Air PocketAir Pocket8th problem8th problem

A vertical air jet from a straw produces A vertical air jet from a straw produces a cavity on a water surface. What a cavity on a water surface. What

parameters determine the volume and parameters determine the volume and the depth of the cavity?the depth of the cavity?

The ProblemThe Problem

The ApparatusThe Apparatus

We used a square-shaped We used a square-shaped aquariumaquarium and and a a compressorcompressor to reproduce the to reproduce the phenomenon.phenomenon.We measured the We measured the depthdepth and the and the width at width at the surfacethe surfaceWe changed the We changed the velocityvelocity of the air jet and of the air jet and the the heightheight of the straw of the strawTwo pipes with different cross-sections Two pipes with different cross-sections were usedwere used

Speed of Air JetSpeed of Air Jet

222211

21 2

1

2

1phgvphgv airairairair

2

2

1vhgp airwaterwater

Neglecting the difference between the heights of the two points:

Bernoulli’s equation is:

air

waterwater hgv

2

The CavityThe Cavity

We approximated the shape of the cavity We approximated the shape of the cavity with a with a paraboloidparaboloid

A paraboloid is a A paraboloid is a parabola rotatedparabola rotated about about the y axisthe y axis

We measured cavity We measured cavity widthwidth, , depthdepth, as the , as the function of function of pipe heightpipe height and and air speedair speed

Cavity Parameters as Functions Cavity Parameters as Functions of Pipe Heightof Pipe Height

Cavity parameters (v=39,09m/s)

0

0,5

1

1,5

2

2,5

3

3,5

0 2 4 6 8 10 12

Height of pipe (cm)

size

(cm

)

Width

Height

Cavity parameters (v=48,38m/s)

0

0,5

1

1,5

2

2,5

3

3,5

4

0 2 4 6 8 10 12

Height of pipe (cm)

size

(cm

) Width

Depth

Cavity parameters (v=56,15m/s)

0

0,5

1

1,5

2

2,5

3

3,5

4

0 2 4 6 8 10 12

Height of pipe (cm)

size

(sm

)

width

depth

Cavity parameters (v=60,34m/s)

0

0,51

1,52

2,53

3,54

4,5

0 2 4 6 8 10 12

Height of pipe (cm)

size

(cm

)width

height

Cavity Parameters as Functions Cavity Parameters as Functions of Air Speedof Air Speed

Cavity parameters (H=10cm)

00,5

11,5

22,5

33,5

44,5

5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)

width

height

Cavity parameters (h=6cm)

0

0,5

1

1,5

2

2,5

3

3,5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)

width

height

Cavity parameters (H=2,5cm)

0

0,5

1

1,5

2

2,5

3

3,5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)

width

height

Cavity parameters (H=1cm)

0

0,5

1

1,5

2

2,5

3

3,5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)width

height

Cavities with different pipe Cavities with different pipe cross-sectionscross-sections

D=1cm, H=2,5cm

0

0,5

1

1,5

2

2,5

3

3,5

0 10 20 30 40 50 60 70

v

size

(cm

)

width

height

D=1cm, H=1cm

0

0,5

1

1,5

2

2,5

3

3,5

0 10 20 30 40 50 60 70

v

size

(m

)

width

height

D=0,5cm, H=2,5c

0

0,5

1

1,5

2

2,5

3

3,5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)

width

height

D=0,5cm, H=1cm

0

0,5

1

1,5

2

2,5

3

3,5

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

size

(cm

)width

height

Volume of the CavityVolume of the Cavity

A paraboloid is a parabola rotated about A paraboloid is a parabola rotated about the y axis.the y axis.

Its volume equals its Its volume equals its inverse function’sinverse function’s volume rotated about the x axisvolume rotated about the x axis

We need to calculate the volume of a We need to calculate the volume of a solid solid of revolutionof revolution

2Axy A

xy

Ayx

2

Inverse function of the

parabola

Volume of the CavityVolume of the Cavity

2)( Axxf A

xxf )(1

Volume of the cavityVolume of the cavity

Parameter ‘Parameter ‘AA’ contains the rate of the ’ contains the rate of the width and height of the cavity, in the width and height of the cavity, in the following way:following way:

0

2

2

2

222

00)0(0

)(

hw

Aw

yw

x

Ayx

Axxy

204

w

hA

Volume of the cavityVolume of the cavity

The volume of a solid of revolution:The volume of a solid of revolution:

The volume of the rotated inverse The volume of the rotated inverse parabola:parabola:

This is the volume of the This is the volume of the original original paraboloid, tooparaboloid, too

dxxyV )(2

20

2

00 22h

Ax

Axdx

AA

xV

hhh

Measured VolumesMeasured Volumes

Cavity volume

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12

H (cm)

V (c

m^3

)

39,09 m/s

48,38 m/s

56,15 m/s

60,34 m/s

64,25 m/s

Cavity volume

0

2

4

6

8

10

12

14

0,00 m/s 10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

v (c

m^3

)

10

8,5

6

4

2,5

1

Work Needed for the Formation Work Needed for the Formation of the Cavity of the Cavity

We calculated this work in We calculated this work in twotwo ways ways

First methodFirst method:: Using that Using that ,,

where F is the where F is the buoyancybuoyancy and and

Like dipping the paraboloid from the surface Like dipping the paraboloid from the surface to hto h0 0 depth continually:depth continually:

dhhFW )(

2

2)()( h

AghgVhF waterwater

000

03

0

2

0 62)(

hwater

h

water

h

hA

gdhh

A

gdhhFW

Work Needed for the Formation Work Needed for the Formation of the Cavityof the Cavity

We getWe get

Substituting backSubstituting back : :

306h

A

gW water

202h

AV

03

1gVhW water

Another Method for Determining Another Method for Determining the Workthe Work

When the cavity is created, the When the cavity is created, the mass mass centrecentre of the water which filled the cavity of the water which filled the cavity will will move to the surfacemove to the surface of the water, of the water, which means that its which means that its potential energypotential energy will will rise:rise:

The needed The needed work equals this change of work equals this change of energy:energy:

hVghmgE water

EW

The problem is then, to determine The problem is then, to determine ΔΔhh

Assuming that Assuming that ρρ is constant, is constant, ΔΔh is the h is the difference of the hdifference of the h00 (height of the water (height of the water

surface) and the hsurface) and the hm m mass centre of the mass centre of the

paraboloid:paraboloid:

hhmm can be calculated from the geometry of can be calculated from the geometry of

the paraboloidthe paraboloid

Another Method for Determining Another Method for Determining the Workthe Work

mhhh 0

We calculated the centre of mass of the We calculated the centre of mass of the paraboloid by this formula:paraboloid by this formula:

We got:We got:

Another Method for Determining Another Method for Determining the Workthe Work

0

0

0

2

0

2

)(

)(

h

h

m

dxxy

dxxxy

h

03

2hhm

Substituting back to the equation with the Substituting back to the equation with the work:work:

Another Method for Determining Another Method for Determining the Workthe Work

00

0

3

2

)(

hhVgW

hhVghVgW

EW

water

mwaterwater

03

1gVhW water

Calcuating EfficiencyCalcuating Efficiency

Efficiency is the rate of input work and and Efficiency is the rate of input work and and useful workuseful work

Input work is the mechanical energy Input work is the mechanical energy change of air slowing down at the surfacechange of air slowing down at the surface

Useful work is the work done while Useful work is the work done while creating the cavitycreating the cavity

2

2

1vmW airin 03

1gVhW wateruse

Calculating EfficiencyCalculating Efficiency

Efficiency is the rate of the two:Efficiency is the rate of the two:

Air fill the cavity continously; its mass isAir fill the cavity continously; its mass is

22 3

2

21

31

mv

ghV

mv

ghVk

W

Wk

use

in

Vm air

EfficienciesEfficiencies

Substituting:Substituting:

Efficiency

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 2 4 6 8 10 12

Height of pipe (cm)

k

39,09 m/s

48,38 m/s

56,15 m/s

60,34 m/s

64,25 m/s

Efficiency

0

0,010,02

0,03

0,04

0,050,06

0,07

0,08

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

k

10

8,5

6

4

2,5

1

23

2

v

ghk

air

cavitywater

Volumes and efficiencies of Volumes and efficiencies of different cross-sectionsdifferent cross-sections

D=0,5cm, Volume

0

2

4

6

8

10

12

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

V(c

m^

3)

2,5cm

1cm

D=1cm, Voulme

0

2

4

6

8

10

12

0 10 20 30 40 50 60 70

v

V(m

^3) 2,5 cm

1 cm

D=0,5cm, Efficiency

0

0,010,02

0,03

0,04

0,050,06

0,07

0,08

0,00m/s

10,00m/s

20,00m/s

30,00m/s

40,00m/s

50,00m/s

60,00m/s

70,00m/s

v

k

2,5cm

1cm

D=1 cm, Efficiency

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 10 20 30 40 50 60 70

v

k2,5 cm

1 cm

ConclusionsConclusions

The measurementsThe measurements

Cavity width, depthCavity width, depth

As the function of:As the function of: Speed of the air jetSpeed of the air jet Height of the pipeHeight of the pipe We measured two types of pipem, with We measured two types of pipem, with

different cross-sectionsdifferent cross-sections