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Chapter 6: Problem 18

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?

Short Answer

Expert verified
The acceleration of the fluid particle, pressure gradient, and its maximum absolute value can be calculated using the equations of motion and Bernoulli principle. By equating the pressure gradient to its given threshold, the required nozzle length can be determined.

Step by step solution

01

Derive the acceleration of the fluid particle

From the equation of motion, \(a = \Delta v/\Delta t\). In this case, the change in speed \(\Delta v = v_{o} - v_{i}\), where \(v_{o}\) is the speed at the outlet and \(v_{i} = 1 m/s\) is the speed at the inlet. Since the fluid is incompressible and the equation of continuity \(\rho v A = const\) applies, we have \(v_{o} = v_{i} (D_{i}/D_{o})^2\). Substituting the known values, we find \(a = (v_{o} - v_{i})/\Delta t\), where \(\Delta t = L/v_{i}\), with \(L = 0.5m\) being the nozzle length. Solving this equation, we obtain the acceleration of the fluid particle.
02

Calculate the pressure gradient and find its maximum value

The pressure gradient can be calculated from the Bernoulli equation, \(p + 0.5 \rho v^2 = const\), which gives \(\delta p/\delta x = -\rho v dv/dx\). Substituting our previously found acceleration \(a\) for \(dv/dx\), we can evaluate the pressure gradient, and its maximum value occurs when the outlet speed \(v_{o}\) is the highest.
03

Determine the length of the nozzle based on the pressure gradient

Equating the maximum pressure gradient to the given threshold of \(5 MPa/m\), we can solve for the nozzle length \(L\) that fulfills this condition.

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Most popular questions from this chapter

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 \(Q=2 \mathrm{L} / \mathrm{min}\)

\(\mathrm{A}\) crude model of a tornado is formed by combining a \(\operatorname{sink},\) of strength \(q=2800 \mathrm{m}^{2} / \mathrm{s},\) and a free vortex, of strength \(K=5600 \mathrm{m}^{2} / \mathrm{s} .\) Obtain the stream function and velocity potential for this flow field. Estimate the radius beyond which the flow may be treated as incompressible. Find the gage pressure at that radius.

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For the flow of Problem 4.123 show that the uniform radial velocity is \(V_{r}=Q / 2 \pi r h .\) Obtain expressions for the \(r\) component of acceleration of a fluid particle in the gap and for the pressure variation as a function of radial distance from the central holes.
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