Force Balance of the Float

A common device used to measure volumetric flow rate is the rotameter. The rotameter consists of a transparent tapered tube with a float in it. When the flow enters the rotameter, the float will rise to a position where both the drag force and buoyant force will be balanced by its own weight. That is,

     F_D + F_B = W
    

Solving for flow velocity, u_m,gives,

Note that u_m is the mean velocity passing through the annular space between the float and the tube; V_b is the volume of the float; ρ_b and ρ_f are the density of the float and fluid, respectively; C_D is the drag coefficient; A_fr is the projected area of the float; and g is the acceleration of gravity.

The volumetric flow rate is then given by

     Q = u_mA_m

where A_m is the area of the annular space.

The discussion of rotameters has taken into account the drag force, which is due to viscous effects. Although this chapter is mainly focused on discussion of incompressible and inviscid flow, for illustration purposes, it is appropriate to introduce the flow measurement technique using the rotameter here, which includes viscous drag effect (modeled as C_D).

Orifice Meter

Nozzle Meter

Venturi Meter

Another method to measure flow is by means of a flow obstruction. Examples of this method include orifices, nozzles and venturi meters. All of these methods are developed based on Bernoulli's equation, hence the fluid flow is assumed to be steady, incompressible and inviscid.

Bernoulli's equation can be applied between point 1 and 2 for any of the diagram shown on the left. This gives,

If the change in the elevation is negligible then the velocity at point 2 is

If the flow is steady and incompressible at both points 1 and 2, the continuity equation gives

     Q = A₁V₁ = A₂V₂

Combining the above two equations yields

Note that the analysis thus far is based on the assumption that the fluid is inviscid. However, friction losses due to viscosity are unavoidable in real engineering practice, hence the actual flow rate would always be smaller than the flow rate given in the above equation. In practice this deviation from theory is corrected with a discharge coefficient.