def Somayajulu(T, Tc, A, B, C):
r'''Calculates air-water surface tension using the [1]_
emperical (parameter-regressed) method. Well regressed, no recent data.
.. math::
\sigma=aX^{5/4}+bX^{9/4}+cX^{13/4}
.. math::
X=(T_c-T)/T_c
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
A : float
Regression parameter
B : float
Regression parameter
C : float
Regression parameter
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Presently untested, but matches expected values. Internal units are mN/m.
Form of function returns imaginary results when T > Tc; None is returned
if this is the case. Function is claimed valid from the triple to the
critical point. Results can be evaluated beneath the triple point.
Examples
--------
Water at 300 K
>>> Somayajulu(300, 647.126, 232.713514, -140.18645, -4.890098)
0.07166386387996758
References
----------
.. [1] Somayajulu, G. R. "A Generalized Equation for Surface Tension from
the Triple Point to the Critical Point." International Journal of
Thermophysics 9, no. 4 (July 1988): 559-66. doi:10.1007/BF00503154.
'''
X = (Tc - T) / Tc
return X * sqrt(sqrt(X)) * (A + X * (B + C * X)) * 1e-3
def Psat_IAPWS(T):
r'''Calculates vapor pressure of water using the IAPWS explicit equation.
.. math::
P^{sat} = 10^6 \left[ \frac{2C}{-B + \sqrt{B^2 - 4AC}} \right]^4
.. math::
A = \nu^2 + n_1 \nu + n_2
.. math::
B = n_3 \nu^2 + n_4\nu + n_5
.. math::
C = n_6\nu^2 + n_7\nu + n_8
.. math::
\nu = T + \frac{n_9}{T - n_{10}}
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
This formulation is quite efficient, and can also be solved backward.
The range of validity of this equation is 273.15 K < T < 647.096 K, the
IAPWS critical point.
Extrapolation to lower temperatures is very poor. The function continues to
decrease until a pressure of 5.7 mPa is reached at 159.77353993926621 K;
under that pressure the vapor pressure increases, which is obviously wrong.
Examples
--------
>>> Psat_IAPWS(300.)
3536.58941301301
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
v = T - 0.23855557567849 / (T - 0.65017534844798E3)
v2 = v * v
A = v2 + 0.11670521452767E4 * v - 0.72421316703206E6
B = -0.17073846940092E2 * v2 + 0.12020824702470E5 * v - 0.32325550322333E7
C = 0.14915108613530E2 * v2 - 0.48232657361591E4 * v + 0.40511340542057E6
x = ((C + C) / (sqrt(B * B - 4.0 * A * C) - B))
x2 = x * x
P = 1E6 * x2 * x2
return P
def Wagner_original(T, Tc, Pc, a, b, c, d):
r'''Calculates vapor pressure using the Wagner equation (3, 6 form).
Requires critical temperature and pressure as well as four coefficients
specific to each chemical.
.. math::
\ln P^{sat}= \ln P_c + \frac{a\tau + b \tau^{1.5} + c\tau^3 + d\tau^6}
{T_r}
.. math::
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature of fluid, [K]
Tc : float
Critical temperature, [K]
Pc : float
Critical pressure, [Pa]
a, b, c, d : floats
Parameters for wagner equation. Specific to each chemical. [-]
Returns
-------
Psat : float
Vapor pressure at T [Pa]
Notes
-----
Warning: Pc is often treated as adjustable constant.
Examples
--------
Methane, coefficients from [2]_, at 100 K.
>>> Wagner_original(100.0, 190.53, 4596420., a=-6.00435, b=1.1885,
... c=-0.834082, d=-1.22833)
34520.44601450499
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
.. [2] McGarry, Jack. "Correlation and Prediction of the Vapor Pressures of
Pure Liquids over Large Pressure Ranges." Industrial & Engineering
Chemistry Process Design and Development 22, no. 2 (April 1, 1983):
313-22. doi:10.1021/i200021a023.
'''
Tr = T / Tc
tau = 1.0 - Tr
tau2 = tau * tau
tau_Tr = tau / Tr
return Pc * exp(((d * tau2 * tau + c) * tau2 + a + b * sqrt(tau)) * tau_Tr)
def dPsat_IAPWS_dT(T):
r'''Calculates the first temperature dervative of vapor pressure of water
using the IAPWS explicit equation. This was derived with SymPy, using the
CSE method.
Parameters
----------
T : float
Temperature of water, [K]
Returns
-------
dPsat_dT : float
Temperature dervative of vapor pressure at T [Pa/K]
Notes
-----
The derivative of this is useful when solving for water dew point.
Examples
--------
>>> dPsat_IAPWS_dT(300.)
207.88388134164282
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
# 1 power, four divisions
x0 = 1.0/(T - 650.175348447980014)
x1 = T - 0.238555575678490006*x0
x2 = x1*x1
x3 = -0.0119059642846224972*T + 0.00284023416392566131*x0 + 0.0000368171193891888733*x2 + 1.0
x4 = 0.00371867596455583913*T
x5 = 0.000887110885486382334*x0
x6 = 5.28184261979789455e-6*x2
x7 = x4 - x5 - x6 - 1.0
x8 = 4668.20858110680001*T - 1113.62718545319967*x0 + 4.0*x2 - 2896852.66812823992
x9 = sqrt(x7*x7 - 9.56991643658934639e-14*x8*(-4823.26573615910002*T + 1150.61693433977007*x0 + 14.9151086135300002*x2 + 405113.405420569994))
x10 = -x4 + x5 + x6 + x9 + 1.0
x10_inv = 1.0/x10
x11 = 0.00153804662447919488*T - 1.0
x11 = 1.0/(x11*x11)
x12 = x1*(1.12864813714895499e-6*x11 + 2.0)
x20 = x10_inv*x10_inv*x10_inv*x10_inv
return (-3946.77829948379076*x3*x3*x3*(2.68752888216023447e-8*x11 - 0.000147268477556755493*x12
+ 0.0476238571384899889 + x3*(-8.39415340011308276e-9*x11 + 0.0000211273704791915782*x12
- 0.0148747038582233565 + 2.0*(x7*(4.19707670005654138e-9*x11
- 0.0000105636852395957891*x12 + 0.00743735192911167825) + x8*(2.60482114645230015e-16*x11
- 1.42736343074136086e-12*x12 + 4.6158250046507185e-10) + (0.00263438245944447809*x11
+ 4.0*x12 + 4668.20858110680001)*(4.6158250046507185e-10*T - 1.10113079121562103e-10*x0
- 1.42736343074136086e-12*x2 - 3.8769014372169964e-8))/x9)*x10_inv)*x20)
def Meybodi_Daryasafar_Karimi(rho_water, rho_oil, T, Tc):
r'''Calculates the interfacial tension between water and a hydrocabon
liquid according to the correlation of [1]_.
.. math::
\gamma_{hw} = \left(\frac{A_1 + A_2 \Delta \rho + A_3\Delta\rho^2
+ A_4\Delta\rho^3} {A_5 + A_6\frac{T^{A_7}}{T_{c,h}} + A_8T^{A_9}}
\right)^{A_{10}}
Parameters
----------
rho_water : float
The density of the aqueous phase, [kg/m^3]
rho_oil : float
The density of the hydrocarbon phase, [kg/m^3]
T : float
Temperature of the fluid, [K]
Tc : float
Critical temperature of the hydrocarbon mixture, [K]
Returns
-------
sigma : float
Hydrocarbon-water surface tension [N/m]
Notes
-----
Internal units of the equation are g/mL and mN/m.
Examples
--------
>>> Meybodi_Daryasafar_Karimi(980, 760, 580, 914)
0.02893598143089256
References
----------
.. [1] Kalantari Meybodi, Mahdi, Amin Daryasafar, and Masoud Karimi.
"Determination of Hydrocarbon-Water Interfacial Tension Using a New
Empirical Correlation." Fluid Phase Equilibria 415 (May 15, 2016):
42-50. doi:10.1016/j.fluid.2016.01.037.
'''
A1 = -1.3687340042E-1
A2 = -3.0391828884E-1
A3 = 5.6225871072E-1
A4 = -3.3074367079E-1
A5 = -3.0050179309E0
A6 = 5.8914210205E-5
A7 = -4.1388901263E0
A8 = 3.0084299030E0
A9 = -3.8203072876E-3
# A10 = 3.5000000000E0
drho = abs(rho_water - rho_oil) * 1e-3 # Correlation in units of g/mL
sigma = ((A1 + drho * (A2 + drho * (A3 + A4 * drho))) /
(A5 + A6 * T**A7 / Tc + A8 * T**A9))
return sigma * sigma * sigma * sqrt(sigma) * 1e-3 # mN/m to N/m
def Miqueu(T, Tc, Vc, omega):
r'''Calculates air-water surface tension using the methods of [1]_.
.. math::
\sigma = k T_c \left( \frac{N_a}{V_c}\right)^{2/3}
(4.35 + 4.14 \omega)t^{1.26}(1+0.19t^{0.5} - 0.487t)
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Vc : float
Critical volume of fluid [m^3/mol]
omega : float
Acentric factor for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Uses Avogadro's constant and the Boltsman constant.
Internal units of volume are mL/mol and mN/m. However, either a typo
is in the article or author's work, or my value of k is off by 10; this is
corrected nonetheless.
Created with 31 normal fluids, none polar or hydrogen bonded. Has an
AARD of 3.5%.
Examples
--------
Bromotrifluoromethane, 2.45 mN/m
>>> Miqueu(300., 340.1, 0.000199, 0.1687)
0.003474100774091376
References
----------
.. [1] Miqueu, C, D Broseta, J Satherley, B Mendiboure, J Lachaise, and
A Graciaa. "An Extended Scaled Equation for the Temperature Dependence
of the Surface Tension of Pure Compounds Inferred from an Analysis of
Experimental Data." Fluid Phase Equilibria 172, no. 2 (July 5, 2000):
169-82. doi:10.1016/S0378-3812(00)00384-8.
'''
Vc = Vc * 1E6
t = 1. - T / Tc
sigma = k * Tc * (N_A / Vc)**(2.0 / 3.0) * (
4.35 + 4.14 * omega) * t**1.26 * (1.0 + 0.19 * sqrt(t) -
0.25 * t) * 10000.0
return sigma
def solubility_parameter(T, Hvapm, Vml):
r'''This function handles the calculation of a chemical's solubility
parameter. Calculation is a function of temperature, but is not always
presented as such. `Hvapm`, `Vml`, `T` are required.
.. math::
\delta = \sqrt{\frac{\Delta H_{vap} - RT}{V_m}}
Parameters
----------
T : float
Temperature of the fluid [k]
Hvapm : float
Heat of vaporization [J/mol/K]
Vml : float
Specific volume of the liquid [m^3/mol]
Returns
-------
delta : float
Solubility parameter, [Pa^0.5]
Notes
-----
Undefined past the critical point. For convenience, if Hvap is not defined,
an error is not raised; None is returned instead. Also for convenience,
if Hvapm is less than RT, None is returned to avoid taking the root of a
negative number.
This parameter is often given in units of cal/ml, which is 2045.48 times
smaller than the value returned here.
Examples
--------
Pentane at STP
>>> solubility_parameter(T=298.2, Hvapm=26403.3, Vml=0.000116055)
14357.68128600315
References
----------
.. [1] Barton, Allan F. M. CRC Handbook of Solubility Parameters and Other
Cohesion Parameters, Second Edition. CRC Press, 1991.
'''
# Prevent taking the root of a negative number
return None if (Hvapm < R * T or Vml < 0.0) else sqrt(
(Hvapm - R * T) / Vml)
def Tsat_IAPWS(P):
r'''Calculates the saturation temperature of water using the IAPWS explicit
equation.
.. math::
T_s = \frac{n_{10} + D - \left[(n_{10}+D)^2 - 4(n_9 + n_{10}D) \right]^{0.5}}{2}
.. math:
D = \frac{2G}{-F - (F^2 - 4EG)^{0.5}}
.. math::
E = \beta^2 + n_3 \beta + n_6
.. math::
F = n_1 \beta^2 + n_4\beta + n_7
.. math::
G = n_2\beta^2 + n_5\beta + n_8
.. math::
\beta = \left(P_{sat} \right)^{0.25}
Parameters
----------
Psat : float
Vapor pressure at T [Pa]
Returns
-------
T : float
Temperature of water along the saturation curve at `Psat`, [K]
Notes
-----
The range of validity of this equation is 273.15 K < T < 647.096 K, the
IAPWS critical point.
The coefficients `n1` to `n10` are (0.11670521452767E4, -0.72421316703206E6,
-0.17073846940092E2, 0.12020824702470E5, -0.32325550322333E7, 0.14915108613530E2,
-0.48232657361591E4, 0.40511340542057E6, -0.23855557567849, 0.65017534844798E3)
Examples
--------
>>> Tsat_IAPWS(1E5)
372.75591861133773
References
----------
.. [1] Kretzschmar, Hans-Joachim, and Wolfgang Wagner. International Steam
Tables: Properties of Water and Steam Based on the Industrial
Formulation IAPWS-IF97. Springer, 2019.
'''
B = sqrt(sqrt(P * 1E-6))
E = B * (B + -0.17073846940092E2) + 0.14915108613530E2
F = B * (0.11670521452767E4 * B + 0.12020824702470E5) + -0.48232657361591E4
G = B * (-0.72421316703206E6 * B +
-0.32325550322333E7) + 0.40511340542057E6
D = 2.0 * G / (-F - sqrt(F * F - 4.0 * E * G))
n10 = 0.65017534844798E3
x0 = (n10 + D)
T = (n10 + D - sqrt(x0 * x0 - 4.0 * (-0.23855557567849 + n10 * D))) * 0.5
return T
def permittivity_IAPWS(T, rho):
r'''Calculate the relative permittivity of pure water as a function of.
temperature and density. Assumes the 1997 IAPWS [1]_ formulation.
.. math::
\epsilon(\rho, T) =\frac{1 + A + 5B + (9 + 2A + 18B + A^2 + 10AB +
9B^2)^{0.5}}{4(1-B)}
.. math::
A(\rho, T) = \frac{N_A\mu^2\rho g}{M\epsilon_0 kT}
.. math::
B(\rho) = \frac{N_A\alpha\rho}{3M\epsilon_0}
.. math::
g(\delta,\tau) = 1 + \sum_{i=1}^{11}n_i\delta^{I_i}\tau^{J_i}
+ n_{12}\delta\left(\frac{647.096}{228}\tau^{-1} - 1\right)^{-1.2}
.. math::
\delta = \rho/(322 \text{ kg/m}^3)
.. math::
\tau = T/647.096\text{K}
Parameters
----------
T : float
Temperature of water [K]
rho : float
Mass density of water at T and P [kg/m^3]
Returns
-------
epsilon : float
Relative permittivity of water at T and rho, [-]
Notes
-----
Validity:
273.15 < T < 323.15 K for 0 < P < iceVI melting pressure at T or 1000 MPa,
whichever is smaller.
323.15 < T < 873.15 K 0 < p < 600 MPa.
Coefficients and constants (they are optimized away in the function itself):
ih = [1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 10]
jh = [0.25, 1, 2.5, 1.5, 1.5, 2.5, 2, 2, 5, 0.5, 10]
Nh = [0.978224486826, -0.957771379375, 0.237511794148, 0.714692244396,
-0.298217036956, -0.108863472196, 0.949327488264E-1,
-.980469816509E-2, 0.165167634970E-4, 0.937359795772E-4,
-0.12317921872E-9]
polarizability = 1.636E-40
dipole = 6.138E-30
Examples
--------
>>> permittivity_IAPWS(373., 958.46)
55.565841872697234
>>> permittivity_IAPWS(650., 40.31090)
1.2659205723606064
References
----------
.. [1] IAPWS. 1997. Release on the Static Dielectric Constant of Ordinary
Water Substance for Temperatures from 238 K to 873 K and Pressures up
to 1000 MPa.
'''
# k = 1.38064852e-23
# actual molecular dipole moment of water, in C*m
# dipole2 = 3.7675044000000003e-59 # dipole = 6.138E-30 # but we square it
# polarizability = 1.636E-40 # actual mean molecular polarizability of water, C^2/J*m^2
# MW = 0.018015268 # molecular weight of water, kg/mol
# N_A = 6.0221367e23
delta = rho * 0.003105590062111801 # 1/322.0
# delta = rho/322.
T_inv = 1.0 / T
tau = 647.096 * T_inv
g = 1.0 + 0.196096504426E-2 * delta * (T * 0.0043859649122807015 -
1.0)**-1.2 # 0.00438.. == 1/228.0
# g = 1.0 + 0.196096504426E-2*delta*(T/228. - 1.0)**-1.2
# ih = [1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 10]
# jh = [0.25, 1, 2.5, 1.5, 1.5, 2.5, 2, 2, 5, 0.5, 10]
# Nh = [0.978224486826, -0.957771379375, 0.237511794148, 0.714692244396,
# -0.298217036956, -0.108863472196, 0.949327488264E-1,
# -.980469816509E-2, 0.165167634970E-4, 0.937359795772E-4,
# -0.12317921872E-9]
#
# for h in range(11):
# g += Nh[h]*delta**ih[h]*tau**jh[h]
tau_rt = sqrt(tau)
tau_15 = tau * tau_rt
tau_25 = tau_15 * tau
tau_2 = tau * tau
tau_5 = tau_25 * tau_25
g += delta * (
-delta *
(delta *
(-delta *
(delta *
(delta *
(delta *
(-1.2317921872000001181e-10 * delta * delta * delta * tau_5 *
tau_5 + 0.000093735979577200004786 * tau_rt) +
0.000016516763497000000026 * tau_5) - 0.0098046981650899995425 *
tau_2) + 0.094932748826400001341 * tau_2) +
0.10886347219599999681 * tau_25 + 0.29821703695599999229 * tau_15) -
0.71469224439599998711 * tau_15) +
0.97822448682600005032 * sqrt(tau_rt) - 0.9577713793750000093 * tau +
0.23751179414799999945 * tau_25)
A = rho * g * T_inv * 10.302249930663786 #N_A*dipole2/(epsilon_0*k*MW)
B = rho * 0.00020588442304500529 #Na*polarizability/1(3.*epsilon_0*MW) simplifies to the constant
epsilon = (1. + A + 5. * B + sqrt(9. + A * (2.0 + A) + B *
(18. + 10. * A + 9. * B))) / (4. -
4. * B)
return epsilon
def RI_IAPWS(T, rho, wavelength=0.5893):
r'''Calculates the refractive index of water at a given temperature,
density, and wavelength.
.. math::
n(\rho, T, \lambda) = \left(\frac{2A + 1}{1-A}\right)^{0.5}
.. math::
A(\delta, \theta, \Lambda) = \delta\left(a_0 + a_1\delta +
a_2\theta + a_3\Lambda^2\theta + a_4\Lambda^{-2}
\frac{a_5}{\Lambda^2-\Lambda_{UV}^2} + \frac{a_6}
{\Lambda^2 - \Lambda_{IR}^2} + a_7\delta^2\right)
.. math::
\delta = \rho/(1000 \text{ kg/m}^3)
.. math::
\theta = T/273.15\text{K}
.. math::
\Lambda = \lambda/0.589 \mu m
.. math::
\Lambda_{IR} = 5.432937
.. math::
\Lambda_{UV} = 0.229202
Parameters
----------
T : float
Temperature of the water [K]
rho : float
Density of the water [kg/m^3]
wavelength : float
Wavelength of fluid [micrometers]
Returns
-------
RI : float
Refractive index of the water, [-]
Notes
-----
This function is valid in the following range:
261.15 K < T < 773.15 K
0 < rho < 1060 kg/m^3
0.2 < wavelength < 1.1 micrometers
Test values are from IAPWS 2010 book.
Examples
--------
>>> RI_IAPWS(298.15, 997.047435, 0.5893)
1.3328581926471605
References
----------
.. [1] IAPWS, 1997. Release on the Refractive Index of Ordinary Water
Substance as a Function of Wavelength, Temperature and Pressure.
'''
delta = rho * 1e-3
theta = T * (1.0 / 273.15)
Lambda = wavelength * (1.0 / 0.589)
LambdaIR = 5.432937
LambdaUV = 0.229202
Lambda2 = Lambda * Lambda
A = delta * (0.244257733 + 0.0097463448 * delta + -0.00373235 * theta +
0.0002686785 * Lambda2 * theta + 0.0015892057 / Lambda2 +
0.0024593426 / (Lambda2 - LambdaUV * LambdaUV) + 0.90070492 /
(Lambda2 - LambdaIR * LambdaIR) -
0.0166626219 * delta * delta)
n = sqrt((2.0 * A + 1.) / (1. - A))
return n
def EQ116(T, Tc, A, B, C, D, E, order=0):
r'''DIPPR Equation #116. Used to describe the molar density of water fairly
precisely; no other uses listed. All 5 parameters are needed, as well as
the critical temperature.
.. math::
Y = A + B\tau^{0.35} + C\tau^{2/3} + D\tau + E\tau^{4/3}
\tau = 1 - \frac{T}{T_c}
Parameters
----------
T : float
Temperature, [K]
Tc : float
Critical temperature, [K]
A-E : float
Parameter for the equation; chemical and property specific [-]
order : int, optional
Order of the calculation. 0 for the calculation of the result itself;
for 1, the first derivative of the property is returned, for
-1, the indefinite integral of the property with respect to temperature
is returned; and for -1j, the indefinite integral of the property
divided by temperature with respect to temperature is returned. No
other integrals or derivatives are implemented, and an exception will
be raised if any other order is given.
Returns
-------
Y : float
Property [constant-specific; if order == 1, property/K; if order == -1,
property*K; if order == -1j, unchanged from default]
Notes
-----
The derivative with respect to T and integral with respect to T are
computed as follows. The integral divided by T with respect to T has an
extremely complicated (but still elementary) integral which can be read
from the source. It was computed with Rubi; the other expressions can
readily be obtained with SymPy.
.. math::
\frac{d Y}{dT} = - \frac{7 B}{20 T_c \left(- \frac{T}{T_c} + 1\right)^{
\frac{13}{20}}} - \frac{2 C}{3 T_c \sqrt[3]{- \frac{T}{T_c} + 1}}
- \frac{D}{T_c} - \frac{4 E}{3 T_c} \sqrt[3]{- \frac{T}{T_c} + 1}
.. math::
\int Y dT = A T - \frac{20 B}{27} T_c \left(- \frac{T}{T_c} + 1\right)^{
\frac{27}{20}} - \frac{3 C}{5} T_c \left(- \frac{T}{T_c} + 1\right)^{
\frac{5}{3}} + D \left(- \frac{T^{2}}{2 T_c} + T\right) - \frac{3 E}{7}
T_c \left(- \frac{T}{T_c} + 1\right)^{\frac{7}{3}}
Examples
--------
Water liquid molar density; DIPPR coefficients normally in kmol/m^3.
>>> EQ116(300., 647.096, 17.863, 58.606, -95.396, 213.89, -141.26)
55.17615446406527
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
tau = 1 - T / Tc
return A + B * tau**0.35 + C * tau**(2 / 3.) + D * tau + E * tau**(4 /
3.)
elif order == 1:
return (-7 * B / (20 * Tc * (-T / Tc + 1)**(13 / 20)) - 2 * C /
(3 * Tc * (-T / Tc + 1)**(1 / 3)) - D / Tc - 4 * E *
(-T / Tc + 1)**(1 / 3) / (3 * Tc))
elif order == -1:
return (A * T - 20 * B * Tc * (-T / Tc + 1)**(27 / 20) / 27 -
3 * C * Tc * (-T / Tc + 1)**(5 / 3) / 5 + D * (-T**2 /
(2 * Tc) + T) -
3 * E * Tc * (-T / Tc + 1)**(7 / 3) / 7)
elif order == -1j:
# 3x increase in speed - cse via sympy
x0 = log(T)
x1 = 0.5 * x0
x2 = 1 / Tc
x3 = T * x2
x4 = -x3 + 1
x5 = 1.5 * C
x6 = x4**0.333333333333333
x7 = 2 * B
x8 = x4**0.05
x9 = log(-x6 + 1)
x10 = sqrt(3)
x11 = x10 * atan(x10 * (2 * x6 + 1) / 3)
x12 = sqrt(5)
x13 = 0.5 * x12
x14 = x13 + 0.5
x15 = B * x14
x16 = sqrt(x13 + 2.5)
x17 = 2 * x8
x18 = -x17
x19 = -x13
x20 = x19 + 0.5
x21 = B * x20
x22 = sqrt(x19 + 2.5)
x23 = B * x16
x24 = 0.5 * sqrt(0.1 * x12 + 0.5)
x25 = x12 + 1
x26 = 4 * x8
x27 = -x26
x28 = sqrt(10) * B / sqrt(x12 + 5)
x29 = 2 * x12
x30 = sqrt(x29 + 10)
x31 = 1 / x30
x32 = -x12 + 1
x33 = 0.5 * B * x22
x34 = -x2 * (T - Tc)
x35 = 2 * x34**0.1
x36 = x35 + 2
x37 = x34**0.05
x38 = x30 * x37
x39 = 0.5 * B * x16
x40 = x37 * sqrt(-x29 + 10)
x41 = 0.25 * x12
x42 = B * (-x41 + 0.25)
x43 = x12 * x37
x44 = x35 + x37 + 2
x45 = B * (x41 + 0.25)
x46 = -x43
x47 = x35 - x37 + 2
return A * x0 + 2.85714285714286 * B * x4**0.35 - C * x1 + C * x11 + D * x0 - D * x3 - E * x1 - E * x11 + 0.75 * E * x4**1.33333333333333 + 3 * E * x6 + 1.5 * E * x9 - x15 * atan(
x14 * (x16 + x17)) + x15 * atan(x14 * (x16 + x18)) - x21 * atan(
x20 *
(x17 + x22)) + x21 * atan(x20 * (x18 + x22)) + x23 * atan(
x24 *
(x25 + x26)) - x23 * atan(x24 * (x25 + x27)) - x28 * atan(
x31 * (x26 + x32)) + x28 * atan(
x31 * (x27 + x32)
) - x33 * log(x36 - x38) + x33 * log(
x36 + x38) + x39 * log(x36 - x40) - x39 * log(
x36 + x40
) + x4**0.666666666666667 * x5 - x42 * log(
x43 + x44) + x42 * log(x46 + x47) + x45 * log(
x43 + x47) - x45 * log(
x44 + x46) + x5 * x9 + x7 * atan(
x8) - x7 * atanh(x8)
else:
raise ValueError(order_not_found_msg)
def Winterfeld_Scriven_Davis(xs, sigmas, rhoms):
r'''Calculates surface tension of a liquid mixture according to
mixing rules in [1]_ and also in [2]_.
.. math::
\sigma_M = \sum_i \sum_j \frac{1}{V_L^{L2}}\left(x_i V_i \right)
\left( x_jV_j\right)\sqrt{\sigma_i\cdot \sigma_j}
Parameters
----------
xs : array-like
Mole fractions of all components, [-]
sigmas : array-like
Surface tensions of all components, [N/m]
rhoms : array-like
Molar densities of all components, [mol/m^3]
Returns
-------
sigma : float
Air-liquid surface tension of mixture, [N/m]
Notes
-----
DIPPR Procedure 7C: Method for the Surface Tension of Nonaqueous Liquid
Mixtures
Becomes less accurate as liquid-liquid critical solution temperature is
approached. DIPPR Evaluation: 3-4% AARD, from 107 nonaqueous binary
systems, 1284 points. Internally, densities are converted to kmol/m^3. The
Amgat function is used to obtain liquid mixture density in this equation.
Raises a ZeroDivisionError if either molar volume are zero, and a
ValueError if a surface tensions of a pure component is negative.
Examples
--------
>>> Winterfeld_Scriven_Davis([0.1606, 0.8394], [0.01547, 0.02877],
... [8610., 15530.])
0.02496738845043982
References
----------
.. [1] Winterfeld, P. H., L. E. Scriven, and H. T. Davis. "An Approximate
Theory of Interfacial Tensions of Multicomponent Systems: Applications
to Binary Liquid-Vapor Tensions." AIChE Journal 24, no. 6
(November 1, 1978): 1010-14. doi:10.1002/aic.690240610.
.. [2] Danner, Ronald P, and Design Institute for Physical Property Data.
Manual for Predicting Chemical Process Design Data. New York, N.Y, 1982.
'''
N = len(xs)
Vms = [0.0]*N
rho = 0.0
for i in range(N):
Vms[i] = 1e3/rhoms[i]
rho += xs[i]*Vms[i]
# rho = 1./rho
rho = 1.4142135623730951/rho # factor out rt2
# For speed, transform the Vms array to contain
# xs[i]*Vms[i]*sigmas_05[i]*rho
tot = 0.0
for i in range(N):
val = sqrt(sigmas[i])*xs[i]*rho*Vms[i]
Vms[i] = val
tot += val*val
tot *= 0.5
for i in range(N):
# Symmetric - can be slightly optimized
for j in range(i):
tot += Vms[i]*Vms[j]
return tot
def SNM0(T, Tc, Vc, omega, delta_SRK=None):
r'''Calculates saturated liquid density using the Mchaweh, Moshfeghian
model [1]_. Designed for simple calculations.
.. math::
V_s = V_c/(1+1.169\tau^{1/3}+1.818\tau^{2/3}-2.658\tau+2.161\tau^{4/3}
.. math::
\tau = 1-\frac{(T/T_c)}{\alpha_{SRK}}
.. math::
\alpha_{SRK} = [1 + m(1-\sqrt{T/T_C}]^2
.. math::
m = 0.480+1.574\omega-0.176\omega^2
If the fit parameter `delta_SRK` is provided, the following is used:
.. math::
V_s = V_C/(1+1.169\tau^{1/3}+1.818\tau^{2/3}-2.658\tau+2.161\tau^{4/3})
/\left[1+\delta_{SRK}(\alpha_{SRK}-1)^{1/3}\right]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Vc : float
Critical volume of fluid [m^3/mol]
omega : float
Acentric factor for fluid, [-]
delta_SRK : float, optional
Fitting parameter [-]
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
73 fit parameters have been gathered from the article.
Examples
--------
Argon, without the fit parameter and with it. Tabulated result in Perry's
is 3.4613e-05. The fit increases the error on this occasion.
>>> SNM0(121, 150.8, 7.49e-05, -0.004)
3.440225640273e-05
>>> SNM0(121, 150.8, 7.49e-05, -0.004, -0.03259620)
3.493288100008e-05
References
----------
.. [1] Mchaweh, A., A. Alsaygh, Kh. Nasrifar, and M. Moshfeghian.
"A Simplified Method for Calculating Saturated Liquid Densities."
Fluid Phase Equilibria 224, no. 2 (October 1, 2004): 157-67.
doi:10.1016/j.fluid.2004.06.054
'''
Tr = T / Tc
m = 0.480 + 1.574 * omega - 0.176 * omega * omega
x0 = (1.0 + m * (1.0 - sqrt(Tr)))
alpha_SRK = x0 * x0
tau = 1. - Tr / alpha_SRK
tau_cbrt = tau**(1 / 3.)
rho0 = 1. + tau_cbrt * (
1.169 + 1.818 * tau_cbrt) - 2.658 * tau + 2.161 * tau * tau_cbrt
V0 = 1. / rho0
if delta_SRK is None:
return Vc * V0
else:
return Vc * V0 / (1. + delta_SRK * (alpha_SRK - 1.0)**(1.0 / 3.0))
