def test_molar_volume(self):
# molar volume of water at [temperature(C), pressure(applied_bar), specific volume] Millero TABLE IV
param = np.array([[5, 0, 1.000036], [30, 0, 1.004369],
[75, 0, 1.025805], [35, 100, 1.001597],
[55, 300, 1.001642]])
# converting param to [temperature(K), pressure(atm), molar volume]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
# 1 atm = 1 applied_atm + 1, according to Millero
param[:, 1] = (param[:, 1] / un.atm_2_bar(1)) + 1
param[:,
2] = (param[:, 2] /
1e6) * fm.WaterPropertiesFineMillero(300).MolecularWeight
# testing molar volume up to a precision of 10^-11
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(wfm.molar_volume(), param[:, 2], 0, 1e-11)
self.assertTrue(test_vals)
def test_free_energy_dp_excess(self):
# parameters in [temperature (K), concentration of ions (M), ion size (A), x_pair, free energy excess]
param = np.array([[273.15, 1.0, 8, 0.4, -0.0036],
[298.15, 1.0, 8, 0.4, -0.0038],
[300.00, 1.0, 9, 0.6, -0.0064],
[298.15, 2.0, 7, 0.4, -0.0074],
[300.00, 4.0, 5, 0.5, -0.0086],
[300.00, 3.5, 5, 0.7, -0.0071]])
obj_water_bp = wbp.WaterPropertiesFineMillero(param[:, 0])
obj_water_aw = waw.WaterPropertiesFineMillero(param[:, 0])
obj_bj = bj.Bjerrum(obj_water_bp)
obj_bj_2 = bj.Bjerrum(obj_water_aw)
obj_dp = dp.Dipole(obj_bj)
obj_dp_2 = dp.Dipole(obj_bj_2)
# testing params up to a precision of 10^-4
test_1 = np.allclose(
obj_dp.free_energy_dp_excess(param[:, 1], param[:, 2],
param[:, 3]), param[:, 4], 0, 1e-4)
test_2 = np.allclose(
obj_dp_2.free_energy_dp_excess(param[:, 1], param[:, 2],
param[:, 3]), param[:, 4], 0, 1e-4)
self.assertTrue(test_1)
self.assertTrue(test_2)
def test_molarity_conversion(self):
# 1 molar NaCl = 1.021411855 molar NaCl at 25C
c = 1
water = wfm.WaterPropertiesFineMillero(298.15)
salt = nacl.NaClPropertiesRogersPitzer(298.15)
convert = con.molarity_2_molality(c, water.molar_volume(), salt.molar_vol(1.021411855), water.MolecularWeight)
# test precision up to 10^-6
test = np.allclose(convert, 1.021411855, 0, 1e-2)
self.assertTrue(test)
def test_pot_chem_dp_excess(self):
# parameters in [temperature (K), concentration of ions (M), ion size (A), x_pair, [mu_p, mu_p, mu_0]]
param = np.array([[
273.15, 1.0, 8, 0.4, -1.39225705e-05, -1.39225705e-05,
-5.82590279e-02
],
[
298.15, 1.0, 8, 0.4, -1.47196945e-05,
-1.47196945e-05, -6.08761754e-02
],
[
300.00, 1.0, 9, 0.6, -2.21908764e-05,
-2.21908764e-05, -4.87829304e-02
],
[
298.15, 2.0, 7, 0.4, -4.60404423e-05,
-4.60404423e-05, -8.97120601e-02
],
[
300.00, 4.0, 5, 0.5, -3.07170181e-05,
-3.07170181e-05, -1.14731930e-01
],
[
300.00, 3.5, 5, 0.7, -9.91702493e-06,
-9.91702493e-06, -7.65120554e-02
]])
# testing params [mu_p, mu_p, mu_0]
for i in range(len(param)):
for j in range(3):
obj_water_bp = wbp.WaterPropertiesFineMillero(param[i, 0])
obj_water_aw = waw.WaterPropertiesFineMillero(param[i, 0])
obj_bj = bj.Bjerrum(obj_water_bp)
obj_bj_2 = bj.Bjerrum(obj_water_aw)
obj_dp = dp.Dipole(obj_bj)
obj_dp_2 = dp.Dipole(obj_bj_2)
dp_method = obj_dp.pot_chem_dp_excess(param[i, 1], param[i, 2],
param[i, 3])
dp_method_2 = obj_dp_2.pot_chem_dp_excess(
param[i, 1], param[i, 2], param[i, 3])
# testing params [mu_p, mu_p] up to a precision of 10^-7
test_1 = np.allclose(dp_method[j], param[i, 4 + j], 0, 1e-7)
test_2 = np.allclose(dp_method_2[j], param[i, 4 + j], 0, 1e-7)
# testing params [mu_0] up to a precision of 10^-4
if j == 2:
test_1 = np.allclose(dp_method[j], param[i, 4 + j], 0,
1e-4)
test_2 = np.allclose(dp_method_2[j], param[i, 4 + j], 0,
1e-4)
self.assertTrue(test_1)
self.assertTrue(test_2)
def test_a_v(self):
# apparent molal volume [temperature(C), Pressure(bar), molal vol (cc/mol)] Bradley & Pitzer TABLE IV
param = np.array([[10, 100, 1.61], [60, 100, 2.55], [120, 100, 4.91],
[70, 400, 2.59], [25, 600, 1.66]])
# converting param to [temperature(K), pressure(atm), molal vol (cc/mol)]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
param[:, 1] = param[:, 1] / un.atm_2_bar(1)
# testing apparent molal volume up to a precision of 10^-2
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(wfm.a_v(), param[:, 2], 0, 1e-2)
self.assertTrue(test_vals)
def molar_vol(self, m):
"""
Molar volume of electrolyte solution according to Pitzer :cite:`Pitzer1973a`
.. math::
:label: molar_vol
\\upsilon_{\\phi}(m)=\\upsilon^{\\infty}_1+\\nu|z_{+}z_{-}|A_v h(I) \\cr +2RT(\\nu_{+}\\nu_{-})m\\left[
\\frac{\\partial \\beta^{(0)}_{\\pm}}{\\partial P}+2\\frac{\\partial \\beta^{(1)}_{\\pm}}{\\partial P}
g_{p1}(x)+2\\frac{\\partial \\beta^{(2)}_{\\pm}}{\\partial P}g_{p1}(x)\\right] \\cr
+2RT(\\nu_{+}\\nu_{-})^{\\frac{3}{2}}m^{2}\\left[\\frac{\\partial C^{(0)}_{\\pm}}{\\partial P}+2\\frac{
\\partial C^{(1)}_{\\pm}}{\\partial P}g_{p1}(x) + 2\\frac{\\partial C^{(2)}_{\\pm}}{\\partial P}
g_{p1}(x)\\right] \\cr +2RT(\\nu_{+}\\nu_{-})^2 m^{3}\\left[\\frac{\\partial D^{(0)}_{\\pm}}
{\\partial P} + 2\\frac{\\partial D^{(1)}_{\\pm}}{\\partial P}g_{p1}(x)) + 2\\frac{\\partial
D^{(2)}_{\\pm}}{\\partial P}g_{p1}(x)\\right]
functions are defined in Eq. :eq:`wang_function_1`
:return: Molar volume of electrolyte solution in SI (float)
"""
# the factor 10 is the conversion from J to bar cm^3
ct = 10 * un.r_gas() * self.tk
# stoichiometric_coefficients
nu, nu_prod, z_prod, nz_prod_plus = self.mat
# ionic strength
i_str = self.ionic_strength(m)
# Pitzer parameters pressure derivative and Vp
vp, beta0_der_p, beta1_der_p, beta2_der_p, C0_der_p, C1_der_p, C2_der_p, D0_der_p, D1_der_p, D2_der_p = self.params_der_p
BV = (beta0_der_p +
2 * beta1_der_p * self.g_fun_phi(self.alpha_b1, i_str**0.5) +
2 * beta2_der_p * self.g_fun_phi(self.alpha_b1, i_str**0.5))
CV = (C0_der_p + 2 * C1_der_p * self.g_fun_phi(self.alpha_c1, i_str) +
2 * C2_der_p * self.g_fun_phi(self.alpha_c2, i_str))
DV = (D0_der_p +
2 * D1_der_p * self.g_fun_phi(self.alpha_d1, i_str**1.5) +
2 * D2_der_p * self.g_fun_phi(self.alpha_d2, i_str**1.5))
val_1 = vp + nu * z_prod * wp.WaterPropertiesFineMillero(
self.tk, self.pa).a_v() * self.h_fun(i_str)
val_2 = 2 * nu_prod * ct * m * BV
val_3 = 2 * nu_prod**1.5 * ct * m**2 * CV
val_4 = 2 * nu_prod**2 * ct * m**3 * DV
# molar volume in cm^3/mol
val = val_1 + val_2 + val_3 + val_4
# return in m^3/mol
molar_vol = 1e-6 * val
return molar_vol
def test_a_phi(self):
# osmotic coefficient of water [temperature(C), Pressure(bar), osmotic coefficient] Bradley & Pitzer TABLE II
param = np.array([[10, 100, 3.80e-1], [60, 100, 4.17e-1],
[120, 100, 4.82e-1], [70, 400, 4.19e-1],
[25, 600, 3.81e-1]])
# converting param to [temperature(K), pressure(atm), osmotic coefficient]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
param[:, 1] = param[:, 1] / un.atm_2_bar(1)
# testing osmotic coefficient up to a precision of 10^-3
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(wfm.a_phi(), param[:, 2], 0, 1e-3)
self.assertTrue(test_vals)
def test_dielectric_constant(self):
# dielectric constant of water at [temperature(K), pressure(MPa), dielectric constant]
param = np.array([[338.15, 0.1, 65.20777], [318.15, 0.1, 71.50373],
[353.15, 0.1, 60.84250], [273.15, 1, 87.89296],
[278.15, 60, 88.17593]])
# converting param to [temperature(K), pressure(atm), dielectric constant]
param[:, 1] = param[:, 1] * 1e6 / un.atm_2_pascal(1)
# testing dielectric constant up to a precision of 10^-5
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(wfm.dielectric_constant(), param[:, 2], 0,
1e-5)
self.assertTrue(test_vals)
def molar_vol_infinite_dilution(self):
"""
Partial molal volume of solute at infinite dilution according to Pitzer :cite:`Pitzer1973a`
.. math::
:label: mol_infty_dilution
\\upsilon_1^{\\infty}=\\frac{\\upsilon(m_r)}{m_r}-Y \\upsilon_0^{\\infty}-\\nu|z_{+}z_{-}|A_v h(I_r)
-2\\nu_+\\nu_{-} RT\\left[ m_r B^{\\nu}_{\\pm}+\\nu_{+}z_{+}m^2_rC^{\\nu}_{\\pm}\\right]
:return: Partial molal volume of solute at infinite dilution in SI (float)
"""
# the factor 10 is the conversion from J to bar cm^3
ct = 10 * un.r_gas() * self.tk
# molar volume water in cm^3/mol
vol_water = 1e6 * wp.WaterPropertiesFineMillero(
self.tk, self.pa).molar_volume()
# stoichiometric_coefficients
nu, nu_prod, z_prod, nz_prod_plus = self.mat
m_r = self.p_ref[1]
y_r = self.p_ref[2]
vp, beta0_der_p, beta1_der_p, beta2_der_p, C0_der_p, C1_der_p, C2_der_p, D0_der_p, D1_der_p, D2_der_p = self.params_der_p
i_str = self.ionic_strength(m_r)
mv_i_0 = vp / m_r - y_r * vol_water
mv_i_1 = -nu * z_prod * wp.WaterPropertiesFineMillero(
self.tk, self.pa).a_v() * self.h_fun(i_str)
mv_i_2 = -2 * nu_prod * ct * (m_r * beta0_der_p +
nz_prod_plus * m_r**2 * C0_der_p)
# this is in cm^3/mol
mv_i = mv_i_0 + mv_i_1 + mv_i_2
# return in SI m^3
molar_vol_infinite_dilution = 1e-6 * mv_i
return molar_vol_infinite_dilution
def test_compressibility(self):
# compressibility of water at [temperature(C), pressure(applied_bar), compressibility (10^6 bar-1)] Millero TABLE V
param = np.array([[5, 0, 49.175], [30, 0, 44.771], [75, 0, 45.622],
[35, 100, 43.305], [55, 300, 40.911]])
# converting param to [temperature(K), pressure(atm), compressibility (atm-1)]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
# 1 atm = 1 applied_atm + 1, according to Millero
param[:, 1] = (param[:, 1] / un.atm_2_bar(1)) + 1
param[:, 2] = (param[:, 2] / 1e6) * un.atm_2_bar(1)
# testing compressibility up to a precision of 10^-9
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(wfm.compressibility(), param[:, 2], 0, 1e-9)
self.assertTrue(test_vals)
def test_density(self):
# density of water at [temperature(C), pressure(applied_bar), specific volume] Millero TABLEIV
param = np.array([[5, 0, 1.000036], [30, 0, 1.004369],
[75, 0, 1.025805], [30, 100, 0.999939],
[55, 300, 1.001642]])
# converting param to [temperature(K), pressure(atm), specific volume]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
# 1 atm = 1 applied_atm + 1, according to Millero
param[:, 1] = (param[:, 1] / un.atm_2_bar(1)) + 1
# testing density up to a precision of 10^-6
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(1e3 / wfm.density(), param[:, 2], 0, 1e-6)
self.assertTrue(test_vals)
def test_a_h(self):
# enthalpy coefficient [temperature(C), Pressure(bar), enthalpy coeff (AH / RT)] Bradley & Pitzer TABLE III
param = np.array([[10, 100, 0.641], [60, 100, 1.180], [120, 100, 2.05],
[70, 400, 1.24], [25, 600, 0.736]])
# converting param to [temperature(K), pressure(atm), enthalpy coeff (AH / RT)]
param[:, 0] = un.celsius_2_kelvin(param[:, 0])
param[:, 1] = param[:, 1] / un.atm_2_bar(1)
# testing enthalpy coeff up to a precision of 10^-2
wfm = fm.WaterPropertiesFineMillero(param[:, 0], param[:, 1])
test_vals = np.allclose(
((2 / 3) * wfm.a_h()) / (un.r_gas() * param[:, 0]), param[:, 2], 0,
1e-2)
self.assertTrue(test_vals, (wfm.a_h()) / (un.r_gas() * param[:, 0]))
def density_sol(self, m):
"""
Density of electrolyte solution according to Pitzer :cite:`Pitzer1973a`
:return: Density of electrolyte solution in SI (float)
"""
mw = wp.WaterPropertiesFineMillero(self.tk, self.pa).MolecularWeight
# convert to cm^3/mol
v_water = 1e6 * wp.WaterPropertiesFineMillero(self.tk,
self.pa).molar_volume()
# convert to cm^3/mol
v_salt = 1e6 * self.molar_vol(m)
mw_salt = self.p_ref[0]
# density in g/cm^3
dens = mw * (1 + 1e-3 * m * mw_salt) / (v_water +
1e-3 * m * mw * v_salt)
# return density in kg/m3
density_sol = 1e3 * dens
return density_sol
def apparent_molal_enthalpy(self, m):
"""
Apparent molal enthalpy according to Wang and Pitzer :cite:`Wang1998`
.. math::
L_{\\phi} = \\frac{\\nu|z_{+}z_{-}|A_{H}}{2b} \\ln(1+bI^{\\frac{1}{2}}) \\cr -2RT^{2}(\\nu_{+}
\\nu_{-})m \\left[\\frac{\\partial \\beta^{(0)}_{\\pm}}{\\partial T}+2\\frac{\\partial
\\beta^{(1)}_{\\pm}}{\\partial T}g_{p1}(x)+2\\frac{\\partial \\beta^{(2)}_{\\pm}}{\\partial T}
g_{p1}(x)\\right] \\cr -2RT^{2}(\\nu_{+}\\nu_{-})^{\\frac{3}{2}}m^{2}\\left[\\frac{\\partial
C^{(0)}_{\\pm}}{\\partial T}+2\\frac{\\partial C^{(1)}_{\\pm}}{\\partial T}g_{p1}(x)+2\\frac{\\partial
C^{(2)}_{\\pm}}{\\partial T}g_{p1}(x)\\right] \\cr -2RT^{2}(\\nu_{+}\\nu_{-})^{2}m^{3}\\left[
\\frac{\\partial D^{(0)}_{\\pm}}{\\partial T}+2\\frac{\\partial D^{(1)}_{\\pm}}{\\partial T}g_{p1}(x)
+2\\frac{\\partial D^{(2)}_{\\pm}}{\\partial T}g_{p1}(x)\\right]
functions are defined in Eq. :eq:`wang_function_1`
:return: apparent molal enthalpy in SI (float)
"""
# stoichiometric_coefficients
nu, nu_prod, z_prod, nz_prod_plus = self.mat
# ionic strength
i_str = self.ionic_strength(m)
# Pitzer Parameters temperature derivative
beta0_der_t, beta1_der_t, beta2_der_t, C0_der_t, C1_der_t, C2_der_t, D0_der_t, D1_der_t, D2_der_t = self.params_der_t
BL = (beta0_der_t +
2 * beta1_der_t * self.g_fun_phi(self.alpha_b1, i_str**0.5) +
2 * beta2_der_t * self.g_fun_phi(self.alpha_b1, i_str**0.5))
CL = (C0_der_t + 2 * C1_der_t * self.g_fun_phi(self.alpha_c1, i_str) +
2 * C2_der_t * self.g_fun_phi(self.alpha_c2, i_str))
DL = (D0_der_t +
2 * D1_der_t * self.g_fun_phi(self.alpha_d1, i_str**1.5) +
2 * D2_der_t * self.g_fun_phi(self.alpha_d2, i_str**1.5))
a_h = wp.WaterPropertiesFineMillero(self.tk, self.pa).a_h()
val_1 = nu * z_prod * (
a_h /
(2 * self.b_param)) * np.log(1 + self.b_param * np.sqrt(i_str))
val_2 = -2 * un.r_gas() * self.tk**2 * nu_prod * m * BL
val_3 = -2 * un.r_gas() * self.tk**2 * nu_prod**1.5 * m**2 * CL
val_4 = -2 * un.r_gas() * self.tk**2 * nu_prod**2 * m**3 * DL
l_phi = val_1 + val_2 + val_3 + val_4
return l_phi
def test_bjerrum_length(self):
# parameters to test (water objects at param (K) temperatures)
param = np.array([273.15, 298.15, 300, 310, 320])
obj_water_bp = wbp.WaterPropertiesFineMillero(param)
obj_water_aw = waw.WaterPropertiesFineMillero(param)
obj_bj = bj.Bjerrum(obj_water_bp)
obj_bj_2 = bj.Bjerrum(obj_water_aw)
# testing params up to a precision of 10^-2
bj_length_param = np.array(
[6.95955102, 7.15043775, 7.16686699, 7.26135158, 7.36525138])
test_1 = np.allclose(obj_bj.bjerrum_length, bj_length_param, 0, 1e-2)
test_2 = np.allclose(obj_bj_2.bjerrum_length, bj_length_param, 0, 1e-2)
self.assertTrue(test_1)
self.assertTrue(test_2)
def log_gamma(self, m):
"""
Activity coefficient according to Wang & Pitzer :cite:`Wang1998`
.. math::
:label: pitzer_activity
\\ln \\gamma_{\\pm} = -|z_{+}z_{-}|A_{\\gamma}f_{p2}(I) \\cr
+ \\frac{2(\\nu_{+}\\nu_{-})}{\\nu} m \\left[2\\beta^{(0)}_{\\pm} + 2\\beta^{(1)}_{\\pm}
f_{p3}(\\alpha_{B1}, I) + 2\\beta^{(2)}_{\\pm}f_{p3}(\\alpha_{B2}, I)\\right]\\cr
+ \\frac{2(\\nu_{+}\\nu_{-})^{\\frac{3}{2}}}{\\nu} m^{2}\\left[3C^{(0)}_{\\pm} + 2C^{(1)}_{\\pm}
f_{p4}(\\alpha_{C1}, I) + 2C^{(2)}_{\\pm}f_{p4}(\\alpha_{C2}, I)\\right] \\cr
+ \\frac{2(\\nu_{+}\\nu_{-})^{2}}{\\nu} m^{3}\\left[4D^{(0)}_{\\pm} + 2D^{(1)}_{\\pm}
f_{p5}(\\alpha_{D1}, I) + 2D^{(2)}_{\\pm}f_{p5}(\\alpha_{D2}, I)\\right]
functions are defined in Eq. :eq:`pitzer_function_2`, :eq:`pitzer_function_3`, :eq:`pitzer_function_4`, :eq:`pitzer_function_5`
:return: activity coefficient (float)
"""
# pressure is 1 atm
press = 1
# stoichiometric_coefficients
nu, nu_prod, z_prod, nz_prod_plus = self.mat
# ionic strength
i_str = self.ionic_strength(m)
# Pitzer Parameters
beta0, beta1, beta2, C0, C1, C2, D0, D1, D2 = self.params
BY = (2 * beta0 + 2 * beta1 * self.p_fun_gamma(self.alpha_b1, i_str) +
2 * beta2 * self.p_fun_gamma(self.alpha_b2, i_str))
CY = (3 * C0 + 2 * C1 * self.p_fun_gamma_2(self.alpha_c1, i_str) +
2 * C2 * self.p_fun_gamma_2(self.alpha_c2, i_str))
DY = (4 * D0 + 2 * D1 * self.p_fun_gamma_3(self.alpha_d1, i_str) +
2 * D2 * self.p_fun_gamma_3(self.alpha_d2, i_str))
val_1 = -z_prod * wp.WaterPropertiesFineMillero(
self.tk, press).a_phi() * self.h_fun_gamma(i_str)
val_2 = 2 * m * (nu_prod / nu) * BY
val_3 = (2 * nu_prod**1.5 / nu) * m**2 * CY
val_4 = (2 * nu_prod**2 / nu) * m**3 * DY
log_gamma = val_1 + val_2 + val_3 + val_4
return log_gamma
def test_temp_star(self):
# parameters in [temperature (K), radius (A), reduced temperature]
param = np.array([[273.15, 2, 0.28737486], [298.15, 2, 0.27970316],
[300, 2, 0.27906197], [298.15, 5, 0.69925789],
[300, 0.5, 0.06976549], [300, 1, 0.13953098]])
obj_water_bp = wbp.WaterPropertiesFineMillero(param[:, 0])
obj_water_aw = waw.WaterPropertiesFineMillero(param[:, 0])
obj_bj = bj.Bjerrum(obj_water_bp)
obj_bj_2 = bj.Bjerrum(obj_water_aw)
# testing params up to a precision of 10^-3
test_1 = np.allclose(obj_bj.temp_star(param[:, 1]), param[:, 2], 0,
1e-3)
test_2 = np.allclose(obj_bj_2.temp_star(param[:, 1]), param[:, 2], 0,
1e-3)
self.assertTrue(test_1)
self.assertTrue(test_2)
def osmotic_coeff(self, m):
"""
Osmotic coefficient according to Wang & Pitzer :cite:`Wang1998`
.. math::
\\phi = 1 - |z_{+}z_{-}|A_{\\phi}\\frac{I^{\\frac{1}{2}}}{\\left(1 + bI^{\\frac{1}{2}}\\right)}\\cr
+\\frac{2\\left(\\nu_{+}\\nu_{-}\\right)}{\\nu}m\\left[\\beta_{\\pm}^{(0)}+\\beta_{\\pm}^{(1)}e^{
-\\alpha_{B1}I^{\\frac{1}{2}}}+ \\beta_{\\pm}^{(2)}e^{-\\alpha_{B2}I^{\\frac{1}{2}}}\\right] \\cr
+\\frac{2\\left(\\nu_{+}\\nu_{-}\\right)^{\\frac{3}{2}}}{\\nu}m^{2}\\left[2\\left[C_{\\pm}^{(0)}+
C_{\\pm}^{(1)}e^{-\\alpha_{C1}I}+ C_{\\pm}^{(2)}e^{-\\alpha_{C2}I}\\right]\\right] \\cr
+\\frac{2\\left(\\nu_{+}\\nu_{-}\\right)^{2}}{\\nu}m^{3}\\left[3\\left[D_{\\pm}^{(0)}+D_{\\pm}^{(1)}e^{
-\\alpha_{D1}I^{\\frac{3}{2}}}+ D_{\\pm}^{(2)}e^{-\\alpha_{D2}I^{\\frac{3}{2}}}\\right]\\right] \\cr
:return: osmotic coefficient in SI (float)
"""
# pressure is 1 atm
press = 1
# stoichiometric_coefficients
nu, nu_prod, z_prod, nz_prod_plus = self.mat
# ionic strength
i_str = self.ionic_strength(m)
# Pitzer Parameters
beta0, beta1, beta2, C0, C1, C2, D0, D1, D2 = self.params
B = (beta0 + beta1 * np.exp(-self.alpha_b1 * i_str**0.5) +
beta2 * np.exp(-self.alpha_b2 * i_str**0.5))
C = (2 * (C0 + C1 * np.exp(-self.alpha_c1 * i_str) +
C2 * np.exp(-self.alpha_c2 * i_str)))
D = (3 * (D0 + D1 * np.exp(-self.alpha_d1 * i_str**1.5) +
D2 * np.exp(-self.alpha_d2 * i_str**1.5)))
val_1 = -z_prod * wp.WaterPropertiesFineMillero(
self.tk, press).a_phi() * np.sqrt(i_str) / (
1 + self.b_param * np.sqrt(i_str))
val_2 = 2 * m * (nu_prod / nu) * B
val_3 = (2 * nu_prod**1.5 / nu) * m**2 * C
val_4 = (2 * nu_prod**2 / nu) * m**3 * D
osmotic_coefficient = 1 + val_1 + val_2 + val_3 + val_4
return osmotic_coefficient
def test_bjerrum_constant(self):
# parameters in [temperature (K), radius (A), Bjerrum constant]
param = np.array([[273.15, 2, 177.15962555], [298.15, 2, 199.08793975],
[300, 2, 201.07303174], [298.15, 5, 4.48179914],
[300, 1, 5697.97522674]])
obj_water_bp = wbp.WaterPropertiesFineMillero(param[:, 0])
obj_water_aw = waw.WaterPropertiesFineMillero(param[:, 0])
obj_bj = bj.Bjerrum(obj_water_bp)
obj_bj_2 = bj.Bjerrum(obj_water_aw)
# testing params up to a precision of 10
test_1 = np.allclose(obj_bj.bjerrum_constant(param[:, 1]), param[:, 2],
0, 1e1)
test_2 = np.allclose(obj_bj_2.bjerrum_constant(param[:, 1]),
param[:, 2], 0, 1e1)
self.assertTrue(test_1)
self.assertTrue(test_2)
def test_bjerrum_constant_approx(self):
# parameters in [temperature (K), radius (A), Bjerrum constant (approx)]
param = np.array([[273.15, 2, 175.6241625002495],
[298.15, 2, 197.37695820415016],
[300, 2, 199.34627081662805],
[298.15, 5, 4.437998539403452],
[300, 1, 5670.345869397299]])
# testing params up to a precision of 10^2
for i in range(len(param)):
obj_water_bp = wbp.WaterPropertiesFineMillero(param[i, 0])
obj_water_aw = waw.WaterPropertiesFineMillero(param[i, 0])
obj_bj = bj.Bjerrum(obj_water_bp).bjerrum_constant_approx(param[i,
1])
obj_bj_2 = bj.Bjerrum(obj_water_aw).bjerrum_constant_approx(
param[i, 1])
test_1 = np.allclose(obj_bj[0], param[i, 2], 0, 1e2)
test_2 = np.allclose(obj_bj_2[0], param[i, 2], 0, 1e2)
self.assertTrue(test_1)
self.assertTrue(test_2)
def __init__(self, tk, pa=1):
"""
constructor
:param tk: temperature in kelvin
:param pa: pressure in atmospheres
:instantiate: temperature, pressure, stoichiometry coefficients, Pitzer Parameters
"""
self.tk = tk
self.pa = pa
# Calculations nacl parameters and coefficients
self.mat_stoich = np.array([[1, 1], [1, -1]])
self.m_ref = 5.550825
self.y_ref = 10
self.m_weight = 58.4428
self.p_ref = np.array([self.m_weight, self.m_ref, self.y_ref])
# values for ion strength dependence and ion size constants in the extended Pitzer model
self.alpha_b1 = 2.0
self.alpha_b2 = 0
self.alpha_c1 = 0
self.alpha_c2 = 0
self.alpha_d1 = 0
self.alpha_d2 = 0
self.b_param = 1.2
self.ion_param = np.array([
self.alpha_b1, self.alpha_b2, self.alpha_c1, self.alpha_c2,
self.alpha_d1, self.alpha_d2, self.b_param
])
self.cm = np.zeros(28)
self.cm[0] = 1.0249125e3
self.cm[1] = 2.7796679e-1
self.cm[2] = -3.0203919e-4
self.cm[3] = 1.4977178e-6
self.cm[4] = -7.2002329e-2
self.cm[5] = 3.1453130e-4
self.cm[6] = -5.9795994e-7
self.cm[7] = -6.6596010e-6
self.cm[8] = 3.0407621e-8
self.cm[9] = 5.3699517e-5
self.cm[10] = 2.2020163e-3
self.cm[11] = -2.6538013e-7
self.cm[12] = 8.6255554e-10
self.cm[13] = -2.6829310e-2
self.cm[14] = -1.1173488e-7
self.cm[15] = -2.6249802e-7
self.cm[16] = 3.4926500e-10
self.cm[17] = -8.3571924e-13
# exponent in 18 (19 in reference) is barely visible, it is 5 (confirmed)
self.cm[18] = 3.0669940e-5
self.cm[19] = 1.9767979e-11
self.cm[20] = -1.9144105e-10
self.cm[21] = 3.1387857e-14
self.cm[22] = -9.6461948e-9
self.cm[23] = 2.2902837e-5
self.cm[24] = -4.3314252e-4
self.cm[25] = -9.0550901e-8
self.cm[26] = 8.6926600e-11
self.cm[27] = 5.1904777e-4
self.qm = np.zeros(19)
self.qm[0] = 0.0765
self.qm[1] = -777.03
self.qm[2] = -4.4706
self.qm[3] = 0.008946
self.qm[4] = -3.3158e-6
self.qm[5] = 0.2664
# this value is not provided
self.qm[6] = 0
# this value is not provided
self.qm[7] = 0
self.qm[8] = 6.1608e-5
self.qm[9] = 1.0715e-6
self.qm[10] = 0.00127
self.qm[11] = 33.317
self.qm[12] = 0.09421
self.qm[13] = -4.655e-5
# this value is not provided
self.qm[14] = 0
self.qm[15] = 41587.11
self.qm[16] = -315.90
self.qm[17] = 0.8514
self.qm[18] = -8.3637e-4
# Pitzer Parameters
self.tc = 298.15
self.beta0_1 = self.qm[0] + self.qm[1] * (
1 / self.tk - 1 / self.tc) + self.qm[2] * np.log(self.tk / self.tc)
self.beta0_2 = self.qm[3] * (self.tk - self.tc) + self.qm[4] * (
self.tk**2 - self.tc**2)
self.beta0 = self.beta0_1 + self.beta0_2
self.beta1 = self.qm[5] + self.qm[8] * (
self.tk - self.tc) + self.qm[9] * (self.tk**2 - self.tc**2)
self.beta2 = 0
self.c_phi_1 = self.qm[10] + self.qm[11] * (
1 / self.tk - 1 / self.tc) + self.qm[12] * np.log(
self.tk / self.tc)
self.c_phi = self.c_phi_1 + self.qm[13] * (self.tk - self.tc)
self.C0 = self.c_phi / 2
self.C1 = 0
self.C2 = 0
self.D0 = 0
self.D1 = 0
self.D2 = 0
self.params = np.array([
self.beta0, self.beta1, self.beta2, self.C0, self.C1, self.C2,
self.D0, self.D1, self.D2
])
# Pitzer Parameters pressure derivative
self.pr = un.atm_2_bar(self.pa)
self.pr_atm = un.atm_2_bar(1)
self.vp_0 = self.cm[0] + self.cm[1] * self.tk + self.cm[
2] * self.tk**2 + self.cm[3] * self.tk**3
self.vp_1 = (self.pr - self.pr_atm) * (
self.cm[4] + self.cm[5] * self.tk + self.cm[6] * self.tk**2)
self.vp_2 = (self.pr - self.pr_atm)**2 * (self.cm[7] +
self.cm[8] * self.tk)
self.vp_sum = self.vp_0 + self.vp_1 + self.vp_2
self.bp_0 = self.cm[9] + self.cm[10] / (self.tk - 227) + self.cm[
11] * self.tk + self.cm[12] * self.tk**2 + self.cm[13] / (680 -
self.tk)
self.bp_1_1 = self.cm[14] + self.cm[15] / (
self.tk - 227) + self.cm[16] * self.tk + self.cm[17] * self.tk**2
self.bp_1 = (self.bp_1_1 + self.cm[18] /
(680 - self.tk)) * (self.pr - self.pr_atm)
self.bp_2_1 = self.cm[19] + self.cm[20] / (
self.tk - 227) + self.cm[21] * self.tk + self.cm[22] / (680 -
self.tk)
self.bp_2 = self.bp_2_1 * (self.pr - self.pr_atm)**2
self.beta0_der_p = self.bp_0 + self.bp_1 + self.bp_2
self.beta1_der_p = 0
self.beta2_der_p = 0
self.cq = self.cm[23] + self.cm[24] / (self.tk - 227) + self.cm[
25] * self.tk + self.cm[26] * self.tk**2 + self.cm[27] / (680 -
self.tk)
self.cp = 0.5 * self.cq
self.C0_der_p = self.cp
self.C1_der_p = 0
self.C2_der_p = 0
self.D0_der_p = 0
self.D1_der_p = 0
self.D2_der_p = 0
# calculates the molar volume at infinite dilution (vp)
# the factor 10 is the conversion from J to bar cm^3
self.ct = 10 * un.r_gas() * self.tk
# molar volume water in cm^3/mol
self.vol_water = 1e6 * bp.WaterPropertiesFineMillero(
self.tk, self.pa).molar_volume()
self.mv_i_0 = self.vp_sum / self.m_ref - self.y_ref * self.vol_water
self.mv_i_1 = -2 * bp.WaterPropertiesFineMillero(self.tk, self.pa).a_v(
) * (0.5 * np.log(1 + self.b_param * np.sqrt(self.m_ref)) /
self.b_param)
self.mv_i_2 = -2 * self.ct * (self.m_ref * self.beta0_der_p +
self.m_ref**2 * self.C0_der_p)
# this is in cm^3/mol
self.mv_i = self.mv_i_0 + self.mv_i_1 + self.mv_i_2
# return in SI m^3
self.molar_vol_infinite_dilution = 1e-6 * self.mv_i
# infinite molar volume, convert to cm^3/mol
self.vp = 1e6 * self.molar_vol_infinite_dilution
self.params_der_p = np.array([
self.vp, self.beta0_der_p, self.beta1_der_p, self.beta2_der_p,
self.C0_der_p, self.C1_der_p, self.C2_der_p, self.D0_der_p,
self.D1_der_p, self.D2_der_p
])
# Pitzer Parameters temperature derivative
self.beta0_der_t = 2 * self.qm[4] * self.tk + self.qm[
2] / self.tk - self.qm[1] / (self.tk**2) + self.qm[3]
self.beta1_der_t = 2 * self.qm[9] * self.tk + self.qm[8]
self.beta2_der_t = 0
self.c_phi_der_t = self.qm[12] / self.tk - self.qm[11] / (
self.tk**2) + self.qm[13]
self.C0_der_t = self.c_phi_der_t / 2
self.C1_der_t = 0
self.C2_der_t = 0
self.D0_der_t = 0
self.D1_der_t = 0
self.D2_der_t = 0
self.params_der_t = np.array([
self.beta0_der_t, self.beta1_der_t, self.beta2_der_t,
self.C0_der_t, self.C1_der_t, self.C2_der_t, self.D0_der_t,
self.D1_der_t, self.D2_der_t
])
super().__init__(tk, pa)
def __init__(
self, param_poly, param_salt, temp, df_w, x_ini, p_ini, n_k, chi_p, chi_e,
param_s):
"""
The constructor, with the following parameters
:param param_poly: polymer parameters :math:`(\\phi_p, \\frac{\\upsilon_w} \
{\\upsilon_p}, \\Delta F_p)`
:param param_salt: salt parameters (see definition below)
:param temp: temprature in Kelvin
:param df_w: free energy change upon formation of hydrogen bond \
in water (in :math:`k_BT` units)
:param x_ini: fraction of polymer hydrogen bonds
:param p_ini: fraction of water hydrogen bonds
:param n_k: number of Kuhn lengths for the polymer
:param chi_p: Flory Huggins parameter between water and polymer
:param chi_e: Flory Huggins parameter between polymer and salt
:param param_s: microscopic salt parameters \
:math:`(h_+, h_-, d_+, d_-, m_+, m_-, \\nu_+, \\nu_-, log(gamma), osm)` \
(number of water molecules \
forming the hydration shell, diameter, maximum number of water \
molecules that maybe bound to each ion, the number of ions per salt,\
activity coefficient, osmotic coefficient)
the parameter param_salt is given by
:math:`(m_s, \\frac{\\upsilon_w}{\\upsilon_+}, \
\\frac{\\upsilon_w}{\\upsilon_-}, \\Delta F_a, \\Delta F_b)`
where :math:`m_s` is the concentration in mol/kg
"""
#concentration of salt (NaCl)
# concentration in mol/kg
self.conc_l = param_salt[0]
# concentration in mol/L
self.T = temp
#salt_nacl = nacl.NaClPropertiesRogersPitzer(self.T)
#param_saltolvent = obj_water_bp.molar_volume()
#m_solvent = obj_water_bp.MolecularWeight
#param_saltolute = salt_nacl.molar_vol(self.conc_l)
#self.conc = con.molality_2_molarity(self.conc_l,
#param_saltolvent,
#param_saltolute,
#m_solvent)
obj_water_bp = wbp.WaterPropertiesFineMillero(self.T)
# molecular volumes
self.u_p = param_poly[1]
self.u_a = param_salt[1]
self.u_b = param_salt[2]
self.u_s = 1 / (1 / self.u_a + 1 /self.u_b)
v_w = obj_water_bp.molar_volume() # molar volume of water
den = obj_water_bp.density() # density of water in SI unit from Ref[4]
self.D_w = 1 / den / v_w #55.509 # mol/kg water
# volume fractions
self.phi_p = param_poly[0]
self.param_salt = self.conc_l / self.u_s / self.D_w
self.V_w = 1
self.V_all = (self.param_salt + self.V_w) / (1 - self.phi_p)
self.phi_s = self.param_salt / self.V_all
self.phi_w = self.V_w / self.V_all
# polymer and polymer interaction parameters
self.n = n_k
self.chi_p = chi_p
self.chi_e = chi_e
# hydration numbers
self.h_a = param_s[0]
self.h_b = param_s[1]
self.m_a = param_s[4]
self.m_b = param_s[5]
# number of ions per salt
self.nu_a = param_s[6]
self.nu_b = param_s[7]
self.nu = self.nu_a + self.nu_a
# fraction of hydration bonds (in water:x, in polymer:y)
self.x = x_ini
self.p = p_ini
# relative free energies of water association
self.df_w = df_w
self.df_p = param_poly[2]
self.df_a = param_salt[3]
self.df_b = param_salt[4]
self.i_size_a = param_s[2]
self.i_size_b = param_s[3]
self.i_size = 0.5 * (self.i_size_a + self.i_size_b)
# fraction of water in each ion size
self.f_a = self.h_a * self.nu_a * self.u_s * self.phi_s / self.phi_w
self.f_b = self.h_b * self.nu_b * self.u_s * self.phi_s / self.phi_w
# Derivative of parameters
self.df_adp = self.f_a / self.phi_w
self.df_ads = self.f_a * (1 / self.phi_s + 1 / self.phi_w)#
self.df_bdp = self.f_b / self.phi_w
self.df_bds = self.f_b * (1 / self.phi_s + 1 / self.phi_w)
self.dxdp = - self.x / self.phi_p
self.dxds = 0
self.dydp = self.p / self.phi_w
self.dyds = self.p / self.phi_w
self.gamma = param_s[8]
self.osm = param_s[9]
#S_para = nacl.NaClPropertiesRogersPitzer(tk=self.T, pa=1)
#self.gamma = S_para.log_gamma(self.conc_l)
#self.osm = S_para.osmotic_coeff(self.conc_l)
self.A_a = self.nu_a * self.m_a * self.conc_l / self.D_w
self.B_b = self.nu_b * self.m_b * self.conc_l / self.D_w
def test_potential_df(self):
"""
checks if chemical potentials add up give free energy
"""
num_pnts = 10
phi_val = np.linspace(1e-1, 0.8, num_pnts)
potential_w = np.zeros_like(phi_val)
potential_p = np.zeros_like(phi_val)
potential_s = np.zeros_like(phi_val)
f_com = np.zeros_like(phi_val)
F_com = np.zeros_like(phi_val)
v_p = np.array([0.4, 1 / 3, 10 / 3])
v_s = np.array([2, 1, 1, -100 / 3, -100 / 3])
temp = 298
df_w = 10 / 3
x_ini = 0.1
p_ini = 0.2
n_k = 100
chi_p = 0.5
chi_s = 0.4
#param_s = np.array([7, 7, 1, 1, 8, 8, 1, 1, 0., 0.])
param_s = np.array([7., 7., 1., 1., 8., 8., 1., 1., 0., 0.])
obj_water_bp = wbp.WaterPropertiesFineMillero(temp)
v_w = obj_water_bp.molar_volume() # molar volume of water
den = obj_water_bp.density() # density of water in SI unit from Ref[4]
D_w = 1 / den / v_w #55.509 # mol/kg water
for ind, phi_p in enumerate(phi_val):
S_para = nacl.NaClPropertiesRogersPitzer(tk=temp, pa=1)
param_s[8] = S_para.log_gamma(v_s[0])
param_s[9] = S_para.osmotic_coeff(v_s[0])
polymer_sol = Pss.PolymerSolutionSalts(
np.array([phi_p, 1 / 3, 10 / 3]), v_s, temp, df_w, x_ini,
p_ini, n_k, chi_p, chi_s, param_s)
potential_w[ind] = polymer_sol.chem_potential_w_full()
potential_p[ind] = polymer_sol.chem_potential_p_full()
potential_s[ind] = polymer_sol.chem_potential_s_full()
f_com[ind] = polymer_sol.free()
# concentration in mols/litre
conc = v_s[0]
# molecular volumes
u_p = v_p[1]
u_a = v_s[1]
u_b = v_s[2]
u_s = 1 / (1 / u_a + 1 / u_b)
# volume fractions
V_s = conc / u_s / D_w
V_w = 1
V_all = (V_s + V_w) / (1 - phi_p)
phi_s = V_s / V_all
phi_w = 1 - phi_s - phi_p
F_com[ind] = (u_p / n_k * phi_p * potential_p[ind] +
u_s * phi_s * potential_s[ind] +
phi_w * potential_w[ind])
self.assertTrue(np.allclose(F_com, f_com, rtol=0.0, atol=1e-14))