Some industrial processes seem to resist optimization. Filters impervious to the slings and arrows of accumulated “gunk” – the particles, minerals and organic buildup that foul water treatment facilities, nuclear power plants and even breweries – are prime among them.

And that’s where the mathematicians come in.

“Several of my colleagues at NJIT are designing new membrane technologies. I focus on how membrane filters foul up,” says Linda Cummings, a professor of applied mathematics. “The goal is to design a membrane microstructure that will maximize a filter’s lifetime and minimize the cost of maintaining it.”

She looks in particular at ways to optimize the size, shape and distribution of pores within the membrane. “They need to be big enough that the pressure required to force the filtrate through the membrane is not too high, but not so big they don’t filter out the gunk.”

As well as the internal structure of the membrane, fouling depends on factors such as the flow dynamics of the feed solution, and the shape, size and chemistry of particles in the feed. “But the broad engineering challenge is the same: to achieve finely controlled separation at low power consumption.”

So far, her team has determined that the internal patterning of pores within the membrane, and the connections between them, are important. In particular, they can predict how pore size and connectivity should vary across the membrane in the direction of flow in order to maximize the filter lifetime — since fouling begins at the upstream surface, pores there foul fastest, so they need to be larger than those downstream, for example.

Cummings first took up membrane modeling at a “Mathematical Problems in Industry” workshop funded by the National Science Foundation (NSF) that annually brings together academics and representatives from industry to tackle longstanding industrial problems in fluid mechanics, data analysis and mathematical finance, among other areas.

“The industrial problems are presented on Monday, and the academics then break up into groups to work intensively on them for three days. On Friday, we report our progress,” says Cummings, who has led three of these workshops at NJIT since 2011. “In this case, the progress we made was sufficiently promising that NSF awarded further funding to take a deeper look.