Surface tension

The Parachor Method

The Parachor method is a semi-empirical correlation used to estimate the surface tension of a liquid–vapor system from liquid and vapor densities and a substance-specific constant called the Parachor.
It is commonly used in petroleum thermodynamics and phase-equilibrium modeling when experimental surface tension data are unavailable.

Fundamental Idea

The Parachor hypothesis relates surface tension $\sigma$ to the density difference between liquid and vapor raised to the m-th power:

\sigma^{1/m} = P (\rho_\ell - \rho_\text{g})

where:

$\sigma$ = surface tension (N·m⁻¹)
$\rho_\ell$ = liquid-phase density (mol·m⁻³)
$\rho_\text{g}$ = vapor-phase density (mol·m⁻³)
$P$ = Parachor constant (characteristic of the fluid)
$m$ = Parachor exponent (TP-Cloud uses $m$ = 3.87).

Rearranged, the surface tension is given by the MacLeod–Sugden equation ¹, ²:

\sigma = [ P (\rho_\ell - \rho_\text{g}) ]^m

Mixtures

The mixture rule includes an interaction parameter:

P_{\text{mix}} = \sum_i \sum_j (1 - k_{ij}) x_i x_j (P_i + P_j)/2

where:

$P_{\text{mix}}$ = mixture Parachor
$x$ = phase mole fractions
$P_i$ = pure-component Parachor
$k_{ij}$ = Interaction parameter (Default value is 0.0)

Then the mixture surface tension is computed as ³:

\sigma_{\text{mix}} = [ P_{\text{mix}}^\ell \rho_{\ell,\text{mix}} - P_{\text{mix}}^\text{g}\rho_{\text{g},\text{mix}} ]^m

Determination of Pure-Component Parachor

Pure-component Parachors are calculated from empirical correlations of surface-tension, $\sigma_i$ :

P_i = \frac{\sigma_i^{1/m}}{\rho_{\ell,i} - \rho_{\text{g},i}}

Assumptions and Limitations

The Parachor method assumes $\sigma^{1/m} \propto \Delta\rho$ (Macleod–Sugden type relationship).
Works best for nonpolar or slightly polar fluids near ambient conditions.
Accuracy decreases near the critical point where $\rho_\ell \to \rho_\text{g}$ .
For strongly interacting mixtures (e.g., water + alcohol) interaction parameters are needed to get good results.

References

Sugden, S., The Parachor and Valency, J. Chem. Soc., 1924. ↩
Macleod, D. B., On a Relation Between Surface Tension and Density, Trans. Faraday Soc., 1923. ↩
Weinaug C.F. and Kautz, D.L., Surface Tensions of Methane–n-Butane Mixtures, Ind. Eng. Chem. 35, 1943. ↩