Wouldn’t they be the same shape as each other for maximum efficiency?
WINDMILLS DO NOT WORK THAT WAY!
Oh, and just to add a bit of substance – their purposes are vastly different, almost opposite. A fan is trying to move the most amount of air with the least amount of rotation possible. A wind turbine wants the most possible rotation (to drive a generator) from the least possible amount of air movement.
Windmills are big freaking things. Those blades can be 100 feet long. If you made one the shape of a box fan it would be much heavier and much more expensive due to the extra material. Fans are the way they are because it’s cheaper to make them short and fat. Longer means thicker material to withstand the torque from moving the air and to prevent. Thicker means it’s more difficult and more expensive to manufacture them because you can’t just stamp them out.
Exactly: small ones can be very fan-like. See http://en.wikipedia.org/wiki/Image:WindmillMuseumLoeriesfontein03.jpg
OK, but why are they long and not short and fat? Woudn’t the force from the wind be related to the square area of the blade?
A long-bladed windmill, once moving, has more serious torque and greater inertia, does it not? This would be good, 'cause once moving, you want it to keep moving.
Might as well go ahead and help the OP out, too: Windmills look like plane propellers, but fans do not. Fans look like boat propellers. Hope that adds to the confusion.
I’d guess that maximizing torque would have a lot to do with it.
When you let the wind drive a wind turbine, you’re making it turn against the resistance of a generator (or a mill it it’s actually a windmill). A given section of blade will generate a particular amount of aerodynamic force whether it’s located far from or near the hub. But for parts located far from the hub, the torque they can apply to the shaft is increased. Longer blades let you put more working surface where the moment arm is long. To generate a given amount of power, you can use lower rpms if you provide more torque. Lower rpms would be desireable for big fans in outdoor areas (noise, mechanical stresses, etc.)
Conversely, when you use an electric motor to drive a fan, keeping the blades short lowers the amount of torque the motor has to apply. You accept higher rpms to get the lower torque requirement. Assuming similar efficiencies, changing the length of the blades won’t change the the power required to accelerate a certain mass of air. But the fan which requires less torque has an advantage, because electric motors are generally happier driving low loads at high speeds than the other way around.
Philster makes me think of an interesting bit of support for my assertion above. Aircraft propellers, which are generally driven by piston engines, are high-aspect ratio (long and skinny). Piston engines prefer higher torque loads at relatively lower rpm for maximum efficiency.
Turbofan engines (like passenger jets), have lots of lower-aspect ratio blades, rather than just a few long ones. Jet engines prefer high rpms and low torque loads for maximum efficiency.
Turboprops (propellers driven by gas turbine engines) are a different animal. I imagine the significant gear reduction in those engines is what allows the turbine to drive a high-load, low-rpm propeller happily.
A windmill’s efficiency (the percentage of the wind’s power which it captures) is closely related to its tip speed ratio - the ratio of the speed of the blade tips to the speed of the wind. For a high tip speed ratio you need aerodynamically efficient blades (high lift to drag ratio). As with efficient aircraft (e.g. sailplanes), this calls for a high aspect ratio (ratio of span to chord) to yield low induced drag. Thus, long thin blades
This site has a reasonably useful and not too technical discussion. Related to the OP, it offers this: