Rotameter is the most widely used variable area flowmeter. Its principle of operation is discussed below. A movable vane meter has similar principle of operation, except the "moving piston (float)" of rotameter is now a swing open valve and the weight of the float is replaced by the spring force applied on the valve.
A rotameter is mounted vertically with the narrow end at the bottom and the tube tappers into a wider top. The flow comes from the bottom and pushes the float inside the rotameter up to a point that the weight of the float is in balance with the force exerted by the flow. The annular area between the float and the tube wall is then related to the volume flow rate.
As long as the fluid speed is substantially subsonic (V < mach 0.3), the incompressible Bernoulli's equation applies.
where g is the gravity acceleration constant (9.81 m/s2 or 32.2 ft/s2), V is the velocity of the fluid, and z is the height above an arbitrary datum. C remains constant along any streamline in the flow, but varies from streamline to streamline. If the flow is irrotational, then C has the same value for all streamlines.
Applying this equation to a streamline traveling up the axis of the vertical tube gives,
where subscript a represents the position right below the float, b is the balanced point of the float, usually the top of the float, V is the flow velocity, p is pressure, and is the density. A shorter form of the above equationi is
where hf is the hight of the float or the distance from the bottom to the indicator of the float that depends on the float design.
From continuity, the volume flow rate at a is the same as the volume flow rate at b, i.e., , which implies
Please note that is the annular area between the float and the tube wall, not the whole cross section area at b.
Hence, the velocity Vb can be substituted out of the Bernoulli's equation to give,
The pressure drop is mostly resulting from the weight of the float
where the subscript f represents the float, Vf is the volume, Af is the cross section area, and f is the density of the float.
Solving for the volumetric flow rate Q, we have
Ideal, inviscid fluids would obey the above equation. The small amount of energy converted into heat within viscous boundary layers tends to somewhat lower the actual velocity of real fluids. A discharge coefficient C is typically introduced to account for the viscosity of fluids,
C is found to depend on the Reynolds Number of the flow.
For a given design, the cross section areas Aa(z) and Ab(z) of the rotameter are functions of the hight z, and the geometry (hf, Af, Vf) and the density (f) of the float are also known. If the density of the fluid is measured and the readout of the position (z) of the float in the rotameter is available, the volume flow rate Q can be calculated from this formula:
The mass flow rate can be easily found by multiplying Q with the fluid density ,