/*! 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 121 A stream of incompressible liqui... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 121

A stream of incompressible liquid moving at low speed leaves a nozzle pointed directly downward. Assume the speed at any cross section is uniform and neglect viscous effects. The speed and area of the jet at the nozzle exit are \(V_{0}\) and \(A_{0},\) respectively. Apply conservation of mass and the momentum equation to a differential control volume of length \(d z\) in the flow direction. Derive expressions for the variations of jet speed and area as functions of z. Evaluate the distance at which the jet area is half its original value. (Take the origin of coordinates at the nozzle exit.)

Short Answer

Expert verified
The distance at which the jet area is half its original value is \(\frac{3V_0^2}{8g}\)

Step by step solution

01

Applying the principle of conservation of mass

On a small control volume of length \(dz\), the principle of conservation of mass says that the mass flow rate in the stream must remain constant. So, it follows then that \(A_0 V_0 = A V\) where \(A\) and \(V\) are the area and speed at a distance \(z\) from the nozzle exit.
02

Derive expression for speed as a function of z

To derive an expression for speed as a function of z, first express \(A\) as a function of z from the mass conservation derived in Step 1: \(A = V_0 A_0/ V\). Now apply the momentum equation \(p + 1/2 \rho V^2 + \rho g z = constant\), where \(p\) is pressure, \(\rho\) is the fluid density, \(V\) is flow velocity and \(g\) is the acceleration due to gravity. Given the nozzle exit is the reference point and the pressure there is atmospheric pressure, you can define the constant value as \(p_0 + 1/2 \rho V_0^2\), where \(p_0\) is atmospheric pressure. Rewrite this momentum equation, solve it for velocity \(V\) and simplify to get \(V = \sqrt{V_0^2 + 2g z}\).
03

Derive expression for area as a function of z

Substitute the expression for \(V\) derived in Step 2 into the equation \(A = A_0 V_0/ V\) to get the area as a function of z, which is \( A = \frac{A_0 V_0}{\sqrt{V_0^2 + 2g z}}\).
04

Find z when area is halved

To find the value of \(z\) when the area is halved we will set \(A = A_0/2\) and solve for \(z\). Substituting \(A_0/2\) into \(A = \frac{A_0 V_0}{\sqrt{V_0^2 + 2g z}}\) and solving for \(z\) gives \(z = \frac{3V_0^2}{8g}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A rocket sled with initial mass of \(900 \mathrm{kg}\) is to be accelerated on a level track. The rocket motor burns fuel at constant rate \(m=13.5 \mathrm{kg} / \mathrm{s}\). The rocket exhaust flow is uniform and axial. Gases leave the nozzle at \(2750 \mathrm{m} / \mathrm{s}\) relative to the nozzle, and the pressure is atmospheric. Determine the minimum mass of rocket fuel needed to propel the sled to a speed of \(265 \mathrm{m} / \mathrm{s}\) before burnout occurs. As a first approximation, neglect resistance forces.

A porous round tube with \(D=60 \mathrm{mm}\) carries water. The inlet velocity is uniform with \(V_{1}=7.0 \mathrm{m} / \mathrm{s}\). Water flows radially and axisymmetrically outward through the porous walls with velocity distribution \\[ v=V_{0}\left[1-\left(\frac{x}{L}\right)^{2}\right] \\] where \(V_{0}=0.03 \mathrm{m} / \mathrm{s}\) and \(L=0.950 \mathrm{m} .\) Calculate the mass flow rate inside the tube at \(x=L\)

\( \mathrm{A}\) small round object is tested in a 0.75 -m diameter wind tunnel. The pressure is uniform across sections (D and (2). The upstream pressure is \(30 \mathrm{mm} \mathrm{H}_{2} \mathrm{O}\) (gage), the downstream pressure is \(15 \mathrm{mm} \mathrm{H}_{2} \mathrm{O}\) (gage), and the mean air speed is \(12.5 \mathrm{m} / \mathrm{s}\). The velocity profile at section (2) is linear; it varies from zero at the tunnel centerline to a maximum at the tunnel wall. Calculate (a) the mass flow rate in the wind tunnel, (b) the maximum velocity at section \((2),\) and \((\mathrm{c})\) the drag of the object and its supporting vane. Neglect viscous resistance at the tunnel wall.

A cylindrical tank, \(0.3 \mathrm{m}\) in diameter, drains through a hole in its bottom. At the instant when the water depth is \(0.6 \mathrm{m}\) the flow rate from the tank is observed to be \(4 \mathrm{kg} / \mathrm{s}\). Determine the rate of change of water level at this instant.

A university laboratory that generates \(15 \mathrm{m}^{3} / \mathrm{s}\) of air flow at design condition wishes to build a wind tunnel with variable speeds. It is proposed to build the tunnel with a sequence of three circular test sections: section 1 will have a diameter of \(1.5 \mathrm{m},\) section 2 a diameter of \(1 \mathrm{m},\) and section 3 a diameter such that the average speed is \(75 \mathrm{m} / \mathrm{s}\) (a) What will be the speeds in sections 1 and \(2 ?\) (b) What must the diameter of section 3 be to attain the desired speed at design condition?
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Medical Physics

Read Explanation

Thermodynamics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.