/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Heavy weights can be moved with ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 24

Heavy weights can be moved with relative ease on air cushions by using a load pallet as shown. Air is supplied from the plenum through porous surface \(A B .\) It enters the gap vertically at uniform speed, \(q\). Once in the gap, all air flows in the positive \(x\) direction (there is no flow across the plane at \(x=0\) ). Assume air flow in the gap is incompressible and uniform at each cross section, with speed \(u(x),\) as shown in the enlarged view. Although the gap is narrow \((h \ll L)\) neglect frictional effects as a first approximation. Use a suitably chosen control volume to show that \(u(x)=q x / h\) in the gap. Calculate the acceleration of a fluid particle in the gap. Evaluate the pressure gradient, \(\partial p / \partial x,\) and sketch the pressure distribution within the gap. Be sure to indicate the pressure at \(x=L\)

Short Answer

Expert verified
The linear velocity of the fluid within the gap is given by \(u(x) = \frac{q}{h}\), the acceleration of a fluid particle is given by \(\frac{q^2 x}{h^2}\), and the pressure gradient within the gap is \(- \rho \frac{q^2}{h^2} x \). The pressure decreases linearly from \(x=0\) to \(x=L\).

Step by step solution

01

Establishing the rate at which air leaves the system

An area is selected as control volume in the region \(x\) to \(x+dx\). The air is assumed to be incompressible and the velocity at each cross-section is uniform. Based on conservation of mass, the air volume entering from the left side should equal the air volume leaving from the right. So, we have \(q dx = u(x) h dx\). Solving this, \(u(x) = \frac{q}{h}\). Thus, the speed of the air in the gap is linear with respect to \(x\).
02

Calculating the acceleration of the fluid particle

The acceleration of a fluid particle along the streamline can be given as the change in velocity divided by the change in time. Here, however, we are considering steady flow, hence the local acceleration is zero and the only form of acceleration is convective acceleration. In one dimension, the convective acceleration given as \(a = \frac{du}{dt} = u \frac{du}{dx} = \frac{q^2 x}{h^2}\).
03

Calculating the pressure gradient

To calculate the pressure gradient, apply the linear momentum equation in \(x\) direction and assume steady flow: \(\rho u \frac{du}{dx} = - \frac{dp}{dx}\). Substitute the value of \(\frac{du}{dx}\) from Step 1 and rearrange to obtain \(\frac{dp}{dx} = - \rho \frac{q^2}{h^2} x \). Thus, the pressure gradient is decreasing linearly with respect to length \(x\).
04

Sketching the pressure distribution

Pressure at \(x=0\) is highest and decreases linearly along the length of the gap to \(x=L\). Therefore, the sketch will be a straight line starting from highest pressure at \(x=0\) to lowest pressure at \(x=L\).

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Most popular questions from this chapter

A nozzle for an incompressible, inviscid fluid of density \(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\) consists of a converging section of pipe. At the inlet the diameter is \(D_{i}=100 \mathrm{mm},\) and at the outlet the diameter is \(D_{o}=20 \mathrm{mm} .\) The nozzle length is \(L=500 \mathrm{mm}\) and the diameter decreases linearly with distance \(x\) along the nozzle. Derive and plot the acceleration of a fluid particle, assuming uniform flow at each section, if the speed at the inlet is \(V_{i}=1 \mathrm{m} / \mathrm{s}\). Plot the pressure gradient through the nozzle, and find its maximum absolute value. If the pressure gradient must be no greater than \(5 \mathrm{MPa} / \mathrm{m}\) in absolute value, how long would the nozzle have to be?

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You (a young person of legal drinking age) are making homemade beer. As part of the process you have to siphon the wort (the fermenting beer with sediment at the bottom) into a clean tank using a \(5-\mathrm{mm}\) ID tubing. Being a young engineer, you're curious about the flow you can produce. Find an expression for and plot the flow rate \(Q\) (liters per minute \()\) versus the differential in height \(h\) (millimeters) between the wort free surface and the location of the hose exit. Find the value of \(h\) for which \(Q=2 \mathrm{L} / \mathrm{min}\)
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