Kelvin's vapor pressure-curvature equation has a defined equilibrium condition
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Peer-reviewed literature and reference materials confirm that the Kelvin equation mathematically describes the saturation vapor pressure across a curved liquid-vapor interface at thermodynamic equilibrium.
Capillary condensation is the first-order vapor-to-liquid phase transition taking place in confined geometries. Such heterogeneous nucleation has been well described by thermodynamic laws such as the Kelvin equation, but the equation’s applicability at the nanoscale is still unresolved. Here, we show that the Kelvin equation is valid down to approximately 0.5 nm radius of curvature when the curvature dependence of surface tension is taken into account. By the shear-mode atomic force microscopy, we have measured directly and accurately the critical tip-surface distance (dc) at which the water meniscus is capillary condensed in ambient condition; e.g., dc ≈ 1.2 nm at 10% relative humidity. In particular, we can determine the Tolman length, the unique characteristic of the curvature-dependent surface tension, as the single fitting parameter (δ 1⁄4 0.21 0.05 nm). Our results that unify the validity of the Kelvin equation at molecular scale and the characterization of the curvature effect of surface tension may provide a better understanding of general nucleation phenomena in nature, including the role of nanometric aerosols in cloud formation.
The thermodynamics of fluids in confined (capillary) media is different from the bulk conditions due to the effects of the surface tension, wettability, and pore radius as described by the classical Kelvin equation. This study provides experimental data showing the deviation of propane vapour pressures in capillary media from the bulk conditions. Comparisons were also made with the vapour pressures calculated by the Peng-Robinson equation-of-state (PR-EOS). While the propane vapour pressures measured using synthetic capillary medium models (Hele-Shaw cells and microfluidic chips) were comparable with those measured at bulk conditions, the measured vapour pressures in the rock samples (sandstone, limestone, tight sandstone, and shale) were 15% (on average) less than those modelled by PR-EOS.
If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . Abstract The thermodynamics of fluids in confined (capillary) media is different from the bulk conditions due to the effects of the surface tension, wettability, and pore radius as described by the classical Kelvin equation. This study provides experimental data showing the deviation of propane vapour pressures in capillary media from the bulk conditions.
Quantitatively, the study demonstrated that overestimation and underestimation issues were noticed with the Kelvin equation and EOS-P cap when predicting condensation and evaporation pressures of propane in pores below 8 nm , comparing with the density function theory (DFT) predicted outcomes. However, reasonable accuracy of vapour pressure prediction could be achieved with the thermodynamical models when the pore size is above 8 nm . The curvature effect on fluids increases as capillary size becomes tighter due to the change of interfacial properties, such as surface tension and pressure difference at the liquid–gas interface.
Background The Kelvin equation 11 is a theoretical modelling approach that describes the influence of curvature radius at vapour-liquid interface on saturation pressures. The general Kelvin equation can be expressed as 1 R T ln P v P ∞ = - 2 σ LV v L r + v L P v - P ∞ , where T is the fluid temperature, R is the universal gas constant, r is the droplet (or capillary) radius, v L is the molar volume of the liquid, σ LV is the vapour-liquid interfacial tension, P ∞ is the vapour pressure at the flat surface, and P v is the vapour pressure at the curved interface. The term P v - P ∞ on the right side of Eq.
( 1 ) can be neglected owing to its small value comparing to the first term ( - 2 σ LV v L \ r ) 12 . Hence, the approximated form of the equation is as follows: 2 R T ln P v P ∞ = - 2 σ LV v L r . When the medium is liquid wet (concave curvature), the vapour pressure of the liquid reduces along with the reduction of pore size thus leading the vapour pressure at confined spaces ( P v ) to be lower than the vapour pressure at the flat surface ( P ∞ ) or bulk condition. One of the major limitations of the Kelvin equation is that it is not applicable for computing the shift of vapour pressure of multicomponent fluids in hydrocarbon reservoirs, due to their complexity.
The Kelvin equation demonstrates the relationship between vapour pressures in capillary and bulk conditions P r = P ∞ exp 2 σ v L rRT . According to this equation, the vapour pressure of fluids becomes lower than those in bulk scenarios when medium sizes are tighter due to the change of surface tension, pressure drop at the interface, and contact angle. As a result, fluids in highly confined spaces tend to have higher viscosities and capillary pressures. The objective of this paper was to experimentally investigate the vapour pressure of propane in different
Figure 11 ( a ) Initiation of vapour-phase (nucleation) in Berea sandstone; ( b ) significant propane vaporization in Berea sandstone. Quantitative analysis Per the Kelvin equation, vapour pressure could be altered by medium size if it is 1000 nm or less. The vaporization pressure gets lower as the pore throat gets tighter. In Hele–Shaw and microfluidic experiments, the recorded vapour pressures were relativity close to the phase-change pressures measured in bulk conditions. Due to their inner medium sizes, the existed capillary effects in the silicate glass models were not sufficient to result in shifted vapour or condensation pressures since they were larger than 100 nm.
Also, it compares the outcomes with the calculated phase-change pressures from the Kelvin equation. The average pore sizes of rocks were the median pore sizes, obtained from the pore size distribution analysis. On average, the vapour pressures in the rocks were 7% less than the bulk and calculated vapour pressures of propane. Figure 12 Condensation pressures of propane in Hele–Shaw cells and micromodels during the pressure build-up process. Each point for the bulk condition represents the average of 3 trials, and each point for the Hele–Shaw cell and microfluidic chip represents the average of 2 trials. All the pressure values in this figure were measured at 20 °C (293.15 K).
The main objective of this paper was to compare our experimental observations with calculated vapour pressures from the Kelvin equation and Peng–Robinson EOS. Phase-change pressures of propane were investigated in capillary/porous media using Hele–Shaw glass cells, microfluidic silica-glass chips, and reservoir rock samples. The vapour pressures, measured in bulk conditions, were considered as benchmarks. The propane vapour pressures measured with Hele–Shaw cells and microfluidic chips were comparable with those measured at bulk conditions. Additionally, they were identical with the computed saturation pressures from the Kelvin equation (Fig. 13 ).
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