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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
BrazilProblem 3 - Dancing Coin
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
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Relevant principles
Figure 2: Scheme of heat conduction in the system.
Conduction
4
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Figure 3: Scheme of heat convection in the system.
External Convection
Internal Convection
Convection
5
Relevant principles
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Perfect black body:
Real case model:
Figure 4: Scheme of heat radiation on the system.
Radiation
6
Relevant principles
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Gas: atmosphere
Heat
Heat
Figure 5: Scheme of heat transfer on the system.
Convection:
Conduction:
Radiation:
7
Relevant principles
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Internal Gas Ideal Gas
External Heating
1° Law of Thermodynamics
Heat Flux
8
Theoretical model and analysis
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Thermal Cap. of gas.
Thermal Cap. of the bottle.
Room Temperature.
Bottle’s Initial Temperature
Bottle’s Temperature as a function of time.
9
Theoretical model and analysis
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Figure 6: Example plot.
10
Theoretical model and analysis
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
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
Theoretical model and analysis
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
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
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Intervals between consecutive jumps as simulated and as calculated.
SimulationAnalytical
Jump number
Inte
rval
(s)
13
Theoretical model and analysis: Simulations
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
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
Theoretical model and analysis
BrazilProblem 3 - Dancing Coin
Introduction Relevant Principles Theoretical Analysis Relevant Parameters
Bottle Coefficient b
Shape
Material
The coin and the opening
Area
Mass
Heat Sources
Temperature
Medium
15
Relevant Parameters
BrazilProblem 3 - Dancing Coin
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
Bottle is in freezer
Removed quickly
Connected to computer
The coin is positioned
Data is collectedData is analyzed
19
Experimental set-up: The procedure
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
Parameter variation: Temperature
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
Parameter variation: Temperature
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
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
BrazilProblem 3 - Dancing Coin
Experimental Set-up Parameter Variation Sound
The coin falls
Sound in a tube
Pressure rises
The coin lifts
Figure 22: Graphical visualization of the jump sound.
27
Sound
BrazilProblem 3 - Dancing Coin
SimulatedAnalytical
Jump number
Inte
rval
(s)
Inte
rnal
Pre
ssu
re (
Pa)
Time (s) 29
Temperature
Medium
Material
Shape
Mass
Area
Summary: Theory
BrazilProblem 3 - Dancing Coin
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
BrazilProblem 3 - Dancing Coin
[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
BrazilProblem 3 - Dancing Coin
Appendix 1: Measuring the ΔP Internal Pressure versus Time
Time (s)
Inte
rnal
Pre
ssu
re (
Pa)
BrazilProblem 3 - Dancing Coin
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
BrazilProblem 3 - Dancing Coin
Appendix 4: Sound frequency analysis
Sound spectrum given through fourier transform using Audacity