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Chapter 4: Problem 117

A Venturi meter installed along a water pipe consists of a convergent section, a constant-area throat, and a divergent section. The pipe diameter is \(D=100 \mathrm{mm},\) and the throat diameter is \(d=50 \mathrm{mm}\). Find the net fluid force acting on the convergent section if the water pressure in the pipe is \(200 \mathrm{kPa}(\text { gage })\) and the flow rate is \(1000 \mathrm{L} / \mathrm{min}\). For this analysis, neglect viscous effects.

Short Answer

Expert verified
The net fluid force acting on the convergent section is 376.99 N.

Step by step solution

01

Conversion

Firstly, convert the flow rate from liters per minute to cubic meters per second: \(Q = 1000 \, L/min = \frac{1000}{1000} \, m^3/min = \frac{1}{60} \, m^3/sec = 0.01667 \, m^3/sec\)
02

Calculation of velocities

Calculate the velocities in the pipe and in the throat using the formula: \(V = \frac{Q}{A}\), where \(A = 蟺/4 \, D^2\) is the cross-section area, D is the diameter and Q is the flow rate. So, \(V1 = \frac{Q}{A1} = \frac{0.01667 \, m^3/sec}{蟺/4 \, (0.1 \, m)^2} = 0.67 \, m/sec\) in the pipe, and \(V2 = \frac{Q}{A2} = \frac{0.01667 \, m^3/sec}{蟺/4 \, (0.05 \, m)^2} = 2.68 \, m/sec\) in the throat.
03

Use Bernoulli鈥檚 theorem

According to Bernoulli's theorem, the pressure in the throat is given by: \(P2 = P1 + \frac{1}{2} \, 蟻 \, (V1^2 - V2^2)\)Substituting the values, \(P2 = 200 \, kPa + \frac{1}{2} \, 1000 \, kg/m^3 \, (0.67^2 - 2.68^2) \, kPa = 152 \, kPa\)
04

Net Fluid Force

The net fluid force acting on the convergent section is the difference in pressures times the area: \(F = (P1 - P2) \, A1\)Substituting the values, \(F = (200 kPa - 152 kPa) \, 蟺/4 \, (0.1m)^2 = 376.99 N\)

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