Free And Force Vortex 1w2ri

FLUID MECHANICS

FREE and FORCED VORTEX mod. HB14/EV

THEORY AND EXERCISES

© COPYRIGHT BY ELETTRONICA VENETA SPA

HB14$$$101E0.DOC

CONTENTS

Page 1. SAFETY RULES

“ 1

2. INTRODUCTION

“ 2

3. THEORY 3.1 Free vortex 3.2 Spiral vortex 3.3 Forced vortex

3 “ 3 “ 5 “ 7

4. TECHNICAL SPECIFICATIONS

“ 8

5. INSTALLATION

“ 9

6. OPERATING INSTRUCTIONS

“ 10

6.1 Free vortex 6.2 Forced vortex

7. EXERCISES 7.1 Free vortex 7.2 Forced vortex

“ 10 “ 11

“ 12 “ 12 “ 14

1. SAFETY RULES

Before installing and setting this equipment at work, read and understand the contents of this handbook. Any maintenance operation must be carried out by a skilled technician. Keep attention to the surface measuring probes and vortex diameter gauge: they are very sharp

1

2. INTRODUCTION

The apparatus allows the study of the shape of "free and forced vortices" and consists of a transparent cylindrical vessel (D1) having two pairs of diametrically opposed inlet tubes of 3.0 mm and 12.5mm diameter. The 12.5 diameter inlet tubes form an angle of 15° with the diameter, so that a swirling motion is imparted to the liquid entering the vessel; they are used as entry tubes for the free vortex experiment. An outlet with interchangeable orifices (8, 12, 16, 24) is centrally positioned in the base of the vessel. The profile of the free vortex formed at the top of the vessel is determined by a gauge mounted diametrically which measures the diameter of the vortex at various depths. This gives the co-ordinate points required to plot the vortex profile. The velocity at 3 different radii may be measured using the pitot tubes supplied. Measure the total head at different radii with the Pitot tubes and calculate the correspondent linear velocity: The forced vortex is created in the vessel by using as the inlet the 3.0 mm bore tubes which are angled at 60° to the diameter. The inlet water hits against a four blades paddle which acts as a stirrer. The water leaves the vessel via the 12.5 mm diameter angled tubes which are used as inlet tubes for the free vortex experiment. The four blades paddle rotates on a vertical shaft ed by a plug in the hole used as the outlet for the free vortex experiment. A bridge piece with needles probes allows to determine the coordinates of the vortex profile to be measured.

2

3. THEORY

rph v xd

Radius of pitot head arm Linear velocity Pitot reading above datum i.e. peripheral water height Angular velocity

mm mm/s mm radians/s

3.1 Free vortex

When water flows out of a vessel through a central hole in the base, a free vortex is formed; the sense of rotation being dependent on initial disturbance. The water moves spirally towards the centre with stream line motion, so that, neglecting losses caused by viscosity, the energy per unit mass remains constant. If, while the mass of water is rotating, the central exit hole is plugged, the flow of water in the vertical plane ceases and the motion becomes one of simple rotation in the horizontal plane, and is known as a Free Cylindrical Vortex. Since stream line motion applies, Bernoulli’s theorem holds, and

P W

v2 2 g

z

cost.

in any horizontal plane

P W

v2 2 g

cost.

3

Differentiating with respect to r. 1 dP W dr

v dv g dr

0

(Eq. 3.1.1)

Consider a pair of stream lines a distance dr apart, lying on the same horizontal plane, and connected by a column of fluid or area da. The centrifugal force on the column is balanced by the difference in pressure between the two ends, i.e.

v2 g r

W da dr

W

v2 g r

dP da dr dr

dP dr

(Eq. 3.1.2)

Combining Eq. 3.1.1 and Eq. 3.1.2

v 2 v dv g r g dr dv v 0 dr r

0

and dr r

dv v

Integrating, ln r + ln r = const

that is v r = const = k

v

k r 4

Then, in a free cylindrical vortex the velocity varies inversely as the distance from the axis of rotation. To determine the equation governing the surface profile, the equation for the curve of equal pressure (atmosphere) is derived from Bernoulli’s theorem

v2 2 g v

z

const. c

k r

k2 2 g r2

z

c

k2 2 g r2

z

c

c z

k2 2 g r2

which is equation to a hyperbolic curve of nature yx2 = A which is asymptotic to the axis of rotation and to the horizontal through z = C.

3.2 Free spiral vortex The motion in a free spiral vortex differs from that in a free cylindrical vortex in that in the former there is a radial flow towards the centre. The equation governing radial flow towards the centre is derived as follows. Consider the flow of water across a segment of circle towards its diameter; then the energy through any stream tube is constant so that

5

v2 2g

P W

z

const.

If A is area of channel at some point where the velocity is v,

A v

const. A 1 v1

where A1 and v1 are the area and velocity at some point distant r, from the centre of the circular plane. Putting A = k r Then:

v

r1 v1 r

and if z is constant:

2

2

P W

r1 v1 2g r2

P W

c

c

2

2

r1 v1 2g r2

and

P1 W P - P1 W

2

v1 2g

c 2

v1 2g

2

2

r1 v1 2g r2

2

2

v1 r 1 12 2g r

A free spiral vortex may be considered as a case of cylindrical vortex and radial motion combined. In each case the velocity is inversely proportional to the radius. The angle between the stream lines and the corresponding radius vector at any point will be constant, the stream lines forming a series of spirals.

6

3.3 Forced vortex Since angular velocity is constant: v=

r

Increase in radial pressure is given by:

W v2 g r

dP dr P2

W 2 ω r g r

W 22 ω rdr g r1

dP P1

P2

P1

W 2 2 ω r2 g

r1

2

r = 0 when P = P 2

P - P0 W

ω1 2 r 2g

or since

P W

h h0

ω1 2 r 2g

h

ω1 2 r 2g

h 2

2

h0

which is the equation of a parabola.

7

4. TECHNICAL SPECIFICATIONS

The main technical characteristics of this unit are indicated here below:

ing framework of AISI 304 stainless steel Cylindrical vessel, code D1, made of transparent methacrylate with 2 12.5 mm inlet tube at 15° to the diameter and with 2 3 mm inlet tube at 60° to the diameter Set of orifice d = 8, 12, 16, 24 mm Stirrer made of AISI 304 stainless steel

8

5. INSTALLATION

Place the apparatus on the hydraulics bench working surface mod. HB/EV so that the central outlet in the base is located over the channel Level the apparatus using the adjustable feet Using the pipe with quick connection provided, connect the inlet of the equipment mod. HB14/EV with the bench mod. HB/EV Position the outlet pipe of the equipment mod. HB14/EV over the tank of the hydraulic bench

9

6. OPERATING INSTRUCTIONS

6.1 Free vortex

Select the orifice and place this into the central outlet located in the base of the apparatus Screw the quick connection onto the discharge available on the bottom of the flow channel of the bench mod. HB/EV Close the apparatus outlet valve V3 Close valve V1 and open partially valve V2 of mod. HB14/EV so that water flows into the cylindrical vessel via the two inlet ports set at 15° to the diameter Close the pump outlet valve V1 of hydraulic bench and start the bench pump G1 Slowly open valve V1 of hydraulic bench and adjust the valve V2 of mod. HB14/EV until water just begins to flow out of the cylindrical vessel’s overflow cut outs; maintain the water at this level by regulating the water flow by means of the valve V2 of mod. HB14/EV When stable conditions are attained, the profile of the vortex is obtained by measuring the vortex diameter at a number of planes, the distance of the planes from the fixed datum being also measured Pitot tubes can be used to obtain measurements of the velocity of the fluid at a number of difference radii: 15, 25, 30 mm. Replace the profile measuring gauge with the 15mm radius arm pitot tube. Immerse the tube until the "nose" is approximately 5mm from the vortex core profile surface. Note these scale readings. Repeat the test using the 25mm and 30mm pitot tube. Switch off pump G1

10

6.2 Forced vortex

Position the blanking plug with shaft in the central hole located in the base of the vessel Open the apparatus outlet valve V3 Close valve V2 and open partially valve V1 of the apparatus so that water flows into the cylindrical vessel via the two inlet ports set at 60° to the diameter Close the pump outlet valve V1 and start the bench pump G1 Slowly open valve V1 of hydraulic bench and adjust the valve V1 and V3 of mod. HB14/EV until water just begins to flow out of the cylindrical vessel’s overflow cut outs; maintain the water at this level by regulating the water flow by means of the valve V3 of mod. HB14/EV. Water will now flow through the ports at 60° and impinge on the paddle wheel before flowing out of the apparatus via the two ports set at 15° and not in use Ensure that the flexible outlet pipe is completely filled with water for maintaining a syphonic action and hence increasing the discharge capacity through the outlet valve The speed of rotation of the paddle wheel is determined by the rate of flow of water into the apparatus that is proportional to the degree of opening of valve V1 of mod. HB14/EV. For each value of flow rate the outlet valve of the apparatus should be adjusted until water just flows out of the overflow cut outs The profile of the water surface is determined by the surface measuring probes (needles) which are adjusted until each probe just breaks the water surface. After speed of rotation of the paddle wheel has been measured by timing a number of red paddle rotations, the measuring probe bridge piece is removed from the apparatus and the length of each probe is measured using the metallic ruler supplied

11

7. EXERCISES

7.1 Free vortex The co-ordinate points for the vortex profile should be plotted using the depth gauge Measure the hydraulic head using the 3 Pitot tubes Repeat the experiment changing the diameter of the orifice

Radius, r (mm) 40.0 35.0 30.0 25.0 20.0 15.0

Measured Depth, x (mm) 14 19 22 31 43 75

1/r2 (mm) 0,000625 0,000816 0,0011 0,0016 0,0025 0,0044

Table 1: Free Vortex with orifice diameter = 24 mm Plotting x vs. 1/r2, we obtain a straight line with slope:

m

k2 2 g

12

m

k2 2 g

17432

k = (2 98100 17432) = 18493 Measure the total head at different radii with the Pitot tubes and calculate the correspondent linear velocity: v Pitot v

2 g h 2 g h

2 9810 h

140 h

Compare this velocity with the velocity calculated with the formula v = k/r.

Radius of Pitot

Pitot head

v = k/r

mm

mm

mm/s

15

76

1232

1220

25

42

740

907

30

33

616

804

13

v Pitot

140 h

mm/s

7.2 Forced Vortex Plot the co-ordinate points experimentally obtained for the vortices at various speeds of rotation using the depth gauges (needles) Calculate the angular velocity using stopwatch and rotating paddle Plot and compare the theoretically obtained curves of vortex surface profile with the experimentally determined forced vortices

Number

Time

rps

revolution (sec.) 50

35

1.43

110

90

r (mm) 70 50

190

203

218

231

236

238

xmeas.

49.7

33.3

20.1

10.3

3.7

0

hcalc.

188.3

204.7

217.9

227.3

234.3

238

xcalc. = (238-hcalc)

30

0

2

h

h0

h calc.

ω1 2 r 2g h h0

ω2 2 r 2g

Example of calculation 2

h calc.

2 1.43 2 r 2 9810

4.1 10

3

r2

For r = 0, hcalc. = 0 and xcalc. = 238 mm For r = 30, hcalc. = 3.7 and xcalc. = 234.3 mm

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ELETTRONICA VENETA spa - 31045 Motta di Livenza (Treviso) ITALY Via Postumia. 16 – Tel. +39 0422 7657 r.a. – Fax +39 0422 861901 www.elettronicaveneta.com

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