The maximum possible air velocity through the vent pipe is the sonic velocity of air:
Vs = (kgRT)^0.5 = (kg144PV)^0.5
Where,
vs = Sonic velocity of air, ft/sec
k = Ratio of specific heats for air
g = acceleration of gravity, 32.2 ft/sec2
P = Absolute pressure, psia
V = Specific volume of air, ft3/lb
From Cranes Technical Paper #410
The sonic velocity has been determined to be approximately 1,129 ft/sec.
Flow input into or output from the tank is Q in cu. ft./sec. That would be the worst case scenario or a total of your 3 water lines entering or the large pipe draining. Assume that they are operating at the worst case scenario which is the maximum flow into or out of the tank.
The minimum area of your vent pipe would be Q/1,129 ft/sec = sq. ft.
The recommended pipe area is at least one pipe size larger than the minimum.
If your vent pipe is more than a few feet long, then you will have to account for the pressure drop in the vent pipe. The pressure rating of the tank must be greater than the pressure drop in the vent pipe. Otherwise, increase the size of the vent pipe to decrease the pressure drop in the vent pipe.
Pressure drop in the vent pipe can be estimated from Cranes Technical Paper #410, Page B-15 flow of air though steel pipe.