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Darcy’s Permeability

Posted on by Jim

Darcy’s Law relates the volumetric flow rate Q through a porous media in response to a pressure gradient \Delta P. …

Q = \frac{\kappa A}{\mu}\frac{\Delta P}{L}

Where \kappa is the intrinsic permeability of the media, \mu is the fluid viscosity, L is the length of the media that fluid must travel through to exit, and A is the cross section normal to the fluid flow direction.

File:Darcy's Law.png From Wikipedia

The permeability \kappa in this equation is truly an intrinsic property of the media. Although the flow rate will vary in proportion to the media cross section, and inversely with fluid viscosity, and length; these quantities are excluded from \kappa so that \kappa is strictly a function of the ultrastructure of the media (porosity, pore size, etc.).

It is important for us to distinguish Darcy’s permeability from our commonly used  Hydraulic Permeability:

\epsilon = \frac{Q}{\Delta P \, A_m}

The relationship between the two permeabilities is

\epsilon = \frac{\kappa }{L \, \mu}

From this relationship it is clear that \epsilon is not an intrinsic material property because changing the value of L (the membrane thickness) or \mu will change its value. Because by “hydraulic” permeability we exclusively mean the passage of water \mu will always have the same value and this dependence is of no consequence. The dependence of \epsilon on length however, is key. We get higher flow rates for the same pressure and area precisely because of membrane thinness and so we want the permeability values that we use for comparisons to other materials to capture this important material difference.

About

Professor McGrath holds a BS degree in Mechanical Engineering from Arizona State and a MS degree in Mechanical Engineering from MIT. He earned a PhD in Biological Engineering from Harvard/MIT's Division of Health Sciences and Technology. He then trained as a Distinguished Post-doctoral Fellow in the Department of Biomedical Engineering at the Johns Hopkins University. Professor McGrath has been on the Biomedical Engineering faculty at the University of Rochester since 2001 where he also served as the director of the graduate program in BME for more than a decade and currently serves as Associate Director of the URNano microfab and metrology core. Professor McGrath also has faculty affiliations with many other programs at UR including the Material Research Program, the Environmental Health and Sciences Center, the Biochemistry and Biophysics program, and the Musculoskeletal Research Center. McGrath's graduate, post-doctoral, and early faculty research was focused on quantitative experiments and mathematical modeling of cell migration covering molecular, cellular, and multi-cellular phenomena. This was true until 2007 when he, along with Professor Philippe Fauchet (now Dean at Vanderbilt) and PhD students Tom Gaborski (RIT) and Chris Streimer (Adarza), discovered a means to self-assembled nanopores in 15 nm thick free-standing silicon and demonstrated the remarkable transport properties of the new material in a Nature paper. This seminal discovery led to the creation of the multidisciplinary Nanomembrane Research Group (NRG) and the founding of SiMPore Inc. in the same year. The NRG and SiMPore have been dedicated to the advancement of ultrathin membrane technologies and exploring all of their potential applications ever since. This blog also dates back to 2007 and has had contributions from more than 100 students, faculty, scientists, engineers, and entrepreneurs. It contains over 2,500 pages and posts logging progress large and small over all these years. Yet somehow it feels like we are just getting started.

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