Experiment 9
First Law of Thermodynamics – Bicycle Braking
Experiment 9
First Law of Thermodynamics – Bicycle Braking
Object
The object of this experiment is to verify the first law of thermodynamics through the use of a
bicycle brake calorimeter and to simultaneously compute Joule’s constant, J = 778.17 (ft
lbf)/Btu.
Introduction
The First Law of Thermodynamics is verified yet again – this time with a bicycle. A bicycle
front caliper brake is removed and replaced with a lever-mounted, copper calorimeter friction
pad shown in Figure 9.1. The calorimeter friction pad rubs on the front tire, heats-up, brings
the bicycle to a stop, and verifies the first law of thermodynamics. The loss in kinetic energy
of the bicycle and rider is equated to the gain in internal energy of the copper calorimeter. A
ratio of these two energies gives Joule’s constant. The data reduction analysis accounts for
bicycle aerodynamic drag, rolling friction, and heat loss into the front tire.
thermocouple wire
pivoting beam
copper calorimeter friction pad
balsa wood insulation
Figure 9.1 The copper calorimeter friction pad used for braking a bicycle.
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Experiment 9
First Law of Thermodynamics – Bicycle Braking
Historical Background
Heating by friction has been know by man for millennia, but quantified only as late as the 19 th
century. Count Rumford [1] performed experiments involving the convertibility of mechanical
work into heat as a result of friction. These experiments involved the boring of cannon with a
dull tool bit on a boring bar, which was driven by two draft horses. His calorimeter was
comprised of a semi-isolated end of a cannon encased in a wooden box containing water with a
thermometer – see Figure 9.2.
Being engaged lately in superintending the boring of cannon in the workshops
of the military arsenal at Munich, I was struck with the very considerable
degree of heat which a brass gun acquires in a short time, in being bored; . . .
Benjamin Count of Rumford
1798
Figure 9.2 Rumford’s cannon boring experimental equipment. The sketch is
from Rumford [1].
In 1850 James Joule [2] analyzed Rumford’s 1798 data and arrived at a value of 1034 (ft lbf)/Btu
for the convertibility of work into heat. This value favorably compared with the experimental
results Joule himself had recently obtained. Joule's analysis on the convertibility of work into
heat came from experiments reported in 1843, 1845, and 1850. This work provided the
empirical basis for our contemporary first law of thermodynamics.
One of Joule’s experiments involved friction between a cast iron disk rotating in a cast iron seat
all immersed in a mercury bath. The result was a reported convertibility value of 774.987 (ft
lbf)/Btu – a value only 0.4% lower than today’s accepted Joule’s constant value of 778.17 (ft
lbf)/Btu.
67
Experiment 9
First Law of Thermodynamics – Bicycle Braking
The Experiments
The current experiment employs a bicycle with a copper calorimeter friction pad that is used as a
brake on the front tire as shown in Figure 9.1. One end of the pivoting beam of this apparatus is
connected to the hand-operated brake cable, the other end carries the copper calorimeter friction
pad, which makes contact with the front tire. The copper pad is about 25 mm wide, 75 mm long
and, 6.4 mm thick and has a type K thermocouple silver-soldered in a small hole drilled in the
back surface. The copper pad is insulated from the pivoting beam with a 6.4 mm thick piece of
balsa wood. Furthermore, phenolic washers thermally isolate the four mounting machine screws
from the pivoting beam. Insulating balsa wood pieces, removed and not shown in Figure 9.1, fit
around the exposed edges of the copper pad. A digital thermometer, audio tape recorder, and
speedometer are mounted on the bicycle handlebar (also not shown).
Coast-down runs were performed with no braking to determine losses due to aerodynamic drag
and rolling friction. Speeds from the bicycle speedometer were read into the audio tape recorder
and later transcribed to provide speed-versus-time data. The braking runs were of short duration
(~ 5 s) so only initial speed was recorded and the final speed was always zero. Temperature data
at the beginning and end of the braking runs were also recorded. All of the tests were performed
on a level road during early mornings when the wind was calm.
Data
The data for this experiment are given in “Experiment 9 Data.xls.” Two tables are provided in
this file: Table 1 contains “Bicycle coast-down without braking” and Table 2 contains “Bicycle
braking.” The mass of the bicycle and rider and the copper friction pad mass are also given.
Analysis
First law of thermodynamics analysis
We apply the first law of thermodynamics to a control volume that contains the bicycle starting at
some initial velocity (state 1) and terminating at rest (state 2) – see Figure 9.3.
Q12 rubber 
(1)
 m
 mg

2
2 
( V2  V1 )  
( z 2  z1 )  ( U 2  U 1 )Cu  W12

 2g c
 bike  g c
 bike
(2)
(3)
(4)
(5)
2
1
Figure 9.3 The closed system used for the first law of thermodynamics analysis.
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(9.1)
Experiment 9
First Law of Thermodynamics – Bicycle Braking
In Equation 9.1:
 term 1 is the heat loss from the copper brake pad to the surroundings (we will show that
for all of the heat loss paths the rubber tire path dominates – thus the subscript “rubber”
on term 1)
 term 2 is the difference in kinetic energies for the bike + rider
 term 3 is the difference in gravitational potential energies for the bike + rider
 term 4 is the difference in internal energy of the copper calorimeter friction pad
 term 5 represents work done by the bike + rider on the surroundings.
Terms 1 and 5 will be examined in detail.
Heat loss (term 1)
During the braking process the copper friction pad heats up with a resulting heat loss to the
surroundings. ( Q12 will be negative since heat is transferred out of the system.) Convection
losses are small due to the small exposed surface area of the copper and the short braking period.
The conduction losses through the balsa wood insulation and the phenolic washers that isolate
the four small machine screws (see Figure 9.1) are also small. However, heat generated at the
copper brake pad–rubber tire interface diffuses into each of these materials, so there is heat loss
into the bicycle tire. Analysis for the heat loss into the tire is complicated by the transient
generation of the heat and transient tire rotation. However, as an approximation, an analysis for
the simpler case of a non-rotating tire may be used.
Our heat loss analysis begins with a solution by Carslaw and Jaeger [3] who provide the transient
temperature T, as a function of time t, at the surface of a semi-infinite solid with zero initial
temperature and with constant surface heat flux q / A
2 ( q/A )   t 
T
 
k  
1/ 2
(9.2)
The semi-infinite solid has thermal conductivity k, and thermal diffusivity  . For the short
duration braking we may take the rubber tire and copper friction pad as semi-infinite solids with
one-dimensional heat diffusion perpendicular to their contact surfaces. The initial temperature of
the tire and copper brake pad are not zero as stipulated but they are the same and will cancel in
the analysis. The rubbing friction provides the common surface heat flux q/A. Furthermore, the
copper and rubber have the same temperature at their common surface contact location. From
Equation 9.2 this means
 2 ( q/A )  t 1 / 2 
 2 ( q/A )  t 1 / 2 
   
  

 k     Cu  k     rubber
which reduces to
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Experiment 9
 ( k  c )Cu 
qCu

 
q rubber  ( k  c )rubber 
First Law of Thermodynamics – Bicycle Braking
1/ 2
 51
(9.3)
The heat transfer rate into the copper qCu , and the heat transfer rate into the rubber qrubber , are
integrated over time to give the net heat transferred Q 1 2 , into each material. By definition
Q12 Cu
and
 t12 qCu dt
t
Q12 rubber  tt
2
1
(9.4)
(9.5)
qrubber dt
Furthermore, by the first law of thermodynamics applied to just the copper friction pad, the net
heat transferred into the copper causes an increase in copper internal energy
Q12 Cu  ( U 2  U1 )Cu
(9.6)
Finally, Equations 9.3-9.6 reduce to
Q12 rubber  1 ( U 2  U 1 )Cu
(9.7)
51
We should note that Equation 9.7 under predicts the heat transferred into the rubber tire because
of the non-rotating tire simplification.
Work due to aerodynamic drag and rolling friction (term 5)
Aerodynamic drag and rolling friction will cause the bicycle to coast-down, without braking, due
to energy transferred from the moving bicycle to the surroundings. For both coast-down and for
braking, speeds were read from a speedometer into an audiocassette recorder mounted on the
bicycle handlebar and later transcribed with a stopwatch to give velocity-versus-time data.
During coast-down the velocity-versus-time data are fitted with the curve
V  c1e c2t
(9.8)
and for the short duration braking the velocity-versus-time data are fitted with a linear function
V  c 3 t  c 4
(9.9)
For the decelerating bicycle we write Newton’s second law
F
1
ma
gc
(9.10)
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Experiment 9
First Law of Thermodynamics – Bicycle Braking
During coast-down the aerodynamic drag and rolling friction are forces acting on the combined
mass of bicycle and rider that produces a negative acceleration (deceleration). This acceleration
may be expressed as a  dV / dt which is obtained by taking the derivative of Equation 9.8.
Using this result in Equation 9.10 we get
F 
m
c2 V
gc
(9.11)
Equation 9.11 expresses the force of the surroundings acting on the bicycle. For the force of the
bicycle acting on the surroundings we have
F
m
c2 V
gc
(9.12)
This force is a function of velocity only and is present during both coast-down and braking. The
work associated with this force during braking is
W12  x12 F dx  t12 F V dt
x
t
(9.13)
Substituting Equations 9.9 and 9.12 into 9.13 and integrating produces
W12 
mc2
gc
 2 t23
2
2 
 c3 c 4 t 2  c 4 t 2 
c 3
3


(9.14)
Joule’s constant analysis
We now return to the first law of thermodynamics (Equation 9.1), rewritten here
Q12 rubber 
 m
2
2
( V2  V1

 2g c
(1)

 mg

)  
( z 2  z1 )  ( U 2  U 1 )Cu  W1 2
 bike
 gc
 bike
(2)
(3)
(4)
(9.1)
(5)
The run from state 1 to state 2 was conducted over level ground thus there was no change in
gravitational potential energy, eliminating term 3. The bicycle comes to rest at state 2 thus
V2  0 . We note that (U 2  U 1 ) Cu  m c (T2  T1 )Cu . And we substitute Equation 9.7 for term 1
and Equation 9.14 for term 5 to obtain

1
mV12
2g c
(1)

bike
3
mc 2  2 t 2
52
2
2 
m c( T2  T1 )Cu

 c3 c 4 t 2  c 4 t 2  
c 3
gc 
3
 51
(2)
(3)
71
(9.15)
Experiment 9
First Law of Thermodynamics – Bicycle Braking
In Equation 9.15 the first term represents the initial kinetic energy of the rolling bicycle; the
second term is due to aerodynamic drag and rolling friction losses represented as work; and the
third term represents the gain in internal energy of the copper friction pad calorimeter plus heat
loss into the tire, both due to friction. Using English Engineering units* the ratio of the left-side
over the right-side of this equation directly gives Joule’s constant in (ft lbf)/Btu.
Required
1. Using the data in “Experiment 9 Data.xls” Table 1, plot (with linear scales) time (s) on the
abscissa and velocity (ft/s) on the ordinate. Fit these data with the curve V  c1e c2t using
TRENDLINE in EXCEL. In this same figure plot run 7 found in Table 2.
2. Construct a table with the following columns (left-to-right): run, V1 , t 2 , ( T2  T1 ) ,
m
2
( V1 ) , W12 , Q12 rubber , ( U 2  U1 )Cu , J. Include the appropriate units for each
2g c
column. Using the data in Table 2 fill in or compute the values for each column. Comment
m
2
( V1 ) , W12 , Q12 rubber ,
on the relative contributions of each of the following terms:
2gc
and ( U 2  U1 )Cu .
3. Plot, using linear scales, the gain in internal energy of the copper calorimeter friction pad
plus heat loss into the tire (Equation 9.15, term 3) on the abscissa and the initial kinetic
energy of the rolling bicycle minus the aerodynamic drag and rolling friction losses
represented as work (Equation 9.15, terms 1 & 2) on the ordinate. Compute the slope of
this line, which is Joule’s constant. Hint: the origin is a datum point because, when
V1  0 , ( U 2  U1 )Cu = 0. Furthermore, this point is known with 100% certainty.
Accordingly, in EXCEL using TRENDLINE, force your line through the origin. Also in
this graph plot the known J = 778.17 (ft lbf)/Btu. Find the percent deviation between your
result and the published value.
* English units are used to distinguish the kinetic energy and work (both in ft lbf) from the internal energy and
heat transfer (in Btu). SI units would give Joules (J) for all of the terms and would require conversion back to
English units to obtain Joule’s constant. It is worthwhile to perform a units check on Equation 9.15. Terms 1 and
2 should each reduce to ft-lbf and term 3 should reduce to Btu. The units for g c , m, V, t, c, and T are given in the
Nomenclature. However, the units for the constants c 2 , c3 and c4 are not immediately obvious and require
examination of their origins, which are found in Equations 9.8 and 9.9. Since both Equations 9.8 and 9.9 give
velocity with units (ft/s) it follows that c2 has units (1/s), c3 has units (ft/s2), and c4 has units (ft/s).
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Experiment 9
First Law of Thermodynamics – Bicycle Braking
Nomenclature
English
a
acceleration, ft / s 2
c1 , c2 , c3 , c4 constants
c
specific heat, Btu/(lbm°F)
F
force, lbf
gc
k
m
Q
q/A
T
t
U
V
W
x
z
lbm ft
lbf s 2
thermal conductivity, Btu/(s ft °F)
mass, lbm
heat loss, Btu
heat flux, Btu / ( s ft 2 )
temperature, °F
time, s
internal energy, Btu
velocity, ft/s
work, ft lbf
distance, ft
elevation, ft
proportionality constant, 32.2
Greek


thermal diffusivity, ft 2 / s
density, lbm / ft 3
Subscripts
Cu
copper
rubber rubber tire
1, 2
states
References
1. Brown, S.C., Editor, “Collected Works of Count Rumford,” Vol. II, Practical Applications of
Heat, The Belknap Press, Harvard University Press, Cambridge, Mass., 1969.
2. Joule, J. P., "The Scientific Papers of James Prescott Joule," Vol. I & II, Dawsons of Pall
Mall, London, first published 1887, reprinted 1963.
3. Carslaw, H. S. & J. C. Jaeger, “Conduction of Heat in Solids,” second edition, Oxford, at the
Clarendon Press, 1959. p. 75.
\ © 2005 by Ronald S. Mullisen \ Physical Experiments in Thermodynamics \ Experiment 9 \
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