Answer: Because they’re small. That’s it. It’s just basic physics. And there’s nothing that can be done about it.
Background
In Post #1843, I figured out and explained why ceiling fans are vastly more efficient that box fans. Where efficiency is measured by cubic feet of air moved per minute, per watt of power used (CFM/watt).
The answer turned out to be remarkably simple: To move the same volume of air, a smaller fan blade has to move that air much faster. That’s just arithmetic. (If the area swept by a 20″ box fan blade is one-seventh the area swept by a big ceiling fan, the box fan has to move the air seven times faster, to keep up with the volume moved by the ceiling fan.)
Moving air faster takes much more energy than moving it slowly. Not due to the energy-wasting turbulence that might create ( though that can be a factor), but merely because it takes more pressure to move air faster, and overcoming that pressure takes more energy.
Roughly speaking, CFM/watt should scale inversely with the size of the fan. Given identical designs and motors, a box fan that is one-seventh the size of a ceiling fan should take seven times the wattage to move the same amount of air. Roughly.
That’s all laid out in Post #1843.
And now on to bath fans
Today the penny dropped, and I realized that this same phenomenon explains the poor performance of bathroom vent fans. Seems like bath fans take forever to clear a bathroom. And I include all bath fans, almost regardless of make or quality. Where a box fan stuck in a window could clear the air in a bathroom in a couple of minutes, an in-ceiling bath fan might take half an hour.
At best, a bathroom vent fan might have 6″ blades, feeding a 6″ diameter duct. (Although 4″ duct for bath fans is far more common). Since the area of a circle goes as the square of the radius, the area swept by the blades of a 6″ bath fan would be about ( 3-squared / 10-squared = ) 9% of the area swept by the blades of a 20″ box fan. And so, to move the same volume of air as a box fan, a hypothetical 6″ bath fan would require (1 / .09 =) 11 times the wattage.
Let me now put that to the test, via virtual shopping at Home Depot.
And, sure enough, the median bath fan from Home Depot moves about one-tenth as much air per watt, compared to a box fan.
Bottom line: A bath fan that could clear as bathroom as fast as a box fan would draw ten times the wattage of the box fan. If you could squeeze that much air, that fast, through the ducts, you’d need to have a 500-watt bath fan*, in order to clear a bathroom as fast as a box fan sitting in the window. That, before we even consider whether or not you could move that much air through a small duct without undue losses due to turbulence. That, before we consider how much noise that would make.
* That’s 2/3rds of a horsepower, more or less. A big electric motor, in this application.
And so, the apparent poor performance of bathroom fans is not a figment of my imagination. Bath fans move air quite slowly, compared to (e.g.) common box fans. It’s not a design flaw, or an intentional choice. It’s just physics. The smaller the fan, the more power it takes to move a given amount of air. And bath fans — typically restricted to 4″ ducts — can only move a tenth of the amount of air that box fans can move, per watt of power.
