How to calculate CFM/pressure needed to move water vapor through duct?
Propeller fans aren't really good at producing any static pressure. They can move air quietly but as soon as they see any restriction air flow stops. Your basically making a remote mount fan assist humidifier.

I'm not sure if it would fit your application but if you have return air present, why not use it's draw to pull the vapour out of your box? 

 

Use a short duct length connected positively to the return air register by using a boot to maximize surface area. 

I know you said the vent is far away so another method would be to pipe a supply duct into the box and pipe the other side to the return. 

Otherwise you will probably need a  squirrel cage type of blower. Noise moving air increases with increasing static pressure, think Dyson air blade hand dryers. 

I answered this in a hurry this morning. To follow up. Calculating output cfm requires specialty tools. Basically it measures feet per minute, does a little math based on technician inputs and spits out a number.

The math is as follows 

cfm=fpm x pi r² 

Assume 10 feet per minute 

Pi =3.14

r=radius for a 6" diameter pipe r = 3" 

So (3x3)3.14=28.26

1 square foot 12"x12"=144

144/28.26=5.09

So 5.09x10(feet per minute) =50.9cfm