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Chapter 8: Problem 9

For fully developed laminar pipe flow in a circular pipe, the velocity profile is given by \(u(r)=2\left(1-r^{2} / R^{2}\right)\) in \(\mathrm{m} / \mathrm{s},\) where \(R\) is the inner radius of the pipe. Assuming that the pipe diameter is \(4 \mathrm{cm},\) find the maximum and average velocities in the pipe as well as the volume flow rate.

Short Answer

Expert verified
The maximum velocity in the pipe is 2 m/s, the average velocity is 1.33 m/s and the volume flow rate is 0.00000534 cubic metres per second.

Step by step solution

01

Determine Maximum Velocity

The maximum velocity \( u_{max} \) occurs at the center of the pipe where \( r = 0 \). So, substitute \( r = 0 \) into the velocity equation: \( u(r)=2\left(1-0^{2} / R^{2}\right) = 2 \, m/s \).
02

Calculate Average Velocity

The average velocity \( u_{avg} \) is given by: \( u_{avg} = {2 \over R^2 } * {2 \over 3} * R^2 = {4 \over 3} \, m/s\).
03

Find the Volume Flow Rate

The volume flow rate \( Q \) is got by multiplying the average velocity by the pipe's cross-sectional area (\( πR^{2} \)): \( Q = u_{avg} * πR^{2} = {4 \over 3} * π * (0.02)^2 = 0.00000534 \, m^3/s \).

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Most popular questions from this chapter

Water is to be moved from a large, closed tank in which the air pressure is 20 psi into a large, open tank through 2000 ft of smooth pipe at the rate of \(3 \mathrm{ft}^{3} / \mathrm{s}\). The fluid level in the open tank is \(150 \mathrm{ft}\) below that in the closed tank. Determine the required diameter of the pipe. Neglect minor losses.

A 2.5 -in.-diameter flow nozzle meter is installed in a 3.8 -in.- diameter pipe that carries water at \(160^{\circ} \mathrm{F}\). If the air-water manometer used te measure the pressure difference across the meter indicates a reading of \(3.1 \mathrm{ft}\), determine the flowrate

A motor-driven centrifugal pump delivers \(15^{\circ} \mathrm{C}\) water at the rate of \(10 \mathrm{m}^{3} / \mathrm{min}\) from a reservoir, through a 2500 -m-long, \(30-\mathrm{cm} 1 . \mathrm{D} .\) plastic pipe, to a second reservoir. The water level in the second reservoir is \(40 \mathrm{m}\) above the water level in the first reservoir. The pump efficiency is \(75 \% .\) Find the motor output power. The pipe entrance is square edged.

For a given head loss per unit length, what effect on the flowrate does doubling the pipe diameter have if the flow is (a) laminar, or (b) completely turbulent?

Water flows downward through a vertical 10 -mm-diameter galvanized iron pipe with an average velocity of \(5.0 \mathrm{m} / \mathrm{s}\) and exits as a free jet. There is a small hole in the pipe \(4 \mathrm{m}\) above the outlet. Will water leak out of the pipe through this hole, or will air enter into the pipe through the hole? Repeat the problem if the average velocity is \(0.5 \mathrm{m} / \mathrm{s}\)
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