# What happens to the flow rate when a water pump is pumping water through a tube that changes size

I'm going to apologize in advance if this is a rudimentary question. It's been 20 years since I took a fluid mechanics class. We are building a waterfall in my backyard. The water pump has a head size calculation which determines how much water will come out the end of the pipe. The waterfall doesn't have nearly as much water coming out of it as we expected.
The pump is in a pond. It has a 3" pipe that travels 30 ft. horizontally and then 11 ft vertically before it dumps into a spill basin at the top. As it enters the spill basin it passes through a 3" to 2" reducer. From what I understand, the flow rate itself shouldn't change when the pipe diameter drops because the pressure would just increase and the velocity would increase to maintain the same flow rate. However, when calculating the head size of the pump, from what I gather, an increase in pressure will drastically reduce the gallons per hour of water that flows out so if by making the water pass through a reducer, that would dramatically increase the pressure and therefore decrease the water flow right? What am I missing?

• You should also post the head size calculation.
– Gert
Jun 12 '16 at 19:52

The total volumetric throughput a pump can deliver is usually dependent on the head pressure it needs to deliver. An example of a series of so-called pump characteristics (for a line of commercially available pumps) is: (Source.)

• Volumetric throughput
• Pipe length and state of pipe (smooth or corroded)
• Changes in height of the flow line
• Local resistances like bends, valves or sudden restrictions in pipe cross section.

A restriction like the one you describe presents a head increase approximated by:

$$h_{restriction} \approx \frac{v^2}{2g}z$$ Where $v$ is the outflow speed (end of the flow line) and $z$ depends on $\frac{d_2^2}{d_1^2}$, with $d_2$ and $d_1$ the pipe diameters resp. after and before the restriction. In your case $d_2=2"$ and $d_1=3"$. For these values: $$h_{restriction} \approx 0.34\frac{v^2}{2g}$$

It depends on how significant that increase is with respect to the others listed higher up but it's entirely possible that this added head pressure may reduce the volumetric throughput considerably, as you observed.

• @PremMukherjee: Thank you! Upvote, maybe? :-)
– Gert
Jun 13 '16 at 14:32