Post on 21-Nov-2015
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FLUID MECHANICS
COPYRIGHT BY ELETTRONICA VENETA SPA
FREE and FORCED
VORTEX
mod. HB14/EV
THEORY AND EXERCISES
HB14$$$101E0.DOC
CONTENTS
Page
1. SAFETY RULES 1
2. INTRODUCTION 2
3. THEORY 3 3.1 Free vortex 3 3.2 Spiral vortex 5 3.3 Forced vortex 7
4. TECHNICAL SPECIFICATIONS 8
5. INSTALLATION 9
6. OPERATING INSTRUCTIONS 10 6.1 Free vortex 10 6.2 Forced vortex 11
7. EXERCISES 12 7.1 Free vortex 12 7.2 Forced vortex 14
1
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
2
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 supported 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 co-
ordinates of the vortex profile to be measured.
3
3. THEORY
rph Radius of pitot head arm mm
v Linear velocity mm/s
xd Pitot reading above datum
i.e. peripheral water height
mm
Angular velocity 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, Bernoullis theorem holds, and
cost.zg2
v
W
P 2
in any horizontal plane
cost.g2
v
W
P 2
4
Differentiating with respect to r.
0dr
dv
g
v
dr
dP
W
1 (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.
drdadr
dP
rg
vdrdaW
2
dr
dP
rg
vW
2
(Eq. 3.1.2)
Combining Eq. 3.1.1 and Eq. 3.1.2
0r
v
dr
dv
0dr
dv
g
v
rg
v2
and
v
dv
r
dr
Integrating,
ln r + ln r = const
that is
v r = const = k
r
kv
5
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 Bernoullis theorem
cconst.zg2
v2
r
kv
czrg2
k2
2
czrg2
k2
2
2
2
rg2
kzc
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
6
const.zg2
v
W
P 2
If A is area of channel at some point where the velocity is v,
11 vAconst.vA
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:
r
vrv 11
and if z is constant:
crg2
vr
W
P2
2
1
2
1
2
2
1
2
1
rg2
vrc
W
P
and
cg2
v
W
P2
11
2
2
1
2
1
2
2
1
2
1
2
11
r
r1
g2
v
rg2
vr
g2
v
W
P-P
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.
7
3.3 Forced vortex
Since angular velocity is constant:
v = r
Increase in radial pressure is given by:
2
1
2
2
2
12
r
r
2
P
P
22
rrg
WPP
drrg
WdP
rg
W
r
v
g
W
dr
dP
2
1
2
1
r = 0 when P = P
2
2
10 rg2
W
P-P
or since hW
P
2
2
10 r
g2
hh
2
2
10 r
g2
hh
which is the equation of a parabola.
8
4. TECHNICAL SPECIFICATIONS
The main technical characteristics of this unit are indicated here
below:
Supporting 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
9
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
10
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 vessels 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
11
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 vessels 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
12
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)
Measured Depth,
x (mm)
1/r2
(mm)
40.0 14 0,000625
35.0 19 0,000816
30.0 22 0,0011
25.0 31 0,0016
20.0 43 0,0025
15.0 75 0,0044
Table 1: Free Vortex with orifice diameter = 24 mm
Plotting x vs. 1/r2, we obtain a straight line with slope:
g2
km
2
13
17432g2
km
2
k = (2 98100 17432) = 18493
Measure the total head at different radii with the Pitot tubes and
calculate the correspondent linear velocity:
hg2vPitot
h140h98102hg2v
Compare this velocity with the velocity calculated with the formula
v = k/r.
Radius of Pitot
mm
Pitot head
mm
v = k/r
mm/s
h140vPitot
mm/s
15 76 1232 1220
25 42 740 907
30 33 616 804
14
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
revolution
Time
(sec.)
rps r (mm)
110 90 70 50 30 0
50 35 1.43 190 203 218 231 236 238 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)
2
2
10 r
g2
hh
22
0calc. rg2
hhh
Example of calculation
232
2
calc. r101.4r98102
1.432h
For r = 0, hcalc. = 0 and xcalc. = 238 mm
For r = 30, hcalc. = 3.7 and xcalc. = 234.3 mm
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