Pumps are complicated
Many questions regarding fluid machinery start with these two assumptions:
- A pump or gravity is needed to make the fluid go faster
- High pressure means faster flow
All of these are correct in principle, but none of them are accurate. The first is inaccurate. Due to conservation of mass, fluid flowing into a pipe on one end will be equal to the flow out the other end. While the pump will make the fluid flow faster, the reality is the pump imparts energy on the flow, until the added pressure loss from the higher flow rate is equal to the added energy.
Because of how complicated the energy loss in a pipe can become (Reynolds numbers, Darcy friction factor, hydraulic diameter, etc.), since most engineering problems deal with water and piping, Hazen Williams equations are typically used. In addition to simplifying the pipe, we simplify the pump. Pumps have a curve, which is always convex. By taking the max flow rate (at no pressure), and the maximum pressure (at no flow), a linear equation can be developed - just use the intercept form. This simplifies the pump. Now, a set of equations can be developed called the Hardy Cross Method. My textbook shows a typical setup:
A source of unknown pressure is feeding the system 3 cfs of liquid - via valves, pumps, or who knows what. The exits have 1 and 2 cfs - matching the conservation of mass. Again, pressure at these points is unknown, but not unlike pressure loss across a resistor, the energy difference between each point means that energy is conserved across the loop. This is why Hardy-Cross Method can be seen to be the fluid version of Kirchoff's Voltage Law
As can be seen, the top line (only 6" and a full 3000' + 4000') will get minimal flow, while the bottom line, which is shorter and has larger diameter pipes, should be expected to hold most of the flow. When adding various pumps, it can be seen that even with sufficiently large pumps, circulation starts to become inevitable:
Your Y-Connector method is interesting because you now have an unknown flow rate - but a known pressure. In this case, you need to use the air in the atmosphere to connect the pressures in a loop (See for example, page 15 of this pdf, showing how this is done. Also, don't forget to add in all of the minor loss coefficients
However, all of this is a model. To prevent backflow, always add a check valve to your pumps. Because you never know what could happen, and backflow can be very expensive.