Discrete motion on a micro scale

May 1, 2012
A new model from MIT could prove useful to engineers looking for accurate ways to predict granular material behavior.

Sand in an hourglass may appear to fall simply and smoothly, but its flow — and that of other granular materials — is surprisingly tricky to model. From a distance, the sand resembles liquid, streaming downward like water from a faucet. Closer inspection tells a different story: Individual grains slide against each other, forming a mound that holds its shape, much like a solid.

The mysterious behavior of sand — part fluid, part solid — makes it difficult for researchers to predict. What's the significance? A precise model for not just sand, but granular flow in general, could prove useful in optimizing pharmaceutical manufacturing and grain production, where tiny pills and grains pour through industrial chutes and silos in massive quantities. When not closely controlled, such flows experience blockages that are costly and sometimes dangerous to clear.

Good news is on the horizon. Ken Kamrin of MIT's Department of Mechanical Engineering has developed a model that predicts granular material behavior under a variety of conditions. The model improves on existing methods by considering one important factor — grain size. Kamrin used the new model to predict sand flow in several configurations, including a chute and a circular trough; the model's predictions almost perfectly match actual results.

“The basic equations governing water flow have been known for over a century,” says Kamrin. “There hasn't been anything similar for sand, where I could give you a cupful and tell you which equations are necessary to predict how it will squish around if I squeeze the cup.”

Kamrin explains that his flow model — a continuum model — essentially “blurs out” individual grains. To understand the approach, consider the analysis of a cup of water. Though a computer could be programmed to predict the behavior of every single water molecule in the cup, the exercise would take years. Instead, researchers use continuum models. Here, the cup is mathematically divided into a patchwork of tiny water cubes, each quite small compared to the cup, yet large enough to contain many molecules. With this model, researchers analyze a single cube, and how it deforms under different stresses. Then a differential equation applies the single cube's behavior to every cube in the volume.

The catch: Grains of sand are much larger than water molecules, and an individual grain's size can significantly affect the accuracy of a continuum model. For example, models can only estimate water molecule flow in a cup because the molecule size is so much smaller than the cup itself. For the same relative scale for sand-grain flow, the container would need to be the size of San Francisco.

Kamrin reasons that unlike small-molecule flow, the larger grains involved in granular flow “bleed over” into neighboring cubes, creating cascade effects overlooked in existing models. When applied, Kamrin's modified continuum-model equations, which factor in grain size, successfully predict the behavior of fast-flowing grains as well as slow-moving grains at the edges of each configuration — areas traditional models assume static. The modeling is done on a computer in minutes.

In pharmaceutical and agricultural industries, Kamrin's models could prove useful to engineers optimizing chute and trough geometries. In meteorological applications, understanding granular material flow could also help predict geological events like landslides and avalanches.

For more information, visit mit.edu.

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