*Tr_inv)
dPsat_dT = ((Tr*tau*(-13.50918765*tau4 - 79.38259385)
+ tau2*(-1.80122502*tau4 - 22.6807411)
- 1.84408259)*T_inv*Tr_inv*taurt2*tau
+ iapws95_Tc_inv*Tr_inv*(35.60803943*Tr_inv
+ Tr*(-2.766123885*Tr_inv*taurt2 + Tr*(151.2682746 - 47.8856157*Tr)
- 131.1311805)))
dPsat_dT *= Psat
return dPsat_dT, Psat
Psat_coeffs_iapws95_235_273 = [-0.0404815673828125, 0.0458526611328125, 0.375885009765625, -0.4257164001464844, -1.6131267547607422, 1.8266944885253906, 4.242580890655518, -4.803114175796509, -7.645859479904175, 8.652797102928162, 10.002255886793137, -11.31277510523796, -9.817383006215096, 11.092894461005926, 7.370591592043638, -8.314844283275306, -4.277057046769187, 4.812127415090799, 1.925885249627754, -2.1572029151720926, -0.6719730040349532, 0.7471348317922093, 0.18054040489369072, -0.19828112466348102, -0.03694033686770126, 0.03975118442394887, 0.005655801688135398, -0.005886636653485766, -0.000631581975426343, 0.0006184879482020733, 5.696264696553044e-05, -4.779557498579834e-05, -1.56374487509936e-05, 5.210719629289429e-05, -0.0001258797167196235, 0.00031015444493132094, -0.0012805056242366497, 0.01149752302139273, -0.13780084439128548, 1.6313925771945534, -11.998047252356896]
Psat_coeffs_iapws95_273_460 = [8.147908374667168e-07, -1.3258541002869606e-06, -2.960063284263015e-06, 4.862929927185178e-06, 6.120302714407444e-06, -9.979732567444444e-06, -6.454627509810962e-06, 1.1027086657122709e-05, 5.648329533869401e-06, -1.0190725788561394e-05, -1.3649275842908537e-06, 4.478973778532236e-07, 2.4834392547745665e-05, -5.852100520087333e-05, 0.00011126565522801002, -0.00036335083247251987, 0.0013489128931727379, -0.0047478345366922525, 0.018241801868318586, -0.07380538465894615, 0.2868986774488336, -1.015108797564658, 3.4736507508968284, -5.619649447193197]
Psat_coeffs_iapws95_460_609 = [2.0064180716872215e-07, 2.892629709094763e-07, -8.132046787068248e-07, -1.2402815627865493e-06, 1.2566961231641471e-06, 1.8773480405798182e-06, -1.3796370694763027e-06, -1.8621731214807369e-06, 4.6375271267606877e-07, 1.1436318345658947e-06, 1.6705293433005863e-06, 1.7150333064819279e-06, 1.3039190918107124e-06, 5.15597523786937e-06, 1.185603442621641e-05, 1.694959209075364e-05, 5.210011446887819e-05, 7.716156925408058e-06, 0.0009684942799086382, -0.0037996536704614225, 0.031002148395983753, -0.1716204979856538, 1.2065145338661658, -1.519520204731372]
Psat_coeffs_iapws95_609_643 = [1.0573770850896835e-05, 1.1763535439968109e-05, -6.56498596072197e-05, -7.157295476645231e-05, 0.0001856974558904767, 0.00019913158030249178, -0.0003122607449768111, -0.00033338970388285816, 0.0003383217645023251, 0.00036499839916359633, -0.00024304199087055167, -0.0002661161202013318, 0.00012133384302615013, 0.00013108684390772396, -4.926522541381928e-05, -5.498319717389677e-05, 3.840095160967394e-06, 9.275973251732239e-06, 4.700250627820424e-06, 1.7356479738772634e-05, 3.5141797559923305e-05, 4.994296202665005e-05, 5.6967466662061206e-05, 5.500573225036831e-05, 7.36448261119127e-05, 0.000486011140590506, -0.0032944327011136323, 0.20513406870743833, -0.24748976782381205]
Psat_coeffs_iapws95_643_646 = [9.057896477315808e-08, 1.211502649312024e-07, -3.360720484124613e-07, -4.495756513733795e-07, 5.768298478869838e-07, 7.768647307671017e-07, -5.138533794024625e-07, -6.86156649720715e-07, 3.3836656099239804e-07, 4.158511894836181e-07, -1.3213676086643034e-07, -1.300110328283921e-07, 2.7710538380576466e-07, 8.91687420706555e-07, 1.834557936633563e-06, 2.582291823115257e-06, 1.5351452190368736e-06, 8.540061517871983e-07, 1.7006021977569903e-05, 0.02017998616540387, -0.02472793196736738]
Psat_coeffs_iapws95_near_critical = [-0.007006160914897919, -0.006299436092376709, 0.06787101551890373, 0.06092516239732504, -0.3032621801830828, -0.27187402336858213, 0.8286682133330032, 0.7421519589261152, -1.5481375773670152, -1.3854497998690931, 2.0945177564717596, 1.8733694699149055, -2.1206636809074553, -1.8960203259257469, 1.6377062088922685, 1.4638579778784333, -0.9743636551480677, -0.8708100283029125, 0.44806853431398963, 0.40042604495646117, -0.15886602644877712, -0.14197406009749258, 0.04308180067010703, 0.0385020246406107, -0.008810816846762659, -0.007874311835966985, 0.0013301372745293527, 0.0011886957505600204, -0.00014366835734647143, -0.0001283599871508001, 1.0624248652432722e-05, 9.493371248271015e-06, -4.747152791886172e-07, -4.074279138207085e-07, 7.613360554058896e-08, 1.2227703013014193e-07, 2.2590783514621522e-07, 6.263045703604046e-07, 2.5993277169989305e-06, 0.0022606230644268894, -0.00226439777925055]
# Numba prevents this from being a lookup - the arrays are not the same size
Psat_all_coeffs_iapws95 = [Psat_coeffs_iapws95_235_273, Psat_coeffs_iapws95_273_460, Psat_coeffs_iapws95_460_609,
Psat_coeffs_iapws95_609_643, Psat_coeffs_iapws95_643_646, Psat_coeffs_iapws95_near_critical]
Psat_iapws95_coeff_set_count = len(Psat_all_coeffs_iapws95)
Psat_iapws95_coeff_boundaries = [235.0, 273.15, 460.1225, 609.7005, 643.35555, 646.721055, iapws95_Tc]
Psat_iapws95_coeff_as = [0.052424639580602915, 0.0106967602187487444, 0.0133709502734359297, 0.0594264456597153254, 0.594264456597155322, 5.33411567172122325]
Psat_iapws95_coeff_bs = [254.074999999999989, 366.636250000000018, 534.911500000000046, 626.528025000000071, 645.038302499999986, 646.908527499949969]
[docs]def iapws95_dPsat_dT(T):
r'''Compute the temperature derivative of saturation pressure of the
IAPWS-95 equation using high-
fidelity polynomial fits. The range of the fit is 235 K to
647.096 K, the critical point.
.. math::
P_{sat} = P_c \exp(\text{polynomial}(a (T - b)))
.. math::
\frac{\partial P_{sat}}{\partial T} = a P_c \exp(\text{polynomial}(a (T - b)))
\exp\left(\frac{\partial \text{polynomial}(a (T - b))}{\partial T}\right)
Parameters
----------
T : float
Temperature at which to calculate the saturation condition and
its temperature derivative, [K]
Returns
-------
dPsat_dT : float
First temperature derivative of Saturation vapor pressure, [Pa/K]
Psat : float
Saturation vapor pressure, [Pa]
Notes
-----
`Psat` must be calculated in the calculation of the derivative, so it is
returned as well which may be useful in some applications.
Examples
--------
>>> iapws95_dPsat_dT(400.0)
(7483.62075827, 245769.3455657)
'''
# May need to optimize this in the future without the loop
for i in range(Psat_iapws95_coeff_set_count):
if Psat_iapws95_coeff_boundaries[i] <= T <= Psat_iapws95_coeff_boundaries[i+1]:
coeffs = Psat_all_coeffs_iapws95[i]
a, b = Psat_iapws95_coeff_as[i], Psat_iapws95_coeff_bs[i]
val, der = horner_and_der(coeffs, a*(T - b))
if val > 0.0: val = 0.0
Psat = exp(val)*iapws95_Pc
dPsat_dT = Psat*a*der
return dPsat_dT, Psat
raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
[docs]def iapws95_Psat(T):
r'''Compute the saturation pressure of the IAPWS-95 equation using high-
fidelity polynomial fits. These have a relative accuracy of under 1e-12,
and are generated by solving the saturation equations under the
high-precision environment of mpmath. The range of the fit is 235 K to
647.096 K, the critical point.
.. math::
P_{sat} = P_c \exp(\text{polynomial}(a (T - b)))
Parameters
----------
T : float
Temperature at which to calculate the saturation condition, [K]
Returns
-------
Psat : float
Saturation vapor pressure, [Pa]
See Also
--------
iapws95_saturation
Notes
-----
This method should be used in preference to :obj:`iapws95_saturation`.
Although using mpmath generates slightly different results than using
plain floating point numbers, the requirement for the saturation curve is
to be smooth, and continuous; mpmath makes this easy and the saturation
equations were solved extremely high precision, well under a floating
point's error.
The polynomial coefficients have been carefully chosen to be able to be
evaluated accurately with horner's method, although they are derived as a
Chebyshev approximation originally.
Examples
--------
>>> iapws95_Psat(400.0)
245769.3455
'''
# Using a loop involves a 10% decrease in PyPy speed but a 50% hit to numba
# Maybe in the future we can use this :)
# for i in range(Psat_iapws95_coeff_set_count):
# if Psat_iapws95_coeff_boundaries[i] <= T <= Psat_iapws95_coeff_boundaries[i+1]:
# coeffs = Psat_all_coeffs_iapws95[i]
# a, b = Psat_iapws95_coeff_as[i], Psat_iapws95_coeff_bs[i]
# val = horner(coeffs, a*(T - b))
# if val > 0.0: val = 0.0
# return exp(val)*iapws95_Pc
# else:
# raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
# # Fit to under 1e-12 precision, with the EOS evaluated with mpmath for max
# # precision.
if 235.0 <= T < 273.15:
# Equation solves down to this temperature but not below.
coeffs = Psat_coeffs_iapws95_235_273
val = horner(coeffs, 0.052424639580602915*(T - 254.074999999999989))
elif 273.15 <= T <= 460.1225: # half the points
coeffs = Psat_coeffs_iapws95_273_460
val = horner(coeffs, 0.0106967602187487444*(T - 366.636250000000018))
elif 460.1225 < T <= 609.7005:
coeffs = Psat_coeffs_iapws95_460_609
val = horner(coeffs, 0.0133709502734359297*(T - 534.911500000000046))
elif 609.7005 < T <= 643.35555:
coeffs = Psat_coeffs_iapws95_609_643
val = horner(coeffs, 0.0594264456597153254*(T - 626.528025000000071))
elif 643.35555 < T <= 646.721055:
coeffs = Psat_coeffs_iapws95_643_646
val = horner(coeffs, 0.594264456597155322*(T - 645.038302499999986))
elif 646.721055 < T <= iapws95_Tc:
coeffs = Psat_coeffs_iapws95_near_critical
val = horner(coeffs, 5.33411567172122325*(T - 646.908527499949969))
if val > 0.0: val = 0.0
else:
raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
return exp(val)*iapws95_Pc
Psat_235 = 22.849568234070716 # iapws95_Psat(235)
[docs]def iapws95_Tsat(P):
r'''Compute the saturation temperature of the IAPWS-95 equation.
The range of the fit is 235 K to 647.096 K, the critical point.
Parameters
----------
Psat : float
Saturation vapor pressure specified, [Pa]
Returns
-------
T : float
Temperature at which the saturation pressure occurs, [K]
See Also
--------
iapws95_Psat
Tsat_IAPWS
Notes
-----
This method is quite fast and precise because it starts with great initial
guesses and the equation is well-bounded. The precision of this calculation
should be the same as :obj:`iapws95_Psat`.
Examples
--------
>>> iapws95_Tsat(iapws95_Psat(400.0))
400.0
'''
if P > iapws95_Pc:
raise ValueError("Pressure higher than critical pressure")
elif P < 22.849568234070716: # iapws95_Psat(235)
raise ValueError("Pressure lower than correlation")
T = Tsat_IAPWS(P)
if T < 235.0:
T = 235.0
dT = 100.0
# Very well-behaved solver, no issues found.
while abs(dT) > 1e-10:
dPsat_dT, Psat = iapws95_dPsat_dT(T)
err = Psat - P
dT = -err/dPsat_dT
T = T + dT
return T
rhol_coeffs_iapws95_235_273 = [-17.222183227539062, 18.77007293701172, 160.3718719482422, -174.77803325653076, -689.9691653251648, 751.9200625419617, 1818.6332448720932, -1981.8576150536537, -3283.683091402054, 3578.2904051095247, 4302.269122205675, -4688.138357363641, -4227.204455545172, 4606.233353788964, 3174.7852028068155, -3459.378792709904, -1840.9434151535388, 2005.933489420975, 826.9075718919048, -901.0039060153795, -287.0057090885457, 312.71981305931695, 76.35687688595681, -83.19737945724955, -15.353652797617087, 16.729144014339, 2.283299787289934, -2.4879351324522645, -0.2437405315180854, 0.2654055681062317, 0.01792607828713244, -0.019693974261834057, -0.0004738133771589048, 0.0007854872751575925, -0.0006095267475783039, 0.0022049570385940243, -0.005221612771147854, 0.01096251463881881, -0.026019484252807534, 0.036332822379610566, 3.0872453741478227]
rhol_coeffs_iapws95_273_460 = [1.7102574929594994e-06, -3.5461271181702614e-06, -4.744477337226272e-06, 1.1568277841433883e-05, 1.0390001989435405e-05, -2.5427980290260166e-05, -7.063001248752698e-06, 2.3168839106801897e-05, 1.3829438557877438e-05, -3.149418807879556e-05, 1.2095285001123557e-05, -1.292190836466034e-05, 3.8989083236629085e-05, -6.452956441194146e-05, 9.861656508292072e-05, -0.00016529921067132136, 0.00028210334725997654, -0.0004912796557263732, 0.0008216553207591737, -0.0015053379092586638, 0.002406131617918583, -0.006780978799273418, 0.00908602594184732, -0.06335671046246652, -0.19965126834113162, 2.9904548603014143]
rhol_coeffs_iapws95_460_609 = [2.034008502960205e-06, 2.436339855194092e-06, -1.459755003452301e-05, -1.7838552594184875e-05, 4.556635394692421e-05, 5.651172250509262e-05, -8.218304719775915e-05, -0.00010129279689863324, 9.901652811095119e-05, 0.00011561789142433554, -9.266487904824317e-05, -0.00010660312182153575, 4.007856477983296e-05, 3.6490591810434125e-05, -5.036522770751617e-05, -5.477863123815041e-05, -3.106556266629923e-05, -4.485767993855916e-05, -7.495337143836878e-05, -0.00011451218220770443, -0.0001927638574699131, -0.00035594329908761324, -0.0006906136151680897, -0.0013645509703152925, -0.0026981001401753524, -0.005280856348563123, -0.01113746163068241, -0.024864014260847, -0.07935231088522635, -0.3683075956298905, 2.424989140035793]
rhol_coeffs_iapws95_609_643 = [-0.005523681640625, -0.00545501708984375, 0.0537109375, 0.05365753173828125, -0.2407855987548828, -0.24350690841674805, 0.6603143215179443, 0.6763124465942383, -1.2398340702056885, -1.2859440445899963, 1.6920538246631622, 1.7755838260054588, -1.7406707555055618, -1.845441060140729, 1.3810706129297614, 1.4778835410252213, -0.8551894403062761, -0.9241933110170066, 0.4143857065355405, 0.4525773521454539, -0.158058475019061, -0.17400417243334232, 0.047671283817180665, 0.05314236543927109, -0.010313240202776797, -0.01200570931268885, 0.0015920358180210314, 0.0011764186526761478, -0.0018841396051811898, -0.0023740562720035996, -0.002218709652339612, -0.001974439023143759, -0.0013470066813248138, -0.00040877893032842394, 0.00047615294604597125, 0.0007077648942215298, -0.0006431129948183401, -0.004859799769289075, -0.013341369671968695, -0.027982763294779223, -0.059743791257592174, -0.2308678349483143, 1.7406588358235677]
rhol_coeffs_iapws95_643_646 = [-0.0025634765625, -0.0032825469970703125, 0.022905349731445312, 0.029368877410888672, -0.09517884254455566, -0.12216567993164062, 0.24343305826187134, 0.3127277195453644, -0.42805875837802887, -0.5503446385264397, 0.5477428995072842, 0.7048549354076385, -0.5267451787367463, -0.678729840554297, 0.3876991346478462, 0.5006899123545736, -0.2203993159928359, -0.28577900753589347, 0.09690586793294642, 0.12655803375673713, -0.032833380970259896, -0.043427356979009346, 0.00839818057283992, 0.01135846923398276, -0.0016914226212065842, -0.0023644802264470854, 0.00010811350080075499, 0.00018391106829085402, -0.00024146455655227328, -0.00037838730947470367, -0.0005190572816360373, -0.0007225867524383034, -0.0009020460988393708, -0.0010130621772712622, -0.0011647964785088671, -0.0018405530481413468, -0.004267423867078957, -0.010988133931514904, -0.027298278157160433, -0.0959238698681941, 1.3178562243402536]
rhol_coeffs_iapws95_646_647 = [-0.0229644775390625, -0.03644561767578125, 0.212982177734375, 0.345855712890625, -0.9255542755126953, -1.5350914001464844, 2.500049591064453, 4.224307537078857, -4.699752390384674, -8.060362190008163, 6.527629733085632, 11.303938150405884, -6.944240380078554, -12.053825380280614, 5.790451696142554, 9.975626651197672, -3.842256080592051, -6.484273410635069, 2.048077084153192, 3.329579567653127, -0.8812978177811601, -1.3517990967593505, 0.3061402047842421, 0.432271851645055, -0.08556108074390067, -0.10814184474293143, 0.01886623827419953, 0.02070108844236529, -0.0034511384185407223, -0.003249207452029168, 0.00016951103771134512, -3.8158334660920445e-06, -0.00046348517059513483, -0.000568669470194294, -0.0007047358201628384, -0.000932012451160702, -0.0012536221138755854, -0.0017243174367475023, -0.0024525617113163867, -0.0036673798893029352, -0.005942076328912957, -0.011318687131713495, -0.03982414332624489, 1.1397375425430707]
rhol_coeffs_iapws95_647_64709 = [-1.2898817658424377e-07, -2.3213215172290802e-07, 4.318426363170147e-07, 1.466367393732071e-06, 1.237931428477168e-07, -4.1386374505236745e-06, -3.063873009523377e-06, 6.612863217014819e-06, 7.028365871519782e-06, -6.654002390860114e-06, -8.305772780659026e-06, 4.523619054452865e-06, 5.796317054773681e-06, -3.283294972789008e-06, -5.114287745300317e-06, -2.4331531847110455e-06, -3.3224186708480374e-06, -5.743677981229212e-06, -7.499835216862039e-06, -1.1263549756890257e-05, -2.327157668147173e-05, -5.8298991014504864e-05, -0.00015448293805092606, -0.00041322408313870795, -0.0011335557976513144, -0.003412156416544368, -0.014736997602476914, 1.0571055050155371]
rhol_coeffs_iapws95_647_647095 = [-0.0005059242248535156, -0.00014829635620117188, 0.004393354058265686, 0.0012152865529060364, -0.017519604414701462, -0.004545917734503746, 0.04250476974993944, 0.010283735115081072, -0.07004969473928213, -0.015714182169176638, 0.08289439062355086, 0.01715690310811624, -0.07257122172450181, -0.013804640759190079, 0.04770259846554836, 0.008318820257045445, -0.023639545008336427, -0.0037765424071949383, 0.00878780496532272, 0.001286487733523245, -0.00241815827996561, -0.00032699767544386305, 0.00047761004627489, 5.577576455095823e-05, -7.249498847983205e-05, -1.772932723653753e-05, -1.061598110485562e-05, -2.727912312749936e-05, -4.84062636949556e-05, -8.56389762576848e-05, -0.0001609044043018315, -0.00032300748651784006, -0.0007014603337645309, -0.0019198857946925232, -0.009595223127881816, 1.028978700851531]
rhol_coeffs_iapws95_647_6470959 = [-4.6759843826293945e-05, -5.772896111011505e-05, 0.00029446277767419815, 0.0003508694935590029, -0.0008547466713935137, -0.0009755055652931333, 0.0014907524455338717, 0.001618919734028168, -0.0017289135430473834, -0.0017798669869080186, 0.0013886251217627432, 0.0013546462596423225, -0.0007873586769164831, -0.0007338769426041836, 0.0003066651955805355, 0.0002761181634696186, -8.79955611026162e-05, -8.321968066127283e-05, 2.5915467887216437e-06, -1.7553803992598205e-06, -2.3491923775154078e-05, -3.44673285415098e-05, -4.9372020503035685e-05, -7.664722940048385e-05, -0.0001250530805668726, -0.00021831841691788423, -0.00042691414370452097, -0.001040188531514219, -0.004763499073651319, 1.0120628799127538]
rhol_coeffs_iapws95_647_64709599 = [-0.001290641725063324, -0.0003778710961341858, 0.010535558685660362, 0.00301162526011467, -0.03902981849387288, -0.010885929688811302, 0.08678736770525575, 0.02358410635497421, -0.12912050032173283, -0.034109388157958165, 0.13565029118035454, 0.03472165907442104, -0.10352977068032487, -0.02556207562884083, 0.05817742309318419, 0.013771163636647543, -0.024138506512485947, -0.005434921958681116, 0.007345041980045153, 0.0015556455067553543, -0.0016143303365225847, -0.00032046819030284723, 0.0002439657241879445, 3.995263912059954e-05, -3.2193630974397536e-05, -1.4593553628650113e-05, -1.416486926064664e-05, -2.4128956016555847e-05, -3.973400246815029e-05, -6.928431227809506e-05, -0.0001353896286368922, -0.0003305923160938848, -0.0015796620258456111, 1.0039108886171622]
rhol_coeffs_iapws95_647_647095999 = [-4.733548848889768e-06, -7.60948023525998e-06, 1.7879367078421637e-05, 3.080385795328766e-05, -3.0345471259352053e-05, -5.6281653996848036e-05, 2.7725049449145445e-05, 5.695053056342658e-05, -1.5959582896130087e-05, -3.672514257857529e-05, 3.8304293354940455e-06, 1.2481707166500655e-05, -3.0915939763787037e-06, -6.661048546519055e-06, -4.9108654751606196e-06, -7.2591069035787825e-06, -1.2581070591322074e-05, -2.1996436737614644e-05, -4.2936997922793374e-05, -0.00010488089397681088, -0.0005081514985303057, 1.0012486252556012]
rhol_coeffs_iapws95_647_64709599999 = [-342.1649169921875, -286.70733642578125, 3696.8092041015625, 3062.9874267578125, -18628.276138305664, -15248.527671813965, 58141.86198616028, 46974.429047584534, -125898.94741535187, -100287.08330655098, 200761.28558278084, 157482.6384627223, -244240.7832775116, -188416.60116776824, 231738.97573629767, 175548.21367172152, -173839.0422554016, -129094.15901541244, 103905.44825894013, 75496.35919268127, -49644.42389814614, -35215.162472276075, 18947.322668814828, 13087.622056136344, -5751.365423334828, -3856.5318880818013, 1377.1159714494315, 892.9758781142154, -256.84921118732746, -160.2506746801855, 36.651197303384606, 21.84684986680145, -3.9021237365971047, -2.1983816265822327, 0.2992404034464471, 0.15648404704313634, -0.01571472634215354, -0.007373118351861052, 0.0005188793214245813, 0.00019840994932129874, -1.665223554836448e-05, -1.1892982883288106e-05, -1.36274066943054e-05, -2.2384801104155527e-05, -4.5679905172298085e-05, -0.0001857746544898474, 1.0003799295366436]
rhol_all_coeffs_iapws95 = [rhol_coeffs_iapws95_235_273, rhol_coeffs_iapws95_273_460,
rhol_coeffs_iapws95_460_609, rhol_coeffs_iapws95_609_643,
rhol_coeffs_iapws95_643_646, rhol_coeffs_iapws95_646_647,
rhol_coeffs_iapws95_647_64709, rhol_coeffs_iapws95_647_647095,
rhol_coeffs_iapws95_647_6470959, rhol_coeffs_iapws95_647_64709599,
rhol_coeffs_iapws95_647_647095999,
rhol_coeffs_iapws95_647_64709599999]
rhol_iapws95_coeff_set_count = len(rhol_all_coeffs_iapws95)
rhol_iapws95_coeff_boundaries = [235.0, 273.15, 460.1225, 609.7005, 643.35555, 646.721055, 647.07, 647.09, 647.095, 647.0959, 647.09599, 647.095999, 647.09599999]
rhol_iapws95_coeff_as = [0.052424639580602915, 0.0106967602187487444, 0.0133709502734359297, 0.0594264456597153254, 0.594264456597155322, 5.73156228058745576, 100.0, 400.0, 2222.22222221862921, 22222.2222221862903, 222222.222783279401, 2020202.04154185997]
rhol_iapws95_coeff_bs = [254.074999999999989, 366.636250000000018, 534.911500000000046, 626.528025000000071, 645.038302499999986, 646.895527500000071, 647.080000000000041, 647.092499999999973, 647.095450000000028, 647.095945000000029, 647.095994499999961, 647.095999495000001]
[docs]def iapws95_rhol_sat(T):
r'''Compute the saturation liquid density of the IAPWS-95 equation using high-
fidelity polynomial fits. These have a relative accuracy of under 1e-13,
except near the critical point where it rises to 1e-10,
and are generated by solving the saturation equations under the
high-precision environment of mpmath. The range of the fit is 235 K to
647.096 K, the critical point.
Parameters
----------
T : float
Temperature at which to calculate the saturation condition, [K]
Returns
-------
rhol : float
Saturation liquid density, [kg/m^3]
See Also
--------
iapws92_rhol_sat
Notes
-----
This method should be used in preference to :obj:`iapws92_rhol_sat`.
Examples
--------
>>> iapws95_rhol_sat(400.0)
937.48603939
'''
if 235.0 <= T < 273.15:
val = horner(rhol_coeffs_iapws95_235_273, 0.052424639580602915*(T - 254.074999999999989))
elif 273.15 <= T <= 460.1225: # half the points
val = horner(rhol_coeffs_iapws95_273_460, 0.0106967602187487444*(T - 366.636250000000018))
elif 460.1225 < T <= 609.7005:
val = horner(rhol_coeffs_iapws95_460_609, 0.0133709502734359297*(T - 534.911500000000046))
elif 609.7005 < T <= 643.35555:
val = horner(rhol_coeffs_iapws95_609_643, 0.0594264456597153254*(T - 626.528025000000071))
elif 643.35555 < T <= 646.721055:
val = horner(rhol_coeffs_iapws95_643_646, 0.594264456597155322*(T - 645.038302499999986))
elif 646.721055 < T <= 647.07:
val = horner(rhol_coeffs_iapws95_646_647, 5.73156228058745576*(T - 646.895527500000071))
elif 647.07 < T <= 647.09:
val = horner(rhol_coeffs_iapws95_647_64709, 100.000000000090949*(T - 647.080000000000041))
elif 647.09 < T <= 647.095:
# On 500 points, avg err 1.0048621666898271e-13, stdev 9.80697435140731e-14, max err 5.999286627611623e-13
# Simply can't get better
val = horner(rhol_coeffs_iapws95_647_647095, 400.000000000363798*(T - 647.092499999999973))
elif 647.095 < T <= 647.0959:
# (3.157060421406988e-13, 2.45822459571187e-13, 1.085522247937845e-12)
val = horner(rhol_coeffs_iapws95_647_6470959, 2222.22222221862921*(T - 647.095450000000028))
elif 647.0959 < T <= 647.09599:
# 1e-12 max
val = horner(rhol_coeffs_iapws95_647_64709599, 22222.2222221862903*(T - 647.095945000000029))
elif 647.09599 < T <= 647.095999:
# (3.667783433807127e-12, 2.9131002083651304e-12, 1.512957262282052e-11)
val = horner(rhol_coeffs_iapws95_647_647095999, 222222.222783279401*(T - 647.095994499999961))
elif 647.095999 < T <= 647.09599999:
# (1.301907916559972e-11, 1.5329272639853585e-11, 9.685327631780702e-11)
val = horner(rhol_coeffs_iapws95_647_64709599999, 2020202.04154185997*(T - 647.095999495000001))
elif 647.09599999 < T <= iapws95_Tc:
# Gotta go to linear interp
val = 1.0000546416597242 - 5.464165972424162e-05*(T-647.09599999)/(647.096-647.09599999)
else:
raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
return val*iapws95_rhoc
[docs]def iapws95_drhol_sat_dT(T):
r'''Compute the first temperature derivative of saturation liquid density
of the IAPWS-95 equation using high-fidelity polynomial fits. The actual
saturated liquid density is returned as well.
The range of the fit is 235 K to 647.096 K, the critical point.
Parameters
----------
T : float
Temperature at which to calculate the saturation condition
and its derivative, [K]
Returns
-------
drhol_dT : float
First temperature derivative of saturation liquid density, [kg/(m^3*K)]
rhol : float
Saturation liquid density, [kg/m^3]
Examples
--------
>>> iapws95_drhol_sat_dT(400.0)
(-0.835194603380, 937.486039392)
'''
if rhol_iapws95_coeff_boundaries[-1] < T <= iapws95_Tc:
return (-5464.1616377970422036*iapws95_rhoc,
(1.0000546416597242 - 5.464165972424162e-05*(T-647.09599999)/(647.096-647.09599999))*iapws95_rhoc)
for i in range(rhol_iapws95_coeff_set_count):
if rhol_iapws95_coeff_boundaries[i] <= T <= rhol_iapws95_coeff_boundaries[i+1]:
coeffs = rhol_all_coeffs_iapws95[i]
a, b = rhol_iapws95_coeff_as[i], rhol_iapws95_coeff_bs[i]
val, der = horner_and_der(coeffs, a*(T - b))
rhol = iapws95_rhoc*val
drhol_dT = iapws95_rhoc*a*der
return drhol_dT, rhol
raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
rhog_coeffs_iapws95_235_273 = [0.1376953125, -0.16064453125, -1.46826171875, 1.69775390625, 7.30413818359375, -8.376739501953125, -22.501876831054688, 25.625389099121094, 48.05792236328125, -54.43896484375, -75.48343086242676, 85.25402688980103, 90.2677903175354, -101.96795904636383, -83.95101803541183, 95.22411647439003, 61.49870456755161, -70.39149758219719, -35.72743892297149, 41.51342293806374, 16.49678842537105, -19.59625118598342, -6.0450619522016495, 7.399189752410166, 1.7498202509596013, -2.2248343522369396, -0.3973083387245424, 0.5282683086916222, 0.07019889306320692, -0.09777055511221988, -0.009583181835296273, 0.013834803114605165, 0.0010116799821844324, -0.0014553592153561112, -8.832041203277186e-05, 0.0001166358491389019, 2.63497172170446e-06, -1.855304673625824e-05, 4.9769065153526526e-05, -0.00012574357976014028, 0.000309940894730687, -0.0012701900216018913, 0.011378017130855733, -0.13488259814146453, 1.5565491834362626, -12.535071599332248]
rhog_coeffs_iapws95_273_460 = [1.7136335372924805e-07, -3.762543201446533e-07, -8.121132850646973e-07, 1.8654391169548035e-06, 2.1441373974084854e-06, -4.7711655497550964e-06, -3.3307005651295185e-06, 7.087073754519224e-06, 4.624249413609505e-06, -8.835311746224761e-06, -4.7929461288731545e-06, 1.0012711754825432e-05, 6.726278115820605e-06, -1.6951385987340473e-05, -9.048057336258353e-06, 2.6337079361837823e-05, 4.6732045689168444e-05, -0.00012556672749042264, 6.779015188840276e-05, -0.0002350752717177329, 0.0014799389422037734, -0.004694000666042086, 0.01756919958847014, -0.07219342538345153, 0.2914447799309543, -0.9555489491588713, 3.249393719828254, -6.510294967224288]
rhog_coeffs_iapws95_460_609 = [1.2516975402832031e-06, 1.475214958190918e-06, -6.6943466663360596e-06, -7.795169949531555e-06, 1.6705133020877838e-05, 2.0584091544151306e-05, -2.2356165573000908e-05, -3.1747971661388874e-05, 1.7298210877925158e-05, 3.512369585223496e-05, -7.913477020338178e-07, -2.318583574378863e-05, -3.4064614737872034e-06, 2.451385444146581e-05, 2.2935704691917636e-05, 8.854244697431568e-06, 2.8566514629346784e-05, 6.899303605223395e-05, 9.508152072612575e-05, 0.00014820537592186156, 0.0002821634795964201, 0.0005001793425947199, 0.0008733352662275706, 0.0016571537903367073, 0.003102883759193986, 0.007114612495199513, 0.008911656572993865, 0.05774574971527424, -0.08122819733346776, 1.2692652844232588, -2.57864220544887]
rhog_coeffs_iapws95_609_643 = [0.00463104248046875, 0.00502777099609375, -0.043544769287109375, -0.04686927795410156, 0.19086074829101562, 0.20387530326843262, -0.5165485143661499, -0.548183023929596, 0.964815080165863, 1.0185167044401169, -1.3173626214265823, -1.3854081854224205, 1.3592094220221043, 1.4264855673536658, -1.0805742950178683, -1.1340672622900456, 0.6691341262776405, 0.7039417792693712, -0.32417146407533437, -0.3427537734824, 0.12286763181327842, 0.13096405056785443, -0.03613373166263045, -0.03892437323611375, 0.008327982634000364, 0.009111485479479597, -0.001247552660402107, -0.0013325000208510573, 0.0005273702425085958, 0.0007008950194204999, 0.0006959742996457408, 0.0009990691382952832, 0.0014405016521461533, 0.002019923767587528, 0.0028278011969495864, 0.003998352412761141, 0.005793164525915717, 0.00878643593329359, 0.0143835404055572, 0.026914486982225663, 0.057727814680128706, 0.3827703866445822, -0.9671092101949783]
rhog_coeffs_iapws95_643_646 = [0.0025491714477539062, 0.0032243728637695312, -0.02312445640563965, -0.029504060745239258, 0.09758061170578003, 0.12572473287582397, -0.25367245078086853, -0.3303760141134262, 0.45409584417939186, 0.5982843823730946, -0.59301583096385, -0.7907937308773398, 0.584277902264148, 0.7885866130236536, -0.4432004929985851, -0.6049542285036296, 0.2619635757873766, 0.3609001541044563, -0.12138009762202273, -0.16812836089229677, 0.04421287349669001, 0.061154636978244525, -0.012583687592723436, -0.017190128449101394, 0.0028757095159903656, 0.003809122871132331, -0.00037953205708163296, -0.000465312238773663, 0.00025548248573414867, 0.00033060110598981396, 0.00034544350496545917, 0.00048058662306349476, 0.0006824852614399912, 0.0009674475238095059, 0.0013911249970349204, 0.002053152962603444, 0.0031923245453685384, 0.0054579682937713064, 0.010793275536161086, 0.02619898932537868, 0.1175436782287684, -0.3582341136353306]
rhog_coeffs_iapws95_646_647 = [0.123199462890625, 0.158203125, -1.3163528442382812, -1.6923675537109375, 6.583179473876953, 8.470105171203613, -20.45041513442993, -26.320425033569336, 44.17425847053528, 56.84443873167038, -70.39197111129761, -90.51797074079514, 85.68509554862976, 110.03725126758218, -81.40586972981691, -104.32665408030152, 61.15828255098313, 78.14822597336024, -36.59310451778583, -46.571773829637095, 17.48063129207003, 22.129124334751396, -6.656315470245318, -8.36757723876508, 2.0094127720640245, 2.503043227981834, -0.47626186647039503, -0.5862514379502954, 0.08751130521761752, 0.10607012874112343, -0.012050429573577048, -0.01430659721164318, 0.0014646848589165984, 0.001716607481384358, 0.00024133432185458048, 0.0003234318164695438, 0.0005561289955204884, 0.0007256413509242332, 0.0009554951631469688, 0.0012914852687246903, 0.0017905898928772257, 0.0025716960996110677, 0.0038933437981215763, 0.006426678469222447, 0.012636314161903143, 0.0457263935802698, -0.1509205151800937]
rhog_coeffs_iapws95_647_64709 = [1.1362135410308838e-07, 2.340239007025957e-07, -2.9721559258177876e-07, -1.4278084563557059e-06, -5.44330760021694e-07, 3.959169589506928e-06, 3.7063909985590726e-06, -6.434178658309975e-06, -7.760540825074713e-06, 6.667401123650052e-06, 9.20005382454292e-06, -4.2027016391443794e-06, -6.02054046083822e-06, 3.1985747952489874e-06, 4.7754411447442635e-06, 1.2908908937347974e-06, 1.470565071493013e-06, 3.6390199011915314e-06, 6.01044395243111e-06, 1.1997590871837271e-05, 2.863470914662991e-05, 7.126299425726235e-05, 0.00017804608291233093, 0.00044947042564998443, 0.0011837217855952265, 0.0035016548512316866, 0.015572696876272937, -0.059303026072814916]
rhog_coeffs_iapws95_647_647095 = [5.293695721775293e-07, 7.85010342951864e-07, -2.635018972796388e-06, -4.107998393010348e-06, 6.07685024078819e-06, 1.0305450814485084e-05, -7.676731115680013e-06, -1.5076962313287368e-05, 6.001800272770197e-06, 1.5050820508122342e-05, -1.7475480333928317e-06, -9.158365092076792e-06, 8.243237843430506e-07, 6.079632582256522e-06, 3.107979187788601e-06, 2.4446046791304354e-06, 5.8453898343913124e-06, 1.037312520968392e-05, 1.669938484161193e-05, 2.7831807917377827e-05, 4.787875459497798e-05, 8.539650861641435e-05, 0.0001611623507655497, 0.000322085599065278, 0.0006989106795284314, 0.0019670251355829796, 0.010063354831980193, -0.02970866443009639]
rhog_coeffs_iapws95_647_6470959 = [4.676311800722033e-05, 5.774045712314546e-05, -0.00029468783395714127, -0.000350944756064564, 0.0008561153908885899, 0.0009757264433574164, -0.0014945372481633967, -0.0016193114388443064, 0.0017349855932025093, 0.001780341666858476, -0.0013948050178242966, -0.0013550677709872616, 0.0007914683613172713, 0.0007341590766571926, -0.0003084301365490205, -0.00027626331840480134, 8.84466149857488e-05, 8.327181409639728e-05, -2.645354638453057e-06, 1.7348490884971923e-06, 2.347746964348474e-05, 3.445484224105788e-05, 4.934813092085516e-05, 7.65989071933211e-05, 0.00012495705237293257, 0.00021810427702287794, 0.000426363732766421, 0.0010400672410686839, 0.0048631072585053284, -0.012181982803022946]
rhog_coeffs_iapws95_647_64709599 = [-8.106724635581486e-06, -2.8820204533985816e-05, 6.728227526764385e-05, 0.00020996382590965368, -0.00022842562088953855, -0.000662341274960454, 0.0004379505493830038, 0.0012083686885659972, -0.0005323801630439107, -0.0014185476701698008, 0.00043475850948127004, 0.0011275509107431247, -0.00024214711127346078, -0.0006165865167107043, 9.456764745485735e-05, 0.0002352168423325196, -2.1673211109851653e-05, -5.628390206668943e-05, 9.020296203157696e-06, 1.7097766284966676e-05, 9.995299434569643e-06, 1.4640843533187536e-05, 2.438769459398374e-05, 3.9748737998279715e-05, 6.926660248509221e-05, 0.00013535838466119182, 0.0003304659427916644, 0.0015890204560773113, -0.003922018479616995]
rhog_coeffs_iapws95_647_647095999 = [-6.350732803639403e-06, 3.169606685560211e-06, 4.043474250181589e-05, -1.1164001431751558e-05, -0.00010510194278801066, 1.7335470121615515e-05, 0.00015198191861953703, -1.3636971201158588e-05, -0.0001346982971606181, 6.242079475360374e-06, 7.785105900667832e-05, -3.2741184249895916e-08, -2.767780142779591e-05, 1.2714908340129283e-06, 9.144014199798448e-06, 3.451458830057083e-06, 3.9403172298118835e-06, 7.697128420847176e-06, 1.267204183537352e-05, 2.1967496184305084e-05, 4.2934223307938634e-05, 0.00010487735229099445, 0.0005090347179200651, -0.0012496837143941386]
rhog_coeffs_iapws95_647_6470959999 = [-5.212060354864434e-06, -1.2219827251414017e-05, 2.8798432351706538e-05, 6.585986411966616e-05, -6.636928452752322e-05, -0.00014977469883592498, 8.476677627999862e-05, 0.00018944485286720436, -6.546990376135597e-05, -0.00014559560201167498, 3.2082676010722344e-05, 7.080311448931509e-05, -9.104274236474037e-06, -2.0487195780491234e-05, 2.7732548831052584e-06, 5.378368545767148e-06, 2.284769716569908e-06, 3.625228977934317e-06, 6.971486769632724e-06, 1.3625029664792398e-05, 3.3242598524321547e-05, 0.00016192478118363886, -0.00039652161031166215]
rhog_coeffs_iapws95_647_64709599999 = [0.00012197307423278403, 3.93254984203395e-05, -0.0006863586597720683, -0.00021157731863841178, 0.0016779695676436557, 0.0004919766855269181, -0.0023356411482816086, -0.000646105308648115, 0.00203897414938424, 0.0005266602670174758, -0.0011586472452729257, -0.0002754642919942079, 0.0004306993903883255, 9.281167781887041e-05, -0.00010189150684249311, -1.9063380369572797e-05, 1.512611351263453e-05, 3.227987819904611e-06, 2.394942461261991e-07, 2.148075213501199e-06, 4.4557488727214e-06, 1.0666958153395576e-05, 5.1515776307511835e-05, -0.00012536531858820264]
rhog_all_coeffs_iapws95 = [rhog_coeffs_iapws95_235_273, rhog_coeffs_iapws95_273_460,
rhog_coeffs_iapws95_460_609, rhog_coeffs_iapws95_609_643,
rhog_coeffs_iapws95_643_646, rhog_coeffs_iapws95_646_647,
rhog_coeffs_iapws95_647_64709, rhog_coeffs_iapws95_647_647095,
rhog_coeffs_iapws95_647_6470959, rhog_coeffs_iapws95_647_64709599,
rhog_coeffs_iapws95_647_647095999, rhog_coeffs_iapws95_647_6470959999,
rhog_coeffs_iapws95_647_64709599999]
rhog_iapws95_coeff_boundaries = [235.0, 273.15, 460.1225, 609.7005, 643.35555, 646.721055, 647.07, 647.09, 647.095, 647.0959, 647.09599, 647.095999, 647.0959999, 647.09599999, iapws95_Tc]
[docs]def iapws95_rhog_sat(T):
r'''Compute the saturation vapor density of the IAPWS-95 equation using high-
fidelity polynomial fits. These have a relative accuracy of under 1e-13,
except near the critical point where it rises to 1e-10,
and are generated by solving the saturation equations under the
high-precision environment of mpmath. The range of the fit is 235 K to
647.096 K, the critical point.
Parameters
----------
T : float
Temperature at which to calculate the saturation condition, [K]
Returns
-------
rhol : float
Saturation vapor density, [kg/m^3]
See Also
--------
iapws92_rhog_sat
Notes
-----
This method should be used in preference to :obj:`iapws92_rhog_sat`.
Examples
--------
>>> iapws95_rhog_sat(400.0)
1.3694075410
'''
if 235.0 <= T < 273.15:
# (2.0607069702164786e-14, 1.3923330571480516e-14, 6.141003425167497e-14)
val = horner(rhog_coeffs_iapws95_235_273, 0.052424639580602915*(T - 254.074999999999989))
elif 273.15 <= T <= 460.1225:
# (7.732981035392273e-15, 6.163792716590463e-15, 2.6045979542015134e-14)
val = horner(rhog_coeffs_iapws95_273_460, 0.0106967602187487444*(T - 366.636250000000018))
elif 460.1225 < T <= 609.7005:
val = horner(rhog_coeffs_iapws95_460_609, 0.0133709502734359297*(T - 534.911500000000046))
elif 609.7005 < T <= 643.35555:
val = horner(rhog_coeffs_iapws95_609_643, 0.0594264456597153254*(T - 626.528025000000071))
elif 643.35555 < T <= 646.721055:
# (2.8661497893057246e-15, 2.191016828653218e-15, 1.3187157069714141e-14)
val = horner(rhog_coeffs_iapws95_643_646, 0.594264456597155322*(T - 645.038302499999986))
elif 646.721055 < T <= 647.07:
# (1.0188884426844997e-14, 9.077386901275837e-15, 6.199859542992496e-14)
val = horner(rhog_coeffs_iapws95_646_647, 5.73156228058745576*(T - 646.895527500000071))
elif 647.07 < T <= 647.09:
# (2.862140730804985e-14, 2.7758664688788258e-14, 1.3652452640827781e-13)
val = horner(rhog_coeffs_iapws95_647_64709, 100.000000000090949*(T - 647.080000000000041))
elif 647.09 < T <= 647.095:
# (8.506386636359037e-14, 6.901837250691837e-14, 4.4843586214975857e-13)
val = horner(rhog_coeffs_iapws95_647_647095, 400.000000000363798*(T - 647.092499999999973))
elif 647.095 < T <= 647.0959:
# (3.225343227150995e-13, 2.520400355423129e-13, 1.1131799288371522e-12)
val = horner(rhog_coeffs_iapws95_647_6470959, 2222.22222221862921*(T - 647.095450000000028))
elif 647.0959 < T <= 647.09599:
# (1.0395342782324554e-12, 6.958372973890899e-13, 3.091689548546392e-12)
val = horner(rhog_coeffs_iapws95_647_64709599, 22222.2222221862903*(T - 647.095945000000029))
elif 647.09599 < T <= 647.095999:
# (2.2970197066187306e-12, 2.3796549478868604e-12, 1.5151556217110994e-11)
val = horner(rhog_coeffs_iapws95_647_647095999, 222222.222783279401*(T - 647.095994499999961))
elif 647.095999 < T <= 647.0959999:
# (1.1329579414863485e-11, 9.749776618222061e-12, 3.890628344114018e-11)
val = horner(rhog_coeffs_iapws95_647_6470959999, 2222222.14362032432*(T - 647.095999450000022))
elif 647.0959999 < T <= 647.09599999:
# (2.4630471386324867e-11, 1.849611580813743e-11, 8.350341664755902e-11)
val = horner(rhog_coeffs_iapws95_647_64709599999, 22222232.6645377725*(T - 647.095999945000017))
elif 647.09599999 < T <= iapws95_Tc:
# Linear fit from boundary points with SymPy
val = min(-3387140.78569631511 + 5234.37138492019039*T,0.0)
else:
raise ValueError("Temperature range must be between 273.15 K to 647.096 K")
return exp(val) * iapws95_rhoc
### IAPWS 95 Trho, Prho, TP solvers
def iapws95_rho_err(rho, T, tau, P_spec):
# For solving for a rho while P is specified
# tau added as a paramter to save a division
delta = rho*iapws95_rhoc_inv
dAddelta_res_val = iapws95_dAr_ddelta(tau, delta)
d2Ad2delta_res_val = iapws95_d2Ar_ddelta2(tau, delta)
P_calc = (1.0 + dAddelta_res_val*delta)*rho*iapws95_R*T
err = P_calc - P_spec
derr = T*(rho*(rho*d2Ad2delta_res_val + 644.0*dAddelta_res_val)
+ 103684.0)*iapws95_R_rhoc_inv2
return err, derr
def iapws95_T_err(T, rho, P_spec):
# Use for solving for T
tau = iapws95_Tc / T
delta = rho * iapws95_rhoc_inv
dAddelta_val = iapws95_dAr_ddelta(tau, delta) + 1.0/delta
err = (dAddelta_val*delta)*rho*iapws95_R*T - P_spec
dP_dT = rho*iapws95_R*delta*(dAddelta_val - tau*iapws95_d2Ar_ddeltadtau(tau, delta))
return err, dP_dT
[docs]def iapws95_P(T, rho):
r'''Calculate the pressure of water according to the IAPWS-95
standard given a temperature `T` and mass density `rho`.
Parameters
----------
T : float
Temperature, [K]
rho : float
Mass density of water, [kg/m^3]
Returns
-------
P : float
Pressure, [Pa]
Notes
-----
The IAPWS-95 model is explicit with inputs of temperature and density,
so this is a direct calculation with no iteration required.
Examples
--------
>>> iapws95_P(330.0, iapws95_rho(T=330.0, P=8e5))
8e5
>>> iapws95_P(823.0, 40.393893559703734)
14e6
Not all temperature and density inputs provide a stable solution; for
example anything between the vapor and gas saturation curves. In some but
not all of these cases a negative pressure is returned:
>>> iapws95_P(T=300, rho=300)
-1.526394720e+23
References
----------
.. [1] Wagner, Wolfgang, and Andreas Pruß. "The IAPWS Formulation 1995 for
the Thermodynamic Properties of Ordinary Water Substance for General and
Scientific Use." Journal of Physical and Chemical Reference Data 31, no.
2 (2002): 387-535.
'''
tau = iapws95_Tc/T
delta = rho*iapws95_rhoc_inv
dAddelta_res_val = iapws95_dAr_ddelta(tau, delta)
return (1.0 + dAddelta_res_val*delta)*rho*(iapws95_R*T)
[docs]@mark_numba_uncacheable
def iapws95_T(P, rho):
r'''Calculate the temperature of water according to the IAPWS-95
standard given a density `rho` and pressure `P`.
Parameters
----------
P : float
Pressure, [Pa]
rho : float
Mass density of water, [kg/m^3]
Returns
-------
T : float
Temperature, [K]
Notes
-----
This solution is iterative due to the nature of the equation.
The solution procedure begins with IAPWS-97's equations as an
initial guess, extrapolating when out of range. Newton's method
converges extremely, normally after 2 or 3 iterations.
Due to water's unique density curve, there is a temperature region
spanning 273.15 K to 280.005 K where there are two solutions. No guarantee
is made as to which solution will be returned.
Examples
--------
>>> iapws95_T(P=1e6, rho=995.0)
306.461547194
References
----------
.. [1] Wagner, Wolfgang, and Andreas Pruß. "The IAPWS Formulation 1995 for
the Thermodynamic Properties of Ordinary Water Substance for General and
Scientific Use." Journal of Physical and Chemical Reference Data 31, no.
2 (2002): 387-535.
'''
try:
T = iapws97_T(P, rho)
MAX_T_STEP = 100.0
except:
if P > iapws95_Pc:
T = 700.0
MAX_T_STEP = 500.0
else:
T = 500.0
MAX_T_STEP = 100.0
# if Psat_235 < P < iapws95_Pc:
# Tsat = iapws95_Tsat(P)
T_old = 10000000.0
iterations = 0
while (abs(T_old - T) > abs(1e-9*T)) and iterations < 100:
T_old = T
err, derr = iapws95_T_err(T, rho, P)
dT = - err/derr
if dT < -MAX_T_STEP:
dT = -MAX_T_STEP
elif dT > MAX_T_STEP:
dT = MAX_T_STEP
T = T_old + dT
iterations += 1
# print(T, err)
if iterations == 100:
raise ValueError("Could not converge a temperature solution")
return T
[docs]def iapws95_rho(T, P):
r'''Calculate the density of water according to the IAPWS-95
standard given a temperature `T` and pressure `P`. The phase is determined
in this calculation.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
rho : float
Mass density of water, [kg/m^3]
Notes
-----
There is a sudden transition at the saturation pressure between liquid and
vapor density, by design.
This solution is iterative due to the nature of the equation.
The solution procedure begins with IAPWS-97's explicit equations as an
initial guess, extrapolating when out of range. If the temperature is under
the critical temperature, the saturation density is calculated, and used
to ensure the solver begins in the feasible region. Newton's method
converges extremely, normally after 2 or 3 iterations.
Temperatures under 273.15 K are not officially supported by [1]_, but a
solution is still attempted down to 235 K.
Examples
--------
>>> iapws95_rho(T=300.0, P=1e6)
996.96002269499
1 GPa and 5000 K are suggested as upper limits of [1]_ although there are
no hardcoded limits for temperature and pressure.
>>> iapws95_rho(T=5000.0, P=1e9)
326.79451662743
See Also
--------
iapws95_rhol_sat
iapws95_rhog_sat
References
----------
.. [1] Wagner, Wolfgang, and Andreas Pruß. "The IAPWS Formulation 1995 for
the Thermodynamic Properties of Ordinary Water Substance for General and
Scientific Use." Journal of Physical and Chemical Reference Data 31, no.
2 (2002): 387-535.
'''
a = 1e-20 # Value where error is always negative
b = 5000.0 # value where error is always positive
MAX_RHO_STEP = 200.0 # iapws95_rho(250, 1e9) is a good point showing the advantage of this
rho = iapws97_rho_extrapolated(T, P, True)
#P_inv = 1.0/P
tau = iapws95_Tc / T
if T < iapws95_Tc:
# Experimental investication hasn't revealed any places where the solver
# skips out of the selected region.
Psat = iapws95_Psat(T)
if P < Psat:
b = rho_high = iapws95_rhog_sat(T)
if rho > rho_high:
rho = rho_high
else:
a = rho_low = iapws95_rhol_sat(T)
if rho < rho_low:
rho = rho_low
rho_old = 100000.0 #
# Adding iterations check did not slow anything down.
"""# Points can be debugged with the following code.
import matplotlib.pyplot as plt
from chemicals.iapws import iapws95_rho_err
rhos = linspace(10, 1500, 10000)
T, P = 1749.5356805149597, 1000000000
errs = [abs(iapws95_rho_err(rho, T, P)[0]) for rho in rhos]
plt.semilogy(rhos, errs)
plt.show()
"""
iterations = 0
# or abs(err*P_inv) > 1e-13
# (abs(rho_old - rho) > abs(1e-13*rho)) hand-tuned for maximum precision achievable
while iterations < 2 or ((abs(rho_old - rho) > abs(1e-13*rho)) and iterations < 100):
err, derr = iapws95_rho_err(rho, T, tau, P)
if err < 0.0:
if iterations > 20 and a == rho:
return rho
a = rho
else:
if iterations > 20 and b == rho:
return rho
b = rho
drho = - err/derr
if drho < -MAX_RHO_STEP:
drho = -MAX_RHO_STEP
elif drho > MAX_RHO_STEP:
drho = MAX_RHO_STEP
rho_old = rho
rho = rho + drho
if rho > b or rho < a:
rho = 0.5*(a + b)
iterations += 1
# print(rho, err)
if iterations >= 99:
raise ValueError("Could not converge")
# Note that the derivatives have not been computed at this spot, so we can't save and return them
return rho
[docs]def iapws95_properties(T, P):
r'''Calculate some basic properties of water according to the IAPWS-95
standard given a temperature `T` and pressure `P`.
The properties are density `rho`, internal energy `U`, entropy `S`,
enthalpy `H`, isochoric heat capacity `Cv`, isobaric heat capacity `Cp`,
speed of sound `w`,
Joule-Thomson coefficient `JT`, isothermal throttling coefficient `delta_T`,
isentropic temperature-pressure coefficient `beta_s`, and the derivative of
mass density with respect to pressure at constant temperature `drho_dP`.
This function is intended as a demonstration of how to use the IAPWS-95
equations. For that reason, mass-units are used in all returned variables.
Parameters
----------
T : float
Temperature, [K]
P : float
Pressure, [Pa]
Returns
-------
rho : float
Mass density of water, [kg/m^3]
U : float
Internal energy of water, [J/(kg)]
S : float
Entropy of water, [J/(kg*K)]
H : float
Enthalpy of water, [J/(kg)]
Cv : float
Isochoric heat capacity, [J/(kg*K)]
Cp : float
Isobaric heat capacity, [J/(kg*K)]
w : float
Speed of sound, [m/s]
JT : float
Joule-Thomson coefficient, [K/Pa]
delta_T : float
Isothermal throttling coefficient, [J/(kg*Pa)]
beta_s : float
Isentropic temperature-pressure coefficient, [K/Pa]
drho_dP : float
Derivative of mass density with respect to pressure at constant
temperature, [kg/(m^3*Pa)]
Notes
-----
Hundreds of useful properties can be obtained from the IAPWS-95 model. It
is intended for this function to serve as a useful starting point to those.
Calculating every property with every set of units is beyond the scope of
`chemicals`. The functions like :obj:`iapws95_dAr_ddelta` can be used
directly in your own implementation - where you can calculate only those
properties which are necessary, for maximum speed.
The formulas are as follows:
.. math::
\frac{u(\delta, \tau)}{R T}=\tau\left(\phi_{\tau}^{\mathrm{o}}
+\phi_{\tau}^{\mathrm{r}}\right)
.. math::
\frac{s(\delta, \tau)}{R}=\tau\left(\phi_{\tau}^{\mathrm{o}}
+\phi_{\tau}^{\mathrm{r}}\right)-\phi^{\mathrm{o}}-\phi^{\mathrm{r}}
.. math::
\frac{h(\delta, \tau)}{R T}=1+\tau\left(\phi_{\tau}^{\mathrm{o}}
+\phi_{\tau}^{\mathrm{r}}\right)+\delta \phi_{\delta}^{\mathrm{r}}
.. math::
\frac{c_{v}(\delta, \tau)}{R}=-\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}
+\phi_{\tau \tau}^{\mathrm{r}}\right)
.. math::
\frac{c_{p}(\delta, \tau)}{R}=-\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}
+\phi_{\tau \tau}^{\mathrm{r}}\right)+\frac{\left(1+\delta
\phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}
\right)^{2}}{1+2 \delta \phi_{\delta}^{\mathrm{r}}+\delta^{2}
\phi_{\delta \delta}^{\mathrm{r}}}
.. math::
\frac{w^{2}(\delta, \tau)}{R T}=1+2 \delta \phi_{\delta}^{\mathrm{r}}
+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}-\frac{\left(1+\delta
\phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}
\right)^{2}}{\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau
\tau}^{\mathrm{r}}\right)}
.. math::
\mu R \rho=\frac{-\left(\delta \phi_{\delta}^{\mathrm{r}}+\delta^{2}
\phi_{\delta \delta}^{\mathrm{r}}+\delta \tau \phi_{\delta \tau}^{
\mathrm{r}}\right)}{\left(1+\delta \phi_{\delta}^{\mathrm{r}}-\delta
\tau \phi_{\delta \tau}^{\mathrm{r}}\right)^{2}-\tau^{2}\left(
\phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau \tau}^{\mathrm{r}}\right)
\left(1+2 \delta \phi_{\delta}^{\mathrm{r}}+\delta^{2} \phi_{\delta
\delta}^{\mathrm{r}}\right)}
.. math::
\delta_{T} \rho=1-\frac{1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau
\phi_{\delta \tau}^{\mathrm{r}}}{1+2 \delta \phi_{\delta}^{\mathrm{r}}
+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}}
.. math::
\beta_{S} \rho R=\frac{1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau
\phi_{\delta \tau}^{\mathrm{r}}}{\left(1+\delta \phi_{\delta}^{
\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}\right)^{2}
-\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau \tau}^{
\mathrm{r}}\right)\left(1+2 \delta \phi_{\delta}^{\mathrm{r}}
+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}\right)}
This derivative isn't part of the same table of properties, but it is
needed by the transport calculation routines:
.. math::
\left(\frac{\partial \rho}{\partial P}\right)_{T} = \frac{1}{
R T\left(1+2 \delta \alpha_{\delta}^{\mathrm{r}}+\delta^{2}
\alpha_{\delta \delta}^{\mathrm{r}}\right)}
Examples
--------
>>> iapws95_properties(T=300.0, P=1e6)
(996.96002269, 112478.998245, 392.813902893, 113482.047492, 4127.21730497, 4178.103605593, 1503.035983829, -2.202166728257e-07, 0.000920088074745, 1.985617879134e-08, 4.48108429028e-07)
>>> rho, U, S, H, Cv, Cp, w, JT, delta_T, beta_s, drho_dP = iapws95_properties(T=500.0, P=1e5)
>>> w
548.3138393244
References
----------
.. [1] Wagner, Wolfgang, and Andreas Pruß. "The IAPWS Formulation 1995 for
the Thermodynamic Properties of Ordinary Water Substance for General and
Scientific Use." Journal of Physical and Chemical Reference Data 31, no.
2 (2002): 387-535.
'''
rho = iapws95_rho(T, P)
tau = iapws95_Tc/T
delta = rho*iapws95_rhoc_inv
A0, dA0_dtau, d2A0_dtau2, d3A0_dtau3 = iapws95_A0_tau_derivatives(tau, delta)
Ar = iapws95_Ar(tau, delta)
dAr_ddelta = iapws95_dAr_ddelta(tau, delta)
d2Ar_ddelta2 = iapws95_d2Ar_ddelta2(tau, delta)
dAr_dtau = iapws95_dAr_dtau(tau, delta)
d2Ar_dtau2 = iapws95_d2Ar_dtau2(tau, delta)
d2Ar_ddeltadtau = iapws95_d2Ar_ddeltadtau(tau, delta)
U = iapws95_R*T*tau*(dA0_dtau + dAr_dtau)
S = iapws95_R*(tau*(dA0_dtau + dAr_dtau) - A0 - Ar)
H = iapws95_R*T*(1.0 + tau*(dA0_dtau + dAr_dtau) + delta*dAr_ddelta)
Cv = -iapws95_R*tau*tau*(d2A0_dtau2 + d2Ar_dtau2)
Cp = iapws95_R*(-tau*tau*(d2A0_dtau2 + d2Ar_dtau2) + (1.0 + delta*dAr_ddelta
- delta*tau*d2Ar_ddeltadtau)**2/(1 + 2*delta*dAr_ddelta + delta*delta*d2Ar_ddelta2))
w = sqrt(iapws95_R*T*(1 + 2.0*delta*dAr_ddelta + delta*delta*d2Ar_ddelta2 - (1.0 + delta*dAr_ddelta
- delta*tau*d2Ar_ddeltadtau)**2/(tau*tau*(d2A0_dtau2 + d2Ar_dtau2))))
JT = ( -(delta*dAr_ddelta + delta*delta*d2Ar_ddelta2 + delta*tau*d2Ar_ddeltadtau)/
((1.0 + delta*dAr_ddelta - delta*tau*d2Ar_ddeltadtau)**2 - tau*tau*(d2A0_dtau2+d2Ar_dtau2)*
(1.0 + 2.0*delta*dAr_ddelta + delta*delta*d2Ar_ddelta2) ))/(iapws95_R*rho)
delta_T = (1.0 - (1.0 + delta*dAr_ddelta - delta*tau*d2Ar_ddeltadtau)/
(1.0 + 2.0*delta*dAr_ddelta + delta**2*d2Ar_ddelta2))/rho
denominator1 = (1.0 + delta*dAr_ddelta - delta*tau*d2Ar_ddeltadtau)
denominator1 *= denominator1
denominator2 = tau*tau*(d2Ar_dtau2+d2A0_dtau2)*(1.0 + 2.0*delta*dAr_ddelta + delta*delta*d2Ar_ddelta2)
beta_s = ( 1.0 + delta*dAr_ddelta - delta*tau*d2Ar_ddeltadtau)/(denominator1 - denominator2)/(iapws95_R*rho)
drho_dP = 1.0/(iapws95_R*T*(1.0 + 2.0*delta*dAr_ddelta + delta*delta*d2Ar_ddelta2))
return (rho, U, S, H, Cv, Cp, w, JT, delta_T, beta_s, drho_dP)
[docs]def iapws11_Psub(T):
r'''Compute the sublimation pressure of the frozen water using the IAPWS-11
equation from the Revised Release on the Pressure along the Melting and
Sublimation Curves of Ordinary Water Substance.
.. math::
P_{sub} = P_t\exp\left(
\theta^{-1} \sum_{i=1}^3 a_i \theta^{b_i}
\right)
.. math::
\theta = \frac{T}{T_t}
Parameters
----------
T : float
Temperature at which to calculate the sublimation condition [K]
Returns
-------
Psub : float
Sublimation vapor pressure, [Pa]
Notes
-----
The triple temperature is 273.16 K, and triple pressure 611.657 Pa.
The coefficients are as follows:
ais = [-0.212144006E2, 0.273203819E2, -0.610598130E1]
bis = [0.333333333E-2, 0.120666667E1, 0.170333333E1]
The equation is valid from 50 K to the triple temperature.
Examples
--------
>>> iapws11_Psub(230.0)
8.947352740189151
'''
return 611.657*exp(-5687.566391521397*T**-0.99666666667
+ 8.569600103586199*T**0.20666666999999994
- 0.11807069369846021*T**0.70333333)