def COSTALD_compressed(T, P, Psat, Tc, Pc, omega, Vs):
r'''Calculates compressed-liquid volume, using the COSTALD [1]_ CSP
method and a chemical's critical properties.
The molar volume of a liquid is given by:
.. math::
V = V_s\left( 1 - C \ln \frac{B + P}{B + P^{sat}}\right)
.. math::
\frac{B}{P_c} = -1 + a\tau^{1/3} + b\tau^{2/3} + d\tau + e\tau^{4/3}
.. math::
e = \exp(f + g\omega_{SRK} + h \omega_{SRK}^2)
.. math::
C = j + k \omega_{SRK}
Parameters
----------
T : float
Temperature of fluid [K]
P : float
Pressure of fluid [Pa]
Psat : float
Saturation pressure of the fluid [Pa]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
(ideally SRK) Acentric factor for fluid, [-]
This parameter is alternatively a fit parameter.
Vs : float
Saturation liquid volume, [m^3/mol]
Returns
-------
V_dense : float
High-pressure liquid volume, [m^3/mol]
Notes
-----
Original equation was in terms of density, but it is converted here.
The example is from DIPPR, and exactly correct.
This is DIPPR Procedure 4C: Method for Estimating the Density of Pure
Organic Liquids under Pressure.
Examples
--------
>>> COSTALD_compressed(303., 9.8E7, 85857.9, 466.7, 3640000.0, 0.281, 0.000105047)
9.287482879788505e-05
References
----------
.. [1] Thomson, G. H., K. R. Brobst, and R. W. Hankinson. "An Improved
Correlation for Densities of Compressed Liquids and Liquid Mixtures."
AIChE Journal 28, no. 4 (July 1, 1982): 671-76. doi:10.1002/aic.690280420
'''
a = -9.070217
b = 62.45326
d = -135.1102
f = 4.79594
g = 0.250047
h = 1.14188
j = 0.0861488
k = 0.0344483
e = exp(f + omega*(g + h*omega))
C = j + k*omega
tau = 1.0 - T/Tc
tau13 = tau**(1.0/3.0)
B = Pc*(-1.0 + a*tau13 + b*tau13*tau13 + d*tau + e*tau*tau13)
return Vs*(1.0 - C*log((B + P)/(B + Psat)))
def Zuo_Stenby(T, Tc, Pc, omega):
r'''Calculates air-water surface tension using the reference fluids
methods of [1]_.
.. math::
\sigma^{(1)} = 40.520(1-T_r)^{1.287}
.. math::
\sigma^{(2)} = 52.095(1-T_r)^{1.21548}
.. math::
\sigma_r = \sigma_r^{(1)}+ \frac{\omega - \omega^{(1)}}
{\omega^{(2)}-\omega^{(1)}} (\sigma_r^{(2)}-\sigma_r^{(1)})
.. math::
\sigma = T_c^{1/3}P_c^{2/3}[\exp{(\sigma_r)} -1]
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor for fluid, [-]
Returns
-------
sigma : float
Liquid surface tension, N/m
Notes
-----
Presently untested. Have not personally checked the sources.
I strongly believe it is broken.
The reference values for methane and n-octane are from the DIPPR database.
Examples
--------
Chlorobenzene
>>> Zuo_Stenby(293., 633.0, 4530000.0, 0.249)
0.03345569011871088
References
----------
.. [1] Zuo, You-Xiang, and Erling H. Stenby. "Corresponding-States and
Parachor Models for the Calculation of Interfacial Tensions." The
Canadian Journal of Chemical Engineering 75, no. 6 (December 1, 1997):
1130-37. doi:10.1002/cjce.5450750617
'''
Tc_1, Pc_1, omega_1 = 190.56, 4599000.0 * 1e-5, 0.012
Tc_2, Pc_2, omega_2 = 568.7, 2490000.0 * 1e-5, 0.4
Pc = Pc * 1e-5
Tr = T / Tc
ST_1 = 40.520 * (1.0 - Tr)**1.287 # Methane
ST_2 = 52.095 * (1.0 - Tr)**1.21548 # n-octane
ST_r_1 = log(1.0 + 0.013537770442486932 *
ST_1) # Constant from 1/(Tc_1**(1.0/3.0)*Pc_1**(2.0/3.0))
# ST_r_1 = log(1.0 + ST_1/(Tc_1**(1.0/3.0)*Pc_1**(2.0/3.0)))
ST_r_2 = log(1.0 + 0.014154874587259097 *
ST_2) # Constant from /(Tc_2**(1.0/3.0)*Pc_2**(2.0/3.0))
sigma_r = ST_r_1 + (omega - omega_1) * (ST_r_2 -
ST_r_1) * 2.5773195876288657
# sigma_r = ST_r_1 + (omega-omega_1)/(omega_2 - omega_1)*(ST_r_2-ST_r_1)
sigma = Tc**(1.0 / 3.0) * Pc**(2.0 / 3.0) * (exp(sigma_r) - 1.0)
sigma = sigma * 1e-3 # N/m, please
return sigma
def Campbell_Thodos(T, Tb, Tc, Pc, MW, dipole=0.0, has_hydroxyl=False):
r'''Calculate saturation liquid density using the Campbell-Thodos [1]_
CSP method.
An old and uncommon estimation method.
.. math::
V_s = \frac{RT_c}{P_c}{Z_{RA}}^{[1+(1-T_r)^{2/7}]}
.. math::
Z_{RA} = \alpha + \beta(1-T_r)
.. math::
\alpha = 0.3883-0.0179s
.. math::
s = T_{br} \frac{\ln P_c}{(1-T_{br})}
.. math::
\beta = 0.00318s-0.0211+0.625\Lambda^{1.35}
.. math::
\Lambda = \frac{P_c^{1/3}} { MW^{1/2} T_c^{5/6}}
For polar compounds:
.. math::
\theta = P_c \mu^2/T_c^2
.. math::
\alpha = 0.3883 - 0.0179s - 130540\theta^{2.41}
.. math::
\beta = 0.00318s - 0.0211 + 0.625\Lambda^{1.35} + 9.74\times
10^6 \theta^{3.38}
Polar Combounds with hydroxyl groups (water, alcohols)
.. math::
\alpha = \left[0.690T_{br} -0.3342 + \frac{5.79\times 10^{-10}}
{T_{br}^{32.75}}\right] P_c^{0.145}
.. math::
\beta = 0.00318s - 0.0211 + 0.625 \Lambda^{1.35} + 5.90\Theta^{0.835}
Parameters
----------
T : float
Temperature of fluid [K]
Tb : float
Boiling temperature of the fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
MW : float
Molecular weight of the fluid [g/mol]
dipole : float, optional
Dipole moment of the fluid [debye]
has_hydroxyl : bool, optional
Swith to use the hydroxyl variant for polar fluids
Returns
-------
Vs : float
Saturation liquid volume, [m^3/mol]
Notes
-----
If a dipole is provided, the polar chemical method is used.
The paper is an excellent read.
Pc is internally converted to atm.
Examples
--------
Ammonia, from [1]_.
>>> Campbell_Thodos(T=405.45, Tb=239.82, Tc=405.45, Pc=111.7*101325, MW=17.03, dipole=1.47)
7.347366126245e-05
References
----------
.. [1] Campbell, Scott W., and George Thodos. "Prediction of Saturated
Liquid Densities and Critical Volumes for Polar and Nonpolar
Substances." Journal of Chemical & Engineering Data 30, no. 1
(January 1, 1985): 102-11. doi:10.1021/je00039a032.
'''
Tc_inv = 1.0/Tc
Tr = T * Tc_inv
Tbr = Tb * Tc_inv
Pc = Pc/101325.
s = Tbr*log(Pc)/(1.0 - Tbr)
Lambda = Pc**(1.0/3.)/(MW**0.5*Tc**(5/6.))
beta = 0.00318*s - 0.0211 + 0.625*Lambda**(1.35)
if dipole is None:
alpha = 0.3883 - 0.0179*s
else:
theta = Pc*dipole*dipole/(Tc*Tc)
beta += 9.74E6 * theta**3.38
if has_hydroxyl:
beta += 5.90*theta**0.835
alpha = (0.69*Tbr - 0.3342 + 5.79E-10*Tbr**-32.75)*Pc**0.145
else:
alpha = 0.3883 - 0.0179*s - 130540 * theta**2.41
Zra = alpha + beta*(1.0 - Tr)
p = 1.0 if T == Tc else (1.0 + (1.0 - Tr)**(2.0/7.))
Vs = R*Tc/(Pc*101325.0)*Zra**p
return Vs
def interpolation_property(P):
'''log(P) interpolation transformation by default.
'''
return log(P)
def calculate(self, T, method):
r'''Method to calculate vapor pressure of a fluid at temperature `T`
with a given method.
This method has no exception handling; see :obj:`thermo.utils.TDependentProperty.T_dependent_property`
for that.
Parameters
----------
T : float
Temperature at calculate vapor pressure, [K]
method : str
Name of the method to use
Returns
-------
Psat : float
Vapor pressure at T, [Pa]
'''
if method == POLY_FIT:
if T < self.poly_fit_Tmin:
Psat = (T - self.poly_fit_Tmin
) * self.poly_fit_Tmin_slope + self.poly_fit_Tmin_value
elif T > self.poly_fit_Tmax:
Psat = (T - self.poly_fit_Tmax
) * self.poly_fit_Tmax_slope + self.poly_fit_Tmax_value
else:
Psat = horner(self.poly_fit_coeffs, T)
Psat = exp(Psat)
elif method == BEST_FIT_AB:
if T < self.poly_fit_Tmax:
return self.calculate(T, POLY_FIT)
A, B = self.poly_fit_AB_high_ABC_compat
return exp(A + B / T)
elif method == BEST_FIT_ABC:
if T < self.poly_fit_Tmax:
return self.calculate(T, POLY_FIT)
A, B, C = self.DIPPR101_ABC_high
return exp(A + B / T + C * log(T))
elif method == WAGNER_MCGARRY:
Psat = Wagner_original(T, self.WAGNER_MCGARRY_Tc,
self.WAGNER_MCGARRY_Pc,
*self.WAGNER_MCGARRY_coefs)
elif method == WAGNER_POLING:
Psat = Wagner(T, self.WAGNER_POLING_Tc, self.WAGNER_POLING_Pc,
*self.WAGNER_POLING_coefs)
elif method == ANTOINE_EXTENDED_POLING:
Psat = TRC_Antoine_extended(T, *self.ANTOINE_EXTENDED_POLING_coefs)
elif method == ANTOINE_POLING:
A, B, C = self.ANTOINE_POLING_coefs
Psat = Antoine(T, A, B, C, base=10.0)
elif method == DIPPR_PERRY_8E:
Psat = EQ101(T, *self.Perrys2_8_coeffs)
elif method == VDI_PPDS:
Psat = Wagner(T, self.VDI_PPDS_Tc, self.VDI_PPDS_Pc,
*self.VDI_PPDS_coeffs)
elif method == COOLPROP:
Psat = PropsSI('P', 'T', T, 'Q', 0, self.CASRN)
elif method == BOILING_CRITICAL:
Psat = boiling_critical_relation(T, self.Tb, self.Tc, self.Pc)
elif method == LEE_KESLER_PSAT:
Psat = Lee_Kesler(T, self.Tc, self.Pc, self.omega)
elif method == AMBROSE_WALTON:
Psat = Ambrose_Walton(T, self.Tc, self.Pc, self.omega)
elif method == SANJARI:
Psat = Sanjari(T, self.Tc, self.Pc, self.omega)
elif method == EDALAT:
Psat = Edalat(T, self.Tc, self.Pc, self.omega)
elif method == EOS:
Psat = self.eos[0].Psat(T)
elif method == POLY_FIT:
Psat = exp(horner(self.poly_fit_coeffs, T))
else:
return self._base_calculate(T, method)
return Psat
def EQ100(T, A=0, B=0, C=0, D=0, E=0, F=0, G=0, order=0):
r'''DIPPR Equation # 100. Used in calculating the molar heat capacities
of liquids and solids, liquid thermal conductivity, and solid density.
All parameters default to zero. As this is a straightforward polynomial,
no restrictions on parameters apply. Note that high-order polynomials like
this may need large numbers of decimal places to avoid unnecessary error.
.. math::
Y = A + BT + CT^2 + DT^3 + ET^4 + FT^5 + GT^6
Parameters
----------
T : float
Temperature, [K]
A-G : 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, integral with respect to T, and integral
over T with respect to T are computed as follows. All derivatives and
integrals are easily computed with SymPy.
.. math::
\frac{d Y}{dT} = B + 2 C T + 3 D T^{2} + 4 E T^{3} + 5 F T^{4}
+ 6 G T^{5}
.. math::
\int Y dT = A T + \frac{B T^{2}}{2} + \frac{C T^{3}}{3} + \frac{D
T^{4}}{4} + \frac{E T^{5}}{5} + \frac{F T^{6}}{6} + \frac{G T^{7}}{7}
.. math::
\int \frac{Y}{T} dT = A \log{\left (T \right )} + B T + \frac{C T^{2}}
{2} + \frac{D T^{3}}{3} + \frac{E T^{4}}{4} + \frac{F T^{5}}{5}
+ \frac{G T^{6}}{6}
Examples
--------
Water liquid heat capacity; DIPPR coefficients normally listed in J/kmol/K.
>>> EQ100(300, 276370., -2090.1, 8.125, -0.014116, 0.0000093701)
75355.81000000003
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
return A + T * (B + T * (C + T * (D + T * (E + T * (F + G * T)))))
elif order == 1:
return B + T * (2 * C + T * (3 * D + T * (4 * E + T *
(5 * F + 6 * G * T))))
elif order == -1:
return T * (A + T * (B / 2 + T * (C / 3 + T *
(D / 4 + T *
(E / 5 + T *
(F / 6 + G * T / 7))))))
elif order == -1j:
return A * log(T) + T * (B + T * (C / 2 + T * (D / 3 + T *
(E / 4 + T *
(F / 5 + G * T / 6)))))
else:
raise ValueError(order_not_found_msg)
def Antoine_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2, base=10.0):
r'''Calculates the antoine coefficients `A`, `B`, and `C` from a known
vapor pressure and its first and second temperature derivative.
Parameters
----------
T : float
Temperature of fluid, [K]
Psat : float
Vapor pressure at specified `T` [Pa]
dPsat_dT : float
First temperature derivative of vapor pressure at specified `T` [Pa/K]
d2Psat_dT2 : float
Second temperature derivative of vapor pressure at specified `T` [Pa/K^2]
Base : float, optional
Base of logarithm; 10 by default
Returns
-------
A : float
Antoine `A` parameter, [-]
B : float
Antoine `B` parameter, [K]
C : float
Antoine `C` parameter, [K]
Notes
-----
Coefficients are for calculating vapor pressure in Pascal. This is
primarily useful for interconverting vapor pressure models, not fitting
experimental data.
Derived with SymPy as follows:
>>> from sympy import * # doctest: +SKIP
>>> base, A, B, C, T = symbols('base, A, B, C, T') # doctest: +SKIP
>>> v = base**(A - B/(T + C)) # doctest: +SKIP
>>> d1, d2 = diff(v, T), diff(v, T, 2) # doctest: +SKIP
>>> vk, d1k, d2k = symbols('vk, d1k, d2k') # doctest: +SKIP
>>> solve([Eq(v, vk), Eq(d1, d1k), Eq(d2, d2k)], [A, B, C]) # doctest: +SKIP
Examples
--------
Recalculate some coefficients from a calcualted value and its derivative:
>>> T = 178.01
>>> A, B, C = (24.0989474955895, 4346.793091137991, -18.96968471040141)
>>> Psat = Antoine(T, A, B, C, base=exp(1))
>>> dPsat_dT, d2Psat_dT2 = (0.006781441203850251, 0.0010801244983894853) # precomputed
>>> Antoine_coeffs_from_point(T, Psat, dPsat_dT, d2Psat_dT2, base=exp(1))
(24.098947495155, 4346.793090994, -18.969684713118)
References
----------
.. [1] Poling, Bruce E. The Properties of Gases and Liquids. 5th edition.
New York: McGraw-Hill Professional, 2000.
'''
x0 = 1.0 / log(base)
x1 = Psat * d2Psat_dT2
dPsat_dT_2 = dPsat_dT * dPsat_dT
x3 = 1.0 / (x1 - dPsat_dT_2)
x4 = dPsat_dT_2 + dPsat_dT_2
A = x0 * log(Psat * exp(-x3 * x4))
B = 4.0 * Psat * dPsat_dT * dPsat_dT_2 * x0 / (
Psat * Psat * d2Psat_dT2 * d2Psat_dT2 + dPsat_dT_2 * dPsat_dT_2 -
x1 * x4)
C = -x3 * (2.0 * Psat * dPsat_dT + T * x1 - T * dPsat_dT_2)
return (A, B, C)
def EQ101(T, A, B, C=0.0, D=0.0, E=0.0, order=0):
r'''DIPPR Equation # 101. Used in calculating vapor pressure, sublimation
pressure, and liquid viscosity.
All 5 parameters are required. E is often an integer. As the model is
exponential, a sufficiently high temperature will cause an OverflowError.
A negative temperature (or just low, if fit poorly) may cause a math domain
error.
.. math::
Y = \exp\left(A + \frac{B}{T} + C\cdot \ln T + D \cdot T^E\right)
Parameters
----------
T : float
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 `n`, the `nth` derivative of the property 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]
Notes
-----
This function is not integrable for either dT or Y/T dT.
.. math::
\frac{d Y}{dT} = \left(- \frac{B}{T^{2}} + \frac{C}{T}
+ \frac{D E T^{E}}{T}\right) e^{A + \frac{B}{T}
+ C \log{\left(T \right)} + D T^{E}}
.. math::
\frac{d^2 Y}{dT^2} = \frac{\left(\frac{2 B}{T} - C + D E^{2} T^{E}
- D E T^{E} + \left(- \frac{B}{T} + C + D E T^{E}\right)^{2}\right)
e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{E}}}{T^{2}}
.. math::
\frac{d^3 Y}{dT^3} = \frac{\left(- \frac{6 B}{T} + 2 C + D E^{3} T^{E}
- 3 D E^{2} T^{E} + 2 D E T^{E} + \left(- \frac{B}{T} + C
+ D E T^{E}\right)^{3} + 3 \left(- \frac{B}{T} + C + D E T^{E}\right)
\left(\frac{2 B}{T} - C + D E^{2} T^{E} - D E T^{E}\right)\right)
e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{E}}}{T^{3}}
Examples
--------
Water vapor pressure; DIPPR coefficients normally listed in Pa.
>>> EQ101(300, 73.649, -7258.2, -7.3037, 4.1653E-6, 2)
3537.44834545549
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
T_inv = 1.0 / T
T_E = T**E
expr = trunc_exp(A + B * T_inv + C * log(T) + D * T_E)
if order == 0:
return expr
elif order == 1:
return T_inv * expr * (-B * T_inv + C + D * E * T_E)
elif order == 2:
x0 = (-B * T_inv + C + D * E * T_E)
return expr * (2.0 * B * T_inv - C + D * E * T_E *
(E - 1.0) + x0 * x0) * T_inv * T_inv
elif order == 3:
E2 = E * E
E3 = E2 * E
x0 = (-B * T_inv + C + D * E * T_E)
return expr * (-6.0 * B * T_inv + 2.0 * C + D * E3 * T_E - 3 * D * E2 *
T_E + 2.0 * D * E * T_E + x0 * x0 * x0 + 3.0 *
(-B * T_inv + C + D * E * T_E) *
(2.0 * B * T_inv - C + D * E2 * T_E - D * E * T_E)
) * T_inv * T_inv * T_inv
else:
raise ValueError(order_not_found_pos_only_msg)
def EQ105(T, A, B, C, D, order=0):
r'''DIPPR Equation #105. Often used in calculating liquid molar density.
All 4 parameters are required. C is sometimes the fluid's critical
temperature.
.. math::
Y = \frac{A}{B^{1 + \left(1-\frac{T}{C}\right)^D}}
Parameters
----------
T : float
Temperature, [K]
A-D : 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, 2, and 3, that derivative of the property 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]
Notes
-----
This expression can be integrated in terms of the incomplete gamma function
for dT, however nans are the only output from that function.
For Y/T dT no integral could be found.
.. math::
\frac{d Y}{dT} = \frac{A B^{- \left(1 - \frac{T}{C}\right)^{D} - 1} D
\left(1 - \frac{T}{C}\right)^{D} \log{\left(B \right)}}{C \left(1
- \frac{T}{C}\right)}
.. math::
\frac{d^2 Y}{dT^2} = \frac{A B^{- \left(1 - \frac{T}{C}\right)^{D} - 1}
D \left(1 - \frac{T}{C}\right)^{D} \left(D \left(1 - \frac{T}{C}
\right)^{D} \log{\left(B \right)} - D + 1\right) \log{\left(B \right)}}
{C^{2} \left(1 - \frac{T}{C}\right)^{2}}
.. math::
\frac{d^3 Y}{dT^3} = \frac{A B^{- \left(1 - \frac{T}{C}\right)^{D} - 1}
D \left(1 - \frac{T}{C}\right)^{D} \left(D^{2} \left(1 - \frac{T}{C}
\right)^{2 D} \log{\left(B \right)}^{2} - 3 D^{2} \left(1 - \frac{T}{C}
\right)^{D} \log{\left(B \right)} + D^{2} + 3 D \left(1 - \frac{T}{C}
\right)^{D} \log{\left(B \right)} - 3 D + 2\right) \log{\left(B
\right)}}{C^{3} \left(1 - \frac{T}{C}\right)^{3}}
Examples
--------
Hexane molar density; DIPPR coefficients normally in kmol/m^3.
>>> EQ105(300., 0.70824, 0.26411, 507.6, 0.27537)
7.593170096339237
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
return A * B**(-(1. + (1. - T / C)**D))
elif order == 1:
x0 = 1.0 / C
x1 = 1.0 - T * x0
x2 = x1**D
return A * B**(-x2 - 1.0) * D * x0 * x2 * log(B) / x1
elif order == 2:
x0 = 1.0 - T / C
x1 = x0**D
x2 = D * x1 * log(B)
den = 1.0 / (C * x0)
return A * B**(-x1 - 1.0) * x2 * (1.0 - D + x2) * den * den
elif order == 3:
x0 = 1.0 - T / C
x1 = x0**D
x2 = 3.0 * D
x3 = D * D
x4 = log(B)
x5 = x1 * x4
den = 1.0 / (C * x0)
return A * B**(-x1 - 1.0) * D * x5 * (x0**(2.0 * D) * x3 * x4 * x4 +
x2 * x5 - x2 - 3.0 * x3 * x5 +
x3 + 2.0) * den * den * den
else:
raise ValueError(order_not_found_msg)
def calculate(self, T, method):
r'''Method to calculate heat of vaporization of a liquid at
temperature `T` with a given method.
This method has no exception handling; see :obj:`T_dependent_property <thermo.utils.TDependentProperty.T_dependent_property>`
for that.
Parameters
----------
T : float
Temperature at which to calculate heat of vaporization, [K]
method : str
Name of the method to use
Returns
-------
Hvap : float
Heat of vaporization of the liquid at T, [J/mol]
'''
if method == BESTFIT:
if T > self.poly_fit_Tc:
Hvap = 0
else:
Hvap = horner(self.poly_fit_coeffs,
log(1.0 - T / self.poly_fit_Tc))
elif method == COOLPROP:
Hvap = PropsSI('HMOLAR', 'T', T, 'Q', 1, self.CASRN) - PropsSI(
'HMOLAR', 'T', T, 'Q', 0, self.CASRN)
elif method == DIPPR_PERRY_8E:
Hvap = EQ106(T, *self.Perrys2_150_coeffs)
# CSP methods
elif method == VDI_PPDS:
Hvap = PPDS12(T, self.VDI_PPDS_Tc, *self.VDI_PPDS_coeffs)
elif method == ALIBAKHSHI:
Hvap = Alibakhshi(T=T, Tc=self.Tc, C=self.Alibakhshi_C)
elif method == MORGAN_KOBAYASHI:
Hvap = MK(T, self.Tc, self.omega)
elif method == SIVARAMAN_MAGEE_KOBAYASHI:
Hvap = SMK(T, self.Tc, self.omega)
elif method == VELASCO:
Hvap = Velasco(T, self.Tc, self.omega)
elif method == PITZER:
Hvap = Pitzer(T, self.Tc, self.omega)
elif method == CLAPEYRON:
Zg = self.Zg(T) if hasattr(self.Zg, '__call__') else self.Zg
Zl = self.Zl(T) if hasattr(self.Zl, '__call__') else self.Zl
Psat = self.Psat(T) if hasattr(self.Psat,
'__call__') else self.Psat
if Zg:
if Zl:
dZ = Zg - Zl
else:
dZ = Zg
Hvap = Clapeyron(T, self.Tc, self.Pc, dZ=dZ, Psat=Psat)
# CSP methods at Tb only
elif method == RIEDEL:
Hvap = Riedel(self.Tb, self.Tc, self.Pc)
elif method == CHEN:
Hvap = Chen(self.Tb, self.Tc, self.Pc)
elif method == VETERE:
Hvap = Vetere(self.Tb, self.Tc, self.Pc)
elif method == LIU:
Hvap = Liu(self.Tb, self.Tc, self.Pc)
# Individual data point methods
elif method == CRC_HVAP_TB:
Hvap = self.CRC_HVAP_TB_Hvap
elif method == CRC_HVAP_298:
Hvap = self.CRC_HVAP_298
elif method == GHARAGHEIZI_HVAP_298:
Hvap = self.GHARAGHEIZI_HVAP_298_Hvap
elif method in self.tabular_data:
Hvap = self.interpolate(T, method)
# Adjust with the watson equation if estimated at Tb or Tc only
if method in self.boiling_methods or (self.Tc and method in [
CRC_HVAP_TB, CRC_HVAP_298, GHARAGHEIZI_HVAP_298
]):
if method in self.boiling_methods:
Tref = self.Tb
elif method == CRC_HVAP_TB:
Tref = self.CRC_HVAP_TB_Tb
elif method in [CRC_HVAP_298, GHARAGHEIZI_HVAP_298]:
Tref = 298.15
Hvap = Watson(T, Hvap, Tref, self.Tc, self.Watson_exponent)
return Hvap
def EQ127(T, A, B, C, D, E, F, G, order=0):
r'''DIPPR Equation #127. Rarely used, and then only in calculating
ideal-gas heat capacity. All 7 parameters are required.
.. math::
Y = A+B\left[\frac{\left(\frac{C}{T}\right)^2\exp\left(\frac{C}{T}
\right)}{\left(\exp\frac{C}{T}-1 \right)^2}\right]
+D\left[\frac{\left(\frac{E}{T}\right)^2\exp\left(\frac{E}{T}\right)}
{\left(\exp\frac{E}{T}-1 \right)^2}\right]
+F\left[\frac{\left(\frac{G}{T}\right)^2\exp\left(\frac{G}{T}\right)}
{\left(\exp\frac{G}{T}-1 \right)^2}\right]
Parameters
----------
T : float
Temperature, [K]
A-G : 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, integral with respect to T, and integral
over T with respect to T are computed as follows. All expressions can be
obtained with SymPy readily.
.. math::
\frac{d Y}{dT} = - \frac{B C^{3} e^{\frac{C}{T}}}{T^{4}
\left(e^{\frac{C}{T}} - 1\right)^{2}} + \frac{2 B C^{3}
e^{\frac{2 C}{T}}}{T^{4} \left(e^{\frac{C}{T}} - 1\right)^{3}}
- \frac{2 B C^{2} e^{\frac{C}{T}}}{T^{3} \left(e^{\frac{C}{T}}
- 1\right)^{2}} - \frac{D E^{3} e^{\frac{E}{T}}}{T^{4}
\left(e^{\frac{E}{T}} - 1\right)^{2}} + \frac{2 D E^{3}
e^{\frac{2 E}{T}}}{T^{4} \left(e^{\frac{E}{T}} - 1\right)^{3}}
- \frac{2 D E^{2} e^{\frac{E}{T}}}{T^{3} \left(e^{\frac{E}{T}}
- 1\right)^{2}} - \frac{F G^{3} e^{\frac{G}{T}}}{T^{4}
\left(e^{\frac{G}{T}} - 1\right)^{2}} + \frac{2 F G^{3}
e^{\frac{2 G}{T}}}{T^{4} \left(e^{\frac{G}{T}} - 1\right)^{3}}
- \frac{2 F G^{2} e^{\frac{G}{T}}}{T^{3} \left(e^{\frac{G}{T}}
- 1\right)^{2}}
.. math::
\int Y dT = A T + \frac{B C^{2}}{C e^{\frac{C}{T}} - C}
+ \frac{D E^{2}}{E e^{\frac{E}{T}} - E}
+ \frac{F G^{2}}{G e^{\frac{G}{T}} - G}
.. math::
\int \frac{Y}{T} dT = A \ln{\left (T \right )} + B C^{2} \left(
\frac{1}{C T e^{\frac{C}{T}} - C T} + \frac{1}{C T} - \frac{1}{C^{2}}
\ln{\left (e^{\frac{C}{T}} - 1 \right )}\right) + D E^{2} \left(
\frac{1}{E T e^{\frac{E}{T}} - E T} + \frac{1}{E T} - \frac{1}{E^{2}}
\ln{\left (e^{\frac{E}{T}} - 1 \right )}\right) + F G^{2} \left(
\frac{1}{G T e^{\frac{G}{T}} - G T} + \frac{1}{G T} - \frac{1}{G^{2}}
\ln{\left (e^{\frac{G}{T}} - 1 \right )}\right)
Examples
--------
Ideal gas heat capacity of methanol; DIPPR coefficients normally in
J/kmol/K
>>> EQ127(20., 3.3258E4, 3.6199E4, 1.2057E3, 1.5373E7, 3.2122E3, -1.5318E7, 3.2122E3)
33258.0
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
return (A + B * ((C / T)**2 * exp(C / T) / (exp(C / T) - 1)**2) + D *
((E / T)**2 * exp(E / T) / (exp(E / T) - 1)**2) + F *
((G / T)**2 * exp(G / T) / (exp(G / T) - 1)**2))
elif order == 1:
return (-B * C**3 * exp(C / T) / (T**4 * (exp(C / T) - 1)**2) +
2 * B * C**3 * exp(2 * C / T) / (T**4 * (exp(C / T) - 1)**3) -
2 * B * C**2 * exp(C / T) / (T**3 * (exp(C / T) - 1)**2) -
D * E**3 * exp(E / T) / (T**4 * (exp(E / T) - 1)**2) +
2 * D * E**3 * exp(2 * E / T) / (T**4 * (exp(E / T) - 1)**3) -
2 * D * E**2 * exp(E / T) / (T**3 * (exp(E / T) - 1)**2) -
F * G**3 * exp(G / T) / (T**4 * (exp(G / T) - 1)**2) +
2 * F * G**3 * exp(2 * G / T) / (T**4 * (exp(G / T) - 1)**3) -
2 * F * G**2 * exp(G / T) / (T**3 * (exp(G / T) - 1)**2))
elif order == -1:
return (A * T + B * C**2 / (C * exp(C / T) - C) + D * E**2 /
(E * exp(E / T) - E) + F * G**2 / (G * exp(G / T) - G))
elif order == -1j:
return (A * log(T) + B * C**2 *
(1 / (C * T * exp(C / T) - C * T) + 1 /
(C * T) - log(exp(C / T) - 1) / C**2) + D * E**2 *
(1 / (E * T * exp(E / T) - E * T) + 1 /
(E * T) - log(exp(E / T) - 1) / E**2) + F * G**2 *
(1 / (G * T * exp(G / T) - G * T) + 1 /
(G * T) - log(exp(G / T) - 1) / G**2))
else:
raise ValueError(order_not_found_msg)
def BVirial_Tsonopoulos_extended(T, Tc, Pc, omega, a=0, b=0, species_type='',
dipole=0, order=0):
r'''Calculates the second virial coefficient using the
comprehensive model in [1]_. See the notes for the calculation of `a` and
`b`.
.. math::
\frac{BP_c}{RT_c} = B^{(0)} + \omega B^{(1)} + a B^{(2)} + b B^{(3)}
.. math::
B^{(0)}=0.1445-0.33/T_r-0.1385/T_r^2-0.0121/T_r^3
.. math::
B^{(1)} = 0.0637+0.331/T_r^2-0.423/T_r^3 -0.423/T_r^3 - 0.008/T_r^8
.. math::
B^{(2)} = 1/T_r^6
.. math::
B^{(3)} = -1/T_r^8
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of the fluid [Pa]
omega : float
Acentric factor for fluid, [-]
a : float, optional
Fit parameter, calculated based on species_type if a is not given and
species_type matches on of the supported chemical classes.
b : float, optional
Fit parameter, calculated based on species_type if a is not given and
species_type matches on of the supported chemical classes.
species_type : str, optional
One of .
dipole : float
dipole moment, optional, [Debye]
order : int, optional
Order of the calculation. 0 for the calculation of B itself; for 1/2/3,
the first/second/third derivative of B with respect to temperature; and
for -1/-2, the first/second indefinite integral of B with respect to
temperature. No other integrals or derivatives are implemented, and an
exception will be raised if any other order is given.
Returns
-------
B : float
Second virial coefficient in density form or its integral/derivative if
specified, [m^3/mol or m^3/mol/K^order]
Notes
-----
Analytical models for derivatives and integrals are available for orders
-2, -1, 1, 2, and 3, all obtained with SymPy.
To calculate `a` or `b`, the following rules are used:
For 'simple' or 'normal' fluids:
.. math::
a = 0
.. math::
b = 0
For 'ketone', 'aldehyde', 'alkyl nitrile', 'ether', 'carboxylic acid',
or 'ester' types of chemicals:
.. math::
a = -2.14\times 10^{-4} \mu_r - 4.308 \times 10^{-21} (\mu_r)^8
.. math::
b = 0
For 'alkyl halide', 'mercaptan', 'sulfide', or 'disulfide' types of
chemicals:
.. math::
a = -2.188\times 10^{-4} (\mu_r)^4 - 7.831 \times 10^{-21} (\mu_r)^8
.. math::
b = 0
For 'alkanol' types of chemicals (except methanol):
.. math::
a = 0.0878
.. math::
b = 0.00908 + 0.0006957 \mu_r
For methanol:
.. math::
a = 0.0878
.. math::
b = 0.0525
For water:
.. math::
a = -0.0109
.. math::
b = 0
If required, the form of dipole moment used in the calculation of some
types of `a` and `b` values is as follows:
.. math::
\mu_r = 100000\frac{\mu^2(Pc/101325.0)}{Tc^2}
For first temperature derivative of B:
.. math::
\frac{d B^{(0)}}{dT} = \frac{33 Tc}{100 T^{2}} + \frac{277 Tc^{2}}{1000 T^{3}} + \frac{363 Tc^{3}}{10000 T^{4}} + \frac{607 Tc^{8}}{125000 T^{9}}
.. math::
\frac{d B^{(1)}}{dT} = - \frac{331 Tc^{2}}{500 T^{3}} + \frac{1269 Tc^{3}}{1000 T^{4}} + \frac{8 Tc^{8}}{125 T^{9}}
.. math::
\frac{d B^{(2)}}{dT} = - \frac{6 Tc^{6}}{T^{7}}
.. math::
\frac{d B^{(3)}}{dT} = \frac{8 Tc^{8}}{T^{9}}
For the second temperature derivative of B:
.. math::
\frac{d^2 B^{(0)}}{dT^2} = - \frac{3 Tc}{125000 T^{3}} \left(27500 + \frac{34625 Tc}{T} + \frac{6050 Tc^{2}}{T^{2}} + \frac{1821 Tc^{7}}{T^{7}}\right)
.. math::
\frac{d^2 B^{(1)}}{dT^2} = \frac{3 Tc^{2}}{500 T^{4}} \left(331 - \frac{846 Tc}{T} - \frac{96 Tc^{6}}{T^{6}}\right)
.. math::
\frac{d^2 B^{(2)}}{dT^2} = \frac{42 Tc^{6}}{T^{8}}
.. math::
\frac{d^2 B^{(3)}}{dT^2} = - \frac{72 Tc^{8}}{T^{10}}
For the third temperature derivative of B:
.. math::
\frac{d^3 B^{(0)}}{dT^3} = \frac{3 Tc}{12500 T^{4}} \left(8250 + \frac{13850 Tc}{T} + \frac{3025 Tc^{2}}{T^{2}} + \frac{1821 Tc^{7}}{T^{7}}\right)
.. math::
\frac{d^3 B^{(1)}}{dT^3} = \frac{3 Tc^{2}}{250 T^{5}} \left(-662 + \frac{2115 Tc}{T} + \frac{480 Tc^{6}}{T^{6}}\right)
.. math::
\frac{d^3 B^{(2)}}{dT^3} = - \frac{336 Tc^{6}}{T^{9}}
.. math::
\frac{d^3 B^{(3)}}{dT^3} = \frac{720 Tc^{8}}{T^{11}}
For the first indefinite integral of B:
.. math::
\int{B^{(0)}} dT = \frac{289 T}{2000} - \frac{33 Tc}{100} \ln{\left (T \right )} + \frac{1}{7000000 T^{7}} \left(969500 T^{6} Tc^{2} + 42350 T^{5} Tc^{3} + 607 Tc^{8}\right)
.. math::
\int{B^{(1)}} dT = \frac{637 T}{10000} - \frac{1}{70000 T^{7}} \left(23170 T^{6} Tc^{2} - 14805 T^{5} Tc^{3} - 80 Tc^{8}\right)
.. math::
\int{B^{(2)}} dT = - \frac{Tc^{6}}{5 T^{5}}
.. math::
\int{B^{(3)}} dT = \frac{Tc^{8}}{7 T^{7}}
For the second indefinite integral of B:
.. math::
\int\int B^{(0)} dT dT = \frac{289 T^{2}}{4000} - \frac{33 T}{100} Tc \ln{\left (T \right )} + \frac{33 T}{100} Tc + \frac{277 Tc^{2}}{2000} \ln{\left (T \right )} - \frac{1}{42000000 T^{6}} \left(254100 T^{5} Tc^{3} + 607 Tc^{8}\right)
.. math::
\int\int B^{(1)} dT dT = \frac{637 T^{2}}{20000} - \frac{331 Tc^{2}}{1000} \ln{\left (T \right )} - \frac{1}{210000 T^{6}} \left(44415 T^{5} Tc^{3} + 40 Tc^{8}\right)
.. math::
\int\int B^{(2)} dT dT = \frac{Tc^{6}}{20 T^{4}}
.. math::
\int\int B^{(3)} dT dT = - \frac{Tc^{8}}{42 T^{6}}
Examples
--------
Example from Perry's Handbook, 8E, p2-499. Matches to a decimal place.
>>> BVirial_Tsonopoulos_extended(430., 405.65, 11.28E6, 0.252608, a=0, b=0, species_type='ketone', dipole=1.469)
-9.679718337596426e-05
References
----------
.. [1] Tsonopoulos, C., and J. L. Heidman. "From the Virial to the Cubic
Equation of State." Fluid Phase Equilibria 57, no. 3 (1990): 261-76.
doi:10.1016/0378-3812(90)85126-U
.. [2] Tsonopoulos, Constantine, and John H. Dymond. "Second Virial
Coefficients of Normal Alkanes, Linear 1-Alkanols (and Water), Alkyl
Ethers, and Their Mixtures." Fluid Phase Equilibria, International
Workshop on Vapour-Liquid Equilibria and Related Properties in Binary
and Ternary Mixtures of Ethers, Alkanes and Alkanols, 133, no. 1-2
(June 1997): 11-34. doi:10.1016/S0378-3812(97)00058-7.
'''
Tr = T/Tc
if order == 0:
B0 = 0.1445 - 0.33/Tr - 0.1385/Tr**2 - 0.0121/Tr**3 - 0.000607/Tr**8
B1 = 0.0637 + 0.331/Tr**2 - 0.423/Tr**3 - 0.008/Tr**8
B2 = 1./Tr**6
B3 = -1./Tr**8
elif order == 1:
B0 = 33*Tc/(100*T**2) + 277*Tc**2/(1000*T**3) + 363*Tc**3/(10000*T**4) + 607*Tc**8/(125000*T**9)
B1 = -331*Tc**2/(500*T**3) + 1269*Tc**3/(1000*T**4) + 8*Tc**8/(125*T**9)
B2 = -6.0*Tc**6/T**7
B3 = 8.0*Tc**8/T**9
elif order == 2:
B0 = -3*Tc*(27500 + 34625*Tc/T + 6050*Tc**2/T**2 + 1821*Tc**7/T**7)/(125000*T**3)
B1 = 3*Tc**2*(331 - 846*Tc/T - 96*Tc**6/T**6)/(500*T**4)
B2 = 42.0*Tc**6/T**8
B3 = -72.0*Tc**8/T**10
elif order == 3:
B0 = 3*Tc*(8250 + 13850*Tc/T + 3025*Tc**2/T**2 + 1821*Tc**7/T**7)/(12500*T**4)
B1 = 3*Tc**2*(-662 + 2115*Tc/T + 480*Tc**6/T**6)/(250*T**5)
B2 = -336.0*Tc**6/T**9
B3 = 720.0*Tc**8/T**11
elif order == -1:
B0 = 289*T/2000. - 33*Tc*log(T)/100. + (969500*T**6*Tc**2 + 42350*T**5*Tc**3 + 607*Tc**8)/(7000000.*T**7)
B1 = 637*T/10000. - (23170*T**6*Tc**2 - 14805*T**5*Tc**3 - 80*Tc**8)/(70000.*T**7)
B2 = -Tc**6/(5*T**5)
B3 = Tc**8/(7*T**7)
elif order == -2:
B0 = 289*T**2/4000. - 33*T*Tc*log(T)/100. + 33*T*Tc/100. + 277*Tc**2*log(T)/2000. - (254100*T**5*Tc**3 + 607*Tc**8)/(42000000.*T**6)
B1 = 637*T**2/20000. - 331*Tc**2*log(T)/1000. - (44415*T**5*Tc**3 + 40*Tc**8)/(210000.*T**6)
B2 = Tc**6/(20*T**4)
B3 = -Tc**8/(42*T**6)
else:
raise ValueError('Only orders -2, -1, 0, 1, 2 and 3 are supported.')
if a == 0 and b == 0 and species_type != '':
if species_type == 'simple' or species_type == 'normal':
a, b = 0, 0
elif species_type == 'methyl alcohol':
a, b = 0.0878, 0.0525
elif species_type == 'water':
a, b = -0.0109, 0
elif dipole != 0 and Tc != 0 and Pc != 0:
dipole_r = 1E5*dipole**2*(Pc/101325.0)/Tc**2
if (species_type == 'ketone' or species_type == 'aldehyde'
or species_type == 'alkyl nitrile' or species_type == 'ether'
or species_type == 'carboxylic acid' or species_type == 'ester'):
a, b = -2.14E-4*dipole_r-4.308E-21*dipole_r**8, 0
elif (species_type == 'alkyl halide' or species_type == 'mercaptan'
or species_type == 'sulfide' or species_type == 'disulfide'):
a, b = -2.188E-4*dipole_r**4-7.831E-21*dipole_r**8, 0
elif species_type == 'alkanol':
a, b = 0.0878, 0.00908+0.0006957*dipole_r
Br = B0 + omega*B1 + a*B2 + b*B3
return Br*R*Tc/Pc
def BVirial_Tsonopoulos(T, Tc, Pc, omega, order=0):
r'''Calculates the second virial coefficient using the model in [1]_.
.. math::
B_r=B^{(0)}+\omega B^{(1)}
.. math::
B^{(0)}= 0.1445-0.330/T_r - 0.1385/T_r^2 - 0.0121/T_r^3 - 0.000607/T_r^8
.. math::
B^{(1)} = 0.0637+0.331/T_r^2-0.423/T_r^3 -0.423/T_r^3 - 0.008/T_r^8
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of the fluid [Pa]
omega : float
Acentric factor for fluid, [-]
order : int, optional
Order of the calculation. 0 for the calculation of B itself; for 1/2/3,
the first/second/third derivative of B with respect to temperature; and
for -1/-2, the first/second indefinite integral of B with respect to
temperature. No other integrals or derivatives are implemented, and an
exception will be raised if any other order is given.
Returns
-------
B : float
Second virial coefficient in density form or its integral/derivative if
specified, [m^3/mol or m^3/mol/K^order]
Notes
-----
A more complete expression is also available, in
BVirial_Tsonopoulos_extended.
Analytical models for derivatives and integrals are available for orders
-2, -1, 1, 2, and 3, all obtained with SymPy.
For first temperature derivative of B:
.. math::
\frac{d B^{(0)}}{dT} = \frac{33 Tc}{100 T^{2}} + \frac{277 Tc^{2}}{1000 T^{3}} + \frac{363 Tc^{3}}{10000 T^{4}} + \frac{607 Tc^{8}}{125000 T^{9}}
.. math::
\frac{d B^{(1)}}{dT} = - \frac{331 Tc^{2}}{500 T^{3}} + \frac{1269 Tc^{3}}{1000 T^{4}} + \frac{8 Tc^{8}}{125 T^{9}}
For the second temperature derivative of B:
.. math::
\frac{d^2 B^{(0)}}{dT^2} = - \frac{3 Tc}{125000 T^{3}} \left(27500 + \frac{34625 Tc}{T} + \frac{6050 Tc^{2}}{T^{2}} + \frac{1821 Tc^{7}}{T^{7}}\right)
.. math::
\frac{d^2 B^{(1)}}{dT^2} = \frac{3 Tc^{2}}{500 T^{4}} \left(331 - \frac{846 Tc}{T} - \frac{96 Tc^{6}}{T^{6}}\right)
For the third temperature derivative of B:
.. math::
\frac{d^3 B^{(0)}}{dT^3} = \frac{3 Tc}{12500 T^{4}} \left(8250 + \frac{13850 Tc}{T} + \frac{3025 Tc^{2}}{T^{2}} + \frac{1821 Tc^{7}}{T^{7}}\right)
.. math::
\frac{d^3 B^{(1)}}{dT^3} = \frac{3 Tc^{2}}{250 T^{5}} \left(-662 + \frac{2115 Tc}{T} + \frac{480 Tc^{6}}{T^{6}}\right)
For the first indefinite integral of B:
.. math::
\int{B^{(0)}} dT = \frac{289 T}{2000} - \frac{33 Tc}{100} \ln{\left (T \right )} + \frac{1}{7000000 T^{7}} \left(969500 T^{6} Tc^{2} + 42350 T^{5} Tc^{3} + 607 Tc^{8}\right)
.. math::
\int{B^{(1)}} dT = \frac{637 T}{10000} - \frac{1}{70000 T^{7}} \left(23170 T^{6} Tc^{2} - 14805 T^{5} Tc^{3} - 80 Tc^{8}\right)
For the second indefinite integral of B:
.. math::
\int\int B^{(0)} dT dT = \frac{289 T^{2}}{4000} - \frac{33 T}{100} Tc \ln{\left (T \right )} + \frac{33 T}{100} Tc + \frac{277 Tc^{2}}{2000} \ln{\left (T \right )} - \frac{1}{42000000 T^{6}} \left(254100 T^{5} Tc^{3} + 607 Tc^{8}\right)
.. math::
\int\int B^{(1)} dT dT = \frac{637 T^{2}}{20000} - \frac{331 Tc^{2}}{1000} \ln{\left (T \right )} - \frac{1}{210000 T^{6}} \left(44415 T^{5} Tc^{3} + 40 Tc^{8}\right)
Examples
--------
Example matching that in BVirial_Abbott, for isobutane.
>>> BVirial_Tsonopoulos(510., 425.2, 38E5, 0.193)
-0.00020935295404416802
References
----------
.. [1] Tsonopoulos, Constantine. "An Empirical Correlation of Second Virial
Coefficients." AIChE Journal 20, no. 2 (March 1, 1974): 263-72.
doi:10.1002/aic.690200209.
'''
Tr = T/Tc
if order == 0:
B0 = 0.1445 - 0.33/Tr - 0.1385/Tr**2 - 0.0121/Tr**3 - 0.000607/Tr**8
B1 = 0.0637 + 0.331/Tr**2 - 0.423/Tr**3 - 0.008/Tr**8
elif order == 1:
B0 = 33*Tc/(100*T**2) + 277*Tc**2/(1000*T**3) + 363*Tc**3/(10000*T**4) + 607*Tc**8/(125000*T**9)
B1 = -331*Tc**2/(500*T**3) + 1269*Tc**3/(1000*T**4) + 8*Tc**8/(125*T**9)
elif order == 2:
B0 = -3*Tc*(27500 + 34625*Tc/T + 6050*Tc**2/T**2 + 1821*Tc**7/T**7)/(125000*T**3)
B1 = 3*Tc**2*(331 - 846*Tc/T - 96*Tc**6/T**6)/(500*T**4)
elif order == 3:
B0 = 3*Tc*(8250 + 13850*Tc/T + 3025*Tc**2/T**2 + 1821*Tc**7/T**7)/(12500*T**4)
B1 = 3*Tc**2*(-662 + 2115*Tc/T + 480*Tc**6/T**6)/(250*T**5)
elif order == -1:
B0 = 289*T/2000. - 33*Tc*log(T)/100. + (969500*T**6*Tc**2 + 42350*T**5*Tc**3 + 607*Tc**8)/(7000000.*T**7)
B1 = 637*T/10000. - (23170*T**6*Tc**2 - 14805*T**5*Tc**3 - 80*Tc**8)/(70000.*T**7)
elif order == -2:
B0 = 289*T**2/4000. - 33*T*Tc*log(T)/100. + 33*T*Tc/100. + 277*Tc**2*log(T)/2000. - (254100*T**5*Tc**3 + 607*Tc**8)/(42000000.*T**6)
B1 = 637*T**2/20000. - 331*Tc**2*log(T)/1000. - (44415*T**5*Tc**3 + 40*Tc**8)/(210000.*T**6)
else:
raise ValueError('Only orders -2, -1, 0, 1, 2 and 3 are supported.')
Br = (B0+omega*B1)
return Br*R*Tc/Pc
def BVirial_Pitzer_Curl(T, Tc, Pc, omega, order=0):
r'''Calculates the second virial coefficient using the model in [1]_.
Designed for simple calculations.
.. math::
B_r=B^{(0)}+\omega B^{(1)}
.. math::
B^{(0)}=0.1445-0.33/T_r-0.1385/T_r^2-0.0121/T_r^3
.. math::
B^{(1)} = 0.073+0.46/T_r-0.5/T_r^2 -0.097/T_r^3 - 0.0073/T_r^8
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of the fluid [Pa]
omega : float
Acentric factor for fluid, [-]
order : int, optional
Order of the calculation. 0 for the calculation of B itself; for 1/2/3,
the first/second/third derivative of B with respect to temperature; and
for -1/-2, the first/second indefinite integral of B with respect to
temperature. No other integrals or derivatives are implemented, and an
exception will be raised if any other order is given.
Returns
-------
B : float
Second virial coefficient in density form or its integral/derivative if
specified, [m^3/mol or m^3/mol/K^order]
Notes
-----
Analytical models for derivatives and integrals are available for orders
-2, -1, 1, 2, and 3, all obtained with SymPy.
For first temperature derivative of B:
.. math::
\frac{d B^{(0)}}{dT} = \frac{33 Tc}{100 T^{2}} + \frac{277 Tc^{2}}{1000 T^{3}} + \frac{363 Tc^{3}}{10000 T^{4}}
.. math::
\frac{d B^{(1)}}{dT} = - \frac{23 Tc}{50 T^{2}} + \frac{Tc^{2}}{T^{3}} + \frac{291 Tc^{3}}{1000 T^{4}} + \frac{73 Tc^{8}}{1250 T^{9}}
For the second temperature derivative of B:
.. math::
\frac{d^2 B^{(0)}}{dT^2} = - \frac{3 Tc}{5000 T^{3}} \left(1100 + \frac{1385 Tc}{T} + \frac{242 Tc^{2}}{T^{2}}\right)
.. math::
\frac{d^2 B^{(1)}}{dT^2} = \frac{Tc}{T^{3}} \left(\frac{23}{25} - \frac{3 Tc}{T} - \frac{291 Tc^{2}}{250 T^{2}} - \frac{657 Tc^{7}}{1250 T^{7}}\right)
For the third temperature derivative of B:
.. math::
\frac{d^3 B^{(0)}}{dT^3} = \frac{3 Tc}{500 T^{4}} \left(330 + \frac{554 Tc}{T} + \frac{121 Tc^{2}}{T^{2}}\right)
.. math::
\frac{d^3 B^{(1)}}{dT^3} = \frac{3 Tc}{T^{4}} \left(- \frac{23}{25} + \frac{4 Tc}{T} + \frac{97 Tc^{2}}{50 T^{2}} + \frac{219 Tc^{7}}{125 T^{7}}\right)
For the first indefinite integral of B:
.. math::
\int{B^{(0)}} dT = \frac{289 T}{2000} - \frac{33 Tc}{100} \ln{\left (T \right )} + \frac{1}{20000 T^{2}} \left(2770 T Tc^{2} + 121 Tc^{3}\right)
.. math::
\int{B^{(1)}} dT = \frac{73 T}{1000} + \frac{23 Tc}{50} \ln{\left (T \right )} + \frac{1}{70000 T^{7}} \left(35000 T^{6} Tc^{2} + 3395 T^{5} Tc^{3} + 73 Tc^{8}\right)
For the second indefinite integral of B:
.. math::
\int\int B^{(0)} dT dT = \frac{289 T^{2}}{4000} - \frac{33 T}{100} Tc \ln{\left (T \right )} + \frac{33 T}{100} Tc + \frac{277 Tc^{2}}{2000} \ln{\left (T \right )} - \frac{121 Tc^{3}}{20000 T}
.. math::
\int\int B^{(1)} dT dT = \frac{73 T^{2}}{2000} + \frac{23 T}{50} Tc \ln{\left (T \right )} - \frac{23 T}{50} Tc + \frac{Tc^{2}}{2} \ln{\left (T \right )} - \frac{1}{420000 T^{6}} \left(20370 T^{5} Tc^{3} + 73 Tc^{8}\right)
Examples
--------
Example matching that in BVirial_Abbott, for isobutane.
>>> BVirial_Pitzer_Curl(510., 425.2, 38E5, 0.193)
-0.00020845362479301725
References
----------
.. [1] Pitzer, Kenneth S., and R. F. Curl. "The Volumetric and
Thermodynamic Properties of Fluids. III. Empirical Equation for the
Second Virial Coefficient1." Journal of the American Chemical Society
79, no. 10 (May 1, 1957): 2369-70. doi:10.1021/ja01567a007.
'''
Tr = T/Tc
if order == 0:
B0 = 0.1445 - 0.33/Tr - 0.1385/Tr**2 - 0.0121/Tr**3
B1 = 0.073 + 0.46/Tr - 0.5/Tr**2 - 0.097/Tr**3 - 0.0073/Tr**8
elif order == 1:
B0 = Tc*(3300*T**2 + 2770*T*Tc + 363*Tc**2)/(10000*T**4)
B1 = Tc*(-2300*T**7 + 5000*T**6*Tc + 1455*T**5*Tc**2 + 292*Tc**7)/(5000*T**9)
elif order == 2:
B0 = -3*Tc*(1100*T**2 + 1385*T*Tc + 242*Tc**2)/(5000*T**5)
B1 = Tc*(1150*T**7 - 3750*T**6*Tc - 1455*T**5*Tc**2 - 657*Tc**7)/(1250*T**10)
elif order == 3:
B0 = 3*Tc*(330*T**2 + 554*T*Tc + 121*Tc**2)/(500*T**6)
B1 = 3*Tc*(-230*T**7 + 1000*T**6*Tc + 485*T**5*Tc**2 + 438*Tc**7)/(250*T**11)
elif order == -1:
B0 = 289*T/2000 - 33*Tc*log(T)/100 + (2770*T*Tc**2 + 121*Tc**3)/(20000*T**2)
B1 = 73*T/1000 + 23*Tc*log(T)/50 + (35000*T**6*Tc**2 + 3395*T**5*Tc**3 + 73*Tc**8)/(70000*T**7)
elif order == -2:
B0 = 289*T**2/4000 - 33*T*Tc*log(T)/100 + 33*T*Tc/100 + 277*Tc**2*log(T)/2000 - 121*Tc**3/(20000*T)
B1 = 73*T**2/2000 + 23*T*Tc*log(T)/50 - 23*T*Tc/50 + Tc**2*log(T)/2 - (20370*T**5*Tc**3 + 73*Tc**8)/(420000*T**6)
else:
raise ValueError('Only orders -2, -1, 0, 1, 2 and 3 are supported.')
Br = B0 + omega*B1
return Br*R*Tc/Pc
def EQ104(T, A, B, C, D, E, order=0):
r'''DIPPR Equation #104. Often used in calculating second virial
coefficients of gases. All 5 parameters are required.
C, D, and E are normally large values.
.. math::
Y = A + \frac{B}{T} + \frac{C}{T^3} + \frac{D}{T^8} + \frac{E}{T^9}
Parameters
----------
T : float
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, integral with respect to T, and integral
over T with respect to T are computed as follows. All expressions can be
obtained with SymPy readily.
.. math::
\frac{d Y}{dT} = - \frac{B}{T^{2}} - \frac{3 C}{T^{4}}
- \frac{8 D}{T^{9}} - \frac{9 E}{T^{10}}
.. math::
\int Y dT = A T + B \log{\left (T \right )} - \frac{1}{56 T^{8}}
\left(28 C T^{6} + 8 D T + 7 E\right)
.. math::
\int \frac{Y}{T} dT = A \log{\left (T \right )} - \frac{1}{72 T^{9}}
\left(72 B T^{8} + 24 C T^{6} + 9 D T + 8 E\right)
Examples
--------
Water second virial coefficient; DIPPR coefficients normally dimensionless.
>>> EQ104(300, 0.02222, -26.38, -16750000, -3.894E19, 3.133E21)
-1.1204179007265156
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
T2 = T * T
return A + (B + (C + (D + E / T) / (T2 * T2 * T)) / T2) / T
elif order == 1:
T2 = T * T
T4 = T2 * T2
return (-B + (-3 * C + (-8 * D - 9 * E / T) / (T4 * T)) / T2) / T2
elif order == -1:
return A * T + B * log(T) - (28 * C * T**6 + 8 * D * T +
7 * E) / (56 * T**8)
elif order == -1j:
return A * log(T) - (72 * B * T**8 + 24 * C * T**6 + 9 * D * T +
8 * E) / (72 * T**9)
else:
raise ValueError(order_not_found_msg)
def EQ106(T, Tc, A, B, C=0.0, D=0.0, E=0.0, order=0):
r'''DIPPR Equation #106. Often used in calculating liquid surface tension,
and heat of vaporization.
Only parameters A and B parameters are required; many fits include no
further parameters. Critical temperature is also required.
.. math::
Y = A(1-T_r)^{B + C T_r + D T_r^2 + E T_r^3}
.. math::
Tr = \frac{T}{Tc}
Parameters
----------
T : float
Temperature, [K]
Tc : float
Critical temperature, [K]
A-D : 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, 2, and 3, that derivative of the property 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]
Notes
-----
This form is used by Yaws with only the parameters `A` and `B`.
The integral could not be found, but the integral over T actually could,
again in terms of hypergeometric functions.
.. math::
\frac{d Y}{dT} = A \left(- \frac{T}{T_{c}} + 1\right)^{B + \frac{C T}
{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}} \left(
\left(\frac{C}{T_{c}} + \frac{2 D T}{T_{c}^{2}} + \frac{3 e T^{2}}
{T_{c}^{3}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} - \frac{B
+ \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}
{T_{c}^{3}}}{T_{c} \left(- \frac{T}{T_{c}} + 1\right)}\right)
.. math::
\frac{d^2 Y}{dT^2} = \frac{A \left(- \frac{T}{T_{c}} + 1\right)^{B
+ \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}
{T_{c}^{3}}} \left(2 \left(D + \frac{3 e T}{T_{c}}\right) \log{\left(
- \frac{T}{T_{c}} + 1 \right)} + \left(\left(C + \frac{2 D T}{T_{c}}
+ \frac{3 e T^{2}}{T_{c}^{2}}\right) \log{\left(- \frac{T}{T_{c}}
+ 1 \right)} + \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}}
+ \frac{e T^{3}}{T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right)^{2}
+ \frac{2 \left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}
\right)}{\frac{T}{T_{c}} - 1} - \frac{B + \frac{C T}{T_{c}} + \frac{D
T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\left(\frac{T}{T_{c}}
- 1\right)^{2}}\right)}{T_{c}^{2}}
.. math::
\frac{d^3 Y}{dT^3} = \frac{A \left(- \frac{T}{T_{c}} + 1\right)^{B
+ \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}
{T_{c}^{3}}} \left(\frac{6 \left(D + \frac{3 e T}{T_{c}}\right)}
{\frac{T}{T_{c}} - 1} + \left(\left(C + \frac{2 D T}{T_{c}}
+ \frac{3 e T^{2}}{T_{c}^{2}}\right) \log{\left(- \frac{T}{T_{c}}
+ 1 \right)} + \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}}
+ \frac{e T^{3}}{T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right)^{3}
+ 3 \left(\left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}
\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{B
+ \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}
{T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right) \left(2 \left(D + \frac{3 e T}
{T_{c}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{2
\left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}\right)}
{\frac{T}{T_{c}} - 1} - \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}
{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\left(\frac{T}{T_{c}}
- 1\right)^{2}}\right) + 6 e \log{\left(- \frac{T}{T_{c}} + 1 \right)}
- \frac{3 \left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}
\right)}{\left(\frac{T}{T_{c}} - 1\right)^{2}} + \frac{2 \left(B
+ \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}
{T_{c}^{3}}\right)}{\left(\frac{T}{T_{c}} - 1\right)^{3}}\right)}
{T_{c}^{3}}
Examples
--------
Water surface tension; DIPPR coefficients normally in Pa*s.
>>> EQ106(300, 647.096, 0.17766, 2.567, -3.3377, 1.9699)
0.07231499373541
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
Tr = T / Tc
return A * (1. - Tr)**(B + Tr * (C + Tr * (D + E * Tr)))
elif order == 1:
x0 = 1.0 / Tc
x1 = T * x0
x2 = 1.0 - x1
x3 = E * x1
x4 = C + x1 * (D + x3)
x5 = B + x1 * x4
return A * x0 * x2**x5 * (x5 / (x1 - 1.0) +
(x1 * (D + 2.0 * x3) + x4) * log(x2))
elif order == 2:
x0 = T / Tc
x1 = 1.0 - x0
x2 = E * x0
x3 = C + x0 * (D + x2)
x4 = B + x0 * x3
x5 = log(x1)
x6 = x0 - 1.0
x7 = 1.0 / x6
x8 = x0 * (D + 2.0 * x2) + x3
return (A * x1**x4 *
(-x4 / x6**2 + 2 * x5 * (D + 3.0 * x2) + 2.0 * x7 * x8 +
(x4 * x7 + x5 * x8)**2) / Tc**2)
elif order == 3:
x0 = T / Tc
x1 = 1.0 - x0
x2 = E * x0
x3 = C + x0 * (D + x2)
x4 = B + x0 * x3
x5 = log(x1)
x6 = D + 3.0 * x2
x7 = x0 - 1.0
x8 = 1 / x7
x9 = x7**(-2)
x10 = x0 * (D + 2.0 * x2) + x3
x11 = x10 * x5 + x4 * x8
return (A * x1**x4 * (-3 * x10 * x9 + x11**3 + 3 * x11 *
(2 * x10 * x8 - x4 * x9 + 2 * x5 * x6) +
2 * x4 / x7**3 + 6 * E * x5 + 6 * x6 * x8) /
Tc**3)
else:
raise ValueError(order_not_found_msg)
def EQ107(T, A=0, B=0, C=0, D=0, E=0, order=0):
r'''DIPPR Equation #107. Often used in calculating ideal-gas heat capacity.
All 5 parameters are required.
Also called the Aly-Lee equation.
.. math::
Y = A + B\left[\frac{C/T}{\sinh(C/T)}\right]^2 + D\left[\frac{E/T}{
\cosh(E/T)}\right]^2
Parameters
----------
T : float
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, integral with respect to T, and integral
over T with respect to T are computed as follows. The derivative is
obtained via SymPy; the integrals from Wolfram Alpha.
.. math::
\frac{d Y}{dT} = \frac{2 B C^{3} \cosh{\left (\frac{C}{T} \right )}}
{T^{4} \sinh^{3}{\left (\frac{C}{T} \right )}} - \frac{2 B C^{2}}{T^{3}
\sinh^{2}{\left (\frac{C}{T} \right )}} + \frac{2 D E^{3} \sinh{\left
(\frac{E}{T} \right )}}{T^{4} \cosh^{3}{\left (\frac{E}{T} \right )}}
- \frac{2 D E^{2}}{T^{3} \cosh^{2}{\left (\frac{E}{T} \right )}}
.. math::
\int Y dT = A T + \frac{B C}{\tanh{\left (\frac{C}{T} \right )}}
- D E \tanh{\left (\frac{E}{T} \right )}
.. math::
\int \frac{Y}{T} dT = A \log{\left (T \right )} + \frac{B C}{T \tanh{
\left (\frac{C}{T} \right )}} - B \log{\left (\sinh{\left (\frac{C}{T}
\right )} \right )} - \frac{D E}{T} \tanh{\left (\frac{E}{T} \right )}
+ D \log{\left (\cosh{\left (\frac{E}{T} \right )} \right )}
Examples
--------
Water ideal gas molar heat capacity; DIPPR coefficients normally in
J/kmol/K
>>> EQ107(300., 33363., 26790., 2610.5, 8896., 1169.)
33585.90452768923
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
.. [2] Aly, Fouad A., and Lloyd L. Lee. "Self-Consistent Equations for
Calculating the Ideal Gas Heat Capacity, Enthalpy, and Entropy." Fluid
Phase Equilibria 6, no. 3 (January 1, 1981): 169-79.
doi:10.1016/0378-3812(81)85002-9.
'''
if order == 0:
return A + B * ((C / T) / sinh(C / T))**2 + D * (
(E / T) / cosh(E / T))**2
elif order == 1:
return (2 * B * C**3 * cosh(C / T) / (T**4 * sinh(C / T)**3) -
2 * B * C**2 / (T**3 * sinh(C / T)**2) +
2 * D * E**3 * sinh(E / T) / (T**4 * cosh(E / T)**3) -
2 * D * E**2 / (T**3 * cosh(E / T)**2))
elif order == -1:
return A * T + B * C / tanh(C / T) - D * E * tanh(E / T)
elif order == -1j:
return (A * log(T) + B * C / tanh(C / T) / T - B * log(sinh(C / T)) -
D * E * tanh(E / T) / T + D * log(cosh(E / T)))
else:
raise ValueError(order_not_found_msg)
def EQ115(T, A, B, C=0, D=0, E=0, order=0):
r'''DIPPR Equation #115. No major uses; has been used as an alternate
liquid viscosity expression, and as a model for vapor pressure.
Only parameters A and B are required.
.. math::
Y = \exp\left(A + \frac{B}{T} + C\ln T + D T^2 + \frac{E}{T^2}\right)
Parameters
----------
T : float
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, 2, and 3, that derivative of the property 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]
Notes
-----
No coefficients found for this expression.
This function is not integrable for either dT or Y/T dT.
.. math::
\frac{d Y}{dT} = \left(- \frac{B}{T^{2}} + \frac{C}{T} + 2 D T
- \frac{2 E}{T^{3}}\right) e^{A + \frac{B}{T} + C \log{\left(T \right)}
+ D T^{2} + \frac{E}{T^{2}}}
.. math::
\frac{d^2 Y}{dT^2} = \left(\frac{2 B}{T^{3}} - \frac{C}{T^{2}} + 2 D
+ \frac{6 E}{T^{4}} + \left(\frac{B}{T^{2}} - \frac{C}{T} - 2 D T
+ \frac{2 E}{T^{3}}\right)^{2}\right) e^{A + \frac{B}{T}
+ C \log{\left(T \right)} + D T^{2} + \frac{E}{T^{2}}}
.. math::
\frac{d^3 Y}{dT^3} =- \left(3 \left(\frac{2 B}{T^{3}} - \frac{C}{T^{2}}
+ 2 D + \frac{6 E}{T^{4}}\right) \left(\frac{B}{T^{2}} - \frac{C}{T}
- 2 D T + \frac{2 E}{T^{3}}\right) + \left(\frac{B}{T^{2}}
- \frac{C}{T} - 2 D T + \frac{2 E}{T^{3}}\right)^{3} + \frac{2 \left(
\frac{3 B}{T} - C + \frac{12 E}{T^{2}}\right)}{T^{3}}\right)
e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{2} + \frac{E}{T^{2}}}
References
----------
.. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
DIPPR/AIChE
'''
if order == 0:
return exp(A + B / T + C * log(T) + D * T**2 + E / T**2)
elif order == 1:
x0 = T**2
x1 = 1 / x0
x2 = 1 / T
return (-(B * x1 - C * x2 - 2 * D * T + 2 * E / T**3) *
exp(A + B * x2 + C * log(T) + D * x0 + E * x1))
elif order == 2:
x0 = 1 / T
x1 = T**2
x2 = 1 / x1
x3 = 2 * D
x4 = 2 / T**3
return (B * x4 - C * x2 + 6 * E / T**4 + x3 +
(B * x2 - C * x0 + E * x4 - T * x3)**
2) * exp(A + B * x0 + C * log(T) + D * x1 + E * x2)
elif order == 3:
x0 = 1 / T
x1 = B * x0
x2 = T**2
x3 = 1 / x2
x4 = E * x3
x5 = 2 / T**3
x6 = 2 * D
x7 = B * x3 - C * x0 + E * x5 - T * x6
return (-(x5 * (-C + 3 * x1 + 12 * x4) + x7**3 + 3 * x7 *
(B * x5 - C * x3 + 6 * E / T**4 + x6)) *
exp(A + C * log(T) + D * x2 + x1 + x4))
else:
raise ValueError(order_not_found_msg)
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 Sanjari(T, Tc, Pc, omega):
r'''Calculates vapor pressure of a fluid at arbitrary temperatures using a
CSP relationship by [1]_. Requires a chemical's critical temperature,
pressure, and acentric factor. Although developed for refrigerants,
this model should have some general predictive ability.
The vapor pressure of a chemical at `T` is given by:
.. math::
P^{sat} = P_c\exp(f^{(0)} + \omega f^{(1)} + \omega^2 f^{(2)})
.. math::
f^{(0)} = a_1 + \frac{a_2}{T_r} + a_3\ln T_r + a_4 T_r^{1.9}
.. math::
f^{(1)} = a_5 + \frac{a_6}{T_r} + a_7\ln T_r + a_8 T_r^{1.9}
.. math::
f^{(2)} = a_9 + \frac{a_{10}}{T_r} + a_{11}\ln T_r + a_{12} T_r^{1.9}
Parameters
----------
T : float
Temperature of fluid [K]
Tc : float
Critical temperature of fluid [K]
Pc : float
Critical pressure of fluid [Pa]
omega : float
Acentric factor [-]
Returns
-------
Psat : float
Vapor pressure, [Pa]
Notes
-----
a[1-12] are as follows:
6.83377, -5.76051, 0.90654, -1.16906,
5.32034, -28.1460, -58.0352, 23.57466,
18.19967, 16.33839, 65.6995, -35.9739.
For a claimed fluid not included in the regression, R128, the claimed AARD
was 0.428%. A re-calculation using 200 data points from 125.45 K to
343.90225 K evenly spaced by 1.09775 K as generated by NIST Webbook April
2016 produced an AARD of 0.644%. It is likely that the author's regression
used more precision in its coefficients than was shown here. Nevertheless,
the function is reproduced as shown in [1]_.
For Tc=808 K, Pc=1100000 Pa, omega=1.1571, this function actually declines
after 770 K.
Examples
--------
>>> Sanjari(347.2, 617.1, 36E5, 0.299)
13651.916109552523
References
----------
.. [1] Sanjari, Ehsan, Mehrdad Honarmand, Hamidreza Badihi, and Ali
Ghaheri. "An Accurate Generalized Model for Predict Vapor Pressure of
Refrigerants." International Journal of Refrigeration 36, no. 4
(June 2013): 1327-32. doi:10.1016/j.ijrefrig.2013.01.007.
'''
Tr = T / Tc
Tr_inv = 1.0 / Tr
log_Tr = log(Tr)
Tr_19 = Tr**1.9
f0 = 6.83377 + -5.76051 * Tr_inv + 0.90654 * log_Tr + -1.16906 * Tr_19
f1 = 5.32034 + -28.1460 * Tr_inv + -58.0352 * log_Tr + 23.57466 * Tr_19
f2 = 18.19967 + 16.33839 * Tr_inv + 65.6995 * log_Tr + -35.9739 * Tr_19
return Pc * exp(f0 + omega * f1 + omega * omega * f2)
