Physics reveals the optimal roof ratios for energy efficiency

Or perhaps the ancient Roman builders were on to something. An expert in thermodynamics and the movement and flow of heat, Bejan, the J.A. Jones Distinguished Professor of Mechanical Engineering at Duke, was the perfect person to sleuth out an answer.

Sitting down with pencil and paper, Bejan went through the equations and calculations that govern heat flow and transfer within two similar shapes: a long roof with a triangular cross section and a circular cone. The results, obtained in collaboration with Pezhman Mardanpour, assistant professor of mechanical and materials engineering at Florida International University, were published online March 28 in the journal International Communications of Heat and Mass Transfer. They showed that there are indeed roof shapes that maximize heat retention -- the older generation of Italian architects knew what they were doing.

"Pockets of air aregood insulators, and attics are basically just differently shaped pockets of air," Bejan said. "While energy conservation is a popular buzzword today, in years long past it was a matter of survival."

The details of how squat or tall a roofline determines how the air within it will act. Given a single peak on an A-frame or circular cone, if that peak is less than three feet tall, the air will flow smoothly and uniformly across it like water careening down the side of a sink. But if the peak is more than three feet tall, the air will tumble around chaotically like smoke waving wildly in the wind.

Based on the physics of these airflows and heat transfer, if a roof peak is shorter than roughly three feet, it should be about three or four times wider than it is tall to minimize heat loss. And if a roof peak is taller than three feet, it should be an equilateral triangle with a height-to-width ratio of one.

Perhaps unsurprisingly, these are roughly the same ratios that can be found in countless older, modest dwellings created across the world. And they are pretty close to the rooflines that Bejan saw that day in southern Italy.

"This type of insight is not hard to rationalize, but it's easy to overlook even though there are examples everywhere," Bejan said. "It's important for our students -- and their professors -- to open their imaginations and ask why things are the way that they are."

While Bejan doubts that architects from days gone by were applying thermodynamics to their roof designs, he doesn't think their shapes were accidental, either. It isn't difficult to imagine, he says, discovering that one neighbor's home is warmer than another's and copying its design repeatedly across many years.

It's a lesson, he says, that modern architects could stand to consider as well.

"Today's homes and buildings are being designed to be as energy efficient as possible," Bejan said. "But to my knowledge, nobody is considering the physical shape of the building, or any 'thing' like a vehicle or animal, as a variable that could help with that efficiency, and perhaps we should be."

This work was supported by the U.S. Air Force Office of Scientific Research (FA9550-23-1-0716, FA9550-22-1-0525.