A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are o

Question

A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe, a configuration called a Venturi meter. Consider such an arrangement with a horizontal pipe carrying fluid of density ?. The fluid rises to heights h1 and h2 in the two open-ended tubes (see figure). The cross-sectional area of the pipe is A1 at the position of tube 1, and A2 at the position of tube 2.

A pair of vertical, open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe, a configuration called a Venturi meter. Consider such an arrangement with a horizontal pipe carrying fluid of density ?. The fluid rises to heights h1 and h2 in the two open-ended tubes (see figure). The cross-sectional area of the pipe is A1 at the position of tube 1, and A2 at the position of tube 2. Find p1, the gauge pressure at the bottom of tube 1. (Gauge pressure is the pressure in excess of outside atmospheric pressure.) Express your answer in terms of quantities given in the problem introduction and g, the magnitude of the acceleration due to gravity. Find v1, the speed of the fluid in the left end of the main pipe. Express your answer in terms of h1, h2, g, and either A1 and A2 or ?, which is equal to A1A2.

Explanation / Answer

a)

P1 = ? g h1

b)

P2 = ? g h2

P2 - P1 = 0.5 ? (v1^2 - v2^2)

? g h2 - ? g h1 = 0.5 ? (v1^2 - v2^2)

g h2 - g h1 = 0.5 (v1^2 - v2^2)

2 g (h2 - h1) = v1^2 - v2^2

v1 A1 = v2 A2 >>>>> v2=v1(A1/A2)

2 g (h2 - h1) = v1^2 - v1^2 (A1/A2)^2

2 g (h2 - h1) = v1^2 (1 - (A1/A2)^2)

2 g (h2 - h1) = v1^2 (1 - ?^2)

v1 = ?(2g(h2 - h1)/(1 - ?^2))

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