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I was designing a turbine to be utilized in order to measure the flow rate of a river system. I can measure the angular velocity of the shaft by utilizing a magnet and hall sensor, however, in order to relate the angular velocity with flow rate, I was unable to find an equation.

A rough sketch of the design is demonstrated: Sketch Of Turbine.

The paddles are essentially rectangular in geometry with cross sectional area A. The flow of water causes some rotation of the turbine, thus causing the rotation of the shaft. At this point, I can consider mechanical losses in the shaft or bearings (not fluid mechanical) as negligible. Clearly the final flow velocity ($v_2$) locally would be lower than the initial ($v_1$) due to some energy being converted into angular ($\omega$) kinetic energy.

I would greatly appreciate it if you could please provide some guidance on how I can relate the angular velocity with the flow rate. Approximations or appropriate simplifications are fine, but I am interested in verifying if I can relate the two quantities so I can program it into Arduino.

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  • $\begingroup$ if the cross-section of the channel is constant then I would expect for the steady state the principle of momentum conservation would result in $v_1=v_2$, especially since you are not using the wheel to extract energy from the flow(from what I understand you are using it only for measuring - and I am also making the assumption that there is no friction on the wheel). I'm really looking forward to hear people's view on this. Regarding the calibration aspect, IMHO the best way is to a measurement if possible (because the depth of the channel will have an affect on the mass flow rate) $\endgroup$
    – NMech
    Jun 11 at 9:06
  • $\begingroup$ Thanks for the reply. I agree that I am not extracting energy from the system however, since the mass of the turbine is nonzero and a resisting drag force from the water is present as the paddle moves upwards, the incoming flow must perform some form of work on the paddle (in order to maintain rotational kinetic energy). So I would imagine a decrease in kinetic energy of the flow. $\endgroup$
    – Amit
    Jun 11 at 9:17
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    $\begingroup$ if there is no friction, then I disagree @Amit. The wheel will be happy to continue to rotate if the water (magically) disappeared with the same velocity ad infinitum. The rotational energy changes only when the speed of the flow changes (i.e. on the transient state). The steady state (steady flow), should not have an effect on the speed/energy of the wheel. $\endgroup$
    – NMech
    Jun 11 at 9:25
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    $\begingroup$ yeap, the only problem with gingerbreads method is that if the submerged flap is sufficiently submerged then different point travel with different velocity. So its best to do a calibration. $\endgroup$
    – NMech
    Jun 11 at 11:08
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    $\begingroup$ I would just measure the rotational velocity at different speed of the stream and then create a table (maybe even fit a curve). So then you will know that for a given $\omega$ you have a set velocity. $\endgroup$
    – NMech
    Jun 11 at 11:45
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In general, v=rω

Where v is the velocity of the water, r is the radius of the turbine, and ω is the angular velocity. This is the simplest that I know of that could be quickly programmed

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  • $\begingroup$ Thank-you for the reply. If I were to utilize wr = v, that assumes the paddle is moving at the same velocity as the water, but if that is true, then the relative velocity between the flow and paddle is zero, hence implying the water applies no force. However, this is not the case. $\endgroup$
    – Amit
    Jun 11 at 9:22
  • $\begingroup$ @Amit This isn't true as force is constantly being applied to the paddle. Since you have a steady stream of water, it is always going to be in motion, and will move the paddle along with it. Your observation will only hold true if frictional losses are truly zero (which they aren't) and if the stream maintains the same velocity throughout $\endgroup$ Jun 11 at 9:30
  • $\begingroup$ Thanks, just a question, how would I prove or show the paddle moves at the same velocity as the flow? $\endgroup$
    – Amit
    Jun 11 at 9:35
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    $\begingroup$ You can measure the amount of water discharged per unit time. Use this along with the cross-section of the stream. You can then use these values, along with height or pressure difference if any, and substitute them into Bernoulli's equation to solve for speed. $\endgroup$ Jun 11 at 9:42

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