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BrazilProblem 3 – Dancing Coin

Team of BrazilBRAZILIYPT 2018

Problem 3

Dancing CoinReporter: Victor Cortez

Take a strongly cooled bottle and put a coin on its neck. Over time you will hear a noise and see movements of the coin. Explain this phenomenon and investigate how the relevant parameters affect the dance.

1

BrazilProblem 3 - Dancing Coin

CONTENTS

BrazilProblem 3 – Dancing Coin

1. Theoretical Introduction

Introduction to the phenomenon

Relevant Principles

Theoretical Model and analysis

Relevant Parameters

2. Experiments

Experimental Set-up

Parameter Variation

Sound

3. Conclusion

Summary


Introduction Relevant Principles Theoretical Analysis Relevant Parameters

Place a coin in the mouth of a very cold bottle.

Heat sources.

Sounds and motion:

Figure 1: Coin dancing.

3



Relevant principles

Figure 2: Scheme of heat conduction in the system.

Conduction

4



Figure 3: Scheme of heat convection in the system.

External Convection

Internal Convection

Convection

5

Relevant principles



Perfect black body:

Real case model:

Figure 4: Scheme of heat radiation on the system.

Radiation

6

Relevant principles



Gas: atmosphere

Heat

Heat

Figure 5: Scheme of heat transfer on the system.

Convection:

Conduction:

Radiation:

7

Relevant principles



Internal Gas Ideal Gas

External Heating

1° Law of Thermodynamics

Heat Flux

8

Theoretical model and analysis



Thermal Cap. of gas.

Thermal Cap. of the bottle.

Room Temperature.

Bottle’s Initial Temperature

Bottle’s Temperature as a function of time.

9




Figure 6: Example plot.

10




Analytical temperature as a function of time.Analytical temperature as a function of time during smaller

time intervals.

Time (s)

Tem

per

atu

re (

K)

Time (s)

Tem

per

atu

re (

K)

11




Simulation temperature versus calculated temperature

Simulated Temperature

Calculated Temperature

Time (s)

Tem

per

atu

re (

K)

12

Simulation of pressure versus time

Inte

rnal

Pre

ssu

re (

Pa)

Time (s)

Theoretical model and analysis: Simulations



Intervals between consecutive jumps as simulated and as calculated.

SimulationAnalytical

Jump number

Inte

rval

(s)

13

Theoretical model and analysis: Simulations



Figure 7: Forces acting on the coin.

Pressure needed to raise the coin (theoretical):

Estimated ΔP for each coin.

5 Cents

1 Real

25 Cents50 Cents

14




Bottle Coefficient b

Shape

Material

The coin and the opening

Area

Mass

Heat Sources

Temperature

Medium

15

Relevant Parameters


Experimental Set-up Parameter Variation Sound

1. Various Bottles;

2. Modern BRL coins of 5 Cents, 25 Cents, 50 Cents and 1 Real;

3. Barometric pressure and temperature module BMP180[1] (Error: P ± 1 Pa and T ± 0,1 K)

4. Arduino UNO;

5. Computer and Python 3 for data-logging;

6. A good freezer.

Figure 9: Modern BRL coins used.

Figure 10: BMP 1801 sensor.

Figure 8: Bottle similar to what was used.

16

Experimental set-up: The materials



Sealed orifice

BMP180

Arduino

Computer

Figure 12: Scheme of experimental set-up..

Figure 13: Part of the experimental assembly.

Figure 11: Sensor inside the bottle.

17

Experimental set-up: The assembly



From left to right:

1 - Small Bottle (Glass)

2 - Medium Bottle (Glass)

3 - Plastic Bottle (Plastic)

4 - Reference Bottle (Glass)

5 - Big Bottle (Glass)

6 - Small Cylindrical Bottle (Glass)

7 - Big Cylindrical Bottle (Glass)

1

23 4

5

Figure 14: The bottles used in experiments.

18

6

7

Experimental set-up: The bottles



Bottle is in freezer

Removed quickly

Connected to computer

The coin is positioned

Data is collectedData is analyzed

19

Experimental set-up: The procedure



Medium

Material

Shape

Mass

Area

Temperature

Figure 15: Typical plot from an experiment with no manual contact. The cyclical nature of the internal pressure is noticeable.

ΔP ≈ 260 Pa

Time (s)

Inte

rnal

Pre

ssu

re (

Pa)

Internal Pressure x Time

20

Parameter variation: Medium



Temperature

Medium

Inte

rval

bet

wee

n t

he

jum

p t

he

pre

vio

us

on

e (s

)

Jump Number Jump Number

Figure 17: Plot of the interval between the n-th jump and the previous one. No manual contact.

Figure 18: Plot of the interval between the n-th jump and the previous one. With manual contact.

Intervals: no manual contact Intervals: manual contact

Inte

rval

bet

wee

n t

he

jum

p t

he

pre

vio

us

on

e (s

)

21

Material

Shape

Mass

Area

Parameter variation: Temperature

Theoretical Interval

Experimental Interval



Temperature

22

Medium

Material

Shape

Mass

Area

Temperature as a function of time.

Time (s)

Inte

rnal

Tem

per

atu

re (

K)

b = (2.83 ± 0.01) x 10-3

Theoretical Temperature

Experimental Temperature




Inte

rval

bet

wee

n t

he

jum

ps

(s)

Temperature difference (K)

Time interval between consecutive jumps

Temperature Difference

Interval

23

Temperature

Medium

Material

Shape

Mass

Area




Area

Bottle’s mouth diameter (mm)

ΔP in

jum

p (

Pa)

ΔP as a function of the bottle’s mouth diameter.

Theoretical ΔP

Experimental ΔP

24

Temperature

Medium

Material

Shape

Mass

Parameter variation: Area



Mass

ΔP as a function of the mass of the coin.

Mass of the coin (g)

ΔP (

Pa)

Theoretical ΔP

Experimental ΔP

25

Temperature

Medium

Material

Shape

Area

Parameter variation: Mass



Material

Shape

BottleExperimental b

(s-1 x 10-3)

1 4.50 ± 0.02

2 3.70 ± 0.02

3 10.3 ± 0.4

4 2.83 ± 0.01

5 2.63 ± 0.01

6 3.42 ± 0.01

7 2.77 ± 0.07

Table 1 - Coefficient b measured for each bottle.

1

23 4

5

26

6

7

Temperature

Medium

Mass

Area

Parameter variation: Shape and material



The coin falls

Sound in a tube

Pressure rises

The coin lifts

Figure 22: Graphical visualization of the jump sound.

27

Sound



Lift Fall

28

Sound


SimulatedAnalytical

Jump number

Inte

rval

(s)

Inte

rnal

Pre

ssu

re (

Pa)

Time (s) 29

Temperature

Medium

Material

Shape

Mass

Area

Summary: Theory


Time (s)In

tern

al P

ress

ure

(P

a)

Tem

per

atu

re (

K)

Time (s)

Comparison of temperature as a function of time

Theoretical TemperatureExperimental Temperature

1

2 3 4

5

Inte

rval

bet

wee

n ju

mp

s (s

)

Temperature difference (K)Bottle’s mouth diameter (mm)

ΔP

in ju

mp

(Pa)

TheoreticalExperimental

ΔP as a function of the bottle’s mouth diameter

Sound

Internal Pressure x Time

30

Summary: Experiments


[1] BOSCH, (5 de Abril de 2013), BMP180 Data sheet,

<https://cdn-shop.adafruit.com/datasheets/BST-BMP180-DS000-09.pdf>, Acessed in: 29/01/2018.

[2] KREITH, F. et al. Princípios de Transferência de Calor. 1 ed. São Paulo:Cengage Learning, 2003.

[3] H.Moysés Nussenzveig, Curso de Física Básica, vol 2, Editora Edgard Blücher, LTDA (1999)

[4] Bailey, R. and Elban, W. (2018). Thermal Performance of Aluminum and Glass Beer Bottles.

Bibliography

https://cdn-shop.adafruit.com/datasheets/BST-BMP180-DS000-09.pdf


Thank you!


Appendix 1: Measuring the ΔP Internal Pressure versus Time

Time (s)

Inte

rnal

Pre

ssu

re (

Pa)


Appendix 2: Geometry of the bottle

Area (m²) 0,0306

Mouth diameter (mm) 19,30

=Height (mm) 225

Volume (L) 0,330

Average thickness (mm) 3,54

Table - Bottle dimensions.

Thickness ≈ Constant


Appendix 3: Van der Waals vs Clapeyron


Appendix 4: Sound frequency analysis

Sound spectrum given through fourier transform using Audacity


Appendix 5: Experimental errors

Internal Temperature vs Time Internal Pressure vs Time

Inte

rnal

Tem

per

atu

re (

°C)

Time (s)In

tern

al P

ress

ure

(P

a)Time (s)

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