Image: Lucy Lindsey |
Sand
in an hourglass might seem simple and straightforward, but such granular
materials are actually tricky to model. From far away, flowing sand resembles a
liquid, streaming down the center of an hourglass like water from a faucet. But
up close, one can make out individual grains that slide against each other,
forming a mound at the base that holds its shape, much like a solid.
Sand’s
curious behavior—part fluid, part solid—has made it difficult for researchers
to predict how it and other granular materials flow under various conditions. A
precise model for granular flow would be particularly useful in optimizing
processes such as pharmaceutical manufacturing and grain production, where tiny
pills and grains pour through industrial chutes and silos in mass quantities.
When they aren’t well-controlled, such large-scale flows can cause blockages
that are costly and sometimes dangerous to clear.
Now
Ken Kamrin of Massachusetts Institute of Technology’s (MIT) Department of
Mechanical Engineering has come up with a model that predicts the flow of
granular materials under a variety of conditions. The model improves on
existing models by taking into account one important factor: How the size of a
grain affects the entire flow. Kamrin and Georg Koval, assistant professor of
civil engineering at the National Institute of Applied Sciences in Strasbourg,
France, used the new model to predict sand flow in several configurations—including
a chute and a circular trough—and found that the model’s predictions were a
near-perfect match with actual results. A paper detailing the new model will
appear in Physical Review Letters.
“The
basic equations governing water flow have been known for over a century,” says
Kamrin, the Class of ’56 Career Development Assistant Professor of Mechanical
Engineering. “There hasn’t been something similar for sand, where I can give
you a cupful of sand, and tell you which equations will be necessary to predict
how it will squish around if I squeeze the cup.”
Blurring the lines
Kamrin explains that developing a flow model—also known as a continuum model—essentially
means “blurring out” individual grains or molecules. While a computer may be
programmed to predict the behavior of every single molecule in, say, a cup of
flowing water, Kamrin says this exercise would take years. Instead, researchers
have developed continuum models. They imagine dividing the cup into a patchwork
of tiny cubes of water, each cube small compared to the size of the entire flow
environment, yet large enough to contain many molecules and molecular
collisions. Researchers can perform basic lab experiments on a single cube of
water, analyzing how the cube deforms under different stresses. To efficiently
predict how water flows in the cup, they solve a differential equation that
applies the behavior of a single cube to every cube in the cup’s grid.
Such
models work well for fluids like water, which is easily divisible into
particles that are almost infinitesimally small. However, grains of sand are much
larger than water molecules—and Kamrin found that the size of an individual
grain can significantly affect the accuracy of a continuum model.
For
example, a model can precisely estimate how water molecules flow in a cup,
mainly because the size of a molecule is so much smaller than the cup itself.
For the same relative scale in the flow of sand grains, Kamrin says, the sand’s
container would have to be the size of San
Francisco.
Neighboring chatter
But why exactly does size matter? Kamrin reasons that when modeling water flow,
molecules are so small that their effects stay within their respective cubes.
As a result, a model that averages the behavior of every cube in a grid, and
assumes each cube is a separate entity, gives a fairly accurate flow estimate.
However, Kamrin says in granular flow, much larger grains such as sand can
cause “bleed over” into neighboring cubes, creating cascade effects that are
not accounted for in existing models.
“There’s
more chatter between neighbors,” Kamrin says. “It’s like the basic mechanical
properties of a cube of grains become influenced by the movement of neighboring
cubes.”
Kamrin
modified equations for an existing continuum model to factor in grain size, and
tested his model on several configurations, including sand flowing through a
chute and rotating in a circular trough. The new model not only predicted areas
of fast-flowing grains, but also where grains would be slow moving, at the very
edges of each configuration—areas traditional models assumed would be
completely static. The new model’s predictions matched very closely with
particle-by-particle simulations in the same configurations.
The
model, run on a computer, can produce accurate flow fields in minutes, and
could benefit engineers designing manufacturing processes for pharmaceuticals
and agricultural products. For example, Kamrin says, engineers could test
various shapes of chutes and troughs in the model to find a geometry that
maximizes flow, or mitigates potentially dangerous wall pressure, before ever
actually designing or building equipment to process granular materials.
Kamrin
says understanding how granular materials flow could also help predict
geological phenomena such as landslides and avalanches and help engineers come
up with new ways to generate better traction in sand.
“Granular
material is the second-most-handled material in industry, second only to water,”
Kamrin says. “I’m convinced there are a million applications.”
