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.
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.