Universal expression for the drag on a fluid sphere.

An expression was developed for prediction of drag coefficients for any spherical particle, drop or bubble in an infinite, homogeneous liquid. The formula reproduces the limiting cases for gas bubbles and solid spheres, as well as the exact Hadamard-Rybczynski solution. The accuracy of the expression, which is valid for Reynolds numbers up to a few hundred, was confirmed by comparison with published numerical predictions of the drag coefficient for a range of physical circumstances.

Scale effects in the latent heat of melting in nanopores.

The curvature of a liquid vapor interface has long been known to change the equilibrium vapor pressure. It has also been shown that a capillary structure will affect the temperature at which both freezing and vaporization of a substance will occur. However, describing interfacial effects on the latent heat of a phase change has proven more difficult. Here, we present a classical thermodynamic model for how the latent heat of melting changes as the size of the particles undergoing the transition decreases. The scale dependence for the surface tension is taken into consideration using a Tolman length correction. The resulting model is tested by fitting to published experimental data for the latent heat of melting for benzene, heptane, naphthalene, and water contained in nano-porous glass. In all cases the model fits the data with a R(2) ≥ 0.94.

Comment on "An analytic solution of capillary rise restrained by gravity" by N. Fries and M. Dreyer.

The porous medium model of Green and Ampt, describing flow in porous media, appeared earlier than the capillary model of Washburn, although both lead to mathematically identical models. Here, the model of Green and Ampt was related to the Washburn model by an examination of the parameters involved in each. Fries et al. [N. Fries, M. Dreyer, J. Colloid Interface Sci. 320 (2008) 259-263] presented an explicit solution to this model. This solution is identical to the explicit solution of the Green and Ampt model presented earlier by Barry et al. [D.A. Barry, J.-Y. Parlange, G.C. Sander, M. Sivaplan, J. Hydrol. 142 (1993) 29-46].

Comment on "The effect of evaporation on the wicking of liquids into a metallic weave" by N. Fries, K. Odic, M. Conrath and M. Dreyer.

A recent paper reported capillary rise and evaporation experiments in metallic wicks, as well as a mathematical model. The authors found a consistent discrepancy between the model predictions and data: The model over-predicted the capillary height rise by about 20%. The model used assumes that the porous medium is either fully wet or dry, an assumption that is particularly unsuited to evaporation from the wick surface. An alternative variable-saturation model is proposed that provides a possible explanation for the 20% discrepancy reported by the authors.

Effect of pore structure on capillary condensation in a porous medium.

The Kelvin equation relates the equilibrium vapor pressure of a fluid to the curvature of the fluid-vapor interface and predicts that vapor condensation will occur in pores or irregularities that are sufficiently small. Past analyses of capillary condensation in porous systems with fractal structure have related the phenomenon to the fractal dimension of the pore volume distribution. Recent work, however, suggests that porous systems can exhibit distinct fractal dimensions that are characteristic of both their pore volume and the surfaces of the pores themselves. We show that both fractal dimensions have an effect on the thermodynamics that governs capillary condensation and that previous analyses can be obtained as limiting cases of a more general formulation.

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions.

The relationship between capillary pressure and saturation in a porous medium often exhibits a power-law dependence. The physical basis for this relation has been substantiated by assuming that capillary pressure is directly related to the pore radius. When the pore space of a medium exhibits fractal structure this approach results in a power-law relation with an exponent of 3-D(v), where D(v) is the pore volume fractal dimension. However, larger values of the exponent than are realistically allowed by this result have long been known to occur. Using a thermodynamic formulation for equilibrium capillary pressure we show that the standard result is a special case of the more general exponent (3-D(v))(3-D(s)) where D(s) is the surface fractal dimension of the pores. The analysis reduces to the standard result when D(s)=2, indicating a Euclidean relationship between a pore's surface area and the volume it encloses, and allows for a larger value for the exponent than the standard result when D(s)>2 .

Simplified thermodynamic model for equilibrium capillary pressure in a fractal porous medium.

Defining a relation for equilibrium pressure in a porous medium has been difficult to do in terms of readily measurable parameters. We present a simplified analysis of this problem using the first law of thermodynamics combined with a fractal description of a porous system. The results show that the variation in fluid interfacial area with fluid volume, and the respective interfacial surface tensions, are dominant factors determining equilibrium capillary pressure. Departures from equilibrium are seen to occur when fluid-solid contact lines are in movement. By describing the pore space as fractal we are able to obtain an expression for the change in fluid interfacial area with respect to its volume, and the resulting model shows a strong fit to pressure data obtained from a capillary rise experiment conducted in a coarse-grained SiO2 sand.

A new equation for macroscopic description of capillary rise in porous media.

Capillary rise in porous media is frequently modeled using the Washburn equation. Recent accurate measurements of advancing fronts clearly illustrate its failure to describe the phenomenon in the long term. The observed underprediction of the position of the front is due to the neglect of dynamic saturation gradients implicit in the formulation of the Washburn equation. We consider an approximate solution of the governing macroscopic equation, which retains these gradients, and derive new analytical formulae for the position of the advancing front, its speed of propagation, and the cumulative uptake. The new solution properly describes the capillary rise in the long term, while the Washburn equation may be recovered as a special case.

Modeling pollutant release from a surface source during rainfall runoff.

Though runoff from manure spread fields is recognized as an important mode of nonpoint-source pollution, there are no models that mechanistically describe transport from a field-spread manure-type source. A mechanistic, physically based model for pollutant release from a surface source, such as field-spread manure, was hypothesized, laboratory tested, and field-applied. The primary objective of this study was to demonstrate the potential applicability of a mechanistic model to pollutant release from surface sources. The laboratory investigation used stable sources and a conservative "pollutant" (KCl) so that the dynamic effects of source dissolution and chemical transformations could be ignored and transport processes isolated. The field investigation used runoff and soluble reactive phosphorus (SP) data collected from a dairy-manure-spread field in the Cannonsville watershed in the Catskills region of New York State. The model predictions corroborated well with observations of runoff and pollutant delivery in both the laboratory and the field. "Pollutant" release from surface sources was generally predicted within 11% of laboratory KCl measurements and field SP observations. Laboratory flume runoff predictions with 15 and 26% errors for 25 and 15 mm h(-1) simulated rainfall intensity experiments, respectively, represented root mean square errors of less than 0.2 mLs(-1). A 26% error was calculated for overland flow predictions in the field, which translated into approximately a 39 mLs(-1) error. Results suggest that the hypothesized model satisfactorily represents the primary mechanisms in pollutant release from surface sources.

Water uptake and water diffusivity of seeds.

When pea (Pisum sativum) seeds were wetted, a sharp front separated the wet and dry portions, the seeds swelled, and the water content in the wetted portion continued to increase for a long time. A model was proposed and tested that takes into account these three characteristics and in particular does not postulate a constant diffusivity. The parameters of the model are simply the rate of penetration of the wetting front and a swelling factor.

Ventilation required to entrain small particles from leaves.

Particles are blown from leaves when the wind at the height of the particles exceeds a minimum which is about 5 m/sec for some fungal spores. In the moderate winds typical within a canopy of leaves, the minimum is attained at spore height during brief changes in wind or puffs before the boundary layer grows to particle height. The requisite change in speed to remove spores occurs over a sizeable area only when the speed changes abruptly in a short distance in the direction of the wind.

Water uptake, diameter change, and nonlinear diffusion in tree stems.

A diffusion model for phloem swelling and contraction is proposed in which the rate of water movement changes markedly with moisture content. Good agreement between the actual swelling of the phloem of cotton stems and that predicted by the model was obtained. This result implies that water moves more readily into the phloem when it becomes wetter. This model also explains the lag of shrinkage of pine stems behind the water potential of the foliage and predicts that the lag is related to the thickness of the phloem.

Boundary Layer Resistance and Temperature Distribution on Still and Flapping Leaves: II. Field Experiments.

The forced convection of heat from reed (Phragmites communis) leaves was observed in their natural environment. The leaves were painted with liquid crystals, which displayed or indicated their temperature without any interference with natural air flow. Temperature differences as large as 15 C were observed between the leading and trailing edges of the nontranspiring, painted leaves. The turbulence of the natural wind decreased the boundary layer resistance around the leaf to about 40% of the resistance in a laminar steady wind.

Boundary layer resistance and temperature distribution on still and flapping leaves: I. Theory and laboratory experiments.

If the evaporation is uniform on a flat exposed leaf, forced convection will also be nearly uniform, and the leaf temperature will vary with the square root of the distance from the leading edge. Then the resistance expressed in terms of the proper, i.e., average, temperature has the same value as the resistance of a leaf at uniform temperature. Compared to a steady laminar flow, the turbulence of a realistic wind decreases the resistance by a constant factor of about 2.5. The same constant factor was observed whether the leaf was flapping or not, when the wind velocity was not too low.

Stomatal dimensions and resistance to diffusion.

In the past, relations of diffusive resistance to stomatal geometry have concerned circular pores or pores that are replaced by equivalent circles of the same area. We calculated the resistance for general shapes that include the realistic slit. The resistance comprises two terms. The first is an outer resistance that depends only on ventilation and leaf geometry and is independent of stomata. The second is an inner resistance and is a function of stomatal interference and of stomatal geometry only. If interstomatal spacing is at least three times stomatal length, interstomatal interference is negligible. The inner resistance can then be calculated by adding the resistance of the two ends and the throat of each stoma. In the case of an elongated stoma, the part of the diffusive resistance per square centimeter determined by stomatal geometry is [Formula: see text] where a, b, d, and n are the semilength, semiwidth, depth, and density of the stomata, and D is the diffusivity. This is the familiar Brown and Escombe result applied to slits.