Experiment 5 Polytropic Expansion of Air

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Experiment 5 Polytropic Expansion of Air


33

Experiment 5

Polytropic Expansion of Air




Object


The object of this experiment is to find the relation between
pressure

and
volume
for the
expansion of air in a pressure vessel


this expansion is a
thermodynamic process
.




Introduction


The expansio
n or compression of a gas can be described by the
polytropic relation

c
pv
n

,
where
p
is

pressure, v

is
specific volume
,
c

is a
constant

and the exponent
n

depends on the
thermodynamic process
. In our experiment compressed air in a steel
pressure vessel is
discharged to the atmosphere while the air remaining inside expands. Temperature and
pressure measurements of the air
inside

the vessel are recorded. These two measurements are
used to produce the
polytropic exponent

n

for the expansio
n process.




Historical background


Sadi Carnot (1796
-
1832) [1] in his 1824 "Reflections on the Motive Power of Heat and on
Machines Fitted to Develop This Power," examines a reciprocating, piston
-
in
-
cylinder
engine. Carnot describes a cycle applied t
o the machine appearing in Figure 5.1, which
contains his original sketch. In this figure air is contained in the chamber formed by the
piston

cd

in the cylinder. Two
heat reservoirs

A

and
B
, with
temperature

T
A

greater than
temperature
T
B
, are available to make contact with
cylinder head ab.

The reservoirs
A

and
B

maintain their respective temperatures during heat transfer to or from the cylinder head.


Carnot gives the following six steps for his machine:

1.

The piston

is initially at
cd

when high
-
temperature reservoir
A

is brought into contact
with the cylinder head
ab
.

2.

There is isothermal expansion to
ef

3.

Reservoir
A

is removed and the piston continues to
gh

and so cools to
T
B
.

4.

Reservoir
B

m
akes contact causing isothermal compression from
gh
to
cd
.

5.

Reservoir
B

is removed but continual compression from
cd

to
ik

causes the
temperature to rise to
T
A
.

6.

Reservoir
A

makes contact, isothermally expanding the air to
cd

and thus
completing
the cycle.


Experiment 5 Polytropic Expansion of Air


34

A decade later Clapeyron [2] analyzed Carnot's cycle by introducing a
pressure
-
volume, p
-
v
diagram
. Clapeyron's diagram is reproduced next to Carnot’s engine in Figure 5.1. Claperon
labels his axes
y

and
x
, which correspond to pres
sure and volume, respectively. We will
examine two process paths in this diagram: the
isothermal compression path F
-
K

and the
isothermal expansion path C
-
E
. Since both of these processes are
isothermal,

pv = RT =

constant. This is a special case of the p
olytropic process
c
pv
n

, where, for the isothermal
process,
n

= 1, so we have the same result,
pv = c
.



Figure 5.1

Left sketch
: Carnot's engine, after Carnot [1].
Right sketch
: Clapeyron's
pressure
-
volume, p
-
v diagram
, after Clapey
ron [2]. For the axes in Clapeyron’s diagram
x = v

and
y = p
.










Experiment 5 Polytropic Expansion of Air


35

The Experiments


Photographs of the equipment appear in Figures 5.2 and 5.3, and a sketch of the components
appears in Figure 5.4.




steel pressure vessel


d
ischarge valves thermocouple conduit pressure transducer




Figure 5.2

The polytropic expansion experiment at Cal Poly.



thermocouples thermocouple conduit




Figure 5.3

Two,
T
ype
-
T thermocouples are located
inside

the pressure vessel, at the
geometric center. Only one thermocouple is used


the other is a spare. In the photo
the thermocouple conduit has been removed and held
outside

of the vessel. The
junctions of these ther
mocouples are constructed of extremely fine wires (0.0254
mm

diameter) that provide a fast time response.

Experiment 5 Polytropic Expansion of Air


36



Figure 5.4

The polytropic expansion experiment equipment.



Pressure measurements come from the pressure transducer tapped in to the pressure ve
ssel
shown in Figure 5.4. The transducer is powered by the unit labeled “CD23”, which is a
Validyne [3] carrier demodulator. The fine wire thermocouple is described in the Figure 5.3
caption. Both thermocouple and pressure signals feed into an Omega [4]

flatbed recorder.


The three discharge valves on the right side of the vessel have small, medium, and large
orifices. These orifices allow the air inside the vessel expand at three different rates. The
pressure vessel is first charged with the compress
ed air supply. This causes the air that enters
the vessel to initially rise in temperature. After a few minutes the temperature reaches
equilibrium at which time one of the discharge valves is opened. Temperature and pressure
are recorded for each expans
ion process. These data are then used to compute the
polytropic
exponent

n

for each process. It is important to note that the temperature and pressure of the
air
inside

the vessel are measured, not the air discharging from the vessel.




Data


Pressure

and temperature data, for the three runs, are provided in the EXCEL file
“Experiment 5 Data.xls.”


Experiment 5 Polytropic Expansion of Air


37


Analysis



In many cases the
process path

for a gas expanding or contracting follows the relationship




c
pv
n










(5.1)


The polytr
opic exponent
n

can theoretically range from





n
. However, Wark [5]
reports that the relation is especially useful when
3
/
5
1


n
. For the following simple
processes the
n

values are:


isobaric process

(constant pressure
)


n = 0

isothermal process
(constant temperature)

n = 1

isentropic process

(constant entropy)

n = k ( k=1.4 for air)

isochoric process

(constant volume)


n =



In our experiment the steel pressure vessel is initially charged with compressed air of mass

1
m
. Next, the vessel is discharged and the remaining air mass is
2
m
.

This final mass
2
m

was part of the initial mass
1
m

and occupied part of the volume of the vessel at
the initial
state. Thus
2
m

expanded
within

the vessel with a corresponding change in temperature and
pressure. Therefore mass
2
m

can be considered a
closed

system

with a moving system
boundary and the following f
orm of the first law of thermodynamics applies



W
dU
Q












(5.2)


If the system undergoes an
adiabatic

expansion
0

Q

, and if the work at the moving system
boundary is reversible

dV
p
W


. Furthermore, if we cons
ider the air to be an
ideal gas
with
constant specific heat

dT
c
m
dU
v
2

. With these considerations the first law reduces to



dv
p
dT
c
0
v










(5.3)


Using the ideal gas assumption
p
v
R
T


and differentiating this equation gi
ves



R
dp
v
dv
p
dT










(5.4)

Substituting Equation 5.4 into 5.3 and using the relationships
v
p
c
c
R



and
v
p
c
c
k


gives


dp
v
dv
p
k
0



Separating variables and integrating this equation,




p
dp
v
dv
k
0
, yields


c
v
p
k










(5.5)

Experiment 5 Polytropic Expansion of Air


38


which is a special case of the polytropic relationship given by Equation 5.1, with
n = k.



It is important to note that in the development of Equation 5.5 the expansion of
2
m

inside the
pressure v
essel was assumed to be
reversible

and
adiabatic
, i.e. an
isentropic

expansion. In
our experiment the adiabatic assumption is accurate during initial discharge. However, the
reversible assumption is clearly not applicable because the air expands
irrevers
ibly

from high
pressure to low pressure. Therefore we anticipate our data to yield
4
.
1
k
n


.


Two approaches are used to determined the polytropic exponent
n

from the data:


1.

Equation 5.1 can be written as
n
v
c
p


, which is a
power
law

equation.
In EXCEL,
a plot of
p
versus
v

and a
power law

curve fit using TRENDLINE

will disclose
n
.



2.

Equation 5.6 (subsequently developed) may be used with only two states to determine
n.


Here is the outline of the development of Equation 5.6. We s
tart with
c
v
p
n

, which also
can be expressed as
p
v
p
v
n
n
1
1
2
2


and combine this with the ideal gas law
p
v
R
T


to obtain




1
1
2
2
1
p
p
ln
T
T
ln
1
n








































(5.6)


The temperatures and pressures in Equation 5.6 are all absolute and
the subscripts 1 and 2
represent the initial and final states.




Required


1.

Pressure and temperature data are provided for all three runs in “Experiment 5
Data.xls.” Use the ideal gas law,
pv = RT,

to compute
v

corresponding to each
p
.
Use SI units:
m
3
/k
g

for
v

and
Pa

for
p
.


2.

Plot
p

versus
v

and find
n
:



For each run, on a separate graph, plot
p
[on the ordinate (vertical) axis] versus
v

[on the abscissa (horizontal) axis]. Use linear scales.



Determine the polytropic exponent
n

for each run using a TR
ENDLINE power
curve fit. Also find the correlation coefficient
)
(
2
R

for each curve. (Be aware
Experiment 5 Polytropic Expansion of Air


39

that the TRENDLINE power curve fit will give
b
ax
y

, where
y = p, x = v

and
a

and
b

are constants.)



Plot all three runs on

a single graph and find
n

for the combined data.


3.

Derive Equation 5.6.


4.

Find
n

for each run using Equation 5.6, where states 1 and 2 represent the beginning
and ending states, respectively.


5.

In a single table show all of the
n

values.


6.

Discuss the meaning

of your
n

values, that is, how does your
n

value compare with
n

values for other, known processes?




Nomenclature



c
constant,
N m


c
p

specific heat constant pressure,
kJ/kg K


c
v

specific heat constant volume
,
kJ/kg K


k

specific heat ratio, dimensionless


n

polytropic exponent, dimensionless


p

absolute pressure,
Pa

or
psia


Q

heat transfer,
kJ


R

gas constant,
kJ/kg K

(
R
air

= 0.287
k
J/kg∙K
)


T

temperature,
°C
or
K


U

internal energy,
kJ


v

specific volume,
m
3
/kg


V

volume
m
3


W

work,
kJ


Subscripts


1,2

thermodynamic states




References


1.

Carnot, S., "Réflexions sur la puissance motive du feu et sur les machines
propres à développer cette puissance," Paris, 1824. Reprints in P
aris: 1878,
1912, 1953. English translation by R. H. Thurston, "Reflections on the
Motive Power of Heat and on Machines Fitted to Develop This Power,"
ASME, New York, 1943.

Experiment 5 Polytropic Expansion of Air


40


2.

Clapeyron, E., "Memoir on the Motive Power of Heat,"
Journal de l'École
Polytechn
ic,
Vol. 14, 1834; translation in E. Mendoza (Ed.) "Reflections on
the motive Power of Fire and Other Papers," Dover, New York, 1960.


3.

Validyne Engineering Sales Corp.,

8626 Wilbur Avenue, Northridge, CA. 91324
http://www.validyne.com/


4.

OMEGA Engineering,
INC., One Omega Drive, Stamford, Connecticut 06907
-
0047
http://www.omega.com/


5.

Wark, K. Jr. & D.E. Richards, Thermodynamics, 6
th

Ed, WCB McGraw
-
Hill, Boston,
1999.






























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© 2005 by Ronald S. Mullisen
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Physical Experiments in Ther
modynamics
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Experiment 5
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