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Can anyone help me with this question, I can't get the right answer :-)

A water reservoir with depth D=200m, which is placed H=500m above on a mountain. A pipe is placed from the bottom of the reservoir to the bottom of the mountain. This pipe has two cross-section areas $A_1=10m^2$ and $A_2=5m^2$. The atmospheric pressure in the bottom is $p_2=100kPa$ and the air density down here is $\rho_2=1 kg/m^3$ and waters density is $\rho_w=1000 kg/m^3$.

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What is the atmospheric pressure $p_0$ at the surface of the reservoir?
It has to be 93 kPa



My attempt on a solution:

I know I need to use Bernoulli's equations:

$$ p_{0}+\rho_w g(H+D)+\frac{1}{2} \rho_w v_{0}^{2}=k $$ $$ p_{1}+\rho_w g H+\frac{1}{2} \rho_w v_{1}^{2}=k $$ $$ p_{2}+\rho_w g(0)+\frac{1}{2} \rho_w v_{2}^{2}=k $$

And the continuity equation. $$ A_{1} v_{1}=A_{2} v_{2} $$

Usually I would assume $v_{0} \simeq 0$, but they don't mention that the surface area is much larger than the two cross sections. So can I still assume this, or should I forget about the first equation?

But so far (assuming v0=0), I got 4 equations and 5 unknowns.

We are assuming the water got the same density. And if the water is a standing liquid (v0=0), we could calculate $$ p_{1}=p_{0}+\rho_{w} g D $$

But solving this system of equations gives me: $$ p_{0}=p_{2}-\rho_{w} g(H+D), $$ which is completely wrong, since I get a negative pressure and the cross sections areas becomes irrelevant. How do I solve this exercise? And why do I even need to know the density of the air at the bottom of the mountain?

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  • $\begingroup$ Static water pressure only depends on depth below the surface, so you don't need most of the data in your problem. Also, at the bottom of a 700 m column of water, it's not possible to have a pressure of 100 kPa. The pressure should be at least 70 times that value. Also, note that the pressure $P_0$ should be equal to the local atmospheric pressure. $\endgroup$ Aug 17, 2019 at 21:25
  • $\begingroup$ @DavidWhite But my final result also gives me, that the pressure p2 only depends on the atmospheric pressure p0 and the depth below the surface (and water density + g). But why is the correct result of the pressure $P_0=93kPa$? But the water is not static, so that's probably where the issue are. $\endgroup$
    – mhj
    Aug 17, 2019 at 21:31
  • $\begingroup$ $P_0$ is a boundary condition, not something that you calculate. You start at the value of $P_0$, and calculate pressures going down the mountain from there. $\endgroup$ Aug 17, 2019 at 21:33
  • $\begingroup$ @DavidWhite: "it's not possible to have a pressure of 100 kPa". I think it's meant to be the atmospheric pressure at $P2$. $\endgroup$
    – Gert
    Aug 17, 2019 at 21:34
  • $\begingroup$ @DavidWhite - Well, it's an exam question and I know the correct answer is $P_0=93kPa$ - So it has to be possible to calculate. $\endgroup$
    – mhj
    Aug 17, 2019 at 21:36

1 Answer 1

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Either you are leaving something out or it is a very bad exam question. The problem does not indicate that the water is moving, so presumably it is a hydrostatic problem. In that case the water does not matter and you just need to know how the atmospheric presure varies with height. To know that you have to know the air temperature profile which is not given.

If the water is moving, and you are supposed to use Bernoulli, then you need to know the flow rate at the exit point, which is not given.

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  • $\begingroup$ You are correct: there is no solution here w/o extra information. $\endgroup$
    – Gert
    Aug 17, 2019 at 22:18

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