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I'm working on a project similar to a vacuum cleaner, where I have a suction pump connected to an open ended pipe that has a negative pressure gradient such that things are being sucked into it with a constant flow rate.

Is it possible to determine an approximate maximum distance that an object with mass m needs to be from the endpoint of the pipe before it is no longer able to be sucked in?

I know that there is Poiseuille flow inside the pipe, and just outside the endpoint of the pipe, the velocity profile is constant. However, I also know that the velocity profile changes the farther you go away from the endpoint of the pipe, thus it is dependent on the distance outside of the pipe.

I am not sure how to find the rate of change of the pressure gradient outside of the pipe. I would use that to determine the force at that distance, and then compare that with the force needed to push the object with mass, m horizontally and see where they (suction force at particular point and force to push object)are equal.

I have tried Bernoulli's equation, where this pipe is connected to an infinitely large pipe that approximates an open-ended pipe, but that didn't really work out.

Is there something else I can do?

For clarification, I am an engineering student in fluid mechanics 1.

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  • $\begingroup$ We had a question in our first year fluids about the l/s needed for a vacuum cleaner. Lots of losses to account for - a good exercise, I think it was about 3 or 4 l/s... $\endgroup$
    – Solar Mike
    Apr 11, 2023 at 12:43
  • $\begingroup$ We also had a lab where we measured the velocity profile at the entrance and exit of a pipe - used a pitot tube... Perhaps you should do that at various distances from the pipe to see. $\endgroup$
    – Solar Mike
    Apr 11, 2023 at 12:46
  • $\begingroup$ Figure required lift and drag to counteract friction, weight of object, and whatever holds it in place. From there get a minimum velocity of wind needed, and from there get max distance from what your device creates. $\endgroup$
    – Abel
    Apr 11, 2023 at 13:19
  • $\begingroup$ @Abel well I have the suction force calculated already, and I am not considering a lift force as I am assuming that both the pipe and the object are on the ground. I am having problems finding the change in wind velocity as you move further away from the pipe. $\endgroup$
    – user41396
    Apr 11, 2023 at 13:58
  • $\begingroup$ @Actually I think I can use the drag force equation to find a minimum velocity at the endpoint of the pipe, and then get a minimum pump power rating to suck the object in right at the endpoint. From there I can experiment with different distances and such. $\endgroup$
    – user41396
    Apr 11, 2023 at 14:08

2 Answers 2

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If you can do it, I suggest experiments over calculations.

If you look at noise factors, i.e. things that will change during application, which you can't or don't want to control, here's a preliminary list:

  • (variations) of mass of the succed in particles
  • (variations) of their shape and sizes
  • (variations) of particle coagulations
  • (variations) of friction and sticking coefficients
  • (variations) of flow rate
  • (variations) of gradients accross its cross-section
  • (variations) of in humidity, temperature, pressure
  • (variations) in angles of inclination
  • (variations) of time (particles may pile up in your apparatus)
  • (variations) of environmental vibrations
  • and so on, and so on

If you can't run experiments, you'd need have a model, which reflects these effects, at least accurate enough by tendency.

Anyway, those results will differ from optimistic analytical formulations. So a common bypass is this:

  • use simulation, i.e. make both your analytical and noise model work, i.e. parametrize both
  • simulate for various noise conditions
  • adjust parameters of the analytical model only to fit results

This may lead to:

  • effective parameters, like $m_{eff}$ to account for noise effects fromreal life
  • extra fits to model variability, e.g. $\sigma_{flow-rate} = f(m, shape, distribution etc.)$
  • the latter can be very simple, e.g. polynomials
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  • $\begingroup$ it can be noisy and still not suck enough... But can be quiet and suck well. Perhaps noise is not the priority. $\endgroup$
    – Solar Mike
    Apr 12, 2023 at 19:20
  • $\begingroup$ Noise, as defined above, distorts intended function. If it doesn‘t, you didn‘t identify relevant impact. Products sensitive to noise give trouble after trouble. Products made insensitive to even the strongest disturbances will be experienced as „high quality“. So it‘s not about sound ;-) $\endgroup$
    – MS-SPO
    Apr 12, 2023 at 19:26
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To understand this well would require a combination of experiment and computational fluid dynamics, but creating even a crude model can of help our understanding, give us a start on creating a more refined answer, and serve as a sanity check. Here is a very simple naive model.

If the air is flowing uniformly from all directions towards the vacuum inlet, then we could make a crude hemispherical approximation for the volume of air flowing into the inlet:

$$\dot{V}=\frac{dV}{dt} \sim 2 \pi r^2 \frac{dr}{dt}$$

where $r$ is the distance from the inlet, so the radial velocity would be $$v=\frac{dr}{dt}\sim\frac{1}{2 \pi r^2}\dot{V}$$ The drag force on an object in the air flow would be

$$F_D=\frac{1}{2}\rho v^2 C_D A \sim \frac{\rho C_D A}{16\pi^2 r^4}\dot{V}^2$$

where $C_D$ and $A$ are the drag coefficient and cross-sectional area of the object, and $\rho$ is the air density. Note that the $r$ exponent could be closer to 3 instead of 4, if the geometry is such that the airflow is more two-dimensional.

The geometry of your situation is not completely clear to me, but I believe the object of mass $m$ will begin to move when the suction drag force exceeds the static friction force, $F_f=\mu m g$, holding the object in place. In our naive model this will happen when $$r\lesssim \left(\frac{\rho C_D A}{\mu m g}\right)^{1/4}\sqrt{\frac{\dot{V}}{4\pi}} $$

If you do experimentally measure the "suction distance" vs flow, it would be great for you to post the results as an answer here, to show whether the above simple model is even close to reality.

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