Free And Forced Vortices Report Examples

Abstract

The experiment aims to study the properties of free and forced vortex in water medium. The experimental data should be compared to the theoretical, errors assessed and the conclusion about applicability of the method made. The velocity profile of free vortex is investigated, namely the relation between vortex and angular velocity.

Introduction 3

Equipment 3
Theory 4
Forced Vortex 6
Theory and Calculations 6
Free Vortex 7
Theory and Calculations 7
Experimental procedure 8
Results 8
Forced Vortex 8
Error calculation 9
Free Vortex 10
Conclusions 11
Introduction
The experiment studies free and forced vortex with the special device. The measurements performed are based on vortex radius of free and forced vortex on different heights. Basing on the experiments, the calculations are performed for height and different radiuses, as well as the rotation. The experimental values are compared to theoretically calculated.
The velocity profile is determined for free and forced vortex. The general understanding of angular rotation and calculation techniques obtained. The analysis of the theoretical formula was performed. The experiment has proved that the theoretical heights of vortexes are compared to the actual experimental values.

Equipment

The vortex properties are studied on F1-23 Free and Forced Vortices apparatus (Figure 1). This is a transparent cylinder 1 made from acrylic. The water is supplied to the cylinder by four orifices 4. The water level is controlled by overflow ducts 2.
The 12 mm inlet tubes (there are two) are positioned at 15 degrees, and water exits the orifices in the base of the tank. The other 2 inlet tubes angled at 60 degrees are meant to create the forced vortex.
The two 12mm diameter inlet tubes, which are angled at 15 degrees, are used as entry tubes for the free vortex experiment. In this case, the water exits the vessel via one of the interchangeable orifices in the base of the tank.
The 9mm inlet tubes, which are angled at 60 degrees, are used to create a forced vortex. As water enters the cylinder, the paddle rotates the stud mounted on a plug that is inserted in the central orifice, and the paddle acts as a stirrer.
Figure 1. Free and forced vortices apparatus: 1 – acrylic cylinder; 2 – measuring needles; 3 – liner; 4 – orifices for water supply; 5 - hydraulics bench; 6 – overflow duct.
The forced vortex profile is measured with needles 2 that are set at fixed distances from the center of the cylinder. In case of free vortex study, the needles are replaced with a gauge that measures the vortex depth.

When the pilot tubes are inserted into the measuring bridge, the velocity heads are visualized.

Theory
The velocity profile of free and forced vortex differ by angular velocities about the central axis. The forced vortex is characterized by the same angular velocity, while the velocity profile of free vortex intensely varies depending on the distance from the rotation axis. For minimizing errors, the certain experiment conditions were set. The experimental conditions used for calculations are presented in Table 1. The experimental measurements of the vortex distance of radius for every height and rotation are presented.

The conditions of the experiment

The conventional signs used for the experiment
Forced Vortex
Theory and Calculations
Forced vortex is characterized with the constant angular velocity about the axis of rotation. It is assumed that the particles have similar speed about the center of axis rotation since some areas of the apparatus create pressure and other areas drain pressure.

The calculations for the method are as follow:

q = ωr (1)

The equation for the centripetal force in a vortex is:

(2)

Substituting the q in (2) for formula from (1), and integration provides:

(3)

For the free and forced vortices apparatus (Figure 1), the pressure at free surface is atmospheric and the equation transforms:

(4)
For the boundary conditions z = z0, when r = 0 we obtain:
(5)

Experimental procedure

A divider hose is connected to two flow pipes and 9 mm inlet hole. The hose is equipped with two quick releasing units connected to the fittings. The outlet hose is positioned to the lowest part of the volumetric tank. The experimental steps a follow.
1. Press the bung with central shaft into the orifice in the cylinder base. Locate the paddle on top of the shaft, and the bridge piece on the top of the cylinder. Make sure the measuring needles are inserted.
2. Switch the hydraulics and adjust the control valve until the reasonable flow into the cylinder is achieved, and the outlet is lifted until it gets filled with water.
3. As the water reaches a certain level, the hose is placed back into the volumetric tank. The control valve was slowly turned off when the desired level of water was reached.
4. The position of the measuring needles is adjusted until they contact the surface of the vortex. Lengths of the needles were measured and recorded with respect to the lowest point of the vortex.

Free Vortex

Theory and Calculations
Free vortex velocity profile differs from the forced vortex. It takes place when fluid exits a cylinder through the central hole in the base. The velocity is measured towards the liquid surface. The velocity profile significantly varies depending on the distance from the axis of rotation.

The calculations of the free vortex are based on several derivations from vortex properties.

(6)

The Bernoulli’s equation can be applied, if we assume steady frictionless flow:

(7)
Since the peizometric pressure is constant (taken as atmospheric), the streamline is on the surface of the vortex. Thus, we obtain:
(8)
The equation (8) is the asymptotic equation to the axis of rotation and to the horizontal through z = c.to a hyperbolic curve.

For the pilot tubes, the velocity is given by:

(9)

Experimental procedure

The sequence of the experimental procedures is described in the section.
1. The inlet hose is connected to the quick release fitting within the unit to direct the outflow to the fitting adjacent control valve. The control valves are securely closed.
2. One of the circular orifices is pushed into the hole in cylinder base. The bridge piece is located on top of the cylinder with adjustable gauges fitted. The control valve and hydraulic bench switched on until a certain level of water is achieved.
3. The gauge is used to measure the profile of the orifice, the arms are used to measure the radiuses at various heights with respect to the fluid surface.

Results

Forced Vortex

The results of forced vortex experiment are presented in Table 3.

The forced vortex experiment
The sample calculations for r = 0.05 m
ω= n t= 59 30=1.97 rps
Theoretical depth of the needle: = 1.972∙0.052 2∙9.81+0.052= 0.0525 m .

The calculation results are presented in Table 4.

The forced vortex calculations
Error calculation
The experimental error is calculated:
σz =[ωr2gdω]2 + [rω2gdr]2 +[dzm]2
Taking the angular uncertainty is 0.01 rps, the radius uncertainty 0.001 m, uncertainty in zm 0.005, for angular speed 1.97 rps the error is calculated:
σz =[1.97∙0.0329.81(0.01)]2 + [1.97∙0.0329.81(0.001)]2 +1.97∙0.0329.81[0.005]2 =0.0029 m.

Free Vortex

The sample calculation for 0.01 m diameter hole for a free vortex (Zm = 0.052 m, r = 0.01 m):
q= 2gh= 2∙9.81∙0.052=1.01 m/s

The free vortex results

Conclusions
The vortex properties have been studied. The velocity profiles of free and forced vortex have been calculated. The experimental values were compared to the theoretically calculated. The measurement error is small, and therefore the method is reliable for determination of the velocity profiles.