This is a piping network problem. I know exactly how to solve it, but cannot go into detail in this post (not enough space). Perhaps the moderators can tell me how to do it by private email?

I don't think piping you sketched can function at all. There is no unique path to terminal units, so network can never be balanced.

That has nothing to do with compressed air network, which is not closed loop system.

First I’ll you will need to flow rates for all future taps, min &max flows and dp of the chiller to understand the system dynamics. I would treat as 2 parallel runs for piping in standard hvac use unless the first taps consume a large % of system flow. I prefer a reverse rtn but the piping cost adds up. Unlike compressible fluid flow, cross tying does not lower installed cost unless a unique need is needed such as a leg out of service, odd dp requirements of the terminal valves or no control vlv at therminals (think of a coil header). I have done this in a process with high pressures and the terminal vlv may have control issues with a DP in the main. If cost is the primary concern, go with 2 direct return legs and use dp type flow controls and flow control at the chiller with a drive.

First confirm you have variable flow chiller and that you have mix of two way and 3 way chiller water flow control valve at the AHUs so the 3 way chilled water flow control valve would insure the minimum chilled water flow would bet above the minimum allowable chilled water flow. Also each AHU should have flow balancing valve.

If all AHU chilled water flow control valve are two way, provide a differential pressure control valve between supply and return to pass the minimum allowable flow when the set DP is reached.

You have to make sure you can provide enough gpm for future taps and also allow for the actual delta T across each AHU coil which may not match the chiller delta T. However the net mixed delta T of all should match the chiller delta T.

You can use 2 parallel branch but make sure you get the worst case by trial and error. Make sure you use the worst case branch to AHU pressure drop including coil, control valve, balance valve, isolation valves, strainer, fittings, pipe lengths.

Locate DP sensor to control speed of pump and determine set point.

Quote from Drazen: I don't think piping you sketched can function at all. There is no unique path to terminal units, so network can never be balanced.

That has nothing to do with compressed air network, which is not closed loop system.

It turns out that it can be done. I never said it was easy though. It requires 69 non-linear equations, plus 26 linear equations for a total of 95 simultaneous equations. You would need a math program like TK Solver or MatLab to do it. Large municipal water and gas utilities do this regularly.

trashcan, both water and gas utilities are open-loop systems and they make flow probability assessments to get optimum pipe diameters, aver flow condition etc., not to balance system. open systems can self-equalize to some extent when supply is connected from two sides.

using that logic in close-loop system makes no sense at all. not only that it cannot be predicted which path flow will take in different terminal units opening scenarios, but even reverse flow cannot be predicted. that is mess which does not belong to engineering practice.

Go to a sprinkler dude who has the software to analyze this system.

It's not simple.

I like your mark-up and you're on the right track. If branch "one" of your supply/return has higher pressure loss, "make" it shorter and branch "two" longer until you get near equal, add reasonable safety factor and size the pump. Watch that one of the take-offs doesn't have a disproportionate length. If so, that will become your index run. I'm assuming you know the flow of each tap at peak building load. What is nice about a loop is when the peak shifts, the flow can be provided from the other side which reduces required pipe size. You can also isolate a small length without taking down the whole system. Have at it!