In 1920, a young German engineer named Albert Betz peered down at his calculations. Although interest in renewable energy was a long way from reaching its peak, he’d been exploring how wind turbines capture energy from the air. In the process, he had come up with a calculation for the greatest possible efficiency of any wind turbine: a shockingly tidy sixteen twenty-sevenths, or 59.3 percent. Since then, this number has been known as the Betz limit, serving as a virtually impossible goal for efficiency—until now.
“In order to make that calculation, [Betz] had to make some simplifying assumptions,” says John Dabiri, a professor of mechanical engineering at the California Institute of Technology. In particular, he chose to approximate all wind as perfectly steady. “Now, we know that’s untrue because the wind is turbulent,” says Dabiri. “If I’m standing on a wind farm, there will be gusts so the wind won’t be a fixed value.” Further complicating the situation, popular propeller-style turbines create a swirling effect that propagates behind the turbine as well.
It’s always been understood that the Betz limit was merely an approximation for maximum efficiency, but Dabiri says there’s a reason no one has tried to make a more accurate model in the intervening decades. “Although his model neglects some of the real physics, the assumption has typically been that the physics he neglected would only reduce the efficiency of the system. So people haven’t really found any reason to question the Betz limit as an upper bound.” And since no wind turbine has surpassed 59.3% efficiency*, it seemed to make sense.
That troubled Dabiri as he wondered what effect a flow that is not perfectly steady would actually have on a turbine’s efficiency. “I teach a wind energy class and when I do the Betz limit I’ve always told the students, it’s a simplified version of the full physics,” he admits. “It’s only recently, this past summer that I sat down to try to come up with a clean way to do this analysis.” His resulting paper, published in Physical Review Fluids, shows that in some cases unsteady airflow could actually improve a turbine’s efficiency, not detract from it.
|Figure 1. Wind turbines have been in use for millennia, but they’ve experienced a recent surge in interest as governments shift to more renewable resources. Image credit: Kim Hansen|
If that seems counterintuitive, you’re not alone—after all, it’s taken a century for this problem to be fully explored. Here’s how the math shakes out.
When Betz originally developed his wind turbine model, he realized that the work that wind does on a turbine is the direct result of a pressure difference across the turbine’s body. “The amount of power that the turbine puts out is proportional to both the pressure drop and how much air is passing through the wind turbine—which is roughly equivalent to the airspeed” Dabiri explains. But the Bernoulli equation, a pillar of fluid dynamics, explicitly states that for a steady flow the sum of the pressure and flow speed (squared, technically) is constant, making it impossible to have both high pressure and high airspeed.
“Now in our case, we note that the Bernoulli equation, if you write it out more fully, has another term,” Dabiri says. The third term relates to the unsteady motion of air around a wind turbine, potentially created by the wind turbine itself, and its addition makes it possible to increase both the pressure and the flow speed—as long as some part of the wind turbine is buffeted upwind.
Many lines of algebra later, Dabiri managed to derive an expression for the maximum efficiency of this new model. While the addition of that third term means the expression is not as tidy as Betz’s simple fraction, they found that in some situations it’s possible to exceed a 59.3% efficiency—reaching efficiencies past 90 percent! These enormously high efficiencies are only possible when the entire wind turbine is moving upwind, which obviously can’t continue unchecked, but even when they took an average of the efficiency over time they found that the turbines could still exceed the Betz limit in the long run.
These findings won’t do much good for conventional turbines that don’t move into the wind, like the ever-popular rotary blade style, as the mechanism for higher efficiency requires the turbine to have the freedom to move upwind. Although a few such turbines do currently exist, Dabiri wants to explore new designs that take full advantage of the new model. “The wind turbines you see out there today qualitatively don’t look any different than they did 30 or 40 years ago,” he says. “Real efficiency is approaching 50%, which is close to 59.3% [calculated by Betz]. That’s really one of the reasons why people think of wind energy as being ‘mature technology’—I would argue that in some cases it discourages students from thinking about wind energy as a really exciting place to be.” If, on the other hand, engineers reconsider the way they design these devices to take full advantage of unsteady flow, Dabiri thinks it could open up some exciting new advances.
He does caution that this is still a theoretical project, and it doesn’t lay out a blueprint for this new type of turbine. Nevertheless, he is optimistic about the sector’s future, adding, “One of the reasons the Betz limit has been so enduring is that it didn’t assume a specific design, so it’s been applied to many different types of wind turbines. Likewise, I think that because this analysis doesn’t hypothesize a specific type of turbine, it potentially is applicable to a variety of types.” Now we just have to design them
*However, many people claim to have found ways around the Betz limit by introducing external baffling and other structures—in direct contradiction of Betz’s original assumption of unbounded airflow. “Betz didn’t claim that putting some other structure near the turbine couldn’t improve the performance. It’s really a separate problem in that case,” Dabiri says. In contrast, Dabiri shows that it’s possible to develop wind turbines with efficiencies exceeding the Betz limit without the additional cost or bulk of external structures.
— Eleanor Hook