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Given an inlet wind speed of $v_1$ and nozzle inlet area of $A_1$ and an outlet area $A_2$ what would be the outlet velocity. The nozzle/concentrator is positioned in an open field where the wind is blowing. The shape of the concentrator is conical.

Obviously, the equation $A_1v_1=A_2v_2$ does not apply given the compressibility of the fluid. What way can it be calculated?

The motivation of this question is accesing the possibility of using concentrated air in turbines/wind mills. Air velocities are low in certain places so increasing its speed may be of interest.

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  • $\begingroup$ Assume no compression then calculate the outlet velocity based on the ratio of diameters. Remember the mass flow will be constant. $\endgroup$
    – Solar Mike
    Commented Oct 24, 2022 at 12:53

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If you check out where some wind turbines have been located, valleys have been used as they are, in effect, a tapered concentrator.

Also wind turbines have been located on top of hills - some distance back from the edge as that improves the performance - the "compression" caused by the ground slope also improves the performance.

As I mentioned in the comment, mass flow between inlet and outlet is constant unless you have a leak. So the change in velocity will be according to the change between the diameters and there will be a discharge coefficient as there will be lost air.

There have been ventilation fans using shrouds to reduce the edge losses, but that becomes impractical for wind turbines.

You could set up an experiment using cones to evaluate the velocity increase based on the diameter change. One thing is to then use a pitot tube to measure the velocity profile across the outlet and inlet if you really want to find out what happens. Then you can use that velocity profile to check the mass flow etc and find out which parameters are mostly responsible for the errors. This was a really good lab experiment which was 3 hours work... The analysis and write-up took much longer.

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  • $\begingroup$ Thank you for your answer. Whenever I do this calculation with for example inlet diameter $D_1=5m$ and outlet $D_2=5cm$, a wind inlet speed of $2.2m/s$ would result in $v_2=\frac{{D_1}^2}{{D_2}^2}*v_1=10000m/s$. I can't imagine this velocity will ever be reached. What am I missing? $\endgroup$
    – jessegerritsen
    Commented Oct 25, 2022 at 6:12
  • $\begingroup$ What I'm guessing is that there is some kind of back pressure/force that counteracts the inflow of air. $\endgroup$
    – jessegerritsen
    Commented Oct 25, 2022 at 6:25
  • $\begingroup$ @jessegerritsen so you are going from subsonic to supersonic, the power needed will be massive, the forces on the cone huge and you will need to factor in compressibility, friction heating, pressure drop and other factors. As I said about wind turbines above in valleys, try a change in area that is more relevant and think about the results. $\endgroup$
    – Solar Mike
    Commented Oct 25, 2022 at 6:57
  • $\begingroup$ Okay thank you. So the flow will never naturally go supersonic given this inlet velocity? $\endgroup$
    – jessegerritsen
    Commented Oct 25, 2022 at 6:59
  • $\begingroup$ @jessegerritsen check out throat area for M=1 - many textbooks cover sub and supersonic flow, Prandtl-Meyer expansion, Condi nozzles etc All good terms to search for. Not a topic mastered in 5 minutes... $\endgroup$
    – Solar Mike
    Commented Oct 25, 2022 at 7:01

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