This is absolutely the key question. The flow rate for both pumps, if they are in series as shown, will be equal. Refer to the Law of Conservation of Mass. Unless the diagram is in error this is a questionable system, and closed loop versus open loop is besides the point. Are you sure the second pump isn't intended to boost pressure?
Your post got me thinking...
I tested it multiple times in our test bay last week and I kept seeing the same phenomena. I'd have the system running at a constant speed and temperature and only increase pump 1's speed. When I did, flow meter 1 (directly after pump 1) would see an increase and flow meter 2 (directly after pump 2) would not. Momentarily, flow in was not equalling flow out and I didn't observe any leaks or malfunctions on the system.
But, I think I can explain why now.. As I stated earlier in the thread, pump 2 (the 2 piston PD pump) operates at ~3500 PSI and that pressure can be manually adjusted. When I only increased pump 1's speed, the pressure in pump 2 would bump up slightly. Once this pressure stabilized, flow in did indeed equal flow out.
So I thought of the PD pump like a tank. It should always have a constant value for maximum volume. If you fill a tank, flow in is always going to equal flow out only if the pressure (and volume) in the tank is constant. But, being a PD pump, water within the pump is trapped and not always subjected to the pressure of the water coming into the pump as water within a tank would be. More water comes into a tank, it pushes more water out. More water comes into a PD pump, it takes time for the pressure within the pump to stabilize since water in can't simply push water out. Only when this pressure has stabilized does flow in = flow out.
Does this make sense? Or am I totally off base... I'm just wondering before I go into a meeting Tuesday afternoon about this. When I wrote our functional design spec for this feature, I included that flow in = flow out was conditional (and sited the "snapshot" of the system after a set point change on pump 1 as proof) I instantly met some internal restriction. It now seems that it is conditional upon the "black box" between flow in and flow out having a constant pressure. I'd never really given it much thought since a couple of classes in school, but that thought process applied to PD pumps seems logical. If you increase the set point of pump 1, it takes time for the first "pocket" within the 2nd PD pump to reach the discharge tube. At some point in time you're going to have multiple pockets within the 2nd pump at different pressures causing flow in not to equal flow out.
If I'm totally wrong, then it seems the flaw is in our equipment.