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Please have patience with me as I have not looked at this sort of theory in many years (I am also very new to StackExchange).

A question asks the following:

A sloping swimming pool two-thirds full of sea water provides the supply for a fixed installation protecting a risk situated above the pool.

The pool measures 12m long, 6m wide and 1m at the shallow end sloping to 2.5m at the deep end. A pipe is placed so that its inlet is at the bottom of the deep end of the pool. A pump imparts pressure to the water in the pipe such that a mercury manometer indicates a pressure difference of 681mm Hg between inlet and outlet. The inlet of the pipe is 80mm diameter; the outlet nozzle is 45mm in diameter and is 5m above the inlet. The inlet velocity of the water is 5m/s, the density of sea water is 1050kgm-3 and the density of mercury is 13600kgm-3.

Use Bernoulli’s theorem to calculate the velocity of the water through the outlet nozzle?

By applying the continuity equation the following is found:

$v_1A_1=v_2A_2$

$v_2=15.8m/s$

However, the solution is given as $v_2=10m/s$

If I apply Bernoulli’s equation:

$$ \frac{P_2 -P_1}{ρ}+\frac{1}{2}(v^2_2-v^2_1)+g(z_2-z_1)=0 $$ $$ \frac{-90792.5}{1050}+\frac{1}{2}(v^2_2-25)+9.81(5)=0 $$

$$ v_2=10m/s $$

This is the answer that the solution gives but continuity is not met as shown above. Is this question ill-posed?

I am very confused why continuity is not met when Bernoulli’s equation is used and why the two methods produce different results. A similar question was asked here but this was for a system without a pump and I am not sure if it is applicable.

My question is: Can Bernoulli’s equation be used when the system has a pump or is it no longer valid due to energy not being conserved?

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  • $\begingroup$ model the pump as a pressure difference. $\endgroup$
    – Solar Mike
    Commented Jul 27, 2021 at 14:50
  • $\begingroup$ The pipe is tilted upward with the outlet located 5m above the inlet, doesn't that makes difference? $\endgroup$
    – r13
    Commented Jul 27, 2021 at 18:43

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No, there is a problem with the question. If the flow rate is set at the inlet, then that is the flow through the pipe, period. The pump doesn't add mass, just a motive force. Bernouli would apply if you were doing the flow calcs yourself. In this case, they gave you the answer at the beginning. The flow rate through the exit will be equal to the flow rate through the inlet no matter what the pump did or how high the exit is.

Bernouli probably doesn't work because they made up the pump head.

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    $\begingroup$ Thank you, as mentioned in my comment above I agree that mass continuity must be maintained under steady incompressible flow. $\endgroup$
    – Charl Pienaar
    Commented Jul 28, 2021 at 8:52
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Solar Mike is correct, but requires more explanation why. The Bernoulli equation is an energy conservation law. The pump is imparting energy on the fluid, therefore, needs to be accounted for as a pressure addition term.

When you applied continuity initially, you are assuming the flow is incompressible (correct assumption) with no energy addition between the two flow stations (incorrect assumption). This would work for a flow device with area change only and no energy addition/removal to/from the system (via work or heat transfer).

To directly answer your question: Yes, you can use an extended Bernoulli equation to include the effects of energy added to or removed from a flow by a device. https://engineeringlibrary.org/reference/bernoullis-equation-fluid-flow-doe-handbook

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  • $\begingroup$ Thank you, I agree that an extended Bernoulli equation can be used i.e. the Energy equation. However, I do not agree that continuity needs not be satisfied. Intuitively if the density remains constant volume in must be equal to volume out as mentioned by @TigerGuy. The only cases where this is not true is if there is deformation to the pipe, a leak in the pipe or the fluid is compressible or am I missing something? $\endgroup$
    – Charl Pienaar
    Commented Jul 28, 2021 at 8:15
  • $\begingroup$ This question may be ill-posed. Notice they tell you to use the Bernoulli principle and do not mention continuity. I googled the question and found the same result: ife.org.uk/write/MediaUploads/Exams/L4C1.pdf Also think about the use of the Bernoulli equation and continuity: Bernoulli equation is assumed to apply along a fluid streamline, while the continuity equation refers to bulk fluid movement (which could contain many streamlines) $\endgroup$
    – mechcad
    Commented Jul 28, 2021 at 12:23

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