Surface Tension (chemicals.interface)

This module contains various surface tension estimation routines, dataframes of fit coefficients, fitting model equations, mixing rules, and water-hydrocarbon interfacial tension estimation routines.

For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker.

Pure Component Correlations

chemicals.interface.Brock_Bird(T, Tb, Tc, Pc)

Calculates air-liquid surface tension using the [1] emperical method. Old and tested.

$$\sigma = P_c^{2/3}T_c^{1/3}Q(1-T_r)^{11/9}$$

$$Q = 0.1196 \left[ 1 + \frac{T_{br}\ln (P_c/1.01325)}{1-T_{br}}\right]-0.279$$

Parameters

Tfloat

Temperature of fluid [K]

Tbfloat

Boiling temperature of the fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

Numerous arrangements of this equation are available. This is DIPPR Procedure 7A: Method for the Surface Tension of Pure, Nonpolar, Nonhydrocarbon Liquids The exact equation is not in the original paper. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1

Brock, James R., and R. Byron Bird. “Surface Tension and the Principle of Corresponding States.” AIChE Journal 1, no. 2 (June 1, 1955): 174-77. doi:10.1002/aic.690010208

Examples

p-dichloribenzene at 412.15 K, from DIPPR; value differs due to a slight difference in method.

>>> Brock_Bird(412.15, 447.3, 685, 3.952E6)
0.02208448325192495

Chlorobenzene from Poling, as compared with a % error value at 293 K.

>>> Brock_Bird(293.15, 404.75, 633.0, 4530000.0)
0.032985686413713036

chemicals.interface.Pitzer_sigma(T, Tc, Pc, omega)

Calculates air-liquid surface tension using the correlation derived by [1] from the works of [2] and [3]. Based on critical property CSP methods.

$$\sigma = P_c^{2/3}T_c^{1/3}\frac{1.86 + 1.18\omega}{19.05} \left[ \frac{3.75 + 0.91 \omega}{0.291 - 0.08 \omega}\right]^{2/3} (1-T_r)^{11/9}$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

omegafloat

Acentric factor for fluid, [-]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

The source of this equation has not been reviewed. Internal units of presure are bar, surface tension of mN/m. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1

Poling, Bruce E. The Properties of Gases and Liquids. 5th edition. New York: McGraw-Hill Professional, 2000.

2

Curl, R. F., and Kenneth Pitzer. “Volumetric and Thermodynamic Properties of Fluids-Enthalpy, Free Energy, and Entropy.” Industrial & Engineering Chemistry 50, no. 2 (February 1, 1958): 265-74. doi:10.1021/ie50578a047

3

Pitzer, K. S.: Thermodynamics, 3d ed., New York, McGraw-Hill, 1995, p. 521.

Examples

Chlorobenzene from Poling, as compared with a % error value at 293 K.

>>> Pitzer_sigma(293., 633.0, 4530000.0, 0.249)
0.03458453513446388

chemicals.interface.Sastri_Rao(T, Tb, Tc, Pc, chemicaltype=None)

Calculates air-liquid surface tension using the correlation derived by [1] based on critical property CSP methods and chemical classes.

$$\sigma = K P_c^xT_b^y T_c^z\left[\frac{1-T_r}{1-T_{br}}\right]^m$$

Parameters

Tfloat

Temperature of fluid [K]

Tbfloat

Boiling temperature of the fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

The source of this equation has not been reviewed. Internal units of presure are bar, surface tension of mN/m. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1

Sastri, S. R. S., and K. K. Rao. “A Simple Method to Predict Surface Tension of Organic Liquids.” The Chemical Engineering Journal and the Biochemical Engineering Journal 59, no. 2 (October 1995): 181-86. doi:10.1016/0923-0467(94)02946-6.

Examples

Chlorobenzene from Poling, as compared with a % error value at 293 K.

>>> Sastri_Rao(293.15, 404.75, 633.0, 4530000.0)
0.03234567739694441

chemicals.interface.Zuo_Stenby(T, Tc, Pc, omega)

Calculates air-liquid surface tension using the reference fluids methods of [1].

$$\sigma^{(1)} = 40.520(1-T_r)^{1.287}$$

$$\sigma^{(2)} = 52.095(1-T_r)^{1.21548}$$

$$\sigma_r = \sigma_r^{(1)}+ \frac{\omega - \omega^{(1)}} {\omega^{(2)}-\omega^{(1)}} (\sigma_r^{(2)}-\sigma_r^{(1)})$$

$$\sigma = T_c^{1/3}P_c^{2/3}[\exp{(\sigma_r)} -1]$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

omegafloat

Acentric factor for fluid, [-]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

Presently untested. Have not personally checked the sources. The reference values for methane and n-octane are from the DIPPR database. If Tc is larger than T, 0 is returned as the model would return complex numbers.

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

Examples

Chlorobenzene

>>> Zuo_Stenby(293., 633.0, 4530000.0, 0.249)
0.03345569011871088

chemicals.interface.Hakim_Steinberg_Stiel(T, Tc, Pc, omega, StielPolar=0.0)

Calculates air-liquid surface tension using the reference fluids methods of [1].

$$\sigma = 4.60104\times 10^{-7} P_c^{2/3}T_c^{1/3}Q_p \left(\frac{1-T_r}{0.4}\right)^m$$

$$Q_p = 0.1574+0.359\omega-1.769\chi-13.69\chi^2-0.51\omega^2+1.298\omega\chi$$

$$m = 1.21+0.5385\omega-14.61\chi-32.07\chi^2-1.65\omega^2+22.03\omega\chi$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

omegafloat

Acentric factor for fluid, [-]

StielPolarfloat, optional

Stiel Polar Factor, [-]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

Original equation for m and Q are used. Internal units are atm and mN/m. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1

Hakim, D. I., David Steinberg, and L. I. Stiel. “Generalized Relationship for the Surface Tension of Polar Fluids.” Industrial & Engineering Chemistry Fundamentals 10, no. 1 (February 1, 1971): 174-75. doi:10.1021/i160037a032.

Examples

1-butanol, as compared to value in CRC Handbook of 0.02493.

>>> Hakim_Steinberg_Stiel(298.15, 563.0, 4414000.0, 0.59, StielPolar=-0.07872)
0.02190790257519

chemicals.interface.Miqueu(T, Tc, Vc, omega)

Calculates air-liquid surface tension using the methods of [1].

$$\sigma = k T_c \left( \frac{N_a}{V_c}\right)^{2/3} (4.35 + 4.14 \omega)t^{1.26}(1+0.19t^{0.5} - 0.487t)$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

Vcfloat

Critical volume of fluid [m^3/mol]

omegafloat

Acentric factor for fluid, [-]

Returns

sigmafloat

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%. If Tc is larger than T, 0 is returned as the model would return complex numbers.

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.

Examples

Bromotrifluoromethane, 2.45 mN/m

>>> Miqueu(300., 340.1, 0.000199, 0.1687)
0.003474100774091376

chemicals.interface.Aleem(T, MW, Tb, rhol, Hvap_Tb, Cpl)

Calculates vapor-liquid surface tension using the correlation derived by [1] based on critical property CSP methods.

$$\sigma = \phi \frac{MW^{1/3}} {6N_A^{1/3}}\rho_l^{2/3}\left[H_{vap} + C_{p,l}(T_b-T)\right]$$

$$\phi = 1 - 0.0047MW + 6.8\times 10^{-6} MW^2$$

Parameters

Tfloat

Temperature of fluid [K]

MWfloat

Molecular weight [g/mol]

Tbfloat

Boiling temperature of the fluid [K]

rholfloat

Liquid density at T and P [kg/m^3]

Hvap_Tbfloat

Mass enthalpy of vaporization at the normal boiling point [kg/m^3]

Cplfloat

Liquid heat capacity of the chemical at T [J/kg/K]

Returns

sigmafloat

Liquid-vapor surface tension [N/m]

Notes

Internal units of molecuar weight are kg/mol. This model is dimensionally consistent.

This model does not use the critical temperature. After it predicts a surface tension of 0 at a sufficiently high temperature, it returns negative results. The temperature at which this occurs (the “predicted” critical temperature) can be calculated as follows:

$$\sigma = 0 \to T_{c,predicted} \text{ at } T_b + \frac{H_{vap}}{Cp_l}$$

To handle this case, if Tc is larger than T, 0 is returned as the model would return complex numbers.

Because of its dependence on density, it has the potential to model the effect of pressure on surface tension.

Claims AAD of 4.3%. Developed for normal alkanes. Total of 472 data points. Behaves worse for higher alkanes. Behaves very poorly overall.

References

1

Aleem, W., N. Mellon, S. Sufian, M. I. A. Mutalib, and D. Subbarao. “A Model for the Estimation of Surface Tension of Pure Hydrocarbon Liquids.” Petroleum Science and Technology 33, no. 23-24 (December 17, 2015): 1908-15. doi:10.1080/10916466.2015.1110593.

Examples

Methane at 90 K

>>> Aleem(T=90, MW=16.04246, Tb=111.6, rhol=458.7, Hvap_Tb=510870.,
... Cpl=2465.)
0.01669970230131523

chemicals.interface.Mersmann_Kind_sigma(T, Tm, Tb, Tc, Pc, n_associated=1)

Estimates the surface tension of organic liquid substances according to the method of [1].

$$\sigma^* = \frac{\sigma n_{ass}^{1/3}} {(kT_c)^{1/3} T_{rm}P_c^{2/3}}$$

$$\sigma^* = \left(\frac{T_b - T_m}{T_m}\right)^{1/3} \left[6.25(1-T_r) + 31.3(1-T_r)^{4/3}\right]$$

Parameters

Tfloat

Temperature of the fluid [K]

Tmfloat

Melting temperature [K]

Tbfloat

Boiling temperature of the fluid [K]

Tcfloat

Critical temperature of the fluid [K]

Pcfloat

Critical pressure of the fluid [Pa]

n_associatedfloat

Number of associated molecules in a cluster (2 for alcohols, 1 otherwise), [-]

Returns

sigmafloat

Liquid-vapor surface tension [N/m]

Notes

In the equation, all quantities must be in SI units. k is the boltzman constant. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1

Mersmann, Alfons, and Matthias Kind. “Prediction of Mechanical and Thermal Properties of Pure Liquids, of Critical Data, and of Vapor Pressure.” Industrial & Engineering Chemistry Research, January 31, 2017. https://doi.org/10.1021/acs.iecr.6b04323.

Examples

MTBE at STP (the actual value is 0.0181):

>>> Mersmann_Kind_sigma(298.15, 164.15, 328.25, 497.1, 3430000.0)
0.016744311449290426

chemicals.interface.sigma_Gharagheizi_1(T, Tc, MW, omega)

Calculates air-liquid surface tension using the equation 4 derived in [1] by gene expression programming.

$$\sigma = 8.948226\times 10^{-4}\left[\frac{A^2}{MW}\sqrt{\frac{A\omega}{MW}} \right]^{0.5}$$

$$A = (T_{c} - T - \omega)$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

MWfloat

Molecular weight [g/mol]

omegafloat

Acentric factor for fluid, [-]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

This equation may fail before the critical point. In this case it returns 0.0 If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1(1,2)

Gharagheizi, Farhad, Ali Eslamimanesh, Mehdi Sattari, Amir H. Mohammadi, and Dominique Richon. “Development of Corresponding States Model for Estimation of the Surface Tension of Chemical Compounds.” AIChE Journal 59, no. 2 (2013): 613-21. https://doi.org/10.1002/aic.13824.

Examples

Methane at 93 K, point from [1]’s supporting material:

>>> sigma_Gharagheizi_1(T=95, Tc=190.564, MW=16.04, omega=0.012)
0.0110389739

chemicals.interface.sigma_Gharagheizi_2(T, Tb, Tc, Pc, Vc)

Calculates air-liquid surface tension using the equation 6 derived in [1] by gene expression programming.

$$\frac{\sigma}{\text{N}/\text{m}} = 10^{-4}\left(\frac{P_c}{\text{bar}}\right)^{2/3} \left(\frac{T_c}{\text{K}}\right)^{1/3}(1-T_r)^{11/9} \left[7.728729T_{br} + 2.476318\left(T_{br}^3 + \frac{V_{c}}{\text{m}^3 /\text{kmol}}\right) \right]$$

Parameters

Tfloat

Temperature of fluid [K]

Tbfloat

Boiling temperature of the fluid [K]

Tcfloat

Critical temperature of fluid [K]

Pcfloat

Critical pressure of fluid [Pa]

MWfloat

Molecular weight [g/mol]

Vcfloat

Critical volume of fluid [m^3/mol]

Returns

sigmafloat

Liquid surface tension, N/m

Notes

This expression gives does converge to 0 at the critical point. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1(1,2)

Gharagheizi, Farhad, Ali Eslamimanesh, Mehdi Sattari, Amir H. Mohammadi, and Dominique Richon. “Development of Corresponding States Model for Estimation of the Surface Tension of Chemical Compounds.” AIChE Journal 59, no. 2 (2013): 613-21. https://doi.org/10.1002/aic.13824.

Examples

Methane at 93 K, point from [1]’s supporting material:

>>> sigma_Gharagheizi_2(T=95, Tb=111.66, Tc=190.564, Pc=45.99e5, Vc=0.0986e-3)
0.01674894057

Mixing Rules

chemicals.interface.Winterfeld_Scriven_Davis(xs, sigmas, rhoms)

Calculates surface tension of a liquid mixture according to mixing rules in [1] and also in [2].

$$\sigma_M = \sum_i \sum_j \frac{1}{V_L^{L2}}\left(x_i V_i \right) \left( x_jV_j\right)\sqrt{\sigma_i\cdot \sigma_j}$$

Parameters

xsarray_like

Mole fractions of all components, [-]

sigmasarray_like

Surface tensions of all components, [N/m]

rhomsarray_like

Molar densities of all components, [mol/m^3]

Returns

sigmafloat

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.

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.

Examples

>>> Winterfeld_Scriven_Davis([0.1606, 0.8394], [0.01547, 0.02877],
... [8610., 15530.])
0.02496738845043982

chemicals.interface.Weinaug_Katz(parachors, Vml, Vmg, xs, ys)

Calculates surface tension of a liquid mixture according to mixing rules in [1] and also in [2]. This is based on the Parachor concept. This is called the Macleod-Sugden model in some places.

$$\sigma_M = \left[\sum_i P_i\left( \frac{x_i}{V_{m,l}} - \frac{y_i}{V_{m,g}}\right) \right]^4$$

Parameters

parachorslist[float]

Parachors of each component, [N^0.25*m^2.75/mol]

Vmlfloat

Liquid mixture molar volume, [m^3/mol]

Vmgfloat

Gas mixture molar volume; this can be set to zero at low pressures, [m^3/mol]

xslist[float]

Mole fractions of all components in liquid phase, [-]

xslist[float]

Mole fractions of all components in gas phase, [-]

Returns

sigmafloat

Air-liquid surface tension of mixture, [N/m]

Notes

This expression is efficient and does not require pure component surface tensions. Its accuracy is dubious.

References

1

Weinaug, Charles F., and Donald L. Katz. “Surface Tensions of Methane-Propane Mixtures.” Industrial & Engineering Chemistry 35, no. 2 (February 1, 1943): 239-246. https://doi.org/10.1021/ie50398a028.

2

Pedersen, Karen Schou, Aage Fredenslund, and Per Thomassen. Properties of Oils and Natural Gases. Vol. 5. Gulf Pub Co, 1989.

Examples

>>> Weinaug_Katz([5.1e-5, 7.2e-5], Vml=0.000125, Vmg=0.02011, xs=[.4, .6], ys=[.6, .4])
0.06547479150776776

Neglect the vapor phase density by setting Vmg to a high value:

>>> Weinaug_Katz([5.1e-5, 7.2e-5], Vml=0.000125, Vmg=1e100, xs=[.4, .6], ys=[.6, .4])
0.06701752894095361

chemicals.interface.Diguilio_Teja(T, xs, sigmas_Tb, Tbs, Tcs)

Calculates surface tension of a liquid mixture according to mixing rules in [1].

$$\sigma = 1.002855(T^*)^{1.118091} \frac{T}{T_b} \sigma_r$$

$$T^* = \frac{(T_c/T)-1}{(T_c/T_b)-1}$$

$$\sigma_r = \sum x_i \sigma_i$$

$$T_b = \sum x_i T_{b,i}$$

$$T_c = \sum x_i T_{c,i}$$

Parameters

Tfloat

Temperature of fluid [K]

xsarray_like

Mole fractions of all components

sigmas_Tbarray_like

Surface tensions of all components at the boiling point, [N/m]

Tbsarray_like

Boiling temperatures of all components, [K]

Tcsarray_like

Critical temperatures of all components, [K]

Returns

sigmafloat

Air-liquid surface tension of mixture, [N/m]

Notes

Simple model, however it has 0 citations. Gives similar results to the Winterfeld_Scriven_Davis model.

Raises a ValueError if temperature is greater than the mixture’s critical temperature or if the given temperature is negative, or if the mixture’s boiling temperature is higher than its critical temperature.

[1] claims a 4.63 percent average absolute error on 21 binary and 4 ternary non-aqueous systems. [1] also considered Van der Waals mixing rules for Tc, but found it provided a higher error of 5.58%

References

1(1,2,3)

Diguilio, Ralph, and Amyn S. Teja. “Correlation and Prediction of the Surface Tensions of Mixtures.” The Chemical Engineering Journal 38, no. 3 (July 1988): 205-8. doi:10.1016/0300-9467(88)80079-0.

Examples

>>> Diguilio_Teja(T=298.15, xs=[0.1606, 0.8394],
... sigmas_Tb=[0.01424, 0.02530], Tbs=[309.21, 312.95], Tcs=[469.7, 508.0])
0.025716823875045505

Correlations for Specific Substances

chemicals.interface.sigma_IAPWS(T)

Calculate the surface tension of pure water as a function of . temperature. Assumes the 2011 IAPWS [1] formulation.

$$\sigma = B\tau^\mu(1+b\tau)\\$$

$$\tau = 1-T/T_c\\$$

$$B = 0.2358 \text{N/m}\\$$

$$b = -0.625\\$$

$$\mu = 1.256$$

Parameters

Tfloat

Temperature of liquid [K]

Returns

sigmafloat

Air-liquid surface tension, [N/m]

Notes

This function is valid from the triple temperature to the critical temperature. No effects for pressure are included in the formulation. Test values are from IAPWS 2010 book.

The equation is valid from the triple point to the critical point, 647.096 K; but [1] also recommends its use down to -25°C.

If a value larger than the critical temperature is input, 0.0 is returned.

References

1(1,2)

IAPWS. 2014. Revised Release on Surface Tension of Ordinary Water Substance

Examples

>>> sigma_IAPWS(300.)
0.0716859625271
>>> sigma_IAPWS(450.)
0.0428914991565
>>> sigma_IAPWS(600.)
0.0083756108728

Petroleum Correlations

chemicals.interface.API10A32(T, Tc, K_W)

Calculates the interfacial tension between a liquid petroleum fraction and air, using the oil’s pseudocritical temperature and Watson K Characterization factor.

$$\sigma = \frac{673.7\left[\frac{\left(T_c - T\right)}{T_c}\right]^{1.232}}{K_W}$$

Parameters

Tfloat

Liquid temperature, [K]

Tcfloat

Pseudocritical temperature (or critical temperature if using the equation with a pure component), [K]

K_Wfloat

Watson characterization factor

Returns

sigmafloat

Air-liquid surface tension, [N/m]

Notes

[1] cautions that this should not be applied to coal liquids, and that it will give higher errors at pressures above 500 psi. [1] claims this has an average error of 10.7%.

This function converges to zero at Tc. If Tc is larger than T, 0 is returned as the model would return complex numbers.

References

1(1,2,3)

API Technical Data Book: General Properties & Characterization. American Petroleum Institute, 7E, 2005.

Examples

Sample problem in Comments on Procedure 10A3.2.1 of [1];

>>> from fluids.core import F2K, R2K
>>> API10A32(T=F2K(60), Tc=R2K(1334), K_W=12.4)
29.577333312096968

Oil-Water Interfacial Tension Correlations

chemicals.interface.Meybodi_Daryasafar_Karimi(rho_water, rho_oil, T, Tc)

Calculates the interfacial tension between water and a hydrocabon liquid according to the correlation of [1].

$$\gamma_{hw} = \left(\frac{A_1 + A_2 \Delta \rho + A_3\Delta\rho^2 + A_4\Delta\rho^3} {A_5 + A_6\frac{T^{A_7}}{T_{c,h}} + A_8T^{A_9}} \right)^{A_{10}}$$

Parameters

rho_waterfloat

The density of the aqueous phase, [kg/m^3]

rho_oilfloat

The density of the hydrocarbon phase, [kg/m^3]

Tfloat

Temperature of the fluid, [K]

Tcfloat

Critical temperature of the hydrocarbon mixture, [K]

Returns

sigmafloat

Hydrocarbon-water surface tension [N/m]

Notes

Internal units of the equation are g/mL and mN/m.

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.

Examples

>>> Meybodi_Daryasafar_Karimi(980, 760, 580, 914)
0.02893598143089256

Fit Correlations

chemicals.interface.REFPROP_sigma(T, Tc, sigma0, n0, sigma1=0.0, n1=0.0, sigma2=0.0, n2=0.0)

Calculates air-liquid surface tension using the REFPROP_sigma [1] regression-based method. Relatively recent, and most accurate.

$$\sigma(T)=\sigma_0\left(1-\frac{T}{T_c}\right)^{n_0}+ \sigma_1\left(1-\frac{T}{T_c}\right)^{n_1}+ \sigma_2\left(1-\frac{T}{T_c}\right)^{n_2}$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

sigma0float

First emperical coefficient of a fluid

n0float

First emperical exponent of a fluid

sigma1float, optional

Second emperical coefficient of a fluid.

n1float, optional

Second emperical exponent of a fluid.

sigma1float, optional

Third emperical coefficient of a fluid.

n2float, optional

Third emperical exponent of a fluid.

Returns

sigmafloat

Liquid surface tension, [N/m]

Notes

Function as implemented in [1]. No example necessary; results match literature values perfectly. Form of function returns imaginary results when T > Tc; 0 is returned if this is the case.

When fitting parameters to this function, it is easy to end up with a fit that returns negative surface tension near but not quite at the critical point.

References

1(1,2)

Diky, Vladimir, Robert D. Chirico, Chris D. Muzny, Andrei F. Kazakov, Kenneth Kroenlein, Joseph W. Magee, Ilmutdin Abdulagatov, and Michael Frenkel. “ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept.” Journal of Chemical Information and Modeling 53, no. 12 (2013): 3418-30. doi:10.1021/ci4005699.

Examples

Parameters for water at 298.15 K

>>> REFPROP_sigma(298.15, 647.096, -0.1306, 2.471, 0.2151, 1.233)
0.07205503890847453

chemicals.interface.Somayajulu(T, Tc, A, B, C)

Calculates air-liquid surface tension using the [1] emperical (parameter-regressed) method. Well regressed, no recent data.

$$\sigma=aX^{5/4}+bX^{9/4}+cX^{13/4}$$

$$X=(T_c-T)/T_c$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

Afloat

Regression parameter

Bfloat

Regression parameter

Cfloat

Regression parameter

Returns

sigmafloat

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; 0.0 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.

This function can be accidentally fit to return negative values of surface tension.

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.

Examples

Water at 300 K

>>> Somayajulu(300, 647.126, 232.713514, -140.18645, -4.890098)
0.07166386387996758

chemicals.interface.Jasper(T, a, b)

Calculates surface tension of a fluid given two parameters, a linear fit in Celcius from [1] with data reprinted in [2].

$$\sigma = a - bT$$

Parameters

Tfloat

Temperature of fluid, [K]

afloat

Parameter for equation. Chemical specific.

bfloat

Parameter for equation. Chemical specific.

Returns

sigmafloat

Surface tension [N/m]

Notes

Internal units are mN/m, and degrees Celcius. This function has been checked against several references.

As this is a linear model, negative values of surface tension will eventually arise. 0 is returned in these cases.

References

1

Jasper, Joseph J. “The Surface Tension of Pure Liquid Compounds.” Journal of Physical and Chemical Reference Data 1, no. 4 (October 1, 1972): 841-1010. doi:10.1063/1.3253106.

2

Speight, James. Lange’s Handbook of Chemistry. 16 edition. McGraw-Hill Professional, 2005.

Examples

>>> Jasper(298.15, 24, 0.0773)
0.0220675

chemicals.interface.PPDS14(T, Tc, a0, a1, a2)

Calculates air-liquid surface tension using the [1] emperical (parameter-regressed) method, called the PPDS 14 equation for surface tension.

$$\sigma = a_0 \tau^{a_1}(1 + a_2 \tau)$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

a0float

Regression parameter, [N/m]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

Returns

sigmafloat

Liquid surface tension, [N/m]

Notes

If Tc is larger than T, 0 is returned as the model would return complex numbers.

If this model is fit with a0 and a2 as positive values, it is guaranteed to predict only positive values of sigma right up to the critical point. However, a2 is often fit to be a negative value.

References

1(1,2)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/PPDS14.htm.

2

Frenkel, Michael, Robert D. Chirico, Vladimir Diky, Xinjian Yan, Qian Dong, and Chris Muzny. “ThermoData Engine (TDE): Software Implementation of the Dynamic Data Evaluation Concept.” Journal of Chemical Information and Modeling 45, no. 4 (July 1, 2005): 816-38. https://doi.org/10.1021/ci050067b.

Examples

Benzene at 280 K from [1]

>>> PPDS14(T=280, Tc=562.05, a0=0.0786269, a1=1.28646, a2=-0.112304)
0.030559764256249854

chemicals.interface.Watson_sigma(T, Tc, a1, a2, a3=0.0, a4=0.0, a5=0.0)

Calculates air-liquid surface tension using the Watson [1] emperical (parameter-regressed) method developed by NIST.

$$\sigma = \exp\left[a_{1} + \ln(1 - T_r)\left( a_2 + a_3T_r + a_4T_r^2 + a_5T_r^3 \right)\right]$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

a3float

Regression parameter, [-]

a4float

Regression parameter, [-]

a5float

Regression parameter, [-]

Returns

sigmafloat

Liquid surface tension, [N/m]

Notes

This expression is also used for enthalpy of vaporization in [1]. The coefficients from NIST TDE for enthalpy of vaporization are kJ/mol.

This model is coded to return 0 values at Tr >= 1. It is otherwise not possible to evaluate this expression at Tr = 1, as log(0) is undefined (although the limit shows the expression converges to 0).

This equation does not have any term forcing it to become near-zero at the critical point, but it cannot be fit so as to produce negative values.

References

1(1,2,3)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/HVPExpansion-SurfaceTension.htm

Examples

Isooctane at 350 K from [1]:

>>> Watson_sigma(T=350.0, Tc=543.836, a1=-3.02417, a2=1.21792, a3=-5.26877e-9, a4=5.62659e-9, a5=-2.27553e-9)
0.0138340926605649

chemicals.interface.ISTExpansion(T, Tc, a1, a2, a3=0.0, a4=0.0, a5=0.0)

Calculates air-liquid surface tension using the IST expansion [1] emperical (parameter-regressed) method developed by NIST.

$$\sigma = \sum_i a_i\left(1 - \frac{T}{T_c} \right)^i$$

Parameters

Tfloat

Temperature of fluid [K]

Tcfloat

Critical temperature of fluid [K]

a1float

Regression parameter, [-]

a2float

Regression parameter, [-]

a3float

Regression parameter, [-]

a4float

Regression parameter, [-]

a5float

Regression parameter, [-]

Returns

sigmafloat

Liquid surface tension, [N/m]

Notes

This equation hsa a term term forcing it to become zero at the critical point, but it can easily be fit so as to produce negative values at any reduced temperature.

References

1(1,2)

“ThermoData Engine (TDE103b V10.1) User`s Guide.” https://trc.nist.gov/TDE/Help/TDE103b/Eqns-Pure-SurfaceTension/ISTExpansion-SurfaceTension.htm

Examples

Diethyl phthalate at 400 K from [1]:

>>> ISTExpansion(T=400.0, Tc=776.0, a1=0.037545, a2=0.0363288)
0.02672100905515996

Fit Coefficients

All of these coefficients are lazy-loaded, so they must be accessed as an attribute of this module.

chemicals.interface.sigma_data_Mulero_Cachadina

Data from [5] with REFPROP_sigma coefficients.

chemicals.interface.sigma_data_Jasper_Lange

Data as shown in [4] but originally in [3] with Jasper coefficients.

chemicals.interface.sigma_data_Somayajulu

Data from [1] with Somayajulu coefficients.

chemicals.interface.sigma_data_Somayajulu2

Data from [2] with Somayajulu coefficients. These should be preferred over the original coefficients.

chemicals.interface.sigma_data_VDI_PPDS_11

Data from [6] with chemicals.dippr.EQ106 coefficients.

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.

2

Mulero, A., M. I. Parra, and I. Cachadina. “The Somayajulu Correlation for the Surface Tension Revisited.” Fluid Phase Equilibria 339 (February 15, 2013): 81-88. doi:10.1016/j.fluid.2012.11.038.

3

Jasper, Joseph J. “The Surface Tension of Pure Liquid Compounds.” Journal of Physical and Chemical Reference Data 1, no. 4 (October 1, 1972): 841-1010. doi:10.1063/1.3253106.

4

Speight, James. Lange’s Handbook of Chemistry. 16 edition. McGraw-Hill Professional, 2005.

5

Mulero, A., I. Cachadiña, and M. I. Parra. “Recommended Correlations for the Surface Tension of Common Fluids.” Journal of Physical and Chemical Reference Data 41, no. 4 (December 1, 2012): 043105. doi:10.1063/1.4768782.

6

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

The structure of each dataframe is shown below:

In [1]: import chemicals

In [2]: chemicals.interface.sigma_data_Mulero_Cachadina
Out[2]: 
                                                    Fluid  ...    Tmax
CAS                                                        ...        
60-29-7                                     Diethyl ether  ...  453.15
64-17-5                                          Ethanol   ...  513.15
67-56-1                                         Methanol   ...  508.15
67-64-1                                          Acetone   ...  353.15
71-43-2                                          Benzene   ...  553.15
...                                                   ...  ...     ...
7783-54-2                            Nitrogen trifluoride  ...  206.36
7789-20-0                                             D2O  ...  642.02
10024-97-2                                 Nitrous oxide   ...  293.15
22410-44-2       RE245cb2 (Methyl-pentafluoroethyl ether)  ...  353.41
29118-24-9  R1234ze(E) (trans-1,3,3,3-tetrafluoropropene)  ...  373.14

[115 rows x 10 columns]

In [3]: chemicals.interface.sigma_data_Jasper_Lange
Out[3]: 
                               Name      a       b    Tmin    Tmax
CAS                                                               
55-21-0                  Benzamide   47.26  0.0705  402.15  563.15
55-63-0     Glycerol tris(nitrate)   55.74  0.2504  286.15  433.15
56-23-5       Carbon tetrachloride   29.49  0.1224  250.15  349.85
57-06-7       Allyl isothiocyanate   36.76  0.1074  193.15  425.15
60-29-7              Diethyl ether   18.92  0.0908  157.15  307.75
...                             ...    ...     ...     ...     ...
13952-84-6          sec-Butylamine   23.75  0.1057  169.15  336.15
14901-07-6                ␤-Ionone   35.36  0.0950  401.15  401.15
18854-56-3     1,2-Dipropoxyethane   25.03  0.0972     NaN     NaN
19550-30-2  2,3-Dimethyl-1-butanol   26.22  0.0992  259.15  391.15
40626-78-6          2-Methylhexane   21.22  0.0966  155.15  363.15

[522 rows x 5 columns]

In [4]: chemicals.interface.sigma_data_Somayajulu
Out[4]: 
                       Chemical      Tt      Tc         A         B          C
CAS                                                                           
60-29-7            Ethyl ether   157.00  466.74   61.0417   -6.7908    0.14046
64-17-5                Ethanol   159.00  513.92  111.4452 -146.0229   89.57030
64-19-7            Acetic acid   290.00  592.70   91.9020  -91.7035   77.50720
67-56-1               Methanol   175.59  512.64  122.6257 -199.4044  153.37440
71-23-8              Propanaol   147.00  536.78  107.1238 -133.8128   84.43570
...                         ...     ...     ...       ...       ...        ...
10035-10-6    Hydrogen bromide   187.15  363.20   74.0521   20.1043  -30.25710
10102-43-9        Nitric oxide   112.15  180.00   58.6304   97.8722  -33.67390
13465-07-1  Hydrogen disulfide   185.15  572.00  130.1176  -40.6216    4.77160
17778-80-2              Oxygen    54.35  154.58   38.2261    5.6316   -7.74050
19287-45-7            Diborane   104.15  289.80   38.0417   29.7743  -24.26050

[64 rows x 6 columns]

In [5]: chemicals.interface.sigma_data_Somayajulu2
Out[5]: 
                       Chemical      Tt      Tc         A         B          C
CAS                                                                           
60-29-7            Ethyl ether   157.00  466.74   61.0417   -6.7908    0.14046
64-17-5                Ethanol   159.00  513.92  111.4452 -146.0229   89.57030
64-19-7            Acetic acid   290.00  592.70   91.9020  -91.7035   77.50720
67-56-1               Methanol   175.59  512.64  122.6257 -199.4044  153.37440
71-23-8              Propanaol   147.00  536.78  107.1238 -133.8128   84.43570
...                         ...     ...     ...       ...       ...        ...
10035-10-6    Hydrogen bromide   187.15  363.20   74.0521   20.1043  -30.25710
10102-43-9        Nitric oxide   112.15  180.00   58.6304   97.8722  -33.67390
13465-07-1  Hydrogen disulfide   185.15  572.00  150.6970 -102.9100   56.72580
17778-80-2              Oxygen    54.35  154.58   38.2261    5.6316   -7.74050
19287-45-7            Diborane   104.15  289.80   38.0417   29.7743  -24.26050

[64 rows x 6 columns]

In [6]: chemicals.interface.sigma_data_VDI_PPDS_11
Out[6]: 
                         Chemical      Tm      Tc  ...        C        D        E
CAS                                                ...                           
50-00-0              Formaldehyde  181.15  408.05  ...  0.00000  0.00000  0.00000
56-23-5      Carbon tetrachloride  250.25  556.35  ...  0.00000  0.00000  0.00000
56-81-5                  Glycerol  291.45  850.05  ...  0.00000  0.00000  0.00000
60-29-7             Diethyl ether  156.75  466.63  ...  0.00000  0.00000  0.00000
62-53-3                   Aniline  267.15  699.05  ...  0.00000  0.00000  0.00000
...                           ...     ...     ...  ...      ...      ...      ...
10097-32-2                Bromine  265.85  584.15  ...  0.00000  0.00000  0.00000
10102-43-9           Nitric oxide  112.15  180.15  ...  0.00000  0.00000  0.00000
10102-44-0       Nitrogen dioxide  261.85  431.15  ...  0.00000  0.00000  0.00000
10544-72-6    Dinitrogentetroxide  261.85  431.10  ...  0.00000  0.00000  0.00000
132259-10-0                   Air   63.05  132.53  ...  0.06889  0.17918 -0.14564

[272 rows x 8 columns]