How do I find out which compressor would fulfill my volume flow rate requirements based on Free Air Delivery as the only volume flow rate given in the catalogue?
1. Calculate the mass flow rate of air given your required duty. You can do this by looking up or calculating the density of air, $\rho_\text{out}$, at your outlet conditions ($300 \space \frac{\text{ft}^3}{\text{min}}$ at $18 \space \text{psig}$ and $T_\text{out}$) and then dividing your volumetric flow rate by this density.
$$\dot{m}=\dot{V}\text{out} \cdot \rho\text{out}$$
1. Convert mass flow rate $\dot{m}$ to standard volumetric flowrate, $\dot{V}\text{std}$, (probably $\text{SCFM}$, or Standard Cubic Feet per Minute in your case) since this is the number vendors use to rate their compressors. You can do this by dividing the mass flow rate by air density at standard conditions, $\rho\text{std}$.
$$\dot{V}\text{std}=\frac{\dot{m}}{\rho\text{std}}$$
$$\dot{V}\text{std}=\dot{V}\text{out} \cdot \left( \frac{\rho_\text{out}}{\rho_\text{std}} \right) $$
1. Provide this standard volumetric flowrate along with:
2. available suction pressure and temperature
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required discharge pressure and temperature (note: often air compressor packages come with cooling options since compressing heats up air)
4. utility power details (3-phase 240 VAC? continuous or intermittent duty?)
Here's an example setup worked out:
My process simulator (DWSIM 5.8) gives $\rho_\text{out}=0.163758 \space \frac{\text{lbm}}{\text{ft}^3}$ for $T_\text{out}=80^{\circ}F$ (?) and $P_\text{out}=32.7 \space \text{psi}$ (this is $18\space\text{psig}$ assuming atmospheric pressure is $14.7\space\text{psig}$)
It gives $\rho_\text{std}=0.0763781 \space \frac{\text{lbm}}{\text{ft}^3}$ for $T=60^{\circ}F$ and $P_\text{out}=14.696 \space \text{psi}$ (probably standard conditions in United States)
Plug it all together and you have:
$$\dot{V}_\text{std}=\left( 300 \space \frac{\text{ft}^3}{\text{min}} \right) \cdot \left( \frac{0.163758 \space \frac{\text{lbm}}{\text{ft}^3}}{0.0763781 \space \frac{\text{lbm}}{\text{ft}^3}} \right)=643.21317 \space\frac{\text{ft}^3}{\text{min}}$$
$$\dot{V}_\text{std}=650\space\text{SCFM}$$
Note: This value is only valid for the exact pressures and temperatures I assumed. You didn't provide an outlet temperature so I filled one in. Also, I didn't take pressure drop across the ducting into account (I simply used $18 \space\text{psig}$).
What would be the best system to change the pressure of the compressor (116 psi) to the required pressure (18 psig). How can I control the pressure while achieving the required flow rate in the system. (Is the varying cross sectional duct with pressure regulation valves the best option)?
If you purchase a compressor that has a much higher maximum operating discharge pressure (ex: the $118 \space\text{psig}$ screw compressor you mentioned), then you can use a pressure regulator (which also would have to be sized for your mass flow rate and compressor outlet conditions) to reduce the pressure to that needed by your duct system. You'll also want to make sure a pressure relief valve (sometimes included with compressor packages if you request one) capable of relieving the maximum flow rate of air back to atmosphere in order to prevent equipment damage or injury in case accidental block-in causes dangerous pressure release (especially important with positive displacement compressors like a screw compressor). 842