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FLUID MECHANICS - THEORY

    Flow meters are encountered frequently in our daily lives (e.g., water meters in our homes and gasoline meters at the gas stations). Several common flow measurement methods for pipe flow will be presented in this section. Flow measurement methods, such as the sluice gate and the sharp crested weir will be presented in the later section (open-channel).

Simple Flowrate
Measurement Technique
   
  Simple Flowrate Measurement Technique

  The simplest technique to determine the steady flow rate of a liquid is to measure the amount of liquid collected over a period of time. For example,one could collect the liquid in a container of known size. If the time needed to fill the container is recorded, then the flow rate can be easily determined from the equation

     Q = Volume/Time
   
    Rotameters


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,

     FD + FB = W
    

Solving for flow velocity, um,gives,

     

Note that um is the mean velocity passing through the annular space between the float and the tube; Vb is the volume of the float; ρb and ρf are the density of the float and fluid, respectively; CD is the drag coefficient; Afr is the projected area of the float; and g is the acceleration of gravity.

The volumetric flow rate is then given by

     Q = umAm

where Am 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 CD).

     
    Orifices, Nozzles and Venturi Meters


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.

     
   
 
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