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Chapter 8: Problem 15

For laminar flow in a round pipe of diameter \(D,\) at what distance from the centerline is the actual velocity equal to the average velocity?

Short Answer

Expert verified
The actual flow velocity equals to the average velocity at a radial distance of approximately 0.707 times the pipe radius, or within 70.7% of the pipe radius from the centerline.

Step by step solution

01

Recall the formula for velocity profile for laminar flow

The formula for the velocity profile for laminar flow in a pipe is: \( u(r) = \frac{ΔP}{4L}(\frac{D^2}{4} - r^2) \), where \( ΔP \) is the pressure difference, \( L \) is the length of the pipe, \( D \) is the pipe diameter and \( r \) is the radial distance from the center of the pipe.
02

Recall the formula for average velocity

The average velocity \( \overline{u} \) for laminar flow in a pipe can be calculated as: \( \overline{u} = \frac{ΔP}{4L}(\frac{D^2}{16}) \), by using the equation of motion and the confinement of the boundaries. This equation represents the average velocity across the flow profile.
03

Solve for the radial distance r

We are asked to find the radial distance \( r \) from the center of the pipe where the actual flow velocity equals to the average velocity. Setting the two expressions for velocity equal to each other, we get: \( \frac{ΔP}{4L}(\frac{D^2}{4} - r^2) = \frac{ΔP}{4L}(\frac{D^2}{16}) \). Simplify it, we get: \( (\frac{D^2}{4} - r^2)= \frac{D^2}{16} \). Solving for \( r \), we find: \( r = D/2 \sqrt{1-\frac{1}{2}} \).
04

Apply the solution to a practical context

So for a laminar flow in a pipe, when moving from the centerline towards the pipe wall, the actual velocity equals the average velocity at an approximately 0.707 times the pipe radius. This means within 70.7% of the pipe radius from the centerline, the velocity will be greater or equal to the average velocity.

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Most popular questions from this chapter

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.

The wall shear stress in a fully developed flow portion of a 12-in.-diameler pipe carrying water is 1.85 Ib/ft? Determine the pressure gracient, \(\partial p / \partial x,\) where \(x\) is in the flow direction, if the pipe is (a) horizontal, (b) vertical with flow up, or (c) vertical with flow down.

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 \(0.5 \mathrm{m} / \mathrm{s}\)

Given two rectangular ducts with equal cross-sectional area but different aspect ratios (width/height) of 2 and \(4,\) which will have the greater frictional losses? Explain your answer

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.
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