A venturi meter is a device to measure fluid flow rates, which in its operation resembles the orifice meter (Section 3.2b). It consists of a tapered Constriction in a line, with pressure taps leading to a differential manometer at points upstream of the constriction and at the point of maximum constriction (the throat). The manometer reading is directly related to the flow rate in the line. Suppose the flow rate of an incompressible fluid is to be measured in a venturi meter in which the cross-sectional area at point 1 is four times that at point 2.
(a) Derive the relationship between the velocities u1 and u2 at points 1 and 2.
(b) Write the Bernoulli equation for the system between points 1 and 2, and use it to prove that to the extent friction is negligible where p1 and p2 are the pressures at point 1 and 2, p and V are the density and volumetric flow rate of the flowing fluid, and A1 is the cross-sectional area of the pipe at point 1.
(c) Suppose this meter is used to measure the flow rate of a stream of water, using mercury as the manometer fluid, and a manometer reading h = 38 cm is recorded. What is the volumetric flow rate of the water if the pipe diameter at point 1 is 15 cm? (Recall the differential manometer equation. Equation3.4-6.)