Liquid/vapor interfacial area

The pore-scale model provides the basis for development of a statistical framework for upscaling from a single pore to a sample of variably saturated porous medium. The statistical distribution of pore sizes is modeled as a gamma distribution with the expected values of liquid configuration in pore space calculated from geometrical and chemical potential considerations within the statistical framework. Model predictions compare favorably with measured retention data, yielding similarly close fits to data as the widely used van Genuchten parametric model. Liquid-vapor interfacial area as a function of chemical potential is readily calculated from estimated retention parameters. Comparisons of calculated inter-... 

The review is organized as follows first, we discuss aspects of the unitary approach for combining adsorptive and capillary contributions, and present the new pore scale model of Tuller et al. (1999). The upscaling scheme of Or and Tuller (1999) for representing sample scale retention properties will be presented, followed by illustrative examples with measured characteristic data and a discussion of critical soil parameters. The role of liquid-vapor interfacial area will be highlighted by comparisons of model predictions with limited measurements. Finally, we introduce hydrodynamic considerations of unsaturated flow in films and comers leading to prediction of hydraulic conductivity of rough rock surfaces and unsaturated porous media. 

The spontaneous snap-off processes require two scenarios to be considered—prior to and after slit filling—for calculation of pore saturation and liquid-vapor interfacial area. Following pore snap-off the pore is completely saturated and the liquid-vapor interface is zero. Expressions for calculating saturation, 2>w( x), defined as liquid area per pore cross-sectional area (these are translated to their respective volumes for three-dimensional pores) for all regular polygon-shaped central pores are given as ... 

Pore Scale Liquid-Vapor Interfacial Area... 

The liquid-vapor interfacial area relative to the entire pore cross-section is calculated as ... 

Closed-form expressions for liquid-vapor interfacial area are derived in a similar fashion as for liquid saturation. A general expression is given as the sum of two terms subjected to identical limits of integration as depicted in Fig. 1-8 ... 

The first term describes the interfacial area per pore volume following pore emptying (while slits are still liquid filled). Liquid-vapor interfaces are composed of the curved interfaces in the corners (capillary contribution) and film interfaces in the flat areas between comers (adsorptive contribution). The second term is for the expected value of liquid-vapor interfacial area following the formation of liquid-vapor interface in the slits following slit snap-off. These include interfaces in the central pore and flat film interfaces in the slits. Detailed solutions for the integrations are given in Tuller and Or (2001). 

We proceed with illustrative examples for application of the proposed up-scaling scheme to seven soil types with properties listed in Table 1-2. The closed-form solution for degree of saturation (Eq. [23]) was fitted to measured data by optimizing parameters p, go, X, and the chemical potential pd at air entry point (that defines Lmax). Note that the Hamaker constant was estimated beforehand, as described in Estimation of the Effective Hamaker Constant for Solid-Vapor Interactions for Different Soils above. The estimated parameters were then used to calculate the liquid-vapor interfacial area for each soil (Eq. [28]). We used square shaped central pores for all soil types except the artificial sand mixture, where triangular pores were applied to emphasize capillaiy processes over adsorption in sand. I lowcver, the closed-form solutions for retention and interfacial area were derived lo accommodate any regular polygon-shaped central pore. Constants for various shapes are described in Table I-1. The values of primary physical constants employed in (he calculations and (heir units are shown in Table 1-3. 

Liquid-vapor interfacial area per pore volume (Eq. [28]) was calculated as a function of chemical potential (Fig. l-10b) using the parameters estimated from the retention curve. The resulting liquid-vapor interfacial values were then multiplied by the medium porosity to yield interfacial area per bulk sample volume as depicted in Fig. 1-1 Ob. The interfacial area Alv = 0 for p > pd (or until the largest pore is invaded by air/vapor). At the dry end of the A]v curve the interfacial area... 

Knowledge of detailed liquid-vapor configurations enables separation of capillary and adsorptive contributions to the interfacial area as shown in Fig. 1-1 la (note the log-log scale). We denote liquid-vapor interfacial areas associated with menisci (curved interfaces at pore comers) as capillary contributions, and those associated with films as adsorptive contributions. The results in Fig. 1-1 la illustrate the dominant contribution of liquid films to the total liquid-vapor interfacial area of a partially saturated porous medium (Millville silt loam). Note that the flat region in Fig. 1-1 la (changes in SA with no change in p) reflects pore snap-off processes. 

The primary reason for the minor capillary contribution to liquid-vapor interfacial area even in a medium with relatively large (triangular-shaped) pores and small surface area lies in the control exerted by the chemical potential on liquid-vapor menisci. For a given potential, meniscus curvature is constant throughout the porous medium irrespective of pore size. This means that after pore snap-off the capillary contribution to liquid-vapor interfacial area from small and large pores is equal if their shapes (polygon and angularity) are similar. 

The shape of the capillary portion of the liquid-vapor interfacial area for sand (Fig. 1-1 lb) resembles simulation results of Reeves and Celia (1996) of interfacial areas in pore networks due to capillarity only. The discussion illustrates potential limitations in using cylindrical pore network models (Reeves Celia, 1996) especially for studies of volatile liquids and surfactants, and other multiphase transport processes where interfacial areas play a crucial role (Kim et al., 1997 Karkare Si. Fort, 1996). Furthermore, the overwhelming role of adsorbed liquid films casts doubts on several proposed constitutive relationships between capillary pressure (saturation) and interfacia] area (Skopp, 1985 Hassanizadeh Gray, 1993) most of which were based on assumed cylindrical capillary geometry in the absence of adsorption. 

Liquid-vapor interfacial area was determined for improved understanding of microbiological and related processes in unsaturated porous media. 

The dominant role of liquid films in creating liquid-vapor interfacial areas was elucidated (pending experimental confirmation). 

Allv = Hamaker constant for liquid-liquid interactions through intervening vapor (J) A v = Liquid-vapor interfacial area (m2)... 

The liquid/vapor interfacial area ratio is the most difficult parameter to be determined [92]. It depends on porosity, pore radii, pore space connectivity, and saturation. The following form is suggested... 

Intrinsic rate constant of evaporation Evaporation-penetration depth Liquid water viscosity Ratio of the distributed liquid vapor interfacial area to the apparent electrode surface area Active site fraction effective proton conductivity in CCL... 

Because the liquid wets and spreads over the solid surfaces, pores will be formed in the liquid. The reduction of the liquid-vapor interfacial area provides the driving force for shrinkage or densification of the compact. If the pore in the liquid is assumed to be spherical with radius of r, the pressure difference across the curved surface is given by the Young and Laplace equation ... 

The system energy of Fig. 12 can be estimated by using the above solution for the meniscus profile. As stated in the previous section, we consider the potential energy E, the energy of the liquid vapor interfacial area Elv, and the work done by the three-phase contact line wetting the cone surface E f/. The dry cone surface and the horizontal liquid surface (z = 0) are taken for the reference state of system energy. E and which represent the work necessary to form the axisymmetric meniscus shown in Fig. 12, are calculated from... 

Written in the form of Equation 1.47, the rate Qmp is determined by the liquid/vapor interfacial area, the temperature, and the density of water vapor in the CL. It does not explicitly depend on the cell current density. However, the dependence on jo is hidden in the structural parameter which is a function of the current-dependent water accumulation in the CCL. 

The interfacial vaporization exchange area is an important property, which exerts a marked impact on CCL performance. Few experimental and theoretical studies have explored this property. It depends on fine topological details of the pore network. The menisci separating liquid and gas phases in pores contribute to the liquid-vapor interfacial area. The dependence on porosity and pore radii is written in the form... 

Total liquid/vapor interfacial area per total geometric electrode surface area. Equation 1.47... 

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