Halliday, Resnick & Walker — Fluids: Problem Problem_65

A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe (Fig. 14.30); the cross-sectional area \(A\) of the entrance and exit of the meter matches the pipe's cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed \(V\) and then through a narrow "throat" of cross-sectional area \(a\) with speed \(v\). A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's speed is accompanied by a change \(\Delta p\) in the fluid's pressure, which causes a height difference \(h\) of the liquid in the two arms of the manometer. (Here \(\Delta p\) means pressure in the throat minus pressure in the pipe.) (a) By applying Bernoulli’s equation and the equation of continuity to points 1 and 2 in Fig. 14.30, show that \[V = \sqrt{\frac{2a^2 \Delta p}{\rho(a^2 - A^2)}}\] where \(\rho\) is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-sectional areas are \(64 \text{ cm}^2\) in the pipe and \(32 \text{ cm}^2\) in the throat, and that the pressure is \(55 \text{ kPa}\) in the pipe and \(41 \text{ kPa}\) in the throat. What is the rate of water flow in cubic meters per second?