CHAPTER 2  BASIC CONCEPTS RELATED TO FLOWING WATER AND MEASUREMENT
14. Application of Energy Principle to TubeType Flowmeters
The energy equation can be used to derive the venturi meter (figure 25) equation by assuming that the centerline of the meter is horizontal (Z_{1} = Z_{2}); and due to its short length, there is no head loss, h_{f} = 0. Although these assumptions were made to simplify the derivation, the final results will be identical for any orientation of the venturi meter.

Thus:
(241)
By the continuity equation for the approach and throat sections:
(242)
Either V_{1} or V_{2} can be solved for in terms of the other; for example:
(243)
Substituting this result into the energy equation results in:
(244)
Solving for the head difference gives:
(245)
Solving for V_{1}^{2}:
(246)
Taking the square root of both sides and multiplying both sides by A_{1} results in the theoretical discharge equation:
(247)
To obtain actual discharge, a coefficient, C_{d}, added to compensate for velocity distribution and for minor losses not accounted for in the energy equation yields:
(248a)
Some investigators solve for discharge using throat area and velocity, resulting in:
(248b)
However, equations 248a and 248b are identical and can be converted to:
(249)
Equations 248b and 249 also apply to nozzles and orifices in pipes. On figure 25, the hydraulic grade line, hgl, represents the pressure that acts on the walls of the venturi meter. An appreciable drop will be noticed at the narrow throat, and a gradual pressure rise is seen as the flow leaves the throat and smoothly spreads and slows in the expanding portion of the meter.
Figure 26 shows the conditions that occur in a pipe orifice meter. As the flow approaches the orifice plate, the water near the pipe walls is slowed and stopped in the corners formed by the plate and the pipe walls. As a result, the pressure just ahead of the orifice at point B is a little greater than in the pipeline farther upstream at A. As the flow accelerates and passes through the orifice, the pressure drops and is lowest just downstream from the plate where the jet is smallest, and the velocity is highest at point C. Farther downstream, the flow begins to spread out and slow down, and a rise in pressure occurs at points D and E.

In both venturi meters and orifice meters, the pressure difference between the inlet tap and the throat or minimum pressure tap is related to discharge tables or curves using the suitable coefficients with the proper equation. An example discharge curve is shown for an 8inch (in) venturi meter on figure 27. Thus, the meters may serve as reliable flow measuring devices.
