#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 17 16:02:22 2021
@author: adedapo.awolayo and Ben Tutolo, University of Calgary
Copyright (c) 2020 - 2021, Adedapo Awolayo and Ben Tutolo, University of Calgary
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
Functions implemented here include water equation of state and dielectric properties
"""
import numpy as np, math
from scipy.optimize import root_scalar, fsolve, brentq
from scipy.linalg import lu_factor, lu_solve
eps = 2.220446049250313e-16
J_to_cal = 4.184
np.random.seed(4321)
[docs]def convert_temperature(T, Out_Unit = 'C'):
"""
This function converts temperatures from Celsius to Kelvin and vice-versa
Parameters
----------
T : float, vector
Temperature in °C or K \n
Out_Unit : string
Expected temperature unit (C or K) \n
Returns
----------
T : float, vector
Temperature in °C or K
Examples
----------
>>> TC = 100; convert_temperature( TC, Out_Unit = 'K' )
373.15
>>> TK = 520; convert_temperature( TK, Out_Unit = 'C' )
246.85
"""
#Accepted units for input and output are:
unit_markers = ['C', 'K']
if not (Out_Unit in unit_markers):
return None
elif Out_Unit == 'K':
return T + 273.15
elif Out_Unit == 'C':
return T - 273.15
[docs]def derivative(f, a, method = 'central', h = 0.001):
'''Compute the derivative of f, f'(a) with step size h.
Parameters
----------
f : function
Vectorized function of one variable
a : number
Compute derivative at x = a
method : string
Difference formula: 'forward', 'backward' or 'central'
h : number
Step size in difference formula
Returns
-------
float
Difference formula:
central: f(a + h) - f(a - h))/2h
forward: f(a + h) - f(a))/h
backward: f(a) - f(a-h))/h
'''
if method == 'central':
return (f(a + h) - f(a - h))/(2*h)
elif method == 'forward':
return (f(a + h) - f(a))/h
elif method == 'backward':
return (f(a) - f(a - h))/h
else:
raise ValueError("Method must be 'central', 'forward' or 'backward'.")
[docs]def readIAPWS95data():
"""
returns all constants and coefficients needed for the IAPWS95 formulation, packed into a dictionary
"""
# Constants for IAPWS-95-Formulation
R = 461.51805 # R: specific gas constant in J/(kg*K)
Tc = 647.096 # T_c: critical temperature of water in K
rhoc = 322 # rho_c: critical density of water in kg/m^3
Pc = 22.06400000000213 # P_c: critical pressure of water in MPa
# n0(1:8)
n0 = [-8.320446483749615, 6.683210527593193, 3.00632, 0.012436,
0.97315, 1.27950, 0.96956, 0.24873]
# gamma0(1:8)
gamma0 = [0.0, 0.0, 0.0, 1.28728967, 3.53734222, 7.74073708, 9.24437796, 27.5075105]
# n(1:56)
n = [0.12533547935523e-1, 0.78957634722828e+1, -0.87803203303561e+1,
0.31802509345418e+0, -0.26145533859358e+0, -0.78199751687981e-2,
0.88089493102134e-2, -0.66856572307965e+0, 0.20433810950965e+0,
-0.66212605039687e-4, -0.19232721156002e+0, -0.25709043003438e+0,
0.16074868486251e+0, -0.40092828925807e-1, 0.39343422603254e-6,
-0.75941377088144e-5, 0.56250979351888e-3, -0.15608652257135e-4,
0.11537996422951e-8, 0.36582165144204e-6, -0.13251180074668e-11,
-0.62639586912454e-9, -0.10793600908932e+0, 0.17611491008752e-1,
0.22132295167546e+0, -0.40247669763528e+0, 0.58083399985759e+0,
0.49969146990806e-2, -0.31358700712549e-1, -0.74315929710341e+0,
0.47807329915480e+0, 0.20527940895948e-1, -0.13636435110343e+0,
0.14180634400617e-1, 0.83326504880713e-2, -0.29052336009585e-1,
0.38615085574206e-1, -0.20393486513704e-1, -0.16554050063734e-2,
0.19955571979541e-2, 0.15870308324157e-3, -0.16388568342530e-4,
0.43613615723811e-1, 0.34994005463765e-1, -0.76788197844621e-1,
0.22446277332006e-1, -0.62689710414685e-4, -0.55711118565645e-9,
-0.19905718354408e+0, 0.31777497330738e+0, -0.11841182425981e+0,
-0.31306260323435e+2, 0.31546140237781e+2, -0.25213154341695e+4,
-0.14874640856724e+0, 0.31806110878444e+0]
# c(1:51)
c = [0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 3, 3, 3, 3, 4, 6, 6, 6, 6]
# d(1:54)
d = [1, 1, 1, 2, 2, 3, 4, 1, 1, 1, 2, 2, 3, 4, 4,
5, 7, 9, 10, 11, 13, 15, 1, 2, 2, 2, 3, 4, 4, 4,
5, 6, 6, 7, 9, 9, 9, 9, 9, 10, 10, 12, 3, 4, 4,
5, 14, 3, 6, 6, 6, 3, 3, 3]
# t(1:54)
t = [-0.5, 0.875, 1, 0.5, 0.75, 0.375, 1, 4, 6, 12, 1, 5, 4, 2, 13,
9, 3, 4, 11, 4, 13, 1, 7, 1, 9, 10, 10, 3, 7, 10, 10, 6, 10,
10, 1, 2, 3, 4, 8, 6, 9, 8, 16, 22, 23, 23, 10, 50, 44, 46, 50, 0, 1, 4]
# alpha(52:54)
alpha = [20, 20, 20]
# beta(52:56)
beta = [150, 150, 250, 0.3, 0.3]
# gamma(52:54)
gamma = [1.21, 1.21, 1.25]
# epsilon(52:54)
epsilon = [1, 1, 1]
# a(55:56)
a = [3.5, 3.5]
# b(55:56)
b = [0.85, 0.95]
# A(55:56)
A = [0.32, 0.32]
# B(55:56)
B = [0.2, 0.2]
# C(55:56)
C = [28, 32]
# D(55:56)
D = [700, 800]
var = [R,Tc,Pc,rhoc,n0,gamma0,n,c,d,t,alpha,beta,gamma,epsilon,a,b,A,B,C,D]
varname = ['R','Tc','Pc','rhoc','n0','gamma0','n','c','d','t','alpha','beta',
'gamma','epsilon','a','b','A','B','C','D']
coeffs = {}
for i in range(len(var)):
coeffs['%s' % varname[i]] = var[i]
return coeffs
IAPWS95_COEFFS = readIAPWS95data()
[docs]class Dummy(object):
"""
Class of functions to evaluate the IAPWS95 equation of state for calculating thermodynamic
properties of water. \n
"""
def __init__(self):
# vectorize the function above to permit array variables
self.vEOSIAPWS95 = np.vectorize(self.EOSIAPWS95)
self.vauxMeltingTemp = np.vectorize(self.auxMeltingTemp)
self.vapxsatpropT = np.vectorize(self.apxsatpropT)
self.vapxsatpropP = np.vectorize(self.apxsatpropP)
self.vcalcsatpropT = np.vectorize(self.calcsatpropT)
self.vcalcsatpropP = np.vectorize(self.calcsatpropP)
self.vwaterviscosity = np.vectorize(self.waterviscosity)
[docs] def EOSIAPWS95(self, TK, rho, FullEOSppt = False):
"""
This function evaluates the IAPWS basic equation of state to calculate thermodynamic
properties of water, which is written as a function of temperature and density. \n
Parameters
----------
TK : temperature [K] \n
rho : density [kg/m3] \n
FullEOSppt: Option to output all or essential water properties [False or True]
Returns
----------
px : pressure [bar] \n
ax : Helmholtz energy [kJ/kg-K] \n
sx : Entropy [kJ/kg/K] \n
hx : Enthalpy [kJ/kg] \n
gx : Gibbs energy [kJ/kg] \n
vx : Volume [m3/kg] \n
pdx : Derivative of pressure with respect to delta in bar \n
adx : Helmholtz energy derivative with respect to delta \n
ztx : zeta value (needed to calculate viscosity) \n
ptx : Derivative of pressure with respect to tau in bar \n
ktx : Compressibility [/bar] \n
avx : Thermal expansion coefficient (thermal expansivity) \n
ux : Internal energy [kJ/kg] if FullEOSppt is True \n
gdx : Gibbs energy derivative in kJ/kg if FullEOSppt is True \n
bsx : Isentropic temperature-pressure coefficient [K-m3/kJ] if FullEOSppt is True \n
dtx : Isothermal throttling coefficient [kJ/kg/bar] if FullEOSppt is True \n
mux : Joule-Thomsen coefficient [K-m3/kJ] if FullEOSppt is True \n
cpx : Isobaric heat capacity [kJ/kg/K] if FullEOSppt is True \n
cvx : Isochoric heat capacity [kJ/kg/K] if FullEOSppt is True \n
wx : Speed of sound [m/s] if FullEOSppt is True \n
Usage
----------
[px, ax, ux, sx, hx, gx, vx, pdx, adx, gdx, ztx, ptx, ktx, avx, bsx, dtx, mux, cpx, cvx, wx] = EOSIAPW95( TK, rho)
"""
Tc = IAPWS95_COEFFS['Tc']
Pc = IAPWS95_COEFFS['Pc']
rhoc = IAPWS95_COEFFS['rhoc']
# specific and molar gas constants
R = IAPWS95_COEFFS['R']/1000 # kJ kg-1 K-1
delta = rho/rhoc
tau = Tc/TK
# To avoid delta values of zero
epxc = eps*delta
if (delta <= epxc):
delta = epxc
rho = rhoc*delta
# To avoid a singularity at the critical density (at any temperature), i.e.
# if delta is unity, then delta - 1 is zero. This will lead to error when evaluating
# (delta - 1)**n where n is negative, or in any division like (x/(d -1)).
x1 = 1.0 - epxc
x2 = 1.00 + epxc
if (delta > x1) & (delta < x2):
if (delta < 1.0):
delta = x1
else:
delta = x2
rho = rhoc*delta
#%%---------------------------------------------------------
# specific dimensionless Helmholtz free energy (phi) and its derivatives
## IAPWS95.residual
def phir(delta, tau):
"""
residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
# a = np.array(IAPWS95_COEFFS['a']);
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (delta**d[:7] * tau**t[:7]))
y = y + np.dot(n[7:51], (delta**d[7:51] * tau**t[7:51] * np.exp(-delta**c[7:51])))
y = y + np.dot(n[51:54], ( delta**d[51:54] * tau**t[51:54] * np.exp(-alpha*(delta - epsilon)**2 \
- beta[:3]*(tau - gamma)**2) ))
y = y + np.dot(n[54:56], (Delta(delta,tau)**b * delta * Psi(delta,tau)))
return y
def phir_t(delta, tau):
"""
partial derivative for tau of phir
where phir = residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
# a = np.array(IAPWS95_COEFFS['a']);
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (t[:7]*delta**d[:7] * tau**(t[:7]-1)))
y = y + np.dot(n[7:51], (t[7:51]*delta**d[7:51] * tau**(t[7:51]-1) * np.exp(-delta**c[7:51])))
y = y + np.dot(n[51:54], ( delta**d[51:54] * tau**t[51:54] * \
np.exp(-alpha*(delta - epsilon)**2 - beta[:3]*(tau - gamma)**2)* \
(t[51:54]/tau - 2*beta[:3]*(tau - gamma)) ))
y = y + np.dot(n[54:56], (delta * (Delta_b_t(delta,tau) * Psi(delta,tau) + Delta(delta,tau)**b * \
Psi_t(delta,tau))))
return y
def phir_tt(delta, tau):
"""
second partial derivative for tau of phir
where phir = residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
# a = np.array(IAPWS95_COEFFS['a']);
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (t[:7]*(t[:7]-1)*delta**d[:7] * tau**(t[:7]-2)))
y = y + np.dot(n[7:51], (t[7:51]*(t[7:51]-1)*delta**d[7:51] * \
tau**(t[7:51]-2) * np.exp(-delta**c[7:51])))
y = y + np.dot(n[51:54], ( delta**d[51:54] * tau**t[51:54] * \
np.exp(-alpha*(delta - epsilon)**2 - \
beta[:3]*(tau - gamma)**2)* \
( (t[51:54]/tau - 2*beta[:3]*(tau - gamma))**2 - \
t[51:54]/tau**2 - 2*beta[:3] )))
y = y + np.dot(n[54:56], (delta * (Delta_b_tt(delta,tau) * Psi(delta,tau) +\
2*Delta_b_t(delta,tau) * Psi_t(delta,tau) +\
Delta(delta,tau)**b * Psi_tt(delta,tau) )))
return y
def phir_d(delta, tau):
"""
partial derivative for delta of phir
where phir = residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (d[:7]*delta**(d[:7]-1) * tau**t[:7]))
y = y + np.dot(n[7:51], ( np.exp(-delta**c[7:51]) * (delta**(d[7:51]-1) * tau**t[7:51] \
* (d[7:51] - c[7:51]*delta**c[7:51]))))
y = y + np.dot(n[51:54], (delta**d[51:54] * tau**t[51:54] \
* np.exp(-alpha*(delta - epsilon)**2 - beta[:3]*(tau - gamma)**2) \
* (d[51:54]/delta - 2*alpha*(delta - epsilon))))
tPsi = Psi(delta,tau)
y = y + np.dot(n[54:56], (Delta(delta,tau)**b * (tPsi + delta*Psi_d(delta,tau)) \
+ (Delta_b_d(delta,tau)*delta*tPsi)))
return y
def phir_dd(delta, tau):
"""
second partial derivative for delta of phir
where phir = residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (d[:7]*(d[:7]-1)*delta**(d[:7]-2) * tau**t[:7]))
y = y + np.dot(n[7:51], ( np.exp(-delta**c[7:51]) * ( delta**(d[7:51]-2) * tau**t[7:51] * \
((d[7:51] - c[7:51]*delta**c[7:51]) * \
(d[7:51] - 1 - c[7:51]*delta**c[7:51]) - \
(c[7:51])**2*delta**c[7:51]) ) ) )
y = y + np.dot(n[51:54], ( tau**t[51:54]* np.exp(-alpha*(delta - epsilon)**2 \
-beta[:3]*(tau - gamma)**2) \
* (-2*alpha*delta**d[51:54] + 4*alpha**2*delta**d[51:54]*(delta-epsilon)**2 \
- 4*d[51:54]*alpha*delta**(d[51:54]-1)*(delta-epsilon) \
+ d[51:54]*(d[51:54]-1)*delta**(d[51:54]-2))) )
tPsi = Psi(delta,tau)
dPsi = Psi_d(delta,tau)
y = y + np.dot(n[54:56], ( Delta(delta,tau)**b * (2*dPsi + delta* Psi_dd(delta,tau)) \
+ 2*Delta_b_d(delta,tau)*(tPsi + delta*dPsi) \
+ Delta_b_dd(delta,tau)*delta*tPsi) )
return y
def phir_dt(delta, tau):
"""
partial derivative for delta and tau of phir
where phir = residual part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
n = np.array(IAPWS95_COEFFS['n']); c = np.array(IAPWS95_COEFFS['c'])
d = np.array(IAPWS95_COEFFS['d']); t = np.array(IAPWS95_COEFFS['t'])
alpha = np.array(IAPWS95_COEFFS['alpha']); beta = np.array(IAPWS95_COEFFS['beta'])
gamma = np.array(IAPWS95_COEFFS['gamma']); epsilon = np.array(IAPWS95_COEFFS['epsilon'])
b = np.array(IAPWS95_COEFFS['b'])
y = np.dot(n[:7], (d[:7]*t[:7]*delta**(d[:7]-1) * tau**(t[:7]-1)))
y = y + np.dot(n[7:51], (t[7:51]* delta**(d[7:51]-1) * tau**(t[7:51]-1) \
* (d[7:51] - c[7:51]*delta**c[7:51]) *np.exp(-delta**c[7:51]) ))
y = y + np.dot(n[51:54], ( delta**d[51:54] * tau**t[51:54] \
* np.exp(-alpha*(delta - epsilon)**2 - beta[:3]*(tau - gamma)**2) \
* (d[51:54]/delta - 2*alpha*(delta - epsilon)) \
*(t[51:54]/tau - 2*beta[:3]*(tau-gamma) )))
tPsi = Psi(delta,tau)
ttPsi = Psi_t(delta,tau)
y = y + np.dot(n[54:56], ( Delta(delta,tau)**b * \
(Psi_t(delta,tau) + delta*Psi_dt(delta,tau)) \
+ (Delta_b_d(delta,tau)*delta*ttPsi) \
+ Delta_b_t(delta,tau)*(tPsi+delta*Psi_d(delta,tau)) \
+ Delta_b_dt(delta,tau)*delta*tPsi))
return y
## IAPWS95.idealgas
# Equation 6.5
def phi0(delta, tau):
"""
ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# unpack coefficients
n0 = np.asarray(IAPWS95_COEFFS['n0'])
gamma0 = np.asarray(IAPWS95_COEFFS['gamma0'])
y = np.log(delta) + n0[0] + n0[1]*tau + n0[2]*np.log(tau)
y = y + np.dot(n0[3:], np.log(1 - np.exp(-gamma0[3:]*tau)))
return y
# derivatives from Table 6.4
def phi0_t(delta, tau):
"""
partial derivative for tau of phi0
where phi0 = ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# unpack coefficients
n0 = np.asarray(IAPWS95_COEFFS['n0'])
gamma0 = np.asarray(IAPWS95_COEFFS['gamma0'])
y = n0[1] + n0[2]/tau
y = y + np.dot(n0[3:], gamma0[3:]*(1/(1 - np.exp(-gamma0[3:]*tau))- 1.0))
return y
def phi0_tt(delta, tau):
"""
second partial derivative for tau of phi0
where phi0 = ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# unpack coefficients
n0 = np.asarray(IAPWS95_COEFFS['n0'])
gamma0 = np.asarray(IAPWS95_COEFFS['gamma0'])
y = -n0[2]/tau**2
y = y - np.dot(n0[3:], (gamma0[3:]**2*np.exp(-gamma0[3:]*tau)*(1-np.exp(-gamma0[3:]*tau))**-2 ))
return y
def phi0_d(delta, tau):
"""
partial derivative for delta of phi0
where phi0 = ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
y = 1/delta
return y
def phi0_dd(delta, tau):
"""
second partial derivative for delta of phi0
where phi0 = ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
y = -1/delta**2
return y
def phi0_dt(delta, tau):
"""
partial derivative for tau and delta of phi0
where phi0 = ideal gas part of free energy, dimensionless
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
y = 0
return y
# Supporting functions for calculating the ideal-gas and residual parts in the IAPWS-95 formulation
def Delta(delta, tau):
"""
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
a = np.array(IAPWS95_COEFFS['a']); B = np.array(IAPWS95_COEFFS['B'])
return theta(delta, tau)**2 + B*((delta - 1)**2)**a
def Delta_d(delta, tau):
"""
Delta_d = (d Delta)/(d delta)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/T dimensionless inverse temperature
"""
# % unpack coefficients
a = np.array(IAPWS95_COEFFS['a']); beta = np.array(IAPWS95_COEFFS['beta'])
A = np.array(IAPWS95_COEFFS['A']); B = np.array(IAPWS95_COEFFS['B'])
d1 = delta - 1
y = d1 * ( A*theta(delta,tau)*2/beta[-2:]* (d1**2)**(1/(2*beta[-2:])-1)+ 2*B*a*(d1**2)**(a-1) )
return y
def Delta_dd(delta, tau):
"""
Delta_dd = (d2 Delta)/(d delta2)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/T dimensionless inverse temperature
"""
# % unpack coefficients
a = np.array(IAPWS95_COEFFS['a']); beta = np.array(IAPWS95_COEFFS['beta'])
A = np.array(IAPWS95_COEFFS['A']); B = np.array(IAPWS95_COEFFS['B'])
d1 = delta - 1
y = 1/d1 * Delta_d(delta, tau) + \
d1**2*( 4*B*a*(a-1)*(d1**2)**(a-2) + \
2*A**2*beta[-2:]**-2*((d1**2)**(1/(2*beta[-2:])-1))**2 + \
A*theta(delta,tau)*4/beta[-2:]*(1/(2*beta[-2:])-1)* (d1**2)**(1/(2*beta[-2:])-2))
return y
def Delta_b_d(delta, tau):
"""
Delta_b_d = (d Delta^b)/(d delta)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
b = np.array(IAPWS95_COEFFS['b'])
return b * Delta(delta,tau)**(b-1) * Delta_d(delta,tau)
def Delta_b_dd(delta, tau):
"""
Delta_b_dd = (d2 Delta^b)/(d delta2)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
b = np.array(IAPWS95_COEFFS['b'])
y = b * ( Delta(delta,tau)**(b-1)*Delta_dd(delta,tau) + \
(b-1)*Delta(delta,tau)**(b-2)*(Delta_d(delta,tau))**2)
return y
def Delta_b_dt(delta, tau):
"""
Delta_b_dt = (d2 Delta^b)/(d delta tau)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
b = np.array(IAPWS95_COEFFS['b'])
beta = np.array(IAPWS95_COEFFS['beta'])
A = np.array(IAPWS95_COEFFS['A'])
d1 = delta - 1
y = -A*b*2/beta[-2:]*Delta(delta,tau)**(b-1) * d1*(d1**2)**(1/(2*beta[-2:])-1) \
- 2*theta(delta,tau)*b*(b-1)*Delta(delta,tau)**(b-2) * Delta_d(delta,tau)
return y
def Delta_b_t(delta, tau):
"""
Delta_b_t = (d Delta^b)/(d tau)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
b = np.array(IAPWS95_COEFFS['b'])
return -2*theta(delta,tau)*b*Delta(delta,tau)**(b-1)
def Delta_b_tt(delta, tau):
"""
∂2delta/∂x|y
Delta_b_tt = (d2 Delta^b)/(d tau2)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
b = np.array(IAPWS95_COEFFS['b'])
return 2*b*Delta(delta,tau)**(b-1) + 4*(theta(delta,tau))**2*b*(b-1)*Delta(delta,tau)**(b-2)
def theta(delta, tau):
"""
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
beta = np.array(IAPWS95_COEFFS['beta']); A = np.array(IAPWS95_COEFFS['A'])
return (1 - tau) + A*((delta - 1)**2)**(1./(2*beta[3:5]))
def Psi(delta, tau):
"""
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
C = np.array(IAPWS95_COEFFS['C'])
D = np.array(IAPWS95_COEFFS['D'])
return np.exp(-C*(delta-1)**2 - D*(tau - 1)**2)
def Psi_d(delta, tau):
"""
∂Psi/∂delta
Psi_d = (d Psi)/(d delta)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
C = np.array(IAPWS95_COEFFS['C'])
return -2*C*(delta - 1)*Psi(delta, tau)
def Psi_t(delta, tau):
"""
Psi_t = ∂Psi/∂tau = (d Psi)/(d tau)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
D = np.array(IAPWS95_COEFFS['D'])
return -2*D*(tau - 1)*Psi(delta,tau)
def Psi_tt(delta, tau):
"""
Psi_tt = ∂2Psi/∂tau2 = (d2 Psi)/(d tau2)
auxiliary function in IAPWS95 formulation
parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
D = np.array(IAPWS95_COEFFS['D'])
return (2*D*(tau - 1)**2 - 1)*2*D*Psi(delta,tau)
def Psi_dd(delta, tau):
"""
Psi_dd = (d2 Psi)/(d delta2)
auxiliary function in IAPWS95 formulation
Parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
C = np.array(IAPWS95_COEFFS['C'])
return (2*C*(delta - 1)**2 - 1)*2*C*Psi(delta, tau)
def Psi_dt(delta, tau):
"""
Psi_dt = (d Psi)/(d delta tau)
auxiliary function in IAPWS95 formulation
Parameters:
delta = rho/rhoc dimensionless density
tau = Tc/TK dimensionless inverse temperature
"""
# % unpack coefficients
C = np.array(IAPWS95_COEFFS['C'])
D = np.array(IAPWS95_COEFFS['D'])
return 4*C*D*(delta - 1)*(tau - 1)*Psi(delta, tau)
# Calculate thermodynamic functions.
# Helmholtz energy. The value is in J/kg-K.
ax = (R*TK*( phi0(delta, tau) + phir(delta, tau) ) )
# Pressure kpa to bar.
px = rho*R*TK*( 1 + delta*phir_d(delta, tau) ) * 0.010
# Internal energy. kJ/kg
ux = R*TK*tau*( phi0_t(delta, tau) + phir_t(delta, tau) )
# Entropy. kJ/kg/K
sx = R*( tau*(phi0_t(delta, tau) + phir_t(delta, tau)) - phi0(delta, tau) - phir(delta, tau) )
# Enthalpy. kJ/kg
hx = R*TK*( 1 + tau*(phi0_t(delta, tau) + phir_t(delta, tau)) + delta*phir_d(delta, tau) )
# Gibbs energy. kJ/kg
# Alternate formulas for the Gibbs energy. gx = hx - TK*sx = ax + hx - ux
gx = R*TK*( 1 + phi0(delta, tau) + phir(delta, tau) + delta*phir_d(delta, tau) )
# Volume. m3/kg
vx = (1/rho)
if FullEOSppt == True:
# Isochoric heat capacity. kJ/kg/K
cvx = -R*tau**2*( phi0_tt(delta, tau) + phir_tt(delta, tau) )
# Isobaric heat capacity. kJ/kg/K
x1 = ( 1 + delta*phir_d(delta, tau) - delta*tau*phir_dt(delta, tau) )**2
x2 = 1 + 2*delta*phir_d(delta, tau) + delta**2*phir_dd(delta, tau)
cpx = x1*0
cpx = float(cvx + R*(x1/x2)) if (abs(x2) > 1e-15) else 1.0 + 100
# Speed of sound. m/s
x1 = ( 1 + delta*phir_d(delta, tau) - delta*tau*phir_dt(delta, tau) )**2
x2 = tau**2*( phi0_tt(delta, tau) + phir_tt(delta, tau) )
x3 = x1/x2
xxt = R*TK*( 1 + 2*delta*phir_d(delta, tau) + delta**2*phir_dd(delta, tau) - x3 )
wx = np.where(xxt > 0, np.sqrt(xxt), 0)
wx = wx*np.sqrt(1000.00) # convert Speed of sound from sqrt(kJ/kg) to m/s
# Joule-Thomsen coefficient. K-m3/kJ (equivalent to the usual K/MPa)
x1 = delta*phir_d(delta, tau) + delta**2*phir_dd(delta, tau) + delta*tau*phir_dt(delta, tau)
x2 = ( 1 + delta*phir_d(delta, tau) - delta*tau*phir_dt(delta, tau) )**2
x3 = ( phi0_tt(delta, tau) + phir_tt(delta, tau) )*( 1.0 + 2*delta*phir_d(delta, tau) + \
delta**2*phir_dd(delta, tau) )
mux = (( - x1/( x2 - tau**2*x3 ) )/(R*rho) )
# Isothermal throttling coefficient.
x1 = 1 + delta*phir_d(delta, tau) - delta*tau*phir_dt(delta, tau)
x2 = 1 + 2*delta*phir_d(delta, tau) + delta**2*phir_dd(delta, tau)
dtx = (( 1 - ( x1/x2 ) )/rho )
dtx = 0.010*dtx # convert Isothermal throttling coefficient from m3/kg to the usual kJ/kg/bar.
# Isentropic temperature-pressure coefficient. (the same units as for the Joule-Thomson coefficient)
x1 = 1.0 + delta*phir_d(delta, tau) - delta*tau*phir_dt(delta, tau)
x2 = x1**2
x3 = ( phi0_tt(delta, tau) + phir_tt(delta, tau) )*( 1 + 2*delta*phir_d(delta, tau) + \
delta**2*phir_dd(delta, tau) )
bsx = (( x1/( x2 - tau**2*x3 ) )/(R*rho) )
# Gibbs energy derivative.
gdx = (R*TK*( phi0_d(delta, tau) + 2*phir_d(delta, tau) + delta*phir_dd(delta, tau) ))
# Derivative of pressure with respect to delta (needed to perform Newton-Raphson
# iteration to matched desired pressure) in bar.
pdx = ( px/delta ) + delta*rhoc*R*TK*( phir_d(delta, tau) + delta*phir_dd(delta, tau) ) * 0.010
# Derivative of pressure with respect to tau (needed to calculate the thermal expansion
# coefficient) in bar.
ptx = ( -px/tau ) + px*delta*phir_dt(delta, tau)/(1 + delta*phir_d(delta, tau))
# Compressibility. Here the value is in /bar.
ktx = 1.0/(delta*pdx)
# Calculate zeta value (needed to calculate viscosity). Note: ztx is dimensionless
# An alternative formula is: ztx = delta*Pc*ktx
ztx = Pc/pdx
# Thermal expansion coefficient (thermal expansivity). This calculation is based on the Maxwell relation:
avx = ktx*ptx*( -tau/TK )
# Helmholtz energy derivative with respect to delta
adx = (R*TK*( phi0_d(delta, tau) + phir_d(delta, tau) ))
if FullEOSppt == True:
output = px, ax, sx, hx, gx, vx, pdx, adx, ztx, ptx, ktx, avx, ux, gdx, bsx, dtx, mux, cpx, cvx, wx
else:
output = px, ax, sx, hx, gx, vx, pdx, adx, ztx, ptx, ktx, avx
return output
[docs] def auxMeltingPressure(self, TK, P):
"""
This function calculates the melting pressure of ice as a function of temperature.
This model is described in IAPWS R14-08(2011), Revised Release on the Pressure along
the Melting and Sublimation Curves of Ordinary Water Substance, as may be found at:
http://www.iapws.org/relguide/MeltSub.html
Five ice phases are covered here. The melting pressure is not a single-valued function
of temperature as there is some overlap in the temperature ranges of the individual phases.
There is no overlap in the temperature ranges of Ices III, V, VI, and VII, which together
span the range 251.165 - 715K. The melting pressure is continuous and monotonically increasing
over this range, albeit with discontinuities in slope at the triple points where two ice
phases and liquid are in equilibrium. The problem comes in with Ice Ih, whose temperature
range completely overlaps that of Ice III and partially overlaps that of Ice V. For a
temperature in the range for Ice Ih, there are two possible melting pressures.
The possible ambiguity here in the meaning of melting pressure is not present if the
temperature is greater than or equal to the triple point temperature of 273.16K, or
if the pressure is greater than or equal to 2085.66 bar (the triple point pressure
for Ice Ih-Ice III-liquid). If neither of these conditions are satisfied, then the Ice
Ih-liquid curve will be used. To deal with the pressure condition noted above, this function
assumes that an actual pressure is specified.
Parameters
----------
P : pressure [bar]
TK : temperature [K]
Returns
----------
Pmelt : melting pressure [bar]
Usage
----------
[Pmelt] = auxMeltingPressure( TK, P)
"""
# Coefficients for calculating the melting pressure of Ice Ih.
a = np.array([0.119539337e+07, 0.808183159e+05, 0.333826860e+04])
b = np.array([0.300000e+01, 0.257500e+02, 0.103750e+03])
PPa = P*1e5 # convert bar to Pa
if (PPa <= 208.566e6):
if (251.165 <= TK <= 273.16):
# Ice Ih.
theta = TK/273.16
pimelt = 1 + np.sum(a*( 1 - theta**b))
Pmelt = pimelt*611.647 #e-06
else:
Pmelt = np.nan #2085.66
else:
if (251.165 <= TK < 256.164):
# Ice III.
theta = TK/251.165
pimelt = 1 - 0.299948*( 1 - theta**60 )
Pmelt = pimelt*208.566e6
elif (256.164 <= TK < 273.31):
# Ice V.
theta = TK/256.164
pimelt = 1 - 1.18721*( 1 - theta**8 )
Pmelt = pimelt*350.1e6
elif (273.31 <= TK < 355.0):
# Ice VI.
theta = TK/273.31
pimelt = 1 - 1.07476*( 1 - theta**4.6 )
Pmelt = pimelt*632.4e6
elif (355.0 <= TK < 715.0):
# Ice VII.
theta = TK/355.0
px = 0.173683e+01*( 1 - theta**(-1) ) - 0.544606e-01*( 1 - theta**5 ) \
+ 0.806106e-07*( 1 - theta**22 )
pimelt = np.exp(px)
Pmelt = pimelt*2216.0e6
elif (715.0 <= TK <= 2000.0):
# This is out of range. Ice VII, extrapolated.
theta = TK/355.0
px = 0.173683e+01*( 1 - theta**(-1) ) - 0.544606e-01*( 1 - theta**5 ) \
+ 0.806106e-07*( 1 - theta**22 )
pimelt = np.exp(px)
Pmelt = pimelt*2216.0e6
else:
Pmelt = np.nan
return Pmelt*1e-5
[docs] def auxMeltingTemp(self, P):
"""
This function calculates the melting temperature of ice as a function of pressure.
This inverts the model for the melting pressure as a function of temperature.
That model is described in IAPWS R14-08(2011), Revised Release on the Pressure
along the Melting and Sublimation Curves of Ordinary Water Substance as may be found at:
http://www.iapws.org/relguide/MeltSub.html
Inversion of the model for the melting pressure is done here using the secant method.
This is chosen instead of the Newton-Raphson method to avoid potential problems with slope
discontinuites at boundaries between the ice phases for pressures above 208.566 MPa,
which is the equilibrium pressure for Ice Ih-Ice III-liquid. The corresponding equlibrium
temperature is 251.165K. Putative melting temperatures should not be less than this for
pressures above 208.566 Mpa, nor more than this for pressures less than this.
Parameters
----------
P : pressure [bar]
Returns
----------
Tmelt : temperature [K]
Usage
----------
[Tmelt] = auxMeltingTemp( P)
"""
# Calculates the melting temperature that corresponds to the actual pressure.
# The variables t0 and t1 are initial values for the iteration.
PPa = P*1e5 # convert bar to Pa
if (PPa < 208.566e6):
# In the Ice Ih field.
tlim0 = 251.165
tlim1 = 273.16
t0 = tlim0
t1 = tlim1
elif (208.566e6 <= PPa < 350.1e6):
# In the Ice III field.
tlim0 = 251.165
tlim1 = 256.164
t0 = tlim0
t1 = tlim1
elif (350.1e6 <= PPa < 632.4e6):
# In the Ice V field.
tlim0 = 256.164
tlim1 = 273.31
t0 = tlim0
t1 = tlim1
elif (632.4e6 <= PPa < 2216.0e6):
# In the Ice VI field.
tlim0 = 273.31
tlim1 = 355.0
tx = tlim1 - tlim0
t0 = tlim0 + 0.3*tx
t1 = tlim0 + 0.7*tx
elif (2216.0e6 <= PPa <= 10000.0e6):
# In the Ice VII field.
# Note: the upper limit here of 10000 MPa is an arbitrary cutoff suggested
# by Figure 1 from IAPWS R14, but this is not part of the IAPWS R14 standard.
tlim0 = 355.0
tlim1 = 1000.0
tx = tlim1 - tlim0
t0 = tlim0 + 0.3*tx
t1 = tlim0 + 0.7*tx
elif (PPa > 20000.0e6):
Tm = np.nan
funct_melt = lambda t: self.auxMeltingPressure(t, P) - P
Tm = root_scalar(funct_melt, method = 'secant',
bracket=[t0, t1], x0=t0, x1=t1, xtol = 1.0e-10).root
Tmelt = Tm
return Tmelt
[docs] def waterviscosity(self, TC, P, rho):
"""
This function calculates the viscosity of water using
Ref:
(1) "IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance" (IAPWS R12-08).
(2) Huber M.L., Perkins R.A., Laesecke A., Friend D.G., Sengers J.V., Assael M.J., Metaxa I.N.,
Vogel E., Mares R., and Miyagawa K. (2009) New International Formulation for the Viscosity
of H2O. J. Phys. Chem. Ref. Data 38, 101-125. \n
Parameters
----------
TC temperature [°C] \n
P pressure [bar] \n
rho density [kg/m3] \n
Returns
----------
visc viscosity [Pa.s]
Usage
----------
[visc] = waterviscosity( TC, P, rho)
"""
TK = convert_temperature( TC, Out_Unit = 'K' ) #convert to Kelvin
PPa = P*1e5
Tc = IAPWS95_COEFFS['Tc'] # K
rhoc = IAPWS95_COEFFS['rhoc']
H0 = [1.67752e+00, 2.20462e+00, 0.6366564e+00, -0.241605e+00]
I1 = [0, 1, 2, 3, 0, 1, 2, 3, 5, 0, 1, 2, 3, 4, 0, 1, 0, 3, 4, 3, 5]
J1 = [0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6]
H1 = [0.520094, 0.0850895, -1.08374, -0.289555, 0.222531, 0.999115, 1.8879700000000001,
1.26613, 0.120573, -0.281378, -0.9068510000000001, -0.772479, -0.48983699999999997,
-0.25704, 0.161913, 0.257399, -0.0325372, 0.0698452, 0.00872102, -0.00435673, -0.000593264]
delta = rho/rhoc
Tbar = TK/Tc
muref = 1.00e-06 # Pa.s
# Check range of validity.
chk = False
# triple point
Ttr = 273.16 # K
Ptr = 611.657 # Pa
Ttmltx = self.auxMeltingTemp(P)
if (PPa < Ptr):
if (Ttr <= TK <= 1173.15):
chk = True
elif Ttmltx > 0.0:
if (Ptr <= PPa <= 300.0e6):
if (Ttmltx <= TK <= 1173.15):
chk = True
elif (300.0e6 < PPa <= 350.0e6):
if (Ttmltx <= TK <= 873.15):
chk = True
elif (350.0e6 < PPa <= 500.0e6):
if (Ttmltx <= TK <= 433.15):
chk = True
elif (500.0e6 < PPa <= 1000.0e6):
if (Ttmltx <= TK <= 373.15):
chk = True
if (not chk) | (Ttmltx <= 0):
mubar = np.nan
else:
# Calculate the viscosity in the dilute gas limit (mubar0)
mubar0 = 100*np.sqrt(Tbar)/np.sum([x/Tbar**idx for idx, x in enumerate(H0)])
s1=np.zeros([len(H1), 1])
for x in range(len(H1)):
s1[x] = H1[x]*((1/Tbar) -1)**I1[x] * (delta-1)**J1[x]
# Calculate the contribution to viscosity due to finite density
mubar1 = np.exp(delta * np.sum(s1))
# critical enhancement
xmu = 0.068
qc = 1/1.9 #/nm
qD = 1/1.1 #/nm
nvc = 0.63
gamma = 1.239
xicl0 = 0.13
gam0 = 0.06
TR = 1.5
Tkr = TR*Tc
# Estimate the reference zeta value by directly evaluating the EOS model
ztxr = self.EOSIAPWS95(Tkr, rho)[8]
ztx = self.EOSIAPWS95(TK, rho)[8]
# Get delchb (DELTA chibar, equation 21).
delchb = delta*( ztx - (ztxr*TR*Tbar) )
if (delchb < 0.0):
delchb = 0.0
# Get xicl (equation 20).
xicl = xicl0*( (delchb/gam0)**(nvc/gamma) )
# Get psid (equation 17).
psid = math.acos(( 1.0 + (qD*xicl)**2 )**(-0.5))
# Get www (w in equation 19).
w = np.sqrt(abs( ((qc*xicl) - 1.0)/((qc*xicl) + 1.0) )) * math.tan(0.5*psid)
# Get lcap (L(w) in equation 18)
if ((qc*xicl) > 1):
lcap = np.log( (1 + w)/(1 - w) )
else:
lcap = 2*math.atan( abs(w) )
if (xicl <= 0.3817016416):
ycap = 0.2*(qc*xicl)*((qD*xicl)**5)* \
(1 - (qc*xicl) + (qc*xicl)**2 - ( 765.0/504.0 )*(qD*xicl)**2)
else:
ycap = ( 1/12 )*math.sin(3*psid) - ( 1/(4*(qc*xicl)) )*math.sin(2*psid) +\
( 1/(qc*xicl)**2 )*( 1.0 - 1.25*(qc*xicl)**2 ) *math.sin(psid) \
-( 1.0/(qc*xicl)**3 ) *( ( 1.0 - 1.5*(qc*xicl)**2 ) \
- ( abs((qc*xicl)**2 - 1) )**1.5 * lcap)
if (645.91 < TK < 650.77) and (245.8 < rho < 405.3):
mubar2 = np.exp(xmu * ycap)
else:
mubar2 = 1
mubar = mubar0 * mubar1 *mubar2
visc = mubar*muref
return visc
[docs] def apxsatpropT(self, TK):
"""
This function evaluates the approximate pressure (psat) as a function of temperature along
the vapor-liquid equilibrium curve, using equation 2.5 of Wagner and Pruss (2002).
It also calculates the derivative of saturation pressure wrt temperature as well as the
densities of the liquid and vapor phases using equations 2.6 and 2.7
from the same source. \n
Parameters:
----------
TK temperature [K] (saturation temperature) \n
Returns:
----------
Psat saturation pressure [bar] \n
Psat_t Derivative of saturation pressure with respect to temperature \n
rhosl density of liquid [kg/m3] \n
rhosv density of vapor [kg/m3] \n
Usage:
----------
[Psat, Psat_t, rhosl, rhosv] = apxsatpropT( TK)
"""
Tc = IAPWS95_COEFFS['Tc'] # K
Pc = IAPWS95_COEFFS['Pc'] # MPa
a = [-7.85951783, 1.84408259, -11.7866497, 22.6807411, -15.9618719, 1.80122502]
rhoc = IAPWS95_COEFFS['rhoc']
b = [1.99274064, 1.09965342, -0.510839303, -1.75493479, -45.5170352, -6.74694450e5]
c = [-2.03150240, -2.68302940, -5.38626492, -17.2991605, -44.7586581, -63.9201063]
# Check to see that the temperature is in the allowed range.
if (TK < 273.15) | (TK > Tc):
Psat = np.nan
Psat_t = np.nan
rhosl = np.nan
rhosv = np.nan
else:
# Saturation pressure.
th = 1 - TK/Tc
t1 = a[0]*th + a[1]*th**1.5 + a[2]*th**3 + a[3]*th**3.5 + a[4]*th**4 + a[5]*th**7.5
Psat = Pc*np.exp(Tc*t1/TK)
# Derivative of saturation pressure with respect to temperature.
x1 = a[0] + 1.5*a[1]*th**0.5 + 3.0*a[2]*th**2.0 + 3.5*a[3]*th**2.5 + 4.0*a[4]*th**3.0 + 7.5*a[5]*th**6.5
x2 = np.log( Psat/Pc ) + x1
Psat_t = -( Psat/TK )*x2
# Density of liquid.
th = (1 - TK/Tc)**(1/3)
t1 = 1 + b[0]*th + b[1]*th**2 + b[2]*th**5 + b[3]*th**16 + b[4]*th**43 + b[5]*th**110
rhosl = rhoc*t1
# Density of vapor.
th = np.sqrt(th)
t2 = c[0]*th**2 + c[1]*th**4 + c[2]*th**8 + c[3]*th**18 + c[4]*th**37 + c[5]*th**71
rhosv = rhoc*np.exp(t2)
return Psat*10, Psat_t*10, rhosl, rhosv
[docs] def apxsatpropP(self, P):
"""
This function evaluates the approximate temperature (tsat) as a function of pressure along
the vapor-liquid equilibrium curve, using equation 2.5 of Wagner and Pruss (2002).
This is similar to apxsatpropT(TK), but evaluates the inverse problem (Tsat as a function
of pressure instead of psat as a function of temperature). Newton-Raphson iteration is used. \n
Parameters:
----------
P pressure [bar] \n
Returns:
----------
Tsat saturation temperature [K] \n
Usage:
----------
[Tsat] = apxsatpropP( P)
"""
Tc = IAPWS95_COEFFS['Tc'] # K
Pc = IAPWS95_COEFFS['Pc']*10 # bar
# triple point
Ttr = 273.16 # K
Ptr = 611.657e-5 # bar
P1atm = 1.01325e0 # bar
Ts1atm = 373.124 # K
# Check to see that the pressure is in the allowed range.
if (P < Ptr) | (P > Pc):
Tsat = np.nan
else:
# Choose a starting temperature value.
if (P >= P1atm):
# Interpolate between 100C, 1.01325 bar and Tcr, Pcr.
dtdp = (Tc - Ts1atm)/(Pc - P1atm)
TK = Ts1atm + dtdp*(P - P1atm)
else:
# Interpolate between the triple point and 100C, 1.013 bar
dtdp = (Ts1atm - Ttr)/(P1atm - Ptr)
TK = Ttr + dtdp*(P - Ptr)
funct_tsat = lambda T: self.apxsatpropT(T)[0]*0.1 - P*0.1
Tsat = fsolve(funct_tsat, TK, xtol=1.48e-10)[0]
return Tsat
[docs] def calcsatpropT(self, TK):
"""
This function calculates the saturation properties as a function of specified temperature.
This is achieved using Newton-Raphson iteration to refine values of pressure, vapor density,
and liquid density, starting with results obtained using approximate equations included
by Wagner and Pruss (2002) in their description of the IAPWS-95 model. \n
Parameters
----------
TK temperature [K] \n
Returns
----------
Psat saturation pressure [bar] \n
rhosl density of liquid [kg/m3] \n
rhosv density of vapor [kg/m3] \n
Usage
----------
[Psat, rhosl, rhosv] = calcsatpropT( TK)
"""
Tc = IAPWS95_COEFFS['Tc'] # K
rhoc = IAPWS95_COEFFS['rhoc']
bettl1 = 1.0e-8
bettl2 = 1.0e-7
btxtol = 1.0e-10
qxiter = False
alpha = np.zeros([3, 1]); beta = alpha
aamatr = np.zeros([3, 3])
deltas = np.zeros([3, 1]); arelax = 1
if (TK <= 298.15):
btxtol = bettl1
elif (647.090 < TK < Tc):
qxiter = True
elif (TK > 647.090):
btxtol = bettl2
# Calculate approximate saturation pressure and corresponding densities of liquid and vapor.
# These results are not those of the IAPWS-95 model itself, but can serve as starting estimates.
[Psat, _, rhosl, rhosv] = self.apxsatpropT(TK)
# Save the values from the approximation.
delta_svq = rhosv/rhoc
delta_slq = rhosl/rhoc
it = 0
itermx = 50
Psat0 = Psat
deltasv0 = delta_svq
deltasv = delta_svq
deltasl0 = delta_slq
deltasl = delta_slq
while True:
# Below is the return point to refine the saturation curve properties.
# First calculate the vapor properties by calling EOSIAPWS95 with the vapor density.
[Pxv, axv, _, _, _, _, Pdxv, adxv] = self.EOSIAPWS95(TK, rhosv)[:8]
# Now calculate the liquid properties by calling EOSIAPWS95 with the liquid density.
[Pxl, axl, _, _, _, _, Pdxl, adxl] = self.EOSIAPWS95(TK, rhosl)[:8]
# The pdx for vapor cannot be negative. Under-relax to prevent this.
if (Pdxv < 0):
if (it <= 0):
#if (icutv >= 30):
#icutv = icutv + 1
rhosv = 0.995*rhosv
deltasv = rhosv/rhoc
else:
#icutv = icutv + 1
arelax = 0.25
# The revised delta for vapor cannot be less than a good
# fraction of the value obtained from the intial approximation.
if (deltasv < 0.9*delta_svq):
arelax = 0.25
# The pdx for liquid cannot be negative. Under-relax to prevent this.
if (Pdxl < 0):
if (it <= 0):
#if (icutl >= 30):
#icutl = icutl + 1
rhosl = 1.001*rhosl
deltasl = rhosl/rhoc
else:
#icutl = icutl + 1
arelax = 0.25
# The revised delta for liquid cannot be less than a good
# fraction of the value obtained from the intial approximation.
if (deltasl > 1.1*delta_slq):
arelax = 0.25
# The delta for liquid cannot be less than the delta for vapor.
# Corrected delta values must be positive to avoid
# a singularity in the equation-of-state model equations.
# Under-relax to prevent this.
if (deltasl < deltasv) | (deltasv <= 0 or deltasl <= 0):
arelax = 0.25
deltas = deltas*arelax
Psat = Psat0 + deltas[2, 0]
deltasl = deltasl0 + deltas[1, 0]
deltasv = deltasv0 + deltas[0, 0]
rhosl = rhoc*deltasl
rhosv = rhoc*deltasv
# =============================================================================
# Have obtained valid (outside the unstable zone) vapor and liquid properties for the current iteration.
# Improve the calculated saturation properties by solving three equations in three unknowns.
# The equations are all in terms of pressure. The unknowns to be found are Psat, deltasv, and deltasl.
# Calculate the Maxwell crition pressure (Gibbs energy equality expressed through the
# Helmholtz energies and the pressure)
# =============================================================================
dix = (1/deltasl) - (1/deltasv)
# dltx = deltasl - deltasv
if (abs(dix) > 1e-15):
if (abs(axv - axl) > 1e-15):
# Normal calculation, result in kPa.
Pxm = rhoc*( axv - axl )/dix
Pxm = 0.01*Pxm # Convert from kPa to bar.
else:
# There is no difference in the Helmholtz energies of the vapor and the liquid.
Pxm = 0
else:
# Exception intended for the critical point.
if (abs(TK - Tc) <= 1e-10) & (abs(deltasv - 1.0) <= 1e-10) & (abs(deltasl - 1) <= 1e-10):
# Am at the critical point.
Pxm = Pxv
else:
# Not at the critical point, but the vapor and liquid densities have converged.
Pxm = 0
# Calculate residual functions.
alpha[0, 0] = Pxm - Psat
alpha[1, 0] = Pxv - Psat
alpha[2, 0] = Pxl - Psat
beta = abs(alpha/Psat)
betamx = np.max(beta)
# Note: using a convergence tolerance below 1.0d-11
# may lead to non-convergence due to the limitations of 64-bit arithmetic.
# print(it, ' Psat: %.6e' % Psat[0], 'betamx: %.6e' % betamx)
if (betamx <= btxtol) | (it >= itermx):
# Iteration has converged.
# P = Psat
break
elif (qxiter and it >= 5):
break
# Since this matrix is only 3x3, the simultaneous equations have the form:
# aamatr(i,1)*deltas(1) + aamatr(i,2)*deltas(2) + aamatr(i,3)*deltas(3) = -alpha(i), i = 1,3
# The Jacobian matrix J here is aamatr(kdim, kdim).
aamatr[0, :] = [Pxm*(-(1/(dix*deltasv**2)) + (adxv/(axv - axl))),
Pxm*((1/(dix*deltasl**2)) - (adxl/(axv - axl))),
-1]
aamatr[1, :] = [Pdxv,
0,
-1]
aamatr[2, :] = [0,
Pdxl,
-1]
deltas = -lu_solve(lu_factor(aamatr), alpha)
# Save current values.
Psat0 = Psat
deltasv0 = deltasv
deltasl0 = deltasl
Psat = Psat0 + deltas[2, 0]
deltasl = deltasl0 + deltas[1, 0]
deltasv = deltasv0 + deltas[0, 0]
rhosl = rhoc*deltasl
rhosv = rhoc*deltasv
it = it + 1
return Psat, rhosl, rhosv
[docs] def calcsatpropP(self, P):
"""
This function calculates the saturation properties as a function of specified pressure.
This is done by iterating using Newton method on pressure to obtain the desired
temperature. This implementation calls calcsatpropT(TK) to calculate the saturation
pressure, liquid and vapor densities. \n
Parameters
----------
P pressure [bar] \n
Returns
----------
Tsat temperature [K] \n
rhosl liquid density [kg/m3] \n
rhosv vapor density [kg/m3] \n
Usage
----------
[Tsat, rhosl, rhosv] = calcsatpropP( P)
"""
btxtol = 1.0e-10
itermx = 50
Tc = IAPWS95_COEFFS['Tc'] # K
# Calculate approximate saturation temperature to use as a starting estimate
Tsat = convert_temperature( self.apxsatpropP(P), Out_Unit = 'C' ) # convert to C
# Iterate to calculate the saturation temperature
funct_tsat = lambda TC: self.calcsatpropT(convert_temperature( TC, Out_Unit = 'K' ))[0] - P
Tsat = brentq(funct_tsat, 0, convert_temperature( Tc, Out_Unit = 'C' ), xtol=btxtol, maxiter = itermx) + 273.15
#Tsat = newton(funct_tsat, Tsat, fprime=None, args=(), tol=btxtol, maxiter=itermx, fprime2=None)
# Calculate liquid and vapor densities
[_, rhosl, rhosv] = self.calcsatpropT(Tsat)
return Tsat, rhosl, rhosv
[docs] def fluidDescriptor(self, P, TK, *rho):
"""
This function calculates the appropriate description of the H2O fluid at any given
temperature and pressure \n
A problem may occur if the pressure is equal or nearly equal to the saturation pressure.
Here comparing the pressure with the saturation pressure pressure may lead to the wrong
description, as vapor and liquid coexist at the saturation pressure. It then becomes
neccesary to compare the fluid density with the saturated vapor and saturated liquid densities.
If the density is known, it will be used. If it is not known, the results obtained
here will determine the starting density estimate, thus in essence choosing "vapor" or "liquid"
for pressures close to the saturation pressure. \n
Parameters:
----------
P pressure [bar] \n
TK temperature [K] \n
rho density [kg/m3] (optional) \n
Returns:
----------
phase fluid description \n
rhosl liquid density [kg/m3] \n
rhosv vapor density [kg/m3] \n
Usage:
----------
[udescr, rhosl, rhosv] = fluidDescriptor( P, TK)
"""
# P = P*10 # MPa to bar
Tc = IAPWS95_COEFFS['Tc'] # K
Pc = IAPWS95_COEFFS['Pc']*10 # convert MPa to bar
# rhoc = IAPWS95_COEFFS['rhoc']
btxtol = 1e-10
rhotol = 1.0e-8
if (TK < 273.15):
# Note that the allowed temperature range has been extended a bit on the low end to include 0C.
phase = 'unknown'
rhosv = 0; rhosl = 0
elif (TK <= Tc):
# Calculate the saturation curve properties.
[Psat, rhosl, rhosv] = self.calcsatpropT(TK)
if (P > Pc):
phase = 'compressed liquid'
if Psat == np.nan:
# if calcsatpropT(TK) failed, an arbitrary liquid-like density will be assigned
# as a starting value for compressed liquid
rhosl = 1.05
else:
# if calcsatpropT(TK) failed, the vapor and liquid states cannot be distinguished
# from one another. Liquid is assigned arbitrarily
if Psat == np.nan:
phase = 'liquid'
rhosl = 1.05
else:
if (P >= Psat):
phase = 'liquid'
else:
phase = 'vapor'
# Use density (rho) if available and pressure is close to psat.
if len(rho) != 0:
Ptest = (P - Psat)/Psat
btest = 10*btxtol
if (abs(Ptest) <= btest):
# Here press is very close to psat. Use rho to determine vapor or liquid.
rtestl = (rho - rhosl)/rhosl
rtestv = (rho - rhosv)/rhosv
if (abs(rtestl) <= rhotol):
phase = 'liquid'
elif (abs(rtestv) <= rhotol):
phase = 'vapor'
else:
phase = 'unknown'
else:
rhosv = 0; rhosl = 0
if (P > Pc):
phase = 'supercritical fluid'
else:
phase = 'hot vapor'
return phase, rhosl, rhosv
[docs] def calcwaterppt(self, TC, P, *rho0, FullEOSppt = False):
"""
This function evaluates thermodynamic properties of water at given temperature and pressure.
The problem reduces to finding the value of density that is consistent with the desired pressure.
The Newton-Raphson method is employed. Small negative values of calculated pressure are okay.
Zero or negative values for calculated "pdx" (pressure derivative with respect to delta)
imply the unstable zone and must be avoided. \n
Parameters:
----------
T : temperature [°C] \n
P : pressure [bar] \n
rho0 : starting estimate of density [kg/m3] (optional)
FullEOSppt: Option to output all or essential water properties [False or True]
Returns:
----------
rho : density [kg/m3] \n
gx : Gibbs energy [cal/mol] \n
hx : Enthalpy [cal/mol] \n
sx : Entropy [cal/mol/K] \n
vx : Volume [m3/mol] \n
Pout : pressure [bar] \n
Tout : temperature [°C] \n
ux : Internal energy [cal/mol] if FullEOSppt is True \n
ax : Helmholtz energy [cal/mol/K] if FullEOSppt is True \n
cpx : Isobaric heat capacity [cal/mol/K] if FullEOSppt is True \n
Usage:
----------
[rho, gxcu, hxcu, sxcu, vxcu, uxcu, axcu, cpxcu, Pout, Tout] = calcwaterppt(T, P), \n
"""
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
else:
TC = TC.ravel()
if np.ndim(P) == 0:
P = np.array(P).ravel()
else:
P = P.ravel()
rho0 = rho0[0]
Pout = P # cases where 'T' is used as input
Tout = TC # cases where 'P' is used as input
TK = convert_temperature( TC, Out_Unit = 'K' )
Ppa = P * 0.1 #Convert bars to MPa
Tc = IAPWS95_COEFFS['Tc'] # K
Pc = IAPWS95_COEFFS['Pc'] # MPa
rhoc = IAPWS95_COEFFS['rhoc']
R = IAPWS95_COEFFS['R']*1e-3 #KJ/(kg*K)
# if len(rho0) != 0:
# itermx = 80
# else:
# itermx = 100
# btxtol = 1.0e-8
# Obtain a description (udescr) of the H2O fluid.
rho = np.zeros(len(TK)).ravel()
for i in range(len(TK)):
if (np.ndim(rho0) != 0):
[phase, rhosl, rhosv] = self.fluidDescriptor(P[i], TK[i], rho0[i])
else:
[phase, rhosl, rhosv] = self.fluidDescriptor(P[i], TK[i])
if (phase == 'vapor'):
# Vapor: assume ideal gas behavior.
rho[i] = 1000.0*Ppa[i]/(TK[i]*R)
elif (phase == 'liquid'):
# Liquid: use a liquid-like density. The liquid density on the saturation curve.
rho[i] = rhosl
elif (phase == 'compressed liquid'):
# Estimate the density of compressed liquid. The saturated liquid density is a minimum value.
rho[i] = 1.10*rhosl
# Close to the upper limit for this field (T near the triple point, 1000 MPa).
# For higher pressure, a higher value might be needed. rho = 1250.0d0
rho[i] = np.maximum(rho[i], rhosl)
rho[i] = np.minimum(rho[i], 1400)
elif (phase == 'supercritical fluid'):
# Estimate density of supercritical fluid. Twice the ideal gas correction to critical point density.
rho[i] = 2.0*(Ppa[i]/Pc)*(Tc/TK[i])*rhoc
# Close to the upper limit for this P, T field. (T near the critical point, 1000 MPa).
# For higher pressure, a higher value might be needed. rho = 1100.0d0
rho[i] = np.minimum(rho[i], 1100.0)
elif (phase == 'hot vapor'):
# Estimate the density of hot vapor. Ideal gas.
rhoidg = 1000.0*Ppa[i]/(TK[i]*R)
# SUPCRT92 estimate, about 15% higher than ideal gas.
rhosup = 2500.0*Ppa[i]/TK[i]
# Ideal gas correction to critical point density.
rhocpa = (Ppa[i]/Pc)*(Tc/TK[i])*rhoc
# The upper limit for this field, the critical pressure (rhocr), 22.064 MPa. rho = rhoc
if (Ppa[i] <= 1.0):
rho[i] = rhoidg
elif (Ppa[i] <= 18.0):
rho[i] = rhosup
else:
rho[i] = rhocpa
rho[i] = np.minimum(rho[i], rhoc)
else:
# If the H2O fluid type could not be determined. A good starting estimate
# of density could not be established, try three times the critical density.
rho[i] = 4*rhoc
funct_tsat = lambda rho: self.EOSIAPWS95(TK[i], rho, FullEOSppt = FullEOSppt)[0] - P[i]
rho[i] = fsolve(funct_tsat, rho[i])
if FullEOSppt == True:
[Px, ax, sx, hx, gx, vx, _, _, _, _, _, _, ux, _, _, _, _, cpx, _, _] = self.vEOSIAPWS95(TK, rho, FullEOSppt = FullEOSppt)
else:
[Px, ax, sx, hx, gx, vx, _, _, _, _, _, _] = self.vEOSIAPWS95(TK, rho, FullEOSppt = FullEOSppt)
# The following reference state data are in kilogram units.
ds = 3.5156150 # Entropy (kJ/kg/K)
du = -15767.19391 # Internal energy (kJ/kg)
dh = -15970.89538 # Enthalpy (kJ/kg)
da = -11906.84446 # Helmholtz energy (kJ/kg-K)
dg = -12110.54592 # Gibbs energy (kJ/kg)
htripl = 0.000611782 # tripple point Enthalpy (kJ/kg)
mwH2O = 18.01528/1000 # kg/mol
# Calculate the entropy, internal energy, enthalpy, Helmholtz energy, and Gibbs energy
# on the standard thermochemical scale, and volume, heat capacity at constant volume.
# Results are converted from KJ and kilogram units to cal and molar units.
sxcu = sx + ds
if FullEOSppt == True:
uxcu = mwH2O*(ux + du)*1000/J_to_cal
axcu = mwH2O*(ux - TK*sxcu + da)*1000/J_to_cal
cpxcu = mwH2O*cpx*1000/J_to_cal
hxcu = mwH2O*(hx - htripl + dh)*1000/J_to_cal
gxcu = mwH2O*(hx - TK*sxcu + dg)*1000/J_to_cal
vxcu = mwH2O*vx*1000/J_to_cal
sxcu = mwH2O*(sx + ds)*1000/J_to_cal
if FullEOSppt == True:
return rho, gxcu, hxcu, sxcu, vxcu, Pout, Tout, uxcu, axcu, cpxcu
else:
return rho, gxcu, hxcu, sxcu, vxcu, Pout, Tout
[docs] def calcwaterppt_Prho(self, P, rho, FullEOSppt = False):
"""
This function evaluates thermodynamic properties of water at given density and pressure.
The problem reduces to finding the value of temperature that is consistent with the desired pressure. \n
Parameters:
----------
P : pressure [bar] \n
rho : density [kg/m3] \n
FullEOSppt: Option to output all or essential water properties [False or True]
Returns:
----------
rho : density [kg/m3] \n
gx : Gibbs energy [cal/mol] \n
hx : Enthalpy [cal/mol] \n
sx : Entropy [cal/mol/K] \n
vx : Volume [m3/mol] \n
Pout : pressure [bar] \n
Tout : temperature [°C] \n
ux : Internal energy [cal/mol] if FullEOSppt is True \n
ax : Helmholtz energy [cal/mol/K] if FullEOSppt is True \n
cpx : Isobaric heat capacity [cal/mol/K] if FullEOSppt is True \n
Usage:
----------
[rho, gxcu, hxcu, sxcu, vxcu, uxcu, axcu, cpxcu, Pout, Tout] = calcwaterppt_Prho(P, rho), \n
"""
if np.ndim(rho) == 0:
rho = np.array(rho).ravel()
else:
rho = rho.ravel()
if np.ndim(P) == 0:
P = np.array(P).ravel()
else:
P = P.ravel()
Tc = IAPWS95_COEFFS['Tc'] # K
Pc = IAPWS95_COEFFS['Pc']*10 # bar
TK = np.zeros([len(P), 1]).ravel()
for i in range(len(P)):
if P[i] > Pc:
TK[i] = Tc
else:
TK[i] = self.calcsatpropP(P[i])[0]
funct_tsat = lambda TK: self.EOSIAPWS95(TK, rho[i])[0] - P[i]
TK[i] = fsolve(funct_tsat, TK[i])
Pout = P; Tout = convert_temperature( TK, Out_Unit = 'C' )
if FullEOSppt == True:
[Px, ax, sx, hx, gx, vx, _, _, _, _, _, _, ux, _, _, _, _, cpx, _, _] = self.vEOSIAPWS95(TK, rho, FullEOSppt = FullEOSppt)
else:
[Px, ax, sx, hx, gx, vx, _, _, _, _, _, _] = self.vEOSIAPWS95(TK, rho, FullEOSppt = FullEOSppt)
# The following reference state data are in kilogram units.
ds = 3.5156150 # Entropy (kJ/kg/K)
du = -15767.19391 # Internal energy (kJ/kg)
dh = -15970.89538 # Enthalpy (kJ/kg)
da = -11906.84446 # Helmholtz energy (kJ/kg-K)
dg = -12110.54592 # Gibbs energy (kJ/kg)
htripl = 0.000611782 # tripple point Enthalpy (kJ/kg)
mwH2O = 18.01528/1000 # kg/mol
# Calculate the entropy, internal energy, enthalpy, Helmholtz energy, and Gibbs energy
# on the standard thermochemical scale, and volume, heat capacity at constant volume.
# Results are converted from KJ and kilogram units to cal and molar units.
sxcu = sx + ds
if FullEOSppt == True:
uxcu = mwH2O*(ux + du)*1000/J_to_cal
axcu = mwH2O*(ux - TK*sxcu + da)*1000/J_to_cal
cpxcu = mwH2O*cpx*1000/J_to_cal
gxcu = mwH2O*(hx - TK*sxcu + dg)*1000/J_to_cal
hxcu = mwH2O*(hx - htripl + dh)*1000/J_to_cal
vxcu = mwH2O*vx*1000/J_to_cal
sxcu = mwH2O*(sx + ds)*1000/J_to_cal
if FullEOSppt == True:
return rho, gxcu, hxcu, sxcu, vxcu, Pout, Tout, uxcu, axcu, cpxcu
else:
return rho, gxcu, hxcu, sxcu, vxcu, Pout, Tout
[docs] def calcwaterstdppt(self, TK, hx, sx, vx, ux = None, cpx = None, Out_Unit = 'standard'):
"""
This function converts thermodynamic properties of water from kilogram units to standard thermochemical
scale (calorie units) and vice-versa. \n
Parameters:
----------
TK : temperature [K] \n
hx : Enthalpy [kJ/kg] or [cal/mol] \n
sx : Entropy [kJ/kg/K] or [cal/mol/K] \n
vx : Volume [m3/kg] or [m3/mol] \n
ux : Internal energy [kJ/kg] or [cal/mol] \n
cpx : Isobaric heat capacity [kJ/kg/K] or [cal/mol/K] \n
Returns:
----------
gx : Gibbs energy [kJ/kg] or [cal/mol] \n
hx : Enthalpy [kJ/kg] or [cal/mol] \n
sx : Entropy [kJ/kg/K] or[cal/mol/K] \n
vx : Volume [m3/kg] or[m3/mol] \n
ax : Helmholtz energy [kJ/kg/K] or[cal/mol/K] \n
ux : Internal energy [kJ/kg] or [cal/mol] \n
cpx : Isobaric heat capacity [kJ/kg/K] or[cal/mol/K] \n
Usage:
----------
[gx, hx, sx, vx, ux, ax, cpx] = calcwaterstdppt(TK, hx, sx, vx, ux, cpx, Out_Unit = 'kilogram') \n
"""
# The following reference state data are in kilogram units.
ds = 3.5156150 # Entropy (kJ/kg/K)
du = -15767.19391 # Internal energy (kJ/kg)
dh = -15970.89538 # Enthalpy (kJ/kg)
da = -11906.84446 # Helmholtz energy (kJ/kg-K)
dg = -12110.54592 # Gibbs energy (kJ/kg)
htripl = 0.000611782 # tripple point Enthalpy (kJ/kg)
mwH2O = 18.01528/1000 # kg/mol
#Accepted units for input and output are:
unit_markers = ['kilogram', 'standard']
if not (Out_Unit in unit_markers):
return None
elif Out_Unit == 'standard':
# Calculate the entropy, internal energy, enthalpy, Helmholtz energy, and Gibbs energy
# on the standard thermochemical scale, and volume, heat capacity at constant volume.
# Results are converted from KJ and kilogram units to cal and molar units.
sxcu = sx + ds
uxcu = mwH2O*(ux + du)*1000/J_to_cal if ux is not None else 0
hxcu = mwH2O*(hx - htripl + dh)*1000/J_to_cal
axcu = mwH2O*(ux - TK*sxcu + da)*1000/J_to_cal if ux is not None else 0
gxcu = mwH2O*(hx - TK*sxcu + dg)*1000/J_to_cal
vxcu = mwH2O*vx*1000/J_to_cal
cpxcu = mwH2O*cpx*1000/J_to_cal if cpx is not None else 0
sxcu = mwH2O*(sx + ds)*1000/J_to_cal
return gxcu, hxcu, sxcu, vxcu, axcu, uxcu, cpxcu
elif Out_Unit == 'kilogram':
uxcu = ux*J_to_cal/1000/mwH2O - du if ux is not None else 0
hxcu = hx*J_to_cal/1000/mwH2O - dh + htripl
vxcu = vx*J_to_cal/1000/mwH2O
cpxcu = cpx*J_to_cal/1000/mwH2O if cpx is not None else 0
sxcu = sx*J_to_cal/1000/mwH2O - ds
axcu = ux*J_to_cal/1000/mwH2O - da + TK*(sxcu + ds) if ux is not None else 0
gxcu = hx*J_to_cal/1000/mwH2O - dg + TK*(sxcu + ds)
return gxcu, hxcu, sxcu, vxcu, axcu, uxcu, cpxcu
[docs]class iapws95:
"""
Implementation of IAPWS Formulation 1995 for ordinary water substance, revised release of 2016
Notes
------
Temperature and Pressure input limits
* -22 ≤ TC ≤ 1000 and 0 ≤ P ≤ 100,000
Parameters
----------
T : float, vector
Temperature [°C] \n
P : float, vector
Pressure [bar] \n
rho : float, vector
Density [kg/m³] \n
rho0 : float, vector
Starting estimate of density [kg/m³] \n
rhom : float, vector
Molar density [kg/m³] \n
delta : float, vector
Reduced density, rho/rhoc \n
tau : float, vector
Reduced temperature, Tc/T \n
v : float, vector
Specific volume [m³/kg] \n
vm : float, vector
Specific molar volume [m³/mol] \n
Out_Unit : string
Expected units ('standard' or 'kilogram') \n
FullEOSppt : bool
Option to output all or essential water properties [False or True] \n
Returns
----------
The calculated instance has the following potential properties: \n
rho : float, vector
Density [kg/m3] \n
G : float, vector
Gibbs energy [cal/mol] or [kJ/kg] \n
H : float, vector
Enthalpy [cal/mol] or [kJ/kg] \n
S : float, vector
Entropy [cal/mol/K] or [kJ/kg/K] \n
V : float, vector
Volume [m3/mol] or [m3/kg] \n
P : float, vector
Pressure [bar] \n
TC : float, vector
Temperature [°C] \n
TK : float, vector
Temperature [K] \n
U : float, vector
Internal energy [cal/mol] or [kJ/kg] if FullEOSppt is True \n
F : float, vector
Helmholtz energy [cal/mol/K] or [kJ/kg-K] if FullEOSppt is True \n
Cp : float, vector
Isobaric heat capacity [cal/mol/K] \n
rhosl : float, vector
Density of liquid [kg/m3] \n
rhosv : float, vector
Density of vapor [kg/m3] \n
pdx : float, vector
Derivative of pressure with respect to delta in bar \n
adx : float, vector
Helmholtz energy derivative with respect to delta \n
ztx : float, vector
zeta value (needed to calculate viscosity) \n
ptx : float, vector
Derivative of pressure with respect to tau in bar \n
gdx : float, vector
Gibbs energy derivative [kJ/kg] if FullEOSppt is True \n
ktx : float, vector
Compressibility [/bar] \n
avx : float, vector
Thermal expansion coefficient (thermal expansivity) \n
mu : float, vector
viscosity [Pa-s] if FullEOSppt is True \n
bsx : float, vector
Isentropic temperature-pressure coefficient [K-m3/kJ] if FullEOSppt is True \n
dtx : float, vector
Isothermal throttling coefficient [kJ/kg/bar] if FullEOSppt is True \n
mux : float, vector
Joule-Thomsen coefficient [K-m3/kJ] if FullEOSppt is True \n
cvx : float, vector
Isochoric heat capacity [kJ/kg/K] if FullEOSppt is True \n
wx : float, vector
Speed of sound [m/s] if FullEOSppt is True
Usage:
----------
The general usage of iapws95 is as follows: \n
(1) For water properties at any Temperature and Pressure not on steam saturation curve: \n
water = iapws95(T = T, P = P), \n
where T is temperature in celsius and P is pressure in bar
(2) For water properties at any Temperature and Pressure on steam saturation curve: \n
water = iapws95(T = T, P = 'T'), \n
where T is temperature in celsius, followed with a quoted character 'T' to reflect steam saturation pressure \n
water = iapws95(T = 'P', P = P), \n
where P is pressure in bar, followed with a quoted character 'P' to reflect steam saturation temperature
(3) For water properties at any Temperature and density : \n
water = iapws95(T = T, rho = rho), \n
where T is temperature in celsius and rho is density in kg/m³
(4) For water properties at any Pressure and density : \n
water = iapws95(P = P, rho = rho), \n
where P is pressure in bar and rho is density in kg/m³
(5) For water saturation properties at any saturation Temperature : \n
water = iapws95(T = T), \n
where T is temperature in celsius
(6) For water saturation properties at any saturation Pressure : \n
water = iapws95(P = P), \n
where P is pressure in bar
Examples
--------
>>> water = iapws95(T = 200., P = 50, FullEOSppt = True)
>>> water.rho, water.G, water.H, water.S, water.V, water.P, water.T, water.mu
867.2595, -60368.41787, -65091.03895, 25.14869, 4.96478e-03, 50.00000,
200.000, 0.00013546
>>> water = iapws95(T=200, rho=996.5560, Out_Unit='kilogram', FullEOSppt=True)
>>> water.P, water.F, water.S, water.H, water.G, water.V, water.Cp, water.pdx
2872.063, -234.204, 2.051, 1024.747, 53.994, 1.0035e-03, 3.883, 10079.17
>>> water.adx, water.ztx, water.ptx, water.ktx, water.avx, water.mu, water.gdx
93.120, 2.189e-03, -7.348e+03, 3.205e-05, 6.809e-04, 1.914e-04, 1011.40
>>> water = iapws95(T = 350)
>>> water.P, water.rhosl, water.rhosv
165.2942 574.7065 113.6056
>>> water = iapws95(P = 150)
>>> water.TC, water.rhosl, water.rhosv
342.1553, 603.5179, 96.7271
References
----------
(1) Wagner, W., Pruß, A., 2002. The IAPWS formulation 1995 for the thermodynamic properties of
ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31, 387–535.
https://doi.org/10.1063/1.1461829
"""
kwargs = {"T": None,
"P": None,
"rho": None,
"rho0": None, 'FullEOSppt': False,
"v": None, "Out_Unit": 'standard'}
mwH2O = 18.015268 # g/mol
def __init__(self, **kwargs):
self.kwargs = iapws95.kwargs.copy()
self.__checker__(**kwargs)
self.__calc__(**kwargs)
[docs] def __checker__(self, **kwargs):
self.kwargs.update(kwargs)
"""initialization """
self.TC = self.kwargs["T"]
self.rho = self.kwargs["rho"]
self.rho0 = self.kwargs["rho0"]
self.P = self.kwargs["P"]
self.Out_Unit = self.kwargs["Out_Unit"]
self.FullEOSppt = self.kwargs["FullEOSppt"]
# Alternative rho input
if "rhom" in self.kwargs:
self.kwargs["rho"] = self.kwargs["rhom"]*self.mwH2O
elif "delta" in self.kwargs:
self.kwargs["rho"] = self.kwargs["delta"]*IAPWS95_COEFFS['rhoc']
elif self.kwargs.get("v", 0):
self.kwargs["rho"] = 1/self.kwargs["v"]
elif self.kwargs.get("vm", 0):
self.kwargs["rho"] = self.mwH2O/self.kwargs["vm"]
# Alternative T input
if "tau" in self.kwargs:
self.kwargs["T"] = IAPWS95_COEFFS['Tc']/self.kwargs["tau"]
if type(self.P) == str:
if self.P == 'T':
self.P = Dummy().vcalcsatpropT(convert_temperature( self.TC, Out_Unit = 'K' ))[0]
self.P[np.isnan(self.P) | (self.P < 1)] = 1.0133
if type(self.TC) == str:
if self.TC == 'P':
self.TC = Dummy().vcalcsatpropP(self.P)[0]
self.TC = convert_temperature( self.TC, Out_Unit = 'C' ) # convert from K to celcius
if self.rho0 is not None:
if np.size(self.rho0) < np.size(self.TC):
self.rho0 = (self.rho0*np.ones(len(self.TC))).ravel()
"""Check if inputs are enough to define state"""
if type(self.P) != str and type(self.TC) != str:
if np.ravel(self.TC).any() and np.ravel(self.P).any():
self.mode = "T_P"
if np.size(self.P) < np.size(self.TC):
if np.ndim(self.P) == 0:
self.P = np.ravel(self.P)
self.P = (self.P[0]*np.ones(len(self.TC))).ravel()
if np.size(self.TC) < np.size(self.P):
if np.ndim(self.TC) == 0:
self.TC = np.ravel(self.TC)
self.TC = (self.TC[0]*np.ones(len(self.P))).ravel()
elif np.ravel(self.TC).any() and np.ravel(self.rho).any():
self.mode = "T_rho"
if np.size(self.rho) < np.size(self.TC):
if np.ndim(self.rho) == 0:
self.rho = np.ravel(self.rho)
self.rho = (self.rho[0]*np.ones(len(self.TC))).ravel()
if np.size(self.TC) < np.size(self.rho):
if np.ndim(self.TC) == 0:
self.TC = np.ravel(self.TC)
self.TC = (self.TC[0]*np.ones(len(self.rho))).ravel()
elif np.ravel(self.P).any() and np.ravel(self.rho).any():
self.mode = "P_rho"
if np.size(self.P) < np.size(self.rho):
if np.ndim(self.P) == 0:
self.P = np.ravel(self.P)
self.P = (self.P[0]*np.ones(len(self.rho))).ravel()
if np.size(self.rho) < np.size(self.P):
if np.ndim(self.rho) == 0:
self.rho = np.ravel(self.rho)
self.rho = (self.rho[0]*np.ones(len(self.P))).ravel()
elif np.ravel(self.TC).any() and np.ravel(self.P).any() is None:
self.mode = "T_x"
elif np.ravel(self.P).any() and np.ravel(self.TC).any() is None:
self.mode = "P_x"
else:
self.mode = ""
else:
if self.TC and self.P:
self.mode = "T_P"
self.TK = convert_temperature( self.TC, Out_Unit = 'K' ) if self.TC is not None else None
[docs] def __calc__(self, **kwargs):
self.kwargs.update(kwargs)
self.msg = 'Temperature and Pressure input limits: -22 ≤ TC ≤ 1000 and 0 ≤ P ≤ 100,000'
if self.mode == "T_P":
water = Dummy().calcwaterppt(self.TC, self.P, self.rho0, FullEOSppt = self.FullEOSppt)
if self.FullEOSppt is True:
[self.rho, self.G, self.H, self.S, self.V, self.P, self.TC, self.U, self.F, self.Cp] = water
self.mu = Dummy().vwaterviscosity(self.TC, self.P, self.rho)
else:
[self.rho, self.G, self.H, self.S, self.V, self.P, self.TC] = water
if self.Out_Unit.lower() == 'kilogram':
if self.FullEOSppt is True:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, ux = self.U, cpx = self.Cp, Out_Unit = 'kilogram')
[self.G, self.H, self.S, self.V, self.F, self.U, self.Cp] = water
else:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, Out_Unit = 'kilogram')
[self.G, self.H, self.S, self.V] = water[:4]
elif self.mode == "T_rho":
water = Dummy().vEOSIAPWS95(self.TK, self.rho, FullEOSppt = self.FullEOSppt)
if self.FullEOSppt is True:
[self.P, self.F, self.S, self.H, self.G, self.V, self.pdx, self.adx, self.ztx, self.ptx, self.ktx, self.avx, self.U, self.gdx, self.bsx, self.dtx, self.mux, self.Cp, self.cvx, self.wx] = water
self.mu = Dummy().vwaterviscosity(self.TC, self.P, self.rho)
else:
[self.P, self.F, self.S, self.H, self.G, self.V, self.pdx, self.adx, self.ztx, self.ptx, self.ktx, self.avx] = water
if self.Out_Unit.lower() == 'standard':
if self.FullEOSppt is True:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, ux = self.U, cpx = self.Cp, Out_Unit = 'standard')
[self.G, self.H, self.S, self.V, self.F, self.U, self.Cp] = water
else:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, Out_Unit = 'standard')
[self.G, self.H, self.S, self.V] = water[:4]
elif self.mode == "P_rho":
water = Dummy().calcwaterppt_Prho(self.P, self.rho, FullEOSppt = self.FullEOSppt)
if self.FullEOSppt is True:
[self.rho, self.G, self.H, self.S, self.V, self.P, self.TC, self.U, self.F, self.Cp] = water
self.mu = Dummy().vwaterviscosity(self.TC, self.P, self.rho)
else:
[self.rho, self.G, self.H, self.S, self.V, self.P, self.TC] = water
if self.Out_Unit.lower() == 'kilogram':
if self.FullEOSppt is True:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, ux = self.U, cpx = self.Cp, Out_Unit = 'kilogram')
[self.G, self.H, self.S, self.V, self.F, self.U, self.Cp] = water
else:
water = Dummy().calcwaterstdppt(self.TK, self.H, self.S, self.V, Out_Unit = 'kilogram')
[self.G, self.H, self.S, self.V] = water[:4]
elif self.mode == "P_x":
water = Dummy().vcalcsatpropP(self.P)
[self.TK, self.rhosl, self.rhosv] = water
self.TC = convert_temperature( self.TK, Out_Unit = 'C' )
elif self.mode == "T_x":
water = Dummy().vcalcsatpropT(self.TK)
[self.P, self.rhosl, self.rhosv] = water
elif self.mode == "":
return ('For water properties at any Temperature and Pressure \n \
input T = temperature (celsius) and P = pressure (bar) \n \
For water properties at any Temperature and Pressure on steam saturation curve \n' + \
'input T = temperature (celsius) and P = "T" to reflect steam saturation pressure \n' + \
'For water properties at any Temperature and density \n' + \
'input T = temperature (celsius) and rho = density (kg/m³) in kg/m³' +\
'For water properties at any Pressure and density \n' + \
'input P = pressure (bar) and rho = density (kg/m³) \n' + \
'For water saturation properties at any saturation Temperature \n' + \
'input T = temperature (celsius) \n' + \
'For water saturation properties at any saturation Pressure \n' + \
'P = pressure (bar)')
[docs]class ZhangDuan():
"""
Implementation of Zhang & Duan model Formulation for water at higher Temperature and Pressure conditions, i.e, Deep Earth Water - DEW
Notes
------
Temperature and Pressure input limits
* 0 ≤ TC ≤ 1726.85 and 1000 ≤ P ≤ 300,000
Parameters
----------
T : float, vector
Temperature [°C] \n
P : float, vector
Pressure [bar] \n
rho : float, vector
Density [kg/m³] \n
rho0 : float, vector
Starting estimate of density [kg/m³] \n
densityEquation : string
specify either 'ZD05' to use Zhang & Duan (2005) or 'ZD09' to use Zhang & Duan (2009)
Returns
----------
The calculated instance has the following potential properties: \n
rho : float, vector
Density [kg/m3] \n
rhohat : float, vector
Density [g/cm³] \n
G : float, vector
Gibbs energy [cal/mol] \n
drhodP_T : float, vector
Partial derivative of density with respect to pressure at constant temperature \n
drhodT_P : float, vector
Partial derivative of density with respect to temperature at constant pressure
Usage:
----------
The general usage of ZhangDuan is as follows: \n
(1) For water properties at any Temperature and Pressure: \n
deepearth = ZhangDuan(T = T, P = P), \n
where T is temperature in celsius and P is pressure in bar
(2) For water properties at any Temperature and density : \n
deepearth = ZhangDuan(T = T, rho = rho), \n
where T is temperature in celsius and rho is density in kg/m³
Examples
--------
>>> deepearth = ZhangDuan(T = 25, P = 5000)
>>> deepearth.rho, deepearth.G, deepearth.drhodP_T, deepearth.drhodT_P
1145.3065, -54631.5351, 2.3283e-05, -0.0004889
>>> deepearth = ZhangDuan(T = 200, rho = 1100)
>>> deepearth.P, deepearth.G, deepearth.drhodP_T, deepearth.drhodT_P
7167.2231, -57319.0980, 2.3282e-05, -0.0005122
References
----------
(1) Zhang, Z., Duan, Z., 2005. Prediction of the PVT properties of water over wide range of
temperatures and pressures from molecular dynamics simulation. Phys. Earth Planet. Inter.
149, 335–354. https://doi.org/10.1016/j.pepi.2004.11.003.
(2) Zhang, C. and Duan, Z., 2009. “A model for C-O-H fluid in the Earth’s mantle”, Geochimica et
Cosmochimica Acta, vol. 73, no. 7, pp. 2089–2102, doi:10.1016/j.gca.2009.01.021.
(3) Sverjensky, D.A., Harrison, B., Azzolini, D., 2014. Water in the deep Earth: The dielectric
constant and the solubilities of quartz and corundum to 60kb and 1200°C. Geochim. Cosmochim.
Acta 129, 125–145. https://doi.org/10.1016/j.gca.2013.12.019
"""
kwargs = {"T": None, "P": None, "rho": None, "rho0": 1e-5,
"error": 1e-2, "densityEquation": 'ZD05'}
def __init__(self, **kwargs):
self.kwargs = ZhangDuan.kwargs.copy()
self.__calc__(**kwargs)
[docs] def __calc__(self, **kwargs):
self.kwargs.update(kwargs)
self.msg = 'Temperature and Pressure input limits: 0 ≤ TC ≤ 1726.85 and 1000 ≤ P ≤ 300,000'
self.TC = self.kwargs["T"]
self.P = self.kwargs["P"]
self.rho = self.kwargs["rho"]
self.rho0 = self.kwargs["rho0"]
self.method = self.kwargs["densityEquation"]
self.error = self.kwargs["error"]
if type(self.P) == str and np.ravel(self.TC).any():
self.mode = "T_P"
elif np.ravel(self.TC).any() and np.ravel(self.P).any():
self.mode = "T_P"
if np.size(self.P) < np.size(self.TC):
self.P = (self.P*np.ones_like(self.TC)).ravel()
if np.size(self.TC) < np.size(self.P):
self.TC = (self.TC*np.ones_like(self.P)).ravel()
elif np.ravel(self.TC).any() and np.ravel(self.rho).any():
self.mode = "T_rho"
if np.size(self.rho) < np.size(self.TC):
self.rho = (self.rho*np.ones_like(self.TC)).ravel()
if np.size(self.TC) < np.size(self.rho):
self.TC = (self.TC*np.ones_like(self.rho)).ravel()
else:
self.mode = ''
if self.mode == "T_P":
self.rhohat = self.ZD_Density(self.TC, self.P, method = self.method, error = self.error).ravel()
self.rho = self.rhohat*1000
elif self.mode == "T_rho":
self.rhohat = self.rho/1000
self.P = self.ZD_Pressure_drhodP(self.TC, self.rhohat, method = self.method)[0]
self.G = self.GibbsEnergy(self.TC, self.P, method = 'VolumeIntegral').ravel()
if type(self.P) != str:
f_rT = lambda x: self.ZD_Density(x, self.P, method = self.method, error = 1e-4)
self.drhodT_P = derivative(f_rT, self.TC, h = 0.0001).ravel()
f_rP = lambda x: self.ZD_Density(self.TC, x, method = self.method, error = 1e-4)
self.drhodP_T = derivative(f_rP, self.P, h = 0.0001).ravel()
else:
self.drhodT_P = 0; self.drhodP_T = 0
if self.mode not in ["T_P", "T_rho"]:
return ('For water properties at any Temperature and Pressure \n \
input T = temperature (celsius) and P = pressure (bar) \n' + \
'For water properties at any Temperature and density \n' + \
'input T = temperature (celsius) and rho = density (kg/m³) in kg/m³')
[docs] def ZD_Pressure_drhodP(self, TC, rhohat, method = None):
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
if np.ndim(rhohat) == 0:
rhohat = np.array(rhohat).ravel()
mwH2O = 18.01528 # 'Molar mass of water molecule in units of g/mol
TK = convert_temperature(TC, Out_Unit = 'K') # 'Temperature must be converted to Kelvin
P = np.zeros([len(TK), 1]); drhodP_T = np.zeros([len(TK), 1])
for k in range(len(TK)):
if method == 'ZD05':
ZD05_R = 83.14467 # 'Gas Constant in units of cm^3 bar/mol/K
ZD05_Vc = 55.9480373 # 'Critical volume in units of cm^3/mol
ZD05_Tc = 647.25 # 'Critical temperature in units of Kelvin
Vr = mwH2O / rhohat[k] / ZD05_Vc
Tr = TK[k] / ZD05_Tc
cc = ZD05_Vc / mwH2O # 'This term appears frequently in the equation and is defined here for convenience
B = 0.349824207 - 2.91046273 / (Tr**2) + 2.00914688 / (Tr**3)
C = 0.112819964 + 0.748997714 / (Tr**2) - 0.87320704 / (Tr**3)
D = 0.0170609505 - 0.0146355822 / (Tr**2) + 0.0579768283 / (Tr**3)
E = -0.000841246372 + 0.00495186474 / (Tr**2) - 0.00916248538 / (Tr**3)
F = -0.100358152 / Tr
G = -0.00182674744 * Tr
delta = 1 + B / Vr + C / (Vr**2) + D / Vr**4 + E / Vr**5 + (F / (Vr**2) + G / Vr**4) * np.exp(-0.0105999998 / Vr**2)
P[k] = ZD05_R * TK[k] * rhohat[k] * delta / mwH2O
kappa = B * cc + 2 * C * (cc**2) * rhohat[k] + 4 * D * cc**4 * rhohat[k]**3 + 5 * E * cc**5 * rhohat[k]**4 + \
(2 * F * (cc**2) * rhohat[k] + 4 * G * cc**4 * rhohat[k]**3 - (F / (Vr**2) + G / Vr**4) * (2 * 0.0105999998 * (cc**2) * rhohat[k])) * np.exp(-0.0105999998 / (Vr**2))
drhodP_T[k] = mwH2O / (ZD05_R * TK[k] * (delta + rhohat[k] * kappa))
elif method == 'ZD09':
ZD09_R = 0.083145 # 'Gas constant in units of dm^3 bar/mol/K
ZD09_epsilon = 510 # 'Lenard-Jones parameter in units of K
ZD09_omega = 2.88 # 'Lenard-Jones parameter in units of 1E-10 m
ZD09_c1 = ZD09_epsilon / (3.0626 * ZD09_omega**3)
# 'Prefactor calculated from 1000 * pow(ZD09_omega / 3.691, 3)
dm = pow(ZD09_omega / 3.691, 3)*1000*rhohat[k]
# 'Prefactor calculated from 0.001 * pow(3.691 / ZD09_omega, 3)
Vm = pow(3.691 / ZD09_omega, 3) * 0.001 * (mwH2O / rhohat[k])
# 'Prefactor calculated from 154 / ZD09_epsilon
Tm = 154 / ZD09_epsilon * TK[k]
B = 0.029517729893 - 6337.56452413 / (Tm**2) - 275265.428882 / (Tm**3)
C = 0.00129128089283 - 145.797416153 / (Tm**2) + 76593.8947237 / (Tm**3)
D = 2.58661493537E-06 + 0.52126532146 / (Tm**2) - 139.839523753 / (Tm**3)
E = -2.36335007175E-08 + 0.00535026383543 / (Tm**2) - 0.27110649951 / (Tm**3)
F = 25038.7836486 / (Tm**3)
G = 0.73226726041 + 0.015483335997 / (Vm**2)
delta = 1 + B / Vm + C / (Vm**2) + D / pow(Vm, 4) + E / pow(Vm, 5) + F / (Vm**2) * G * \
np.exp(-0.015483335997 / (Vm**2))
Pm = ZD09_R * Tm * delta / Vm
P[k] = Pm * ZD09_c1
kappa = B / mwH2O + 2 * C * dm / (mwH2O**2) + 4 * D * pow(dm, 3) / pow(mwH2O, 4) + \
5 * E * pow(dm, 4) / pow(mwH2O, 5) + (2 * F * dm / (mwH2O**2) * G + \
F / pow(Vm, 2) * (1 - G) * \
(2 * 0.015483335997 * dm / (mwH2O**2))) * \
np.exp(-0.015483335997 / (Vm**2))
drhodP_T[k] = ZD09_c1 * mwH2O / (ZD09_c1 * ZD09_R * Tm * (delta + dm * kappa))
return P, drhodP_T
[docs] def ZD_Density(self, TC, P, method = 'ZD05', error = None):
if type(P) == str:
if P == 'T':
# 'This equation models the density of water as a function of temperature along the Psat curve.
#'It has an R^2 value of 0.9999976885 as compared with Supcrt92 values.
rhohat = -1.01023381581205e-104 * pow(TC, 40) + -1.1368599785953e-27 * pow(TC, 10) + \
-2.11689207168779e-11 * pow(TC, 4) + 1.26878850169523e-08 * pow(TC, 3) + \
-4.92010672693621e-06 * pow(TC, 2) + -3.2666598612692e-05 * TC + 1.00046144613017
else:
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
if np.ndim(P) == 0:
P = np.array(P).ravel()
rhohat = np.zeros([len(TC), 1])
for k in range(len(TC)):
#'Define variables
minGuess = 0.00001
guess = self.rho0
equation = 1 if method == 'ZD05' else 2
maxGuess = 7.5 * equation - 5 #'The maxGuess is dependent on the value of "equation"
calcP = 0
#'Loop through and find the density
for i in range(100):
#'Calculates the pressure using the specified equation
calcP = self.ZD_Pressure_drhodP(TC[k], guess, method = method)[0]
#'If the calculated pressure is not equal to input pressure, this determines a new
#'guess for the density based on current guess and how the calculated pressure
#'relates to the input pressure. In effect, this a form of a bisection method.
# print('count ', i, 'rhohat', guess, 'P', calcP > P)
if abs(calcP - P[k]) > error:
if calcP > P[k]:
maxGuess = guess
guess = (guess + minGuess) / 2 #
else:
minGuess = guess
guess = (guess + maxGuess) / 2 #
else:
rhohat[k] = guess
break
return rhohat
[docs] def GibbsEnergy(self, TC, P, method = 'VolumeIntegral'):
if type(P) == str:
if P == 'T':
dGH2O = -2.72980941772081e-103 * pow(TC, 40) + 2.88918186300446e-25 * pow(TC, 10) +\
-2.21891314234246e-08 * pow(TC, 4) + 3.0912103873633e-05 * pow(TC, 3) +\
-3.20873264480928E-02 * pow(TC, 2) + -15.169458452209 * TC + -56289.0379433809
else:
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
if np.ndim(P) == 0:
P = np.array(P).ravel()
dGH2O = np.zeros([len(TC), 1])
for k in range(len(TC)):
if method == 'DH78': # 'Delaney & Helgeson (1978) equation
coeff = [-56130.073, 0.38101798, -0.0000021167697, 2.0266445e-11, -8.3225572e-17, -15.285559,
0.0001375239, -1.5586868e-09, 6.6329577e-15, -0.026092451, 0.000000035988857,
-2.7916588e-14, 0.000017140501, -1.6860893e-11, -6.0126987e-09]
dGH2O_ = 0
Count = 0
for j in range(4):
for k in range(4 - j):
dGH2O_ = dGH2O_ + coeff(Count) * pow(TC[k], j) * pow(P[k], k)
Count = Count + 1
dGH2O[k] = dGH2O_
elif method == 'VolumeIntegral':
mwH2O = 18.01528 # 'Molar mass of water molecule in units of g/mol
# 'Gibbs Free Energy of water at 1 kb. This equation is a polynomial fit to data as a function of temperature.
# 'It is valid in the range of 100 to 1000 C.
dGH2O_1Kb = 2.6880734e-09*TC[k]**4 + 6.3163061e-07*TC[k]**3 - 0.019372355*TC[k]**2 - 16.945093*TC[k] - 55769.287
if P[k] < 1000: # 'Simply return zero, this method only works at P >= 1000 bars
integral = np.nan
elif P[k] == 1000: # 'Return the value calculated above from the polynomial fit
integral = 0
elif P[k] > 1000: # 'Integrate from 1 kb to P over the volume
integral = 0
# 'Integral is sum of rectangles with this width. This function in effect limits the spacing
# 'to 20 bars so that very small pressures do not have unreasonably small widths. Otherwise the width
# 'is chosen such that there are always 500 steps in the numerical integration. This ensures that for very
# 'high pressures, there are not a huge number of steps calculated which is very computationally taxing.
spacing = 20 if ((P[k] - 1000) / 500 < 20) else (P[k] - 1000) / 500
for i in np.arange(1000, P[k], spacing):
# 'This integral determines the density only down to an error of 100 bars
# 'rather than the standard of 0.01. This is done to save computational
# 'time. Tests indicate this reduces the computation by about a half while
# 'introducing little error from the standard of 0.01.
integral = integral + (mwH2O / self.ZD_Density(TC[k], i, error = 100) / 41.84) * spacing
dGH2O[k] = dGH2O_1Kb + integral
return dGH2O
[docs]class water_dielec():
"""
Class Implementation of Water dielectric constants, the Debye-Huckel "A" and "B" parameters and their derivatives at ambient to deep-earth Temperature and Pressure conditions with three different formulations
Parameters
----------
T : float, vector
Temperature [°C] \n
P : float, vector
Pressure [bar] \n
rho : float, vector
Density [kg/m³] \n
Dielec_method : string
specify either 'FGL97' or 'JN91' or 'DEW' as the method to calculate dielectric constant (optional), if not specified, default - 'JN91' \n
Dielec_DEWoutrange : string
specify either 'FGL97' or 'JN91' as the method to calculate dielectric constant for out of range for 'DEW' method if any
Returns
----------
The calculated instance has the following potential properties:
E : float, vector
dielectric constant of water \n
rhohat : float, vector
density [g/cm³] \n
Ah : float, vector
Debye-Huckel "A" parameters [kg^1/2 mol^-1/2] \n
Bh : float, vector
Debye-Huckel "B" parameters [kg^1/2 mol^-1/2 Angstrom^-1] \n
bdot : float, vector
bdot at any given temperature T \n
Adhh : float, vector
Debye-Huckel "A" parameters associated with apparent molar enthalpy \n
Adhv : float, vector
Debye-Huckel "A" parameters associated with apparent molar volume \n
Bdhh : float, vector
Debye-Huckel "B" parameters associated with apparent molar enthalpy \n
Bdhv : float, vector
Debye-Huckel "B" parameters associated with apparent molar volume \n
dEdP_T : float, vector
Partial derivative of dielectric constant with respect to pressure at constant temperature \n
dEdT_P : float, vector
Partial derivative of dielectric constant with respect to temperature at constant pressure
Notes
------
FGL97 Temperature and Pressure input limits:
* -35 ≤ TC ≤ 600 and 0 ≤ P ≤ 12000
DEW Temperature and Pressure input limits:
* 100 ≤ TC ≤ 1200 and 1000 ≤ P ≤ 60000
JN91 Temperature and Pressure input limits:
* 0 ≤ TC ≤ 1000 and 0 ≤ P ≤ 5000
Usage
----------
The general usage of water_dielec is as follows: \n
(1) For water dielectric properties at any Temperature and Pressure: \n
dielect = water_dielec(T = T, P = P, Dielec_method = 'JN91'), \n
where T is temperature in celsius and P is pressure in bar
(2) For water dielectric properties at any Temperature and density : \n
dielect = water_dielec(T = T, rho = rho, Dielec_method = 'JN91'), \n
where T is temperature in celsius and rho is density in kg/m³
(3) For water dielectric properties at any Temperature and Pressure on steam saturation curve: \n
dielect = water_dielec(T = T, P = 'T', Dielec_method = 'JN91'), \n
where T is temperature in celsius and P is assigned a quoted character 'T' to reflect steam saturation pressure \n
dielect = water_dielec(P = P, T = 'P', Dielec_method = 'JN91'), \n
where P is pressure in bar and T is assigned a quoted character 'P' to reflect steam saturation temperature
Examples
--------
>>> dielect = water_dielec(T = 50, P = 500, Dielec_method = 'JN91')
>>> dielect.E, dielect.rhohat, dielect.Ah, dielect.Bh, dielect.bdot
71.547359, 1.00868586, 0.52131899, 0.33218072, 0.04088528
>>> dielect.Adhh, dielect.Adhv, dielect.Bdhh, dielect.Bdhv
0.64360153, 2.13119279, 15.6936832 , -14.52571678
>>> dielect.dEdP_T, dielect.dEdT_P
0.03293026, -0.32468033
>>> dielect = water_dielec(T = 200, rho = 1100, Dielec_method = 'FGL97')
>>> dielect.E, dielect.rhohat, dielect.Ah, dielect.Bh, dielect.bdot
49.73131404, 1.1, 0.5302338, 0.34384714, 0.04452579
>>> dielect.Adhh, dielect.Adhv, dielect.Bdhh, dielect.Bdhv
1.21317825, 2.21165281, 28.0047878, -34.21216547
>>> dielect.dEdP_T, dielect.dEdT_P
0.01444368, -0.16864644
>>> dielect = water_dielec(T = 250, P = 5000, Dielec_method = 'DEW')
>>> dielect.E, dielect.rhohat, dielect.Ah, dielect.Bh, dielect.bdot, dielect.Adhh
39.46273008, 1.0238784, 0.62248141, 0.35417088, 0.02878662, 0.80688122
>>> dielect.Adhv, dielect.Bdhh, dielect.Bdhv, dielect.dEdP_T, dielect.dEdT_P
3.13101408, 39.76402294, -35.29670957, 0.0129006 , -0.08837842
References
----------
(1) Release on the Static Dielectric Constant of Ordinary Water Substance for
Temperatures from 238 K to 873 K and Pressures up to 1000 MPa" (IAPWS R8-97, 1997).
(2) Fernandez D. P., Goodwin A. R. H., Lemmon E. W., Levelt Sengers J. M. H.,
and Williams R. C. (1997) A Formulation for the Permittivity of Water and
Steam at Temperatures from 238 K to 873 K at Pressures up to 1200 MPa, including
Derivatives and Debye-Hückel Coefficients. J. Phys. Chem. Ref. Data 26, 1125-1166.
(3) Helgeson H. C. and Kirkham D. H. (1974) Theoretical Prediction of the Thermodynamic
Behavior of Aqueous Electrolytes at High Pressures and Temperatures: II.
Debye-Huckel Parameters for Activity Coefficients and Relative Partial Molal Properties.
Am. J. Sci. 274, 1199-1251.
(4) Johnson JW, Norton D (1991) Critical phenomena in hydrothermal systems: State, thermodynamic,
electrostatic, and transport properties of H2O in the critical region. American Journal of Science 291:541-648
(5) D. A. Sverjensky, B. Harrison, and D. Azzolini, "Water in the deep Earth: the dielectric
constant and the solubilities of quartz and corundum to 60 kb and 1200 °C," Geochimica et
Cosmochimica Acta, vol. 129, pp. 125–145, 2014
"""
kwargs = {"T": None,
"P": None,
"rho": None,
"Dielec_method": None,
"Dielec_DEWoutrange": None}
def __init__(self, **kwargs):
self.kwargs = water_dielec.kwargs.copy()
self.__checker__(**kwargs)
self.__calc__(**kwargs)
[docs] def __checker__(self, **kwargs):
self.kwargs.update(kwargs)
self.TC = self.kwargs["T"]
self.P = self.kwargs["P"]
self.rho = self.kwargs['rho']
self.Dielec_method = 'JN91' if self.kwargs["Dielec_method"] is None else self.kwargs["Dielec_method"]
self.Dielec_DEWoutrange = 'JN91' if self.kwargs["Dielec_DEWoutrange"] is None else self.kwargs["Dielec_DEWoutrange"]
if (type(self.P) == str) or (type(self.TC) == str):
if self.P == 'T':
self.P = iapws95(T = self.TC).P
self.P[np.isnan(self.P) | (self.P < 1)] = 1.0133
elif self.TC == 'P':
self.TC = iapws95(P = self.P).TC
if self.rho is None:
self.rho = iapws95(T = self.TC, P = self.P).rho
self.TK = convert_temperature( self.TC, Out_Unit = 'K' ) if self.TC is not None else 0
if np.ndim(self.TC) == 0:
self.TC = np.array(self.TC).ravel()
else:
self.TC = self.TC.ravel()
if np.ndim(self.P) == 0:
self.P = np.array(self.P).ravel()
else:
self.P = self.P.ravel()
if np.ndim(self.rho) == 0:
self.rho = np.array(self.rho).ravel()
else:
self.rho = self.rho.ravel()
[docs] def __calc__(self, **kwargs):
if self.Dielec_method.upper() == 'FGL97':
self.msg = 'Temperature and Pressure input limits: -35 ≤ TC ≤ 600 and 0 ≤ P ≤ 12000 \n'
waterdielec = self.dielec_FGL97(self.TC, self.rho)
elif self.Dielec_method.upper() == 'JN91':
self.msg = 'Temperature and Pressure input limits: 0 ≤ TC ≤ 1000 and 0 ≤ P ≤ 5000 \n'
waterdielec = self.dielec_JN91(self.TC, self.rho)
elif self.Dielec_method.upper() == 'DEW':
self.msg = 'Temperature and Pressure input limits: 100 ≤ TC ≤ 1200 and 1000 ≤ P ≤ 60000 \n'
waterdielec = self.dielec_DEW()
self.E, self.rhohat, self.Ah, self.Bh, self.bdot, self.Adhh, self.Adhv, self.Bdhh, self.Bdhv, self.dEdP_T, self.dEdT_P = waterdielec
[docs] def dielec_FGL97(self, TC, rho):
"""
This function employs the FGL91 formulation to calculate the dielectric constant of water (E), the Debye-Huckel "A" parameters
and Debye-Huckel "B" parameters (3) and their derivatives as a
function of temperature and pressure \n
Notes
------
Temperature and Pressure input limits:
* -35 ≤ TC ≤ 600 and 0 ≤ P ≤ 12000
Parameters
----------
TC : temperature [°C] \n
rho : density [kg/m3]
Returns
----------
E : dielectric constant of water \n
rhohat : density [g/cm³] \n
Ah : Debye-Huckel "A" parameters [kg^1/2 mol^-1/2] \n
Bh : Debye-Huckel "B" parameters [kg^1/2 mol^-1/2 Angstrom^-1] \n
bdot : bdot at any given temperature T \n
Adhh : Debye-Huckel "A" parameters associated with apparent molar enthalpy \n
Adhv : Debye-Huckel "A" parameters associated with apparent molar volume \n
Bdhh : Debye-Huckel "B" parameters associated with apparent molar enthalpy \n
Bdhv : Debye-Huckel "B" parameters associated with apparent molar volume \n
dEdP_T : Partial derivative of dielectric constant with respect to pressure at constant temperature \n
dEdT_P : Partial derivative of dielectric constant with respect to temperature at constant pressure
Usage
----------
[E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P] = dielec_FGL97( TC, rho)
References
----------
(1) Release on the Static Dielectric Constant of Ordinary Water Substance for
Temperatures from 238 K to 873 K and Pressures up to 1000 MPa" (IAPWS R8-97, 1997).
(2) Fernandez D. P., Goodwin A. R. H., Lemmon E. W., Levelt Sengers J. M. H.,
and Williams R. C. (1997) A Formulation for the Permittivity of Water and
Steam at Temperatures from 238 K to 873 K at Pressures up to 1200 MPa, including
Derivatives and Debye-Hückel Coefficients. J. Phys. Chem. Ref. Data 26, 1125-1166.
(3) Helgeson H. C. and Kirkham D. H. (1974) Theoretical Prediction of the Thermodynamic
Behavior of Aqueous Electrolytes at High Pressures and Temperatures: II.
Debye-Huckel Parameters for Activity Coefficients and Relative Partial Molal Properties.
Am. J. Sci. 274, 1199-1251.
"""
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
else:
TC = TC.ravel()
if np.ndim(rho) == 0:
rho = np.array(rho).ravel()
else:
rho = rho.ravel()
TK = convert_temperature( TC, Out_Unit = 'K' )
water = iapws95(T = TC, rho = rho)
ptx, ktx, avx = water.ptx, water.ktx, water.avx
ptx = ptx*0.1 # convert to MPa units
ktx = ktx/0.1 # convert to /MPa units
Tc = IAPWS95_COEFFS['Tc'] # K
# Pc = IAPWS95_COEFFS['Pc'] # MPa
rhoc = IAPWS95_COEFFS['rhoc'] # kg/m3
mwH2O = 18.01528/1000 # kg/mol
R = IAPWS95_COEFFS['R']/1000*mwH2O # kJ mol-1 K-1
Nh = np.array([0.978224486826, -0.957771379375e0, 0.237511794148e0, 0.714692244396e0,
-0.298217036956, -0.108863472196e0, 0.949327488264e-01, -0.980469816509e-02,
0.165167634970e-04, 0.937359795772e-04, -0.123179218720e-09, 0.196096504426e-02])
ih = np.array([1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 10])
jh = np.array([0.25, 1.0, 2.5, 1.5, 1.5, 2.5, 2.0, 2.0, 5.0, 0.5, 10.0])
Nh = Nh.reshape(-1, 1); ih = ih.reshape(-1, 1); jh = jh.reshape(-1, 1)
alphammp = 1.636e-40 # Mean molecular polarizability C^2 J^-1 m^2
xmudp = 6.138e-30 # Molecular dipole moment C.m
e = 1.6021773310e-19 # elementary charge e is 1.602176634 x 10^-19 coulomb (C)
kB = 1.380658e-23 # Boltzmann constant k is 1.380649 x 10^-23 J K^-1
NA = 6.0221367e23 # Avogadro constant NA is 6.02214076 x 10^23 mol^-1
# expression for calculating the permittivity of free space, eps0.
c = 2997924580.0 # the speed of light in classical vacuum m/s
mu0 = 4.0e-07 * np.pi # the permeability of free space N Angstrom^-2
rhom = rho/mwH2O # molar density mol/m3
rhocrm = rhoc/mwH2O # critical molar density mol/m3
# Get the permittivity of free space (eps0). eps0 = 1/(mu0*c^2)
eps0 = 1/( mu0 * (c * 0.1)**2) # C^2 J^-1 m^-1
delta = rhom/rhocrm
tau = Tc/TK
# Get the Harris and Alder g factor (g).
g = 1 + np.sum(Nh[:-1]*(delta**ih)*(tau**jh), 0) + Nh[-1]*delta*((TK/228) - 1)**(-1.2)
# Get A and B.
A = ( NA*(xmudp**2)*rhom*g )/( eps0*kB*TK )
B = ( NA*alphammp*rhom )/( 3*eps0 )
# Get the dielectric constant (E).
E = ( 1 + A + 5*B + np.sqrt(9 + 2*A + 18*B + A**2 + 10*A*B + 9*B**2) )/( 4 - 4*B )
# Calculates the set of Debye-Huckel coefficients, derivatives and bdot
# at given temperature T and pressure P
# Debye-Huckel Agammae, Aphi and Agamma10 parameters, Units are kg^0.5 mol^-0.5
Agammae = np.sqrt(2*np.pi*NA*rho)*((e**2/( 4*np.pi*E*eps0*kB*TK ))**1.5)
Agamma10 = Agammae/np.log(10)
Ah = Agamma10
Aphi = Agammae/3
# Debye-Huckel B(gamma) Bh parameter, Units are kg^1/2 mol^-1/2 Angstrom^-1
kB_erg = kB * 1e7 # converts Boltzmann's constant from J K^-1 to erg K^-1.
rho = rho/1000 # converts rho from kg m^-3 to g cm^-3
rhohat = rho
Bh = 1.0e-08*np.sqrt((8*np.pi*NA*rho*((c*e)**2))/(1000*E*kB_erg*TK))
# Calculates bdot Ref: Helgeson H.C.,1969, American Journal of Science, Vol.267, pp:729-804
b = [0.0374e0, 1.3569e-4, 2.6411e-7, -4.6103e-9]
bdot = np.where(TC>=300, 0, b[0] + b[1]*(TC-25.0) + b[2]*(TC-25.0)**2 + b[3]*(TC-25.0)**3)
# Partial derivative of the g factor with respect to density at constant temperature.
dgdr_T = np.sum((Nh[:-1]*ih/rhocrm)*(delta**(ih - 1))*(tau**jh), 0) + \
(Nh[11]/rhocrm)*((TK/228) - 1)**(-1.2)
# Partial derivative of the g factor with respect to temperature at constant density.
dgdT_r = np.sum(Nh[:-1]*(delta**ih)*jh*(tau**(jh - 1))*(-tau/TK), 0) + \
Nh[11]*delta*(-1.2*((TK/228) - 1)**(-2.2) )/228
# Partial derivative of A with respect to density.
A1 = (A/rhom) + (A/g)*dgdr_T
# Partial derivative of A with respect to temperature.
A2 = -(A/TK) + (A/g)*dgdT_r
# Partial derivative of B with respect to density.
B1 = B/rhom
C = 9 + 2*A + 18*B + A**2 + 10*A*B + 9*B**2
# Partial derivative of E with respect to density at constant temperature.
dEdr_T = (4*B1*E/(4 - 4*B)) + (A1 + 5*B1 + 0.5*C**(-0.5)*\
(2*A1 + 18*B1 + 2*A*A1 + 10*(A1*B + A*B1) + 18*B*B1))/(4 - 4*B)
# Partial derivative of E with respect to temperature at constant density.
dEdT_r = (A2 + 0.5*C**(-0.5)*A2*(2 + 2*A + 10*B))/(4 - 4*B)
# Partial derivative of density with respect to Pressure at constant temperature using the compressibility (ktx)
drdP_T = rhom*ktx
# Partial derivative of E with respect to Pressure at constant temperature
dEdP_T = dEdr_T*drdP_T
# Partial derivative of Pressure with respect to temperature at constant density
# using "ptx" which is the partial derivative of Pressure with respect to tau
dPdT_r = -tau*ptx/TK
# Partial derivative of E with respect to temperature at constant Pressure
dEdT_P = dEdT_r - dEdr_T*dPdT_r*drdP_T
# Debye-Huckel AV constant. Units are cm^3 kg^1/2 mol^-3/2
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Adhv = 2*Aphi*1000*R*TK*( (3*dEdP_T/E) - ktx )
# Debye-Huckel AH constant. Units are kJ kg^1/2 mol^-3/2.
# AH/RT are kg^1/2 mol^-1/2
Ahrt = -6*Aphi*TK*((dEdT_P/E) + (1/TK) + (avx/3))
Adhh = R*TK*Ahrt
# convert Adhh from kJ kg^1/2 mol^-3/2 to kcal kg^1/2 mol^-3/2
Adhh = Adhh/J_to_cal
# Get the Debye-Huckel BV constant. Units are cm^3 kg^1/2 mol^-3/2 Angstrom^-1
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Bdhv = 2*np.log(10)*1000*R*TK*(Bh/2)*( ktx - dEdP_T/E )
# convert Bdhv from cm^3 kg^1/2 mol^-3/2 Angstrom^-1 to cm^2 kg^1/2 mol^-3/2 10^-6
Bdhv = 100*Bdhv
# Get the Debye-Huckel BH constant. Units are kJ kg^1/2 mol^-3/2 Angstrom^-1
# BH/RT parameter is expressed below as kg^1/2 mol^-1/2 Angstrom^-1.
Bhrt = -2*np.log(10)*TK*(Bh/2)*((dEdT_P/E) + (1/TK) + avx )
Bdhh = R*TK*Bhrt #
# convert Bdhh from kJ kg^1/2 mol^-3/2 Angstrom^-1 to cal kg^1/2 mol^-3/2 cm^-1 10^-9
Bdhh = Bdhh*1000/J_to_cal*0.10
# E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P, dEdT_r, dEdr_T, drdP_T
return E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P
[docs] def dielec_JN91(self, TC, rho):
"""
This dielec_JN91 implementation employs the JN91 formulation to calculate the
dielectric properties of water and steam, the Debye-Huckel "A" parameters and
Debye-Huckel "B" parameters and their derivatives
Notes
------
Temperature and Pressure input limits:
* 0 ≤ TC ≤ 1000 and 0 ≤ P ≤ 5000
Parameters
----------
TC : temperature [°C] \n
rho : density [kg/m3]
Returns
----------
E : dielectric constant of water \n
rhohat : density [g/cm³] \n
Ah : Debye-Huckel "A" parameters [kg^1/2 mol^-1/2] \n
Bh : Debye-Huckel "B" parameters [kg^1/2 mol^-1/2 Angstrom^-1] \n
bdot : bdot at any given temperature T \n
Adhh : Debye-Huckel "A" parameters associated with apparent molar enthalpy \n
Adhv : Debye-Huckel "A" parameters associated with apparent molar volume \n
Bdhh : Debye-Huckel "B" parameters associated with apparent molar enthalpy \n
Bdhv : Debye-Huckel "B" parameters associated with apparent molar volume \n
dEdP_T : Partial derivative of dielectric constant with respect to pressure at constant temperature \n
dEdT_P : Partial derivative of dielectric constant with respect to temperature at constant pressure
Usage
----------
[E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P] = dielec_JN91( TC, rho)
References
----------
(1) Johnson JW, Norton D (1991) Critical phenomena in hydrothermal systems: State, thermodynamic,
electrostatic, and transport properties of H2O in the critical region. American Journal of Science 291:541-648
(2) Helgeson H. C. and Kirkham D. H. (1974) Theoretical Prediction of the Thermodynamic
Behavior of Aqueous Electrolytes at High Pressures and Temperatures: II.
Debye-Huckel Parameters for Activity Coefficients and Relative Partial Molal Properties.
Am. J. Sci. 274, 1199-1251.
"""
if np.ndim(TC) == 0:
TC = np.array(TC).ravel()
else:
TC = TC.ravel()
if np.ndim(rho) == 0:
rho = np.array(rho).ravel()
else:
rho = rho.ravel()
TK = convert_temperature( TC, Out_Unit = 'K' )
Tr = 298.15 #K
mwH2O = 18.01528/1000 # kg/mol
R = IAPWS95_COEFFS['R']/1000* mwH2O # kJ mol-1 K-1
# Convert density to dimensionless g/cm3
rhohat = rho/1000 #g/cm^3
# Dielectric constant of water calculation. Johnson et al 1992- SUPCRT92
ae = [0.1470333593*100, 0.2128462733*1000, -0.1154445173*1000,
0.1955210915*100, -0.8330347980*100, 0.3213240048*100,
-0.6694098645*10, -0.3786202045*100, 0.6887359646*100,
-0.2729401652*100]
# Dielectric constants from UEMATSU and FRANCK
# ae = [7.62571, 2.44003E2, -1.40569E2, 2.77841E1, -9.62805E1,
# 4.17909E1, -1.02099E1, -4.52059E1, 8.46395E1, -3.58644]
That = TK/Tr
k = [0]*5
k[0] = [1]*len(That)
k[1] = ae[0]/That
k[2] = ae[1]/That + ae[2] + ae[3]*That
k[3] = ae[4]/That + ae[5]*That + ae[6]*That**2
k[4] = ae[7]*That**(-2) + ae[8]/That + ae[9]
# E = ((k[0]*rhohat**0)+(k[1]*rhohat**1)+(k[2]*rhohat**2)+(k[3]*rhohat**3)+(k[4]*rhohat**4))
E = np.zeros(np.size(TK))
for a in range(len(E)):
for b in range(5):
E[a] = E[a] + k[b][a]*rhohat[a]**b
Ar, Br = 1.8248292380e6, 50.291586490
Ah = Ar*np.sqrt(rhohat)/(E*TK*np.sqrt(E*TK))
Bh = Br*np.sqrt(rhohat)/np.sqrt(E*TK)
b = [0.0374e0, 1.3569e-4, 2.6411e-7, -4.6103e-9]
bdot = np.where(TC>=300, 0, b[0] + b[1]*(TC-25.0) + b[2]*(TC-25.0)**2 + b[3]*(TC-25.0)**3)
water = iapws95(T = TC, rho = rho)
ptx, ktx, avx = water.ptx, water.ktx, water.avx
ptx = ptx*0.1 # convert to MPa units
ktx = ktx/0.1 # convert to /MPa units
Aphi = Ah*np.log(10)/3
dkdT = [0]*5
dkdT[0] = [0]*len(That)
dkdT[1] = -ae[0]*Tr/TK**2
dkdT[2] = -ae[1]*Tr/TK**2 + ae[3]/Tr
dkdT[3] = -ae[4]*Tr/TK**2 + ae[5]/Tr + 2*ae[6]*TK/Tr**2
dkdT[4] = -2*ae[7]*Tr**2/TK**3 - ae[8]*Tr/TK**2
# Partial derivative of E with respect to Pressure at constant temperature
dEdP_T = np.zeros(np.size(TK))
for a in range(len(E)):
for b in range(5):
dEdP_T[a] = dEdP_T[a] + b*k[b][a]*rhohat[a]**b
dEdP_T = dEdP_T*ktx
# Partial derivative of E with respect to temperature at constant Pressure
dEdT_P = np.zeros(np.size(TK))
for a in range(len(E)):
for b in range(5):
dEdT_P[a] = dEdT_P[a] + rhohat[a]**b*(dkdT[b][a] - b*avx[a]*k[b][a])
# Debye-Huckel AV constant. Units are cm^3 kg^1/2 mol^-3/2
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Adhv = 2*Aphi*1000*R*TK*( (3*dEdP_T/E) - ktx )
# Debye-Huckel AH constant. Units are kJ kg^1/2 mol^-3/2.
# AH/RT are kg^1/2 mol^-1/2
Ahrt = -6*Aphi*TK*((dEdT_P/E) + (1/TK) + (avx/3))
Adhh = R*TK*Ahrt
# convert Adhh from kJ kg^1/2 mol^-3/2 to kcal kg^1/2 mol^-3/2
Adhh = Adhh/J_to_cal
# Get the Debye-Huckel BV constant. Units are cm^3 kg^1/2 mol^-3/2 Angstrom^-1
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Bdhv = 2*np.log(10)*1000*R*TK*(Bh/2)*( ktx - dEdP_T/E )
# convert Bdhv from cm^3 kg^1/2 mol^-3/2 Angstrom^-1 to cm^2 kg^1/2 mol^-3/2 10^-6
Bdhv = 100*Bdhv
# Get the Debye-Huckel BH constant. Units are kJ kg^1/2 mol^-3/2 Angstrom^-1
# BH/RT parameter is expressed below as kg^1/2 mol^-1/2 Angstrom^-1.
Bhrt = -2*np.log(10)*TK*(Bh/2)*((dEdT_P/E) + (1/TK) + avx )
Bdhh = R*TK*Bhrt #
# convert Bdhh from kJ kg^1/2 mol^-3/2 Angstrom^-1 to cal kg^1/2 mol^-3/2 cm^-1 10^-9
Bdhh = Bdhh*1000/J_to_cal*0.10
return E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P
[docs] def dielec_DEW(self):
"""
This watercalc implementation employs the DEW formulation embedded in Sverjensky et al. (2014)
to calculate the dielectric properties of water and steam, the Debye-Huckel "A" parameters and
Debye-Huckel "B" parameters and their derivatives. This function has been set up to use
either Johnson and Norton (1991) or Fernandez et al. (1997) formulation below 5000 bar and
Sverjensky et al. (2014) formulation above 5000 bar.
Notes
------
Temperature and Pressure input limits:
* 100 ≤ TC ≤ 1200 and 1000 ≤ P ≤ 60000
Parameters
----------
TC : temperature [°C] \n
P : pressure [bar]
Returns
----------
E : dielectric constant of water \n
rhohat : density [g/cm³] \n
Ah : Debye-Huckel "A" parameters [kg^1/2 mol^-1/2] \n
Bh : Debye-Huckel "B" parameters [kg^1/2 mol^-1/2 Angstrom^-1] \n
bdot : bdot at any given temperature T \n
Adhh : Debye-Huckel "A" parameters associated with apparent molar enthalpy \n
Adhv : Debye-Huckel "A" parameters associated with apparent molar volume \n
Bdhh : Debye-Huckel "B" parameters associated with apparent molar enthalpy \n
Bdhv : Debye-Huckel "B" parameters associated with apparent molar volume \n
dEdP_T : Partial derivative of dielectric constant with respect to pressure at constant temperature \n
dEdT_P : Partial derivative of dielectric constant with respect to temperature at constant pressure
Usage
----------
[E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P] = dielec_DEW()
References
----------
(1) D. A. Sverjensky, B. Harrison, and D. Azzolini, "Water in the deep Earth: the dielectric
constant and the solubilities of quartz and corundum to 60 kb and 1200 °C," Geochimica et
Cosmochimica Acta, vol. 129, pp. 125–145, 2014
(2) Helgeson H. C. and Kirkham D. H. (1974) Theoretical Prediction of the Thermodynamic
Behavior of Aqueous Electrolytes at High Pressures and Temperatures: II.
Debye-Huckel Parameters for Activity Coefficients and Relative Partial Molal Properties.
Am. J. Sci. 274, 1199-1251.
"""
TC = self.TC; P = self.P; Dielec_DEWoutrange = self.Dielec_DEWoutrange
TK = self.TK
mwH2O = 18.01528/1000 # kg/mol
R = IAPWS95_COEFFS['R']/1000* mwH2O # kJ mol-1 K-1
def rho_dielectric(TC, P, Dielec_method = 'JN91'):
deepearth = ZhangDuan(T = TC, P = P)
#[rho, dGH2O, drdP_T, drdT_P] =
drdT_P = deepearth.drhodT_P
drdP_T = deepearth.drhodP_T/0.1 # g cm^-3 MPa^-1
# Convert density to dimensionless g/cm3
# rhohat = deepearth.rho/1000 # g/cm^3
# Power Function - Created by Dimitri Sverjensky and Brandon Harrison
a = [-0.00157637700752506, 0.0681028783422197, 0.754875480393944]
b = [-8.01665106535394E-05, -0.0687161761831994, 4.74797272182151]
A = a[0] * TC + a[1] * np.sqrt(TC) + a[2]
B = b[0] * TC + b[1] * np.sqrt(TC) + b[2]
# Sverjensky et al. (2014) is used for P > 1000 bars and the JN91 or FGL97 Equation for
# P ≤ 1000 bar to calculate the Dielectric Constant.
E = np.zeros(len(TC)); dEdP_T = np.zeros(len(TC)); dEdT_P = np.zeros(len(TC));
for i, j in enumerate(TC):
if (P[i] < 1000) & (j < 100):
# deepearth_P = DEW(T = j, rho = deepearth.rho[i]).P.ravel()
if Dielec_method.upper() == 'FGL97':
[E[i], _, _, _, _, _, _, _, _, dEdP_T[i], dEdT_P[i]] = self.dielec_FGL97(j, deepearth.rho[i])
elif Dielec_method.upper() == 'JN91':
[E[i], _, _, _, _, _, _, _, _, dEdP_T[i], dEdT_P[i]] = self.dielec_JN91(j, deepearth.rho[i])
else:
E[i] = np.exp(B[i]) * deepearth.rhohat[i] ** A[i]
dEdr_T = A[i] * np.exp(B[i]) * deepearth.rhohat[i] ** (A[i] - 1)
# Partial derivative of E with respect to Pressure at constant temperature
dEdP_T[i] = dEdr_T*drdP_T[i] # unit is /MPa
return E, deepearth.rhohat, drdP_T, drdT_P, dEdP_T #.ravel()
E, rhohat, drdP_T, drdT_P, dEdP_T = rho_dielectric(TC, P, Dielec_method = Dielec_DEWoutrange)
Ar, Br = 1.8248292380e6, 50.291586490
Ah = Ar*np.sqrt(rhohat)/(E*TK*np.sqrt(E*TK))
Bh = Br*np.sqrt(rhohat)/np.sqrt(E*TK)
b = [0.0374e0, 1.3569e-4, 2.6411e-7, -4.6103e-9]
bdot = np.where(TC>=300, 0, b[0] + b[1]*(TC-25.0) + b[2]*(TC-25.0)**2 + b[3]*(TC-25.0)**3)
Aphi = Ah*np.log(10)/3
# Partial derivative of E with respect to temperature at constant Pressure
f_ET = lambda x : rho_dielectric(x, P)[0]
dEdT_P = derivative(f_ET, TC, h = 0.0001)
# Debye-Huckel AV constant. Units are cm^3 kg^1/2 mol^-3/2
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Adhv = 2*Aphi*1000*R*TK*( (3*dEdP_T/E) - drdP_T/rhohat )
# Debye-Huckel AH constant. Units are kJ kg^1/2 mol^-3/2.
# AH/RT are kg^1/2 mol^-1/2
Ahrt = -6*Aphi*TK*((dEdT_P/E) + (1/TK) - (drdT_P/rhohat/3))
Adhh = R*TK*Ahrt
# convert Adhh from kJ kg^1/2 mol^-3/2 to kcal kg^1/2 mol^-3/2
Adhh = Adhh/J_to_cal
# Get the Debye-Huckel BV constant. Units are cm^3 kg^1/2 mol^-3/2 Angstrom^-1
# Multiply RT in kJ mol^-1 by 1000 to get R in cm^3 MPa mol^-1
Bdhv = 2*np.log(10)*1000*R*TK*(Bh/2)*( drdP_T/rhohat - dEdP_T/E )
# convert Bdhv from cm^3 kg^1/2 mol^-3/2 Angstrom^-1 to cm^2 kg^1/2 mol^-3/2 10^-6
Bdhv = 100*Bdhv
# Get the Debye-Huckel BH constant. Units are kJ kg^1/2 mol^-3/2 Angstrom^-1
# BH/RT parameter is expressed below as kg^1/2 mol^-1/2 Angstrom^-1.
Bhrt = -2*np.log(10)*TK*(Bh/2)*((dEdT_P/E) + (1/TK) + (drdT_P/rhohat) )
Bdhh = R*TK*Bhrt #
# convert Bdhh from kJ kg^1/2 mol^-3/2 Angstrom^-1 to cal kg^1/2 mol^-3/2 cm^-1 10^-9
Bdhh = Bdhh*1000/J_to_cal*0.10
# E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P, dEdT_r, dEdr_T, drdP_T
return E, rhohat, Ah, Bh, bdot, Adhh, Adhv, Bdhh, Bdhv, dEdP_T, dEdT_P