Turbine meter equations and derivations
Turbine flowmeters make use of the angular velocity of the well-designed blades to determine the flow velocity. The blades need to be suitable for the type of fluid and strong enough for high rotation speed conditions. One way to model the turbine is to describe the fluid driving torque in terms of momentum exchange.

In an ideal situation when the rotor is not slowed down by any forces such as friction, it will rotate at a speed that sustains the fluid velocity vector at the blade surfaces, which are assumed to be flat. The flow into the turbine has a velocity V and this causes the turbine to rotate at an angular velocity ώ. Assuming there is no loss in velocity, the ideal angular velocity ώi can be correlated to the flow velocity V by equation (1) (1)
Where is the angle between the flow direction and the turbine blades and is the root mean square of the inner and outer radii of the blades from the turbine axis (parallel to flow).

Now instead of the no loss in velocity assumption, after the fluid passes by the turbine, its velocity changes to a value of VE, and a torque T is applied from the flow to the turbine, subsequently making it rotate.

A force balance leads to the following integral for the torque over the area of the blade.

(2)

Combining equations (1) and (2) and substituting for the ideal angular velocity, the angular velocity can be related to the velocity by equation (3). (3)

With a steady