Comment pubs.acs.org/Langmuir
Comment on “Source-like Solution for Radial Imbibition into a Homogeneous Semi-infinite Porous Medium”
■
■
INTRODUCTION In a recent article,1 Xiao et al. gave a theoretical analysis of capillary imbibition from a small circular source into a semiinfinite homogeneous porous material. They also reported valuable results from experiments on the imbibition of water from such a source into a packing of glass microspheres. The movement of the wetting front was tracked photographically. The capillary absorption of liquids from finite sources into extended porous media in two and three dimensions is of longstanding practical interest, notably in soil physics2 and construction engineering.3 Understanding diverging flows from finite sources is particularly important in the design of devices such as the disk infiltrometer.4 The principal result of Xiao et al.1 is that at long times the imbibition rate reaches a steady nonzero value, and because the wetted region is hemispherical and the porosity constant, the wetting front advances radially as t1/3, where t is the elapsed time. It should be noted that the radial flow behavior of such a transport model was discussed earlier by Philip, first in 19665 and then more fully in 1969.6 Philip gave the complete solution for the absorption rate from a spherical cavity as a function of time and showed the existence at large times of a steady nonzero absorption rate. Philip’s analysis was used later7 to model the long-time absorption of water through a small circular source into a block of fired-clay ceramic. Experimental data confirmed the longtime steady rate. The Philip analysis has also been applied to the absorption of water from a hemispherical cavity in mortars and brick.8 In all of these analyses, it is assumed that the wetted region is saturated, as is assumed also by Xiao et al.1 These are therefore Green-Ampt or Sharp-Front models and are distinct from unsaturated flow models in which a gradient of liquid−water content exists within the wetted region. However, the fact that there is a large-time steady rate in capillary absorption from finite sources in three dimensions is a general result that holds for all saturated and unsaturated flows2,5,6 (and also for the corresponding diffusion and heat conduction problems).
REFERENCES
(1) Xiao, J.; Stone, H. A.; Attinger, D. Source-like solution for radial imbibition into a homogeneous semi-infinite porous medium. Langmuir 2012, 28, 4208−4212. (2) Warrick, A. W. Soil Water Dynamics; Oxford University Press: New York, 2003. (3) Hall, C.; Hoff, W. D. Water Transport in Brick, Stone and Concrete, 2nd ed.; Spon Press: London, 2012. (4) Smettem, K. R. J.; Parlange, J.-Y.; Ross, P. J.; Haverkamp, R. Three-dimensional analysis of infiltration from the disc infiltrometer: 1. A capillary-based theory. Water Resour. Res. 1994, 30, 2925−2929. (5) Philip, J. R. Absorption and infiltration in two- and threedimensional systems. UNESCO Symposium on Water in the Unsaturated Zone; International Association for Scientific Hydrology: Wageningen, 1996; Vol. 1, pp 503−525. (6) Philip, J. R. Theory of infiltration. Adv. Hydrosci. 1969, 5, 215− 296. (7) Hall, C. Water movement in porous building materials − IV. The initial surface absorption and the sorptivity. Build. Environ. 1981, 16, 201−207. (8) Wilson, M. A.; Hoff, W. D.; Hall, C. Water movement in porous building materials − XI. Capillary absorption from a hemispherical cavity. Build. Environ. 1994, 29, 99−104.
Christopher Hall*
■
School of Engineering, The University of Edinburgh, The King’s Buildings, Edinburgh EH9 3JL, U.K.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS I thank W. D. Hoff for valuable comments. © 2012 American Chemical Society
Received: March 4, 2012 Published: May 22, 2012 8587
dx.doi.org/10.1021/la3009305 | Langmuir 2012, 28, 8587−8587
