Halliday, Resnick & Walker — Fluids: Problem 59

59 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. 14-43); 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. 14-43, show that \[ V = \sqrt{\frac{2 a^2 \Delta p}{\rho(a^2 - A^2)}} \] where \( \rho \) is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-sectional areas are \( 60 \text{ cm}^2 \) 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?