Having a finned type cooler after the tank will do nothing for water separation because there is very little temperature change to draw the moisture out of the air. If the air is shop room temperature there wont be much drop in temp. They would also need a water separator right after to catch the small amount it did separate.
When compressed air emerges from the pump, it’s hot. This temporarily keeps the water in its vapor state. As air cools down, excess water vapor condenses back into a liquid. The first place this happens is in the after-cooler system . Industrial air compressors may be either air-cooled or water-cooled. An air after-cooler brings temperatures for compressed air to within 15-20°F of ambient air temperatures; a water-cooled system can bring air temperature down to 10-15°F above water temperature. In a rotary-screw air compressor , about 70% of excess moisture in compressed air falls out in the aftercooler. (water separator here) This greatly increases the efficiency of the dryers. If liquid water enters a refrigerated dryer it reduces its efficiency by taking its cooling capacity just to cool the water present.
@MechanicJon I hear and understand what you are saying. It is the standard way people see it.
Let us discuss a few of your statements.
yes this does happen but a “shit load” of water would have condensed to the tank anyhow , which it is designed to do.
So are you saying the work of heat loss has already happen by the time it leaves the tank, even without a cooler?
these operate at 100 f less to start with compared to a reciprocating compressor.
Ill pin this article which this statement comes from
How to Prevent & Remove Moisture in Air Compressors – Fluid-Aire Dynamics.
I have a working understanding of psychrometrics it is why I came to the conclusion I did.
Here is a 
Why would you not put all your air drying equipment between the pump and tank??
all of it : the cooler, separator, refrigerated air dryer …
Anyhow I am loving the discord and that everyone has different opinions about the subject. (or I guess it is that I have a different opinion then everyone else)
All I want is a definite answer that is objective not subjective , even if I am completely wrong about this.
I should start a new post for this discussion. I have a “shit ton” of paperwork to do first but I’ll start a topic sometime today and we can continue a in depth discussion there.
I enjoy the productive debate!
Air drying is a rabbit hole for sure and a deep one. I do enjoy see all the great information and opinions about it.
I actually thought I seen someone put all their air drying equipment between the pump and tank.
As for where to put what I can’t say what’s best or if anything is detrimental. I could see different applications using different setups.
I have been thinking the exact same thing. I had thought that I would install mine horizontally and that might work better. But no…it wouldn’t. Last night I woke up and thought of this:
What do you think? No trapped water!