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Mathematical modeling of planar and spherical vapor–liquid phase interfaces for multicomponent fluids
Celný, David ; Vinš, Václav ; Planková, Barbora ; Hrubý, Jan
Methods for accurate modeling of phase interfaces are important for understanding natural processes and application in technology. In particular, prediction of the non-equilibrium phase transition requires the knowledge of the strongly curved phase interfaces of microscopic droplets. In our work, we focus on the spherical vapor–liquid interfaces for binary mixtures. We developed a computational method able to determine the density and concentration profiles. The fundamentals of our approach lie in the gradient theory, allowing to transcribe the functional formulation into a system of Euler-Langrange equations. System is then modified into a shape suitable for iterative computation. For this task, we employ the Newton-Raphson and the shooting methods ensuring a good convergence speed. For the thermodynamic properties, the PC–SAFT EoS is used. We determined the density and concentration profiles of the binary mixture C O 2 & C 9 H 20 for spherical phase interfaces at various saturation factors.
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Density gradient theory combined with the PC-SAFT equation of state used for modeling the surface tension of associating systems
Vinš, Václav ; Planková, Barbora ; Hrubý, Jan ; Celný, D.
The density gradient theory (GT) combined with a SAFT-type (Statistical Associating Fluid Theory) equation of state has been used for modeling the surface tension of associating fluids represented by a series of six alkanols ranging from methanol to 1-pentanol. The effect of nonzero dipole moment of the selected alkanols on the predicted surface tension was investigated in this study. Results of the GT + non-polar Perturbed Chain (PC) SAFT equation of state were compared to predictions of GT combined with the PC-polar-SAFT, i.e. PCP-SAFT, equation. Both GT + PC-SAFT and GT + PCP-SAFT give reasonable prediction of the surface tension for pure alkanols. Results of both models are comparable as no significant difference in the modeled saturation properties and in the predicted surface tension using GT was found. Consideration of dipolar molecules of selected alkanols using PCP-SAFT had only minor effect on the predicted properties compared to the non-polar PC-SAFT model.
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Droplet and Bubble Nucleation Modeled by Density Gradient Theory - Cubic Equation of State versus SAFT Model
Vinš, Václav ; Hrubý, Jan ; Planková, Barbora
The study presents some preliminary results of the density gradient theory (GT) combined with two different equations of state (EoS): the classical cubic equation by van der Waals and a recent approach based on the statistical associating fluid theory (SAFT), namely its perturbed-chain (PC) modification. It has been shown that PC-SAFT is a promising tool for accurate modeling of nucleation using the GT. Besides the basic case of a planar phase interface, the spherical interface was analyzed to model a critical cluster occurring either for nucleation of droplets (condensation) or bubbles (boiling, cavitation).
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The Temperature Dependence of Homogeneous Nucleation Rates by the Gradient Theory
Hrubý, Jan
We demonstrate that the incorrect temperature dependence of the nucleation rates as predicted by the classical nucleation theory is likely a consequence of the ignored dependence of the surface tension on the radius of the droplet parameterized here as a quadratic function of the pressure difference between the gas and liquid phases of the critical cluster. The size-dependent surface tension is obtained using a simplified gradient theory.
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Výpočet hustotních profilů nano-kapiček použitím gradientní teorie: numerické řešení v transformovaných proměnných
Hrubý, Jan
The Cahn-Hilliard gradient theory extends the continuum fluid mechanics and thermodynamics to objects of typical dimensions of 1 nm. The present investigation concerns the computation of the density profile of a equilibrium spherical case (droplet, bubble). The solution for this case reduces to a single second-order ordinary differential equation (ODE) with boundary conditions. To overcome computational difficulties due to the tremendous difference of thermodynamic properties of the liquid and vapor phases at low temperatures, we suggest some mathematical transforms. Using these transforms, the right-hand side of the ODE becomes well-behaved and the numerical solution is converging.
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Aplikace gradientní teorie fázových rozhraní na modelování nano-kapiček
Hrubý, Jan
The gradient theory (GT) is the simplest theory giving a physically sound picture of fluid-fluid phase interfaces. The density of the fluid varies in a continuous manner across the phase interface. In the GT, energy comprises the kinetic energy due to the macroscopic motion, internal energy of homogeneous fluid, and a gradient-square term, representing the effect of phase interfaces. Somewhat surprisingly, attempts to apply GT to nucleation are very scarce. A fundamental problem of the nucleation theory is determination of the so-called work of formation. In this work we solve numerically the differential Euler-Lagrange equations in spherical symmetry simulating a droplet of mixture of n-nonane and methane. The density of n-nonane is monotonously decreasing function of the radius. The density of methane shows a “hump” located on the outer side of the interface, corresponding to the known effect of gas adsorption.
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