The cross section of a turbine looks like the following:
To simplify the derivation of the formula, we assume that the blades of the turbine are straight and thin. The radius of the rotor (the radius at the roots of the blades) is a and the radius of the turbine (radius measured at the outer edges of the blades) is R, the width of of blades is c, and the distance between blades is S. The incoming flow with velocity V causes the turbine to rotate at angular velocity .
If there is no velocity loss, the ideal angular velocity i can be related to the flow velocity V by a simple trigonometric formula
where is the angle between the pipe axis (incoming flow direction) and the blades of the turbine, is the root-mean-square value of the inner and outer radii of blades to represent the average radius
Now, instead of the ideal situation, the flow's velocity changes to VE after it passes the turbine blades, as shown in the above illustration. Because of this change of velocity vector, the flow applies a torque T to the turbine to make it rotate.
The flow velocity V can then be related to the angular velocity of the turbine .
Since the turbine is rotating at a constant speed, the above mentioned torque T must be counteracted by an equal amount of resistance torque. Neglecting all minor factors, the single most important contributor to this resistance torque is the sum of the drag force on each blade Fd
where Cd is the drag coefficient, the ratio of the blade's drag to the drag of a perpendicular flat plate with equal area, and Re is Reynolds number.
The torque T becomes
where n is the number of blades. Using this expression, the to V ratio can be written as:
The volume flow rate Q can then be expressed in terms of the angular velocity of the turbine
In industrial applications, a K factor is commonly introduced to compensate for the neglected factors in the above analysis.