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pressurized tank with exit tube and valve

I have a system with a pressurized nitrogen that pushes water out of a tank. Bernoulli's equation tells me the exit velocity if I suddenly open the valve, but I'm confused on how to account for a potential nozzle diameter? My intuition tells me if I restrict the flow at the valve, the velocity should increase, yet Bernoulli's equation does not account for this, right? I've calculated the exit velocity to be about 57 [m/s], but what would happen if I add a nozzle that is half the area of the exit pipe? Is it wrong to assume flowrate would be fixed?

I'm sorry if this question seems too simple, but I would appreciate help figuring out what a nozzle would do to the flow.

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Bernoulli's Equation is a useful idealisation but one that doesn't take into account any pressure loss (in engineering often referred to as head loss) due to friction in pipes, to bends, constrictions/expansions, valves and other elements commonly encountered in Real Life piping.

It can be adjusted for these losses, as follows:

$$h_1+\frac{p_1}{\rho g}+\frac{v_1^2}{\rho g}+\Delta h_p=h_2+\frac{p_2}{\rho g}+\frac{v_2^2}{\rho g}+ \Delta h_f+\Delta h_m$$

(Ref.)

where $\Delta h_p$ is the pressure contribution from a pump (zero in your case), $\Delta h_f$ is the loss due to pipe friction and $\Delta h_m$ are the so-called 'minor' losses' due to bends, constrictions/expansions, valves and other elements in a pipe system.

You can see how $\Delta h_f$ is calculated in the reference provided above.

For sudden contractions/expansions, $\Delta h_m$ can be calculated as in this wiki entry.

The head losses that make up $\Delta h_m$ are often represented as:

$$\Delta h_i=z_i\frac{v^2}{2g}$$

where $z_i$ is a geometry dependent factor, intrinsic to the specific loss-creating element $i$ and $v$ the flow speed through it.

Then $\Delta h_m=\sum_{i=1}^n \Delta h_i$, for all the loss making elements.

I hope this helps.

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  • $\begingroup$ That was very insightful and helpful. Thanks! $\endgroup$ Mar 14, 2022 at 18:50
  • $\begingroup$ Thanks for accepting! $\endgroup$
    – Gert
    Mar 14, 2022 at 19:11
  • $\begingroup$ does this imply that in a truly frictionless/lossless system that nozzle diameter would not affect output velocity? $\endgroup$ Mar 15, 2022 at 10:29
  • $\begingroup$ Yes, I believe so. $\endgroup$
    – Gert
    Mar 16, 2022 at 17:15
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A larger nozzle allows a greater rate of flow in the exit pipe (or hose). This increases the friction along the pipe and causes a pressure drop along its length. Bernoulli's equation applies just inside and outside the nozzle. Continuity then determines the flow rate in the pipe, and that determines the pressure drop along the pipe (there's a formula for that). You might use simultaneous equations, or loops of calculation. Now that I think about it, Bernoulli's equation also predicts a pressure drop as the fluid enters the pipe from the tank.

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You forgot to consider exit loss coefficient of the fluid leaving the tank. Distance from the tank bottom can also have a significant affect on those losses. Never locate a nozzle adjacent to a tank floor.

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