Simple differential vacuum question

[TommyC]

This conundrum will sound hypothetical, but it represents a real-world problem:

You have a pump sucking water through the same diameter line from the same level in 2 different places in a pool. FWIW, the pump is a Hayward Super Pump Series Model SP2607X102S:
http://www.hayward-pool.com/prd/In-Ground-Pool-Pumps-Super-Pump-_10201_10551_13004_-1_14002__I.htm

One suction line is ~30' long, while the other is ~70' long. The lines meet at a Jandy diversion valve just upstream of the pump which allows the vacuum to be accurately measured by isolating each line.

The steady state vacuum would not be expected to be equal in each suction line but which line should give the higher reading?

It's been many years since I studied physics but the following competing theories emerge in my mind:

Theory 1. The shorter line should provide the greater vacuum reading because it's, well, shorter and thus less resistance (greater flow/greater suction).

Theory 2. The longer line should provide the greater vacuum reading because, since it's longer, it provides the greater resistance and therefore the pump must suck harder to move the water.

Before I announce what vacuum differential I measured, I'd like to see if anyone can please confirm Theory 1, Theory 2, or perhaps some other one.

Thanks in advance for your help.

 

[jambaugh]

I'm not clear on where you are measuring the vacuum, I'll assume your pressure meter is attached to the Jandy valve.

The setup will be very similar to an electrical circuit with the pump acting as resistive battery. Treat pressure as potential (voltage).

Let's see. The longer pipe provides more resistance, the valve presumably provides equal resistance for each setting, and the pump can be modeled by a fixed resistance in series with a potential source.

The outflow of the pump presumably also has constant resistance which we can combine with valve and pump resistances and we can assume the return pressure equals inlet pressure.

In the electrical analogue you have:
pool=ground---R1--*-[Pump=Battery]-x-R--->pool
pool=ground---R2--*-[Pump=Battery]-x-R--->pool

The point * is where I believe you are measuring pressure. The point x is another pressure point and the difference between Px and P* will be the pumping pressure.

The flows will be $I_k = \frac{V}{R+R_k}$.
The (negative) pressures at * will be of magnitude
$V_k = R_k I_k \frac{V}{R/R_k + 1}$
So for the longer piper with bigger $R_k$ you have smaller $R/R_k$, smaller denominator and larger pressure difference (higher relative vacuum).

At the x points we have positive pressures of magnitude:
$V_k = RI_k = \frac{V}{1+R_k/R}$
So the longer pipe will provide lower positive pressure at x.

Now this assumes the resistance to flow is a linear function which is invalid over large variations of flow rate but good for small difference approximations. This should apply to the small resistances of the pipes.

 

[TommyC]

Thanks for your kind reply, Jambaugh. Interesting your use of an electrical circuit analogy, and I think it's apropos. BTW, it strikes me that, to the extent that all pumps - in moving fluid - create both suction (upstream) & pressure (downstream), this is related to Newton's 3rd Law of Motion.

To clarify, the vacuum was independently measured just upstream of the Jandy (diverter) valve.

The answer is Theory 2: the vacuum measured in the longer line would be expected to be greater (coupled w/ less flow). In fact, I measured 14.5"Hg in the longer line, and 9.3"Hg in the shorter one.

This is consistent w/ your electrical analogy, in which the longer conductor poses greater resistance, thus smaller voltage drop.

Thanks again for your helpful reply!

 

[jambaugh]

Thanks for your kind reply, Jambaugh. Interesting your use of an electrical circuit analogy, and I think it's apropos. BTW, it strikes me that, to the extent that all pumps - in moving fluid - create both suction (upstream) & pressure (downstream), this is related to Newton's 3rd Law of Motion.

I find it most helpful to remember that even suction is due to pressure, specifically atmospheric pressure. Only relative pressures can be negative. This is important to recognize especially when using an electronics analogy since with pressure there is an absolute minimum value while voltages are unbounded. The electrical analogue e.g. breaks down for a suction pump or siphon working beyond the 32feet for water at 1 atmosphere.

 

[TommyC]

Duly noted. Thanks again, good sir.