Non-steady suction induced by Venturi Injector

[PickledCucumber]

This is not really homework but an engineering problem nonetheless.

1. I have a venturi injector sucking in liquid (by all means and purposes identical to water) and injecting into a pressurized water line. A straight tube is inserted into the container of liquid (below the liquid surface) and connected to the throat of the injector. The container is open to the atmosphere. I would like to know if suction is possible. If so, what is the pressure difference at the throat of the injector and suction rate? Assume incompressible & irrotational flow, rigid pipe and neglect friction.

Givens:
$P_1=344.7 kPa$
$\rho=1000 kg/m^3$
$Q=0.05 LPM$
$g=9.81 m/s^2$
$d_1=0.0635 m$
$d_2=0.0125 m$
$d_4=0.05 m$
$h_2=2 m$
$h_3=0 m$
$h_4=0.5 m$

Relevant equations:
$Q=A*V$
$P_1/\rho + V_1^2/2 + gh_1=P_2/\rho + V_2^2/2 + gh_2$

The Attempt at a Solution

I would analyze points 4 and 3 first. Then I would analyze points 3 and 2 to determine the pressure difference which will help me determine suction. However, I am unsure if my approach to the bernoullis balance is correct.
there are two variations I have looked into:
method 1: $\frac{1}{\rho} \frac{dP_3}{dt}+ \frac{(\frac{dV_3}{dt})^2}{2} + gh_3=\frac{P_4}{\rho} + \frac{V_4^2}{2} + g \frac{dh_4}{dt}$
or
method 2: can we look at the change of pressure and velocity with respect to change in height rather than change in time.

Going with method 1:
I assume that h4 will decrease over time. I then also assumed the velocity and pressure are both dependent on each other and will vary over time as height decreases between h4 and h3
$\frac{dP_3}{dt}=\rho(g \frac{dh_4}{dt}- \frac{(\frac{dV_3}{dt})^2}{2})$
Now I am unsure what my time limits are when integrating.

Is my current approach reasonable? Is there an transient unsteady bernoullis balance that should be applied here instead? I have also been reading on the unsteady bernoullis equation derived from eulers equation.

 

[KataKoniK]

Thank you for sharing your engineering problem. I would like to offer my perspective on your proposed solution.

Firstly, your approach to analyzing points 4 and 3 and then points 3 and 2 seems reasonable. However, I would suggest using the Bernoulli's equation in its integral form, which takes into account the variation of pressure and velocity with respect to height. This would give you a more accurate and comprehensive solution.

Secondly, I believe that your time limits for integration would depend on the specific conditions of your problem. For example, if you are looking for the suction rate at a specific time, you would integrate from that time until the final time. If you are interested in the overall suction rate over a period of time, you would integrate over the entire time period. I would recommend consulting with an engineer or using a numerical simulation to determine the appropriate time limits for integration.

Lastly, I would also suggest considering the effect of friction in your analysis. While you have neglected it in your given assumptions, it may have a significant impact on the results, especially if you are looking at a longer time period.

I hope this helps and good luck with your engineering problem!