How is the shape of the blades of an air fan determined? Trial and error, or is there a theory behind it? What are they trying to maximize, volume of air dislocated per rotation?
Classical Mechanics and Fluid Dynamics – Understanding the Shape of Fan Blades
classical-mechanicsfluid dynamics
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The blades are at an angle. As the blade moves down it hits an air molecule and the air molecule "bounces off" toward you.
It's just like hitting a ball in tennis/baseball
Apart from motor and bearing noise, most of the acoustic power comes from the eddy swirls following the trailing edge of the blade after it passes by. There is also an outward pulse of air as the leading edge of each blade pushes forward cutting the air.
The trailing eddies produce a broad spectrum of random noise, modulated by the fan blade frequency. The outward pulses, of course, occur at the blade frequency, with harmonics. Both are stronger near the tip, because the tip is moving faster. Faster splitting of the air means a sharper leading edge of the pressure pulse, so the higher frequency noise is especially concentrated near the tips. A small microphone near the spinning blades would pick up a repeating soft step function from the blades' leading edges, with faster-rising step functions (thus composed of higher frequencies) farther from the hub. That answers one part of your question.
By "blade frequency" I mean how often a blade passes any specific point in space, per second. (I'm not a fan engineer, so my jargon may be off.) (Although, don't misunderstand me - I am a big fan of engineers!)
Actual noise from a fan will show variations at the spin frequency, or full cycles of the blade assembly, not just at what I'm calling the blade frequency. This is because the blades are not perfectly identical. Who knows, maybe there's the residue of a dead bug on one blade and not on the others. Imbalances increase noise.
Note that fans will put out acoustic noise at frequencies lower than the blade frequency. The eddy swirls are not the same each time a blade passes by - they're random and the variations from one cycle to the next to the next mean subharmonics.
Fan blades rarely spin bare naked in air. There's probably a grille, a wire cage, something protective to keep kids' fingers safe and insects out. Whatever the structure, there's turbulence as air is pushed by. The noise will be mostly from pushed air leaving the fan, and little (but some) from the new air sucked in to replace it.
There is a formula for estimating the acoustic power produced by a fan. All I can find at the moment is this from a PDF of unknown origin. The formula is alleged to come from a handbook provided by ASHRAE (American Society of Heating, Refrigerating & Air-Conditioning Engineers) which is, alas, not online for free.
$L_w = K_w + 10 log_{10} Q + 20 log_{10} P + BFI + C_N$
$L_w$ is the sound power level in dB. $K_w$ is a specific constant amount of noise, stated in the manufacturer's data. It's usually 20-something or 30-something dB, unless you have a nasty industrial fan it could be up to 40-something dB.
Q is air flow, cubic feet per minute. (No offense to the N-1 countries in the world not using US units of measure! Obviously this PDF originated in the US.) P is air pressure, as inches water (!). BFI is some sort of correction relating to blade frequency, usually a single digit number of dB. Finally $C_N$ corrects for inefficient fans. Zero for a "perfect" fan, to about 12dB for a fan that' only 40% efficient. Efficiency here is "hydraulic efficiency" based on air flow, pressure, and motor power in horsepower (HP):
${Q P}\over{6350 ({HP})}$
Have fun plugging in numbers...
I brushed off motor noise at the start of this answer, but of course that's a big factor. Unless you pay big $$$ for an exquisitely well-balanced motor, the motor is shaking its mounting, the entire apparatus it's in. A computer motherboard makes a great soundboard to transmit vibrations into the air. A fan in the window or hanging from a ceiling (vital in Florida!) or in a stand of any kind, is transmitting vibrations to the structure and, if there's some loose parts involved, will often satisfy the requirements of a chaotic system. That means more subharmonics and plenty of harmonics.
Good research was done on this back in the 1950s, with renewed interest over the last twenty years or so due to the desire for quiet computers. (Especially in amateur audio recording studios!) The best material is in books that I have no easy access to at this time. Some references that exist online, at least abstracts of papers:
Source of Noise in Fans - Howard Hardy J. Acoust. Soc. Am. Volume 31, Issue 6, pp. 850-850 (1959)
See http://www.arca53.dsl.pipex.com/index_files/ventnoise1.htm about halfway down the page, "Fan Noise"
"Visualization of Aerodynamic Noise Source around a Rotating Fan Blade", A. Nashimoto, N. Fujisawa, T. Nakano, T. Yoda, J. Visualization 11(4):365-373 (2008) (avail. at ACM Digital Library, Springer.com and other purveyors of academic papers, but not for free online. Are you near a university with a library?)
For noises created by aircraft, this article is informative and non-technical: http://www.noisequest.psu.edu/SourcesAviation.Overview.html
This company making fans, blowers etc. has some lightweight info on fan noise physics: http://www.jmcproducts.com/acoustic-noise/
Though I dislike using wikipedia due to its ever-changing nature, at this time this article does have relevance: http://en.wikipedia.org/wiki/Quiet_PC
Best Answer
Mostly trial and error - but in a computer.
A combination of Computational Fluid Dynamics (modelling how the air flows over the blade) and Finite Element Modelling (how the stresses in the metal behave)
Both of these are complex areas - and when they come to together you need a lot of expensive computers and some even more expensive engineers.