Problem No. 5: “CAR”
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Transcript of Problem No. 5: “CAR”
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PROBLEM NO. 5: “CAR”Build a model car powered by an engine using an elastic air-filled toy balloon as the energy source. Determine how the distance travelled by the car depends on relevant parameters and maximize the efficiency of the car.
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OVERVIEW balloon observations
piston inflation/deflation model energy input and output rubber stretching losses
construction observed motion parameters
initial conditions (volume, pressure) jets, nozzles maximised travelled distance and efficiency
conclusion
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PISTON MODEL ~ ENERGY piston contains all air later used for
inflation/deflation slowly pushed – balloon inflates released piston – balloon deflates
at every moment ideal gas law equation states:
V20 = Vpiston
V2f = 0V1fV10
p0 = patmpf
𝑝0(𝑉 ¿¿10+𝑉 20)=𝑛𝑅𝑇=𝑐𝑜𝑛𝑠𝑡 ¿
𝑝 (𝑉 ¿¿1+𝑉 2)=𝑛𝑅𝑇 ¿
𝑝 𝑓𝑉 1 𝑓=𝑛𝑅𝑇
𝑑𝑊=− (𝑝−𝑝0 )𝑑𝑉 2
initial state final state
integrating initial to final state
𝑅−𝑔𝑎𝑠𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡=8.314 𝐽𝑚𝑜𝑙𝐾
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PISTON MODEL ~ ENERGY
maximal available energy deflation work
same equations valid
V1f V10
p0 = patmpf
𝑝0(𝑉 ¿¿10+𝑉 20)=𝑛𝑅𝑇=𝑐𝑜𝑛𝑠𝑡 ¿𝑝 𝑓𝑉 1 𝑓=𝑛𝑅𝑇
varies for inflation/deflation – determined experimentally
𝑊=∫𝑝0
𝑝 𝑓
𝑝𝑑𝑉 1+𝑝𝑓 𝑉 𝑓 ¿known conditions
V2f = 0 V2f =
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PRESSURE / VOLUME RELATION INFLATION AND DEFLATION pressure
digital pressure sensor volume
photos taken at known pressureballoon edge coordinates
balloon shape determinedballoon radius/height function
numerically integratedshape rotation – body volume
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height
radius
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TYPICAL PRESSURE/VOLUME RELATION
deflation curve is under the inflation curve
energy partly lost due to rubber deformations
volume [cm3]
0 1000 2000 3000 4000
over
pres
sure
[kP
a]
0
2
4
6
8
10
inflationdeflation
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BALOON STRETCHING – THEORETICAL CURVE pressure (force acting on a surfaca) balloon elastic energy
*
work needed to increase radius from to r+dr under pressure difference
surfaceforce
*Y. Levin, F. L. da Silveira, Two rubber balloons: Phase diagram of air transfer, PHYS. REV. E (2004)
relative extension r/r0
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BALLOON STRETCHING
obtained curve has the expected maximum due to surface/force
relation
explains and confirms the obtained shape of the overpressure/strain curve
∆ 𝑃=𝛼 ( 1𝜆− 1𝜆7 )−𝑠𝑖𝑛𝑔𝑙𝑒𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 (𝛼𝑟𝑢𝑏𝑏𝑒𝑟 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 ) 𝑓𝑖𝑡
relative extension r/r0
0,5 1,0 1,5 2,0 2,5 3,0 3,5
over
pres
sure
[kP
a]
0
1
2
3
4
5
6
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INITIAL VOLUME surface under the curve enlarges with initial volume
determining part of total input and output energy efficiency is determined by the deflation and inflation
surfaces under the curve ratio between (initial + final conditions) + surface under the curve
volume [cm3]
0 1000 2000 3000 4000 5000
over
pres
sure
[kP
a]
0
1
2
3
4
5
6
inflationV1V2V3
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initial volume [m3]
0,000 0,001 0,002 0,003 0,004 0,005
defla
tion/
infla
tion
effic
ienc
y
0,6
0,7
0,8
0,9
1,0
INITIAL VOLUME
𝜂=𝐸𝑑𝐸𝑖
initial volume [m3]
0,000 0,001 0,002 0,003 0,004 0,005
inpu
t ene
rgy
[J]
0
2
4
6
8
10
12
14
input energy inflation
inflation/deflation varies rubber stretching
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CONSTRUCTION
cart /styrofoam
plastic tubeballoon
• plastic cart (lenght ~ 12cm)• styrofoam• plastic tube (d = 2,9 cm)• metal jets / cardboard
noozles• car mass ~120g
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OBSERVED MOTION - SETUP
rails• car moving straight• position determination• lenght 5 m
video • camera 120 fps• placed 10 m from the rails
analysis time / distance coordinates
car
rails – with distance scale
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OBSERVED MOTION
distance/time graph obtained from video coordinates program time-distance graph:
5 data points frame – cubic or linear curve fitted derived: one point in the v-t graph moves one point to the next frame
time [s]0 1 2 3 4 5
velo
city
[m/s
]0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4vmax
deflation accelerated motion
decceleration
time [s]0 1 2 3 4 5
dist
ance
[m]
0
1
2
3
4
5
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TRAVELLED DISTANCE travelled distance:
acceleration path vmax time distance s0 determined from the video
decceleration a determined from the graph linear fit coefficient
decceleration path
time [s]0 1 2 3 4 5
velo
city
[m/s
]
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
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BASIC WORKING PRINCIPLE input energy is used for inflation deflation transforms it into kinetic energy of the air molecules
losses due to resistance to movement (mutual collisions) momentum conservation
simplified model: air behavior mass inside the balloon velocity according to Bernoulli's principle (valid for turbulent
flows)
vcar
vair
Ffr
𝑑(𝑚¿¿𝑎𝑣𝑎)−𝐹 𝑓 𝑟 𝑑𝑡=𝑚𝑐𝑑𝑣𝑐 ¿
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BASIC WORKING PRINCIPLE simplified model:
friction force main friction force source is the air drag friction force duration - flow rate
maximum car velocity vc initial volume relation obtained
vcar
vair
Ffrvcar
vair
Ffr
𝑚𝑎𝑣𝑎−𝐹 𝑓 𝑟 𝑡=𝑚𝑐𝒗𝒄
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INITIAL VOLUME simplified model basic working principle fit
main friction force source is air drag
as the initial volume increases the air drag force gains significance reason of non-linear
progression simplification
𝑚𝑎𝑣𝑎−𝐹 𝑓 𝑟 𝑡=𝑚𝑐𝑣𝑐
initial volume [L]
0 1 2 3 4 5
max
imum
vel
ocity
[m/s
]
0,4
0,5
0,6
0,7
0,8
0,9
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MAXIMISATION – VMAX IMPORTANCE
motion consists of acceleration and decceleration
acceleration to vmax is determined by: initial conditions
volume/pressure jets, noozles travelled distance found from
the video decceleration – same for all
parameters friction
air wheels/rails/shaft
energy input and ouput ratio energy input
work needed to inflate the balloon
piston model energy output
vmax car motion observation
travelled distance efficiency
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PARAMETERS initial volume
total input energy rubber stretching losses
jet diameter volumetric flow of air Q
Reynolds number 105 tubulent flow
exit velocity deflation time
nozzle directs the flow
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INITIAL VOLUME initial volume! – pressure total input energy and deflation energy
critical sizes min
needed to overcome static friction - start car movement
max radius ~ car height the balloon must not touch the floor
range ~ 0.15 dm3 - 4.5 dm3
deflated balloon ~ 0.075 dm3
air drag friction force
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initial volume [m3]0,000 0,001 0,002 0,003 0,004 0,005
trave
lled
dist
ance
[m]
0
2
4
6
8
10
12
INITIAL VOLUME RESULTS
efficiency: output/input energy small
losses on the air kinetic energy maximisation smallest balloon
𝜂=𝑚𝑣𝑚𝑎𝑥 22𝐸𝑖
initial volume [m3]
0,000 0,001 0,002 0,003 0,004 0,005
effic
ienc
y [%
]
-1
0
1
2
3
4
5
travelled distance:
maximisation largest balloon
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different diameters d
jet diameter changes how well the air is directed time needed for deflation
air drag duration amount of resistance the air endures in mutual collisions Reynolds number , kynematic viscosity
turbulent flow Re>4000, 105
JETS - CIRCULAR TUBE OPENINGS
d1 d2
deflation time
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RESULTS (V~4DM3) different vmax for a certain diameter
both travelled distance and efficiency depend on vmax
2
max distance 47.3 m max efficiency 2.7%
best jet diameter 1.2cm-1.4cmjet r2 [cm2]
0 1 2 3 4 5
trave
lled
dist
ance
[m]
0
10
20
30
40
50
jet r2 [cm2]0 1 2 3 4 5
effic
ienc
y [%
]
0,0
0,5
1,0
1,5
2,0
2,5
3,0
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NOZZLES – CONE SHAPED
different nozzles change• angle – how well the air is directed• tube lenght – amount of losses due to friction
6 cones – different angles and thus lenghts (7.5°- 30°)
effective velocity – horizontal component of rapid molecule movements
𝜷
l l
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RESULTS (V~3DM3) D=1CM
max distance 20.5 m max efficiency 1.5%
best cone angle 15°-20°cone angle [°]
5 10 15 20 25 30 35
effic
ienc
y [%
]
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
cone angle [°]5 10 15 20 25 30 35
trave
lled
dist
ance
[m]
0
5
10
15
20
25
different vmax for a certain diameter both travelled distance and efficiency depend on
vmax2
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CONCLUSION
initial V =4.5dm3 max optimal:
jet diameter1.2cm nozzle angle 20°
DISTANCE 70m
initial V=1.5dm3 min optimal:
jet diameter 1.2cm nozzle angle 20°
EFFICIENCY 6,4%
travelled distance efficiency
piston inflation/deflation model – energy evaluation stretching losses – theoretical and experimental curve basic working priciple explained found maximum conditions for jets and noozles
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Thank You!
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PISTON MODEL ~ ENERGYV20 = Vpiston
V2f = 0V1fV10
p0 = patmpf
𝑝0(𝑉 ¿¿10+𝑉 20)=𝑛𝑅𝑇=𝑐𝑜𝑛𝑠𝑡 ¿
𝑝 (𝑉 ¿¿1+𝑉 2)=𝑛𝑅𝑇 ¿
𝑝 𝑓𝑉 1 𝑓=𝑛𝑅𝑇
−𝑑𝑊=(𝑝−𝑝0 )𝑑𝑉 2
varies for inflation/deflation – determined from the graph
𝑊=∫𝑝0
𝑝 𝑓
𝑝𝑑𝑉 1+𝑝𝑓 𝑉 𝑓 ¿
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PISTON MODEL ~ ENERGYV20 = Vpiston
V2f = 0V1fV10
p0 = patmpf
𝑝0(𝑉 ¿¿10+𝑉 20)=𝑛𝑅𝑇=𝑐𝑜𝑛𝑠𝑡 ¿ 𝑝 𝑓𝑉 1 𝑓=𝑛𝑅𝑇
−𝑑𝑊=(𝑝−𝑝0 )𝑑𝑉 2
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PISTON MODEL ~ ENERGY
𝑊=∫𝑝0
𝑝 𝑓
𝑝𝑑𝑉 1+𝑝𝑓 𝑉 𝑓 ¿
𝑊=∫𝑝0
𝑝 𝑓
𝑝𝑑𝑉 1+𝑝𝑓 𝑉 𝑓 ln𝑝𝑓
𝑝0+¿𝑝0(0−𝑉 20)¿
𝑝0(𝑉 ¿¿10+𝑉 20)=𝑝 𝑓𝑉 𝑓 ¿
𝑉 20=𝑝 𝑓 𝑉 𝑓
𝑝0−𝑉 10
)
varies for inflation/deflation – determined from the graph
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elastic energy
work needed to increase radius from to r+dr under pressure difference
BALOON STRETCHING – THEORETICAL CURVE
initial volum
e
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BALOON STRETCHING – THEORETICAL CURVE
final expression
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FRICTION
h - car-surface distances - horizontal travelled distance
s2
l𝒔=𝒔𝟏+ 𝒔𝟐
s1
F1
𝐹 1=𝐹𝑔 𝑠1𝑙Fg
F2