We've all been there: You wash your hands after using the restroom facilities, and turn away from the sink only to find one of those sad, completely ineffectual, old-style hot-air hand dryers bolted to the wall. You know, the kind with the info graphic shown to the right (image credit: nyulocal.com). Why do these things work so poorly compared to paper towels? What insight did Excel and Dyson have that makes their systems so much better?
It all comes down to the physics of trying to dry your hands. At a rough estimate, the surface area of your hands is around 430 cm2. If your hands, when wet, are coated on average by a layer of water 100 microns thick (seems not crazy), that's a total volume of water of 4.3 cm3. How can you get that water off of you? One approach, apparently the one pursued by the original hot air dryers, is to convert that water into vapor. Clearly the idea is not to do this by raising the temperature of your hands to the boiling point of water. Rather, the idea is to flow hot, dry air over your hands, with the idea that the water molecules in question will acquire the necessary latent heat of vaporization (the energy input required to pull water molecules out of the condensed (liquid) phase and into the vapor phase) from their surroundings - the dry air, your hands, etc. This "borrowing" of energy is the principle behind evaporative cooling, why you feel cold when you step out of the shower.
[A digression in fancy thermodynamic language: When liquid water is in contact with dry air, the chemical potential for the water molecules is much higher in the liquid than in the air. While the water molecules are attracted to each other via hydrogen bonds and polar interactions, there are so many many more ways that the water molecules could be arranged if they were diluted out into vapor in the air that they will tend to leave the liquid, provided each molecule can, through a thermal fluctuation of some sort, acquire enough energy to sever its bonds from the liquid. The departing molecules leave behind a liquid with a lower average total energy, cooling it. Note that water molecules can come from the vapor phase and land in the liquid, too, depositing that same latent heat per molecule back into the liquid. When the departure and arrival processes balance, the vapor is said to be at the "saturated vapor pressure", and evaporative cooling stops. This is why sweating a whole bunch on a super humid day does not cool you off.]
Back to your hands. Converting 4.3 cm3 of water into vapor requires about 9700 Joules of energy. If you wanted to do this with the heat supplied by the hot air dryer, and to do it in about a minute (which is far longer than most people are willing to stand there rubbing their hands as some feeble fan wheezes along), the dryer would have to be imparting about 160 W of power into the water. Clearly that's not happening - you just can't get that much power into the water without cooking your hands! Instead, you give up in disgust and wipe your hands discreetly on your pants.
In contrast, paper towels use thermodynamics much more effectively. Rather than trying to convert the water to vapor, paper towels take great advantage of (1) the very large surface area of paper towels, and (2) capillary forces, the fact that the liquid-solid surface interaction between water and paper towel fibers is so attractive that it's energetically favorable for the water to spread out (even at the cost of increasing more liquid-vapor interface) and coat the fibers, soaking into the towel. [Bonus physics lesson: the wet paper towel looks darker because the optical properties of the water layer disfavor the scattering processes on micron-scale bits of fluff that tend to make the towel look white-ish.] Yes, it takes energy to make paper towels, and yes, they must then be disposed. However, they actually get your hands dry!
What about Excel and Dyson? They realized very clearly that trying to vaporize the water on your hands is a fool's errand. Instead, they try to use actual momentum transfer from the air to the water to blow the water off your hands. Basically they accelerate a stream of air up to relatively high velocity (400 miles per hour, allegedly, though that sounds high to me). That air, through its viscosity, transfers momentum to the water and that shear force drives the water off your hands. They seem to have found a happy regime where they can blow the water off your hands in 10-15 seconds without the force from the air hurting you. The awesome spectacle of those good dryers just shows how sad and lame the bad ones are by comparison.