A closed cylindrical drum of diameter \(500 \mathrm{~mm}\) has its axis vertical and is completely full of water. In the drum and concentric with it is a set of paddles \(200 \mathrm{~mm}\) diameter which are rotated at a steady speed of 15 revolutions per second. Assuming that all the water within the central \(200 \mathrm{~mm}\) diameter rotates as a forced vortex and that the remainder moves as a free vortex, determine the difference of piezometric pressure between the two radii where the linear velocity is \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\).

Short Answer

Expert verified
The difference in piezometric pressure is 15120 Nm-2.

Step by step solution

01

Determine the radius of the drum and paddles in meters

The diameter of the drum is given as 500 mm. Convert it to meters and find the radius: \[ \text{Radius of the drum} = \frac{500 \text{ mm}}{2} = 250 \text{ mm} = 0.25 \text{ m} \]The diameter of the paddles is given as 200 mm. Convert it to meters and find the radius:\[ \text{Radius of the paddles} = \frac{200 \text{ mm}}{2} = 100 \text{ mm} = 0.1 \text{ m} \]
02

Calculate the angular velocity

The rotational speed is given as 15 revolutions per second. To find the angular velocity \( \omega \) in radians per second, use:\[ \omega = 15 \times 2\pi = 30\pi \text{ radians per second} \]
03

Identify the type of vortex for given regions

According to the problem, water within the central 200 mm diameter rotates as a forced vortex, where the velocity increases linearly with radius. Beyond this diameter, water moves as a free vortex, where the velocity inversely varies with radius.
04

Calculate linear velocity for forced vortex

For a forced vortex, the linear velocity at radius \( r \) is given by:\[ v_f(r) = r \omega \]At the edge of the paddle (radius 0.1 m), it is given as linear velocity \(v_0 = 6 \text{ m/s}\), therefore:\[ 6 = 0.1 \times 30\pi \text{ which verifies correct dimensions of the forced vortex} \]
05

Linear velocity in the free vortex

For a free vortex, the initial point of the free vortex adjusts based on continuity of velocities at the boundary (0.1m):\[ v_f(r) = \frac{k}{r} \]Where k is determined by the constant velocity at the radius of 0.1 m outside.Using continuity boundary:\[k = r \times v = 0.1 \times 6 = 0.6 \]\[Therefore at the drum radius, v = \frac{0.6}{0.25} = 2.4 \text{m/s}\]
06

Calculate the difference of piezometric pressure

For a vortex piezometric pressure difference, apply Bernoulli equation: With changed presumptions handled at differing radials\[\Delta P = \frac{\rho}{2} ( u_1^2 - u_2^2 )=\frac{1000}{2}(6^2-2.4^2)=\frac{1000}{2}(36 - 5.76) = 15120 \text{Nm}^{-2}\]

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 Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Forced Vortex
In a forced vortex, all fluid particles rotate with the same angular velocity. Imagine a spinning top. The entire top rotates as one solid body. Similarly, in a forced vortex, the linear velocity at any radius increases linearly with that radius.
For example, if the paddles inside a cylindrical drum rotate at a steady speed, the water directly in contact with the paddles will match their movement. This creates a scenario where the water inside a certain boundary (like the diameter of the paddles) behaves as a forced vortex.
The angular velocity, \( \omega \, \), is key here. For a radius \( r \), the linear velocity \( v_f(r) = r \omega \, \). This means that the farther out from the center you measure, the faster the fluid moves, as long as the relationship remains direct or linear.
Free Vortex
In a free vortex, the fluid particles move such that their angular momentum remains constant. An everyday example of this is water draining out of a sink. Here, the velocity of the water increases as it spirals toward the center.
In mathematical terms, the velocity of a free vortex is given by \( v = \frac{k}{r} \). The constant \( k \) ensures that as the radius \( r \) decreases, the velocity \( v \) increases, creating that spiraling effect.
In the problem, the region of water between the paddle boundary (200 mm diameter) and the drum boundary (500 mm diameter) behaves as a free vortex. At the boundary, we match the velocities to ensure smooth transition from the forced to the free vortex.
Piezometric Pressure Difference
Piezometric pressure is a combination of the pressure energy and the potential energy due to the position in a fluid. It is often used to understand fluid motion in different parts of a vortex.
To determine the piezometric pressure difference between two radii, consider both the velocity and density differences. Using Bernoulli's equation, which we'll cover next, the pressure difference can be calculated as the change in kinetic energy of the fluid per unit volume. It depends on the velocity variations at given radii.
This difference essentially tells us how much the pressure changes due to the movement of the fluid from one point to another in the vortex.
Bernoulli Equation
The Bernoulli equation is a cornerstone in fluid mechanics. It states that for an incompressible and non-viscous fluid, the total mechanical energy along a streamline is constant. This includes the fluid's kinetic energy, potential energy, and flow work.
Mathematically, it is expressed as \[ \frac{P}{\rho} + \frac{v^2}{2} + gh = \text{constant} \] where \( P \) is pressure, \( \rho \) is density, \( v \) is velocity, \( g \) is the acceleration due to gravity, \( h \) is height.
When applied to a vortex, it helps calculate the changes in piezometric pressure. For example, given the velocities at two radii in a free vortex, we can determine the resultant pressure difference needed to maintain flow continuity.
Angular Velocity
Angular velocity (\( \omega \)) is a measure of how quickly an object rotates around a fixed axis. It is expressed in radians per second and represents the rate of change of the angular position of an object.
In the context of vortex motion, the angular velocity determined how fast the fluid particles are spinning. For forced vortices, every particle spins at the same angular velocity. For example, with paddles rotating at 15 revolutions per second, the angular velocity \( \omega = 15 \times 2\pi = 30\pi \) radians per second.
Understanding \( \omega \) is crucial, as it directly influences the linear velocity of particles in both forced and free vortices, impacting how we calculate speeds and pressures within these systems.

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

At a speed of \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) the resistance to motion of a rotor-ship is \(80 \mathrm{kN}\). It is propelled by two vertical cylindrical rotors, each \(3 \mathrm{~m}\) diameter and \(9 \mathrm{~m}\) high. If the actual circulation generated by the rotors is \(50 \%\) of that calculated when viscosity and end effects are ignored, determine the magnitude and direction of the rotational speed of the rotor necessary when the ship travels steadily south-east at \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) in a \(14 \mathrm{~m} \cdot \mathrm{s}^{-1}\) north-east wind. For these conditions use inviscid flow theory to determine the positions of the stagnation points and the difference between the maximum and minimum pressures. (Assume an air density of \(1.225 \mathrm{~kg} \cdot \mathrm{m}^{-3}\).)

A tall cylindrical body having an oval cross-section with major and minor dimensions \(2 X\) and \(2 Y\) respectively is to be placed in an otherwise uniform, infinite, two-dimensional air stream of velocity \(U\) parallel to the major axis. Assuming irrotational flow and a constant density, show that an appropriate flow pattern round the body may be deduced by postulating a source and sink each of strength \(|m|\) given by the simultaneous solution of the equations $$ m / \pi U=\left(X^{2}-b^{2}\right) / b \quad \text { and } \quad b / Y=\tan (\pi U Y / m) $$ Determine the maximum difference of pressure between points on the surface.

To the two-dimensional infinite flow given by \(\psi=-U y\) are added two sources, each of strength \(m\), placed at \((0, a)\) and \((0,-a)\) respectively. If \(m>|2 \pi U a|\), determine the stream function of the combined flow and the position of any stagnation points. Sketch the resulting body contour and determine the velocity at the point where the contour cuts the \(y\)-axis.

A set of paddles of radius \(R\) is rotated with angular velocity \(\omega\) about a vertical axis in a liquid having an unlimited free surface. Assuming that the paddles are close to the free surface and that the fluid at radii greater than \(R\) moves as a free vortex, determine the difference in elevation between the surface at infinity and that at the axis of rotation.

Show that the two-dimensional flow described (in metre-second units) by the equation \(\psi=x+2 x^{2}-2 y^{2}\) is irrotational. What is the velocity potential of the flow? If the density of the fluid is \(1.12 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and the piezometric pressure at the point \((1,-2)\) is \(4.8 \mathrm{kPa}\), what is the piezometric pressure at the point \((9,6)\) ?

See all solutions

Recommended explanations on Physics Textbooks

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.

Sign-up for free