I don't understand why pressure ratio is used for fans, and disk loading is used for helicopters. I'm familar with helicopter momentum theory, essentially induced velocity is equal to the square root of (thrust divided by (2 * density * area) ), or, since disk loading is thrust over area, induced velocity is square root of (disk loading / (2 * density) ). How is pressure ratio related to induced velocity, or thrust / power?

I guess it has to do with the parameters that are of interest... different industries use different words to say pretty much the same thing.

 

Regarding fans, the engineer wants to know the pressure of the gas that comes in, and the pressure of the gas that goes out... so it makes sense to work with the pressure ratio. It's almost the only parameter of interest, apart from the power usage of the fan. They often run at a constant rpm. The blades of the fan are fixed on the axis. All-in-all it's a simple machine. Here are a couple of thermodynamic laws that explain compressors and the power required to do that.

 

In helicopter design however, it's also interesting to find these numbers related to the weight of the helicopter (which is a variable, and depends on cargo and passengers), engine power (can be varied by the pilot) and for example the position of the blades of the rotor.