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I'm confused about pumps in fluid dynamics.

As far as I understand, the basic effect of a pump which deliver a power $\mathcal{P}$ can be described with modified bernoulli equation between a point $A$ before the pump and a point $B$ after the pump.

$$(p_A+\frac{1}{2} \rho v_A^2 +\rho g h_A)\cdot Q +\mathcal{P}=( p_B +\frac{1}{2} \rho v_B^2 +\rho g h_B)\cdot Q=\mathrm{constant}\tag{1}$$

Now my specific problem is: does it really matter where the pump is located inside the tube?

In the picture the pump is located ad height $B$, but, if it was located at $A$ or $C$, would something change? That is, would the fluid have different speed at the top when it flows out of the tube?

enter image description here

My answer would be no, since I can place the pump in $B$, but I can also use Bernoulli equation between $A$ and $B$, which says that the $\mathrm{constant}$ in the equation is the same for $A$ and $B$, so the situation in the picture is equivalent to one with the pump in $A$.

So if this is true I can use $(1)$ between any poin before the pump and any point after the pump, regardless the distance from the pump itself.

Can the reasoning be correct?

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In the picture the pump is located ad height BB, but, if it was located at AA or CC, would something change? That is, would the fluid have different speed at the top when it flows out of the tube?

In a nutshell: no.

Bernoulli equation.

Bernoulli's equation, between the points $1$ and $2$, is as follows, where $p$ is the pressure supplied by the pump:

$$p_1+\frac{1}{2} \rho v_1^2 +\rho g h_1+p=p_2 +\frac{1}{2} \rho v_2^2 +\rho g h_2$$

Now it is important to understand the suffixes $1$ and $2$.

At point $1$ (the surface of the lower tank), $p_1=p_0$, where $p_0$ is atmospheric pressure.

Similarly, assuming the point $2$ gives to open air, then $p_2=p_0$.

In addition, is we assume the bottom tank's surface area is much larger than the cross section of the pipe, then $v_1 \ll v_2$.

After minimal reworking, the equation then simplifies to:

$$v_2\approx \sqrt{2\big[\frac{p}{\rho}-g(h_2-h_1)\big]}$$

So the placement of the pump is immaterial, only the pressure it delivers and the difference in height between the points $1$ and $2$ matter. The equation doesn't depended on the distances $|AB|$ or $|BC|$ at all.

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  • $\begingroup$ I think you have misunderstood the OP's equation. P is not a pressure but the power of the pump, and Q is the volume flow rate. $\endgroup$ – sammy gerbil Jun 16 '16 at 0:33
  • $\begingroup$ I've slightly edited the answer. $\endgroup$ – Gert Jun 16 '16 at 0:42
  • $\begingroup$ Thanks for the answer! I still have a doubt about this: if the pump in the picture is located above level $B$, then does it matter if the point used in Bernoulli equation (which you called $1$) is located at height $B$ or at height $A$ (as you did in your answer) or at any other point before the pump? The answer is no if I got that right, but then (again supposing the pump above level $B$) could I write $p_A+\frac{1}{2} \rho v_A^2+\rho g h_A=p_B +\frac{1}{2} \rho v_B^2 +\rho g h_B$? That is "normal" Bernoulli equation between points A and B, ignoring the presence of the pump? $\endgroup$ – Sørën Jun 24 '16 at 8:25
  • $\begingroup$ One the one side it should be possible, since the pump is above ("after") point $B$ and should not have an influence, on the other side of course the pump has the effect to make the fluid go up at a constant rate, so Bernoulli constant would (in my view) be surely different between A and B in the case proposed $\endgroup$ – Sørën Jun 24 '16 at 8:27
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I think it does matter where you place the pump.

Regardless of how powerful the pump is, the minimum pressure it can create at the input is zero - a perfect vacuum. The pressure pushing the fluid up to the pump is then the atmospheric pressure.

If the fluid is water, the maximum distance the atmosphere can pump it up to is about 10m. So the pump must be located no more than 10m above the water level. Of course, this will not matter if $h < 10m$.

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  • $\begingroup$ C'mon Sammy: the OP doesn't specify water. Also, using Bernoulli is an approximation at best. You're right: the pump cannot be higher than 10.33 m but that is hardly in the spirit of the question! :-) $\endgroup$ – Gert Jun 16 '16 at 0:39
  • $\begingroup$ @Gert : Yes, I am being pedantic. Probably this point was not considered by the question-setter. But it might be relevant if the fluid is denser than water, eg thick sludge. $\endgroup$ – sammy gerbil Jun 16 '16 at 0:47
  • $\begingroup$ Dense sludge? Again, Bernoulli in it's basic form is too simplistic too take that accurately into account. We shouldn't expect too much from a formula designed for inviscid fluids. $\endgroup$ – Gert Jun 16 '16 at 0:51
  • $\begingroup$ @Gert : My answer does not invoke the Bernoulli Equation. I am merely pointing out a practical limit to the position of the pump. $\endgroup$ – sammy gerbil Jun 16 '16 at 0:56
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    $\begingroup$ Negative pressure is attainable in an incompressible fluid, and is not a problem as long as there is no bubble nucleation. Sap circulation in trees requires negative pressure to be created at the level of leaves to pull up sap over 10s of meters, this is achieved using Laplace pressure in small leaf pores. When bubble nucleate, this creates an emboly in the sap canals. science.sciencemag.org/content/148/3668/339 $\endgroup$ – Joce Jun 16 '16 at 10:23

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