Simple applications of the Bernoulli equation

There is a rather large range of problems in fluid mechanics to which the perfect fluid model can be successfully applied, yielding solutions adequate for the petroleum engineer. One such problem is that of velocity measurements based on the measurement of the dynamic pressure. The most simple device for this is the Pitot tube. As shown in Fig. 4.4, the opening of the right-angled bend is directed upstream so that a stagnation point occurs at this opening. In front of the opening the fluid is at rest; its stagnation pressure is po. The Bernoulli equation can be written for the streamline leading to the stagnation point. Far from the Pitot tube the flow is undisturbed, its velocity is u and the static pressure is p. Thus

The static pressure can be measured using a piezometer opening. The difference between the stagnation and the static pressure is obtained by connecting both points to the opposite ends of a differential manometer. The hydrostatic equation

for the U-tube is

The most frequently used device for measuring flow rates is the well-known  orifice meter. As shown in Fig. 4.7, this is a thin, flat circular plate with a concentric hole in it, which is clamped between two flanges at any joint in a pipeline. The sharp-edged hole of the orifice plate has a calibrated diameter D2. From the sharp edge of the opening free streamlines separate, forming an axisymmetric stream surface, which represents the boundary between the contracting stream tube and the stagnation region of constant pressure. The static pressure connections are made taking into the vena contracta effect. The upstream tap is commonly located at a distance D, upstream of the orifice plate, while the downstream tap is placed at the minimum-pressure position, which is assumed to be at the vena contracta. The Bernoulli equation between cross-sections 1 and C can be written as