The Liquid-Vapor Interface in the Ideal Soil

Gardner, Willard; Gardner, J.

doi:10.1097/00010694-195308000-00006

Cited by 2 publications

(1 citation statement)

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“…In addition, the volume of a pendular ring is required for the liquid saturation. Several expressions can be found in the literature with discrepancies among them (Dallavalle 1948, p. 288;Gardner and Gardner 1953;Rose 1958;Mayer and Stowe 1966;Gvirtzman and Roberts 1991;Likos and Lu 2004). …”

Section: Pendular Regionmentioning

confidence: 99%

An Analytical Model for Capillary Pressure–Saturation Relation for Gas–Liquid System in a Packed-Bed of Spherical Particles

Lappalainen¹,

Manninen

Alopaeus³

et al. 2008

Transp Porous Med

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Capillary pressure is considered in packed-beds of spherical particles. In the case of gas-liquid flows in packed-bed reactors, capillary pressure gradients can have a significant influence on liquid distribution and, consequently, on the overall reactor performance. In particular, capillary pressure is important for non-uniform liquid distribution, causing liquid spreading as it flows down the packing. An analytical model for capillary pressure-saturation relation is developed for the pendular and funicular regions and the factors affecting capillary pressure in the capillary region are discussed. The present model is compared to the capillary pressure models of Grosser et al. (AIChE J., 34:1850(AIChE J., 34: -1860(AIChE J., 34: , 1988 and Attou and Ferschneider (Chem. Eng. Sci., 55:491-511, 2000) and to the experiments of Dodds and Srivastava (Part Part Syst. Charact., 23:29-39, 2006) and Dullien et al. (J. Colloid Interface Sci., 127:362-372, 1989). The non-homogeneity of real packings is considered through particle size and porosity distributions. The model is based on the assumption that the particles are covered with a liquid film, which provides hydrodynamic continuity. This makes the model more suitable for porous or rough particles than for non-porous smooth particles. The main improvements of the present model are found in the pendular region, where the liquid dispersion due to capillary pressure gradients is most significant. The model can be used to improve the hydrodynamic models (e.g., CFD and cellular automata models) for packed-bed reactors, such as trickle-bed reactors, where gas, liquid, and solid phases are present. Models for such reactors have become quite common lately (Sáez and Carbonell, AIChE J., 31:52-62, 1985; K. Lappalainen (B) · V. Alopaeus Chemical Engineering and Plant Design, Nomenclatures A c Cross-sectional area (m 2 ) d Half spacing between neighboring microcylinders (m) d p Particle diameter (m) d min As defined in Eq. 5 (m) D H Hydraulic diameter (m) f Wetting efficiency J (S L ) Leverett's J -function k Permeability (m 2 ) N c Number of particle contact points per particle p c Capillary pressure (Pa) p i Pressure of phase i (Pa) r 1,2 Scaled radii of curvature R 1,2 Radii of curvature (m) R Particle radius (m) R * Mean curvature of the meniscus (m) s Perimeter (m) S L Liquid saturation V f Volume of fluid in a pendular ring (m 3 )Greek letters ε Void fraction of the packed-bed ϕ Filling angle (see Fig. 3) (rad) (x) Gamma function θ Liquid-solid contact angle (rad) θ i Holdup of phase i ρ i Density of phase i (kg · m −3 ) σ Surface tension between the wetting and the non-wetting phases (N · m −1 ) X(1 − cos ϕ)

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