what I've determined as of now, the following may have wacky scenarios you'd never see in real life in order to demonstrate concepts..
All pumps being identical:
2 pumps in series increases water pressure by a factor of 2.
2 pumps in parallel increases water capacity by a factor of 2 and increases the water pressure by splitting the flow passing through each pump, therefore making the pumps perform better on their curves. if parallel pumps had a completely flat curve, 2 pumps in parallel would not give any added pressure gain compared to 1 pump.
2 pumps in series and 2 pumps in parallel all tied in together, increases water pressure by at least a factor of 2 (more if they have non-flat pump curves) and water capacity by a factor of 2.
When 2 parallel pumps with check valves after the pumps do not have the same churn pressure, and one pump's churn pressure is lower than the other pumps actual operating pressure at any given time, the other pumps operating pressure will keep one of the pump's downstream check valves closed - causing only one leg of a loop to flow water. Parallel pumps need churn pressures to either be equal or to never be below any actual pressure created in one parallel pump at any time. Theoretically, if you had one pump in parallel which had varying pipe sizes (such as 1" feeding into one pump, and 8" feeding into another pump, pressure would be so different for each leg of the loop at the outlet of the pump that one check valve downstream the pump may end up being held closed.)
If there were no check valves downstream parallel pumps, I assume in these situations that water could possibly flow into a pump in the wrong direction, probably causing mayhem.
1 pump with a bypass, the bypass having a check valve, will keep the bypass-check-valve closed (no flow movement there), UNLESS the pressure in the bypass-check-valve pipeline is equal to or greater than the pump-discharge pressure. this situation could theoretically happen if the pump-path choked off pressure by very small pipe sizes, while the bypass-check-valve-path did not, for example.
1 pump with a bypass, the bypass not having a check valve, the water in the bypass-path will be at a lower pressure (city supply pressure) than the pump-discharge pressure, changing the bypass-path flow direction back towards the city-supply and ultimately back into the pump suction flange, making accurate hydraulic calculation impossible, ultimately however any benefit of using the bypass-path would be impossible as the flow direction would be backwards. *This answers the original question/diagram situation.
*** About the hand calcs
I love the theory of pipe networks, one of my favorite things from tech school was getting special hazard systems to work. You had to really balance those things, even flow changing direction (such as tees, bull vs non bull) could mess up your special hazard system requirements. The software used for special hazard systems, at least the software we had access to, made HASS look like space-age stuff (some special hazard systems had no software available and we did it all by hand). For this reason, we quickly learned the best way to make our lives at school easier when doing special hazard systems was to "keep it simple", make very symmetrical networks, etc. It was fun because there wasn't any paper calculation sheets which involved doing the same repetitive procedure (copying data into boxes on a table on a piece of paper without needing to know why or being told why the boxes were significant). In my mind, the hydraulic calculation sheets for sprinkler systems is a humanized way to accomplish a computer problem. A designer should be taught and learn the fundamental principles of fluid dynamics as it relates to sprinkler systems to include learning why splitting flow paths can be useful, why pipe sizing can be useful, the equivalent length of fittings, importance of c-factor, and how all of it relates to hazen-williams or darcy-weisbach equations, the relations of different aspects of darcy-weisbach like velocity, but not antiquated procedures (filling data in boxes) for calculating systems from start to finish, over and over. I've learned how to do that and subsequently forgot how several times over, because the brain doesn't remember such things unless you use it constantly for a long time, like Art had to do.
