© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
Problems
The problems included here have been selected from a much larger set of problems that are available on
the website associated with this book (www.cambridge.org/kleinandnellis).
A. Heat and Work
3.A-1 Cryogenic liquids (e.g., liquid helium or liquid neon) are sometimes used to keep instruments at
cryogenic (i.e., very cold) temperatures for space science missions. As the liquid boils off due to
heat transfer it is vented to space. When all of the cryogenic liquid is gone, the instrument warms
up and the mission is over. Flight operations engineers need to be able to check the amount of
liquid that is left in the tank from the ground while the spacecraft is in orbit. In microgravity, the
mixture of liquid and vapor in the tank is not stratified by gravity in the same way that it is on
earth. Therefore, traditional liquid level measurement techniques do not work. One alternative
technique that has been used by NASA is referred to as mass gauging. In order to accomplish
mass gauging, a heater is activated for a short time and the temperature rise of the neon and the
tank material is measured. The magnitude of the temperature rise can be used to calculate the
mass of neon that is left in the tank. Consider the spherical, aluminum cryogenic tank shown in
Figure 3.A-1.
Rin = 10.0 inch
th = 0.125 inch
neon
T1 = 27 K
m = 40 kg
Q = 1 kJ
Aluminum tank
T1 = 27 K
cAl = 24 J/kg-K
ρAl = 0.098 lbm/in3
Figure 3.A-1: A spherical cryogenic tank.
The tank has an inner radius of Rin = 10 inch and a wall thickness of th = 0.125 inch. The density
and specific heat capacity of aluminum (at the cryogenic temperatures associated with this
problem) are ρAl = 0.098 lbm/in3 and cAl = 24 J/kg-K, respectively. The tank contains a mixture of
saturated liquid and saturated vapor neon at T1 = 27 K. The mass of neon in the tank is m = 40
kg. For these calculations you may assume that no mass leaves the tank during the short time that
it takes to complete the mass gauging process. A heater in the tank is activated and provides Q =
1 kJ of heat in order to accomplish the mass gauging. You may assume that the temperature of
the neon is uniform and that the tank and the neon are at the same temperature. There is no work
done on or by the tank or its contents during this process.
a.) What is the mass of the aluminum?
b.) What is the quality of the neon initially in the tank? What is the pressure of the neon initially
in the tank?
c.) What is the temperature of the neon and aluminum at the conclusion of the mass gauging
process (T2)? What is the increase in temperature detected by the operators?
d.) What are the quality and pressure of the neon in the tank at the conclusion of the mass
gauging process?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
e.) Generate a calibration curve that an operator could use for the mass gauging process. That is,
generate a plot showing the mass of neon in the tank (m) as a function of the temperature rise
(ΔT = T2 - T1) for neon mass ranging from 1 kg to 80 kg.
f.) If your temperature sensors are capable of resolving a temperature difference of
approximately 1 mK (that is, the uncertainty in your measurement of the temperature rise is
δΔT = 0.001 K) then estimate the resolution of the mass gauging process (that is, how well
can you measure the mass in the tank?). Use your engineering judgment to answer this
question; you may want to refer to the calibration curve generated in (e).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.A-2 Figure 3.A-2 is the pressure-volume diagram for a thermodynamic cycle that is executed by m
= 18 lbm of nitrogen gas. Assume that nitrogen behaves according to the ideal gas law.
1250
Pressure (psi)
1000
1,5
2
750
500
250
4
3
0
0
5
10
15
20
Volume (ft3)
Figure 3.A-2: Pressure-volume diagram for a thermodynamic cycle.
a.) Determine the temperature at each of the states shown in Figure 3.A-2.
b.) Determine the work done by the nitrogen gas for the cycle for each of the processes shown in
Figure 3.A-2 (i.e., process 1 to 2, 2 to 3, 3 to 4, and 4 to 5). What is the net work done by the
nitrogen during cycle?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.A-3 A cylinder fitted with a frictionless piston contains m = 5 lbm of water at a pressure of P1 =
200 psia with a volume of V1 = 9.15 ft3. The piston diameter is D = 4 inch. The surroundings
are at Tsur = 77°F and Patm = 14.7 psia. The system is heated at constant pressure until the
quality of the water is x2 = 1.
a.) Determine the mass of the piston.
b.) Calculate the specific volume, temperature and quality (if applicable) of the water in the
cylinder before the heating process is initiated.
c.) Sketch the heating process on a temperature-volume plot. Label the initial and final states.
d.) What is the final volume of the water?
e.) Calculate the work done by the water on its surroundings. Be sure to indicate the direction of
the work.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.A-4 Figure 3.A-4 illustrates a small motor that is being used to lift an m = 10 kg mass.
2
iin = 2.5 amp
+
Ein = 24 V
-
h = 1.5 Btu/hr-ft -R
T∞ = 70°F
τ = 12.8 in-lbf
N = 350 rev/min
motor
As = 0.05 m2
ε = 0.75
m = 10 kg
Figure 3.A-4: Motor being used to lift a mass.
The motor is operating at steady state. The surface area of the motor is As = 0.05 m2 and the
motor is surrounded by air at T∞ = 70ºF. The surface of the motor has an emissivity of ε = 0.75.
The heat transfer coefficient between the motor and the air is approximately h = 1.5 Btu/hr-ft2-R.
The motor radiates to surroundings that are also at T∞. The motor is provided electrical input
power with a voltage of Ein = 24 V and a current iin = 2.5 amp. The motor shaft is rotating at N =
350 rev/min and the torque on the shaft is τ = 12.8 inch-lbf.
a.) Carry out an energy balance on a system that consists of the motor. What is the rate at which
the energy in the system is changing? What is the rate of heat transfer from the motor?
b.) What is the surface temperature of the motor?
c.) Assume that all of the mechanical power carried by the shaft is being used to increase the
potential energy of the mass. What is the velocity at which the mass is rising?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.A-5 Figure 3.A-5 illustrates a piston cylinder device filled with a fluid. The piston is frictionless and
has a mass of mp = 90 kg. The top of piston is exposed to ambient pressure at Patm = 100 kPa.
The cross-sectional area of the piston is Ac = 0.01 m2 and the initial position of the piston is L1 =
10 cm (as shown in Figure 1). The mass of fluid in the piston cylinder is m = 0.2 kg; the piston is
leak-tight so the mass of fluid never changes. The specific internal energy of the fluid is initially
u1 = 5200 J/kg.
Patm = 100 kPa
Ac = 0.01 m2
mp = 90 kg
fluid
m = 0.2 kg
L1 = 10 cm
u1 = 5200 J/kg
Figure 3.A-5: Piston cylinder device.
a.) What is the initial pressure in the fluid?
b.) What is the initial specific volume of the fluid?
In going from state 1 to state 2, a total heat transfer of Q1,2 = 1 kJ is transferred to the fluid. The
specific internal energy of the fluid does not change during this process.
c.) What is the work done by the fluid during this process?
d.) What is the position of the piston at state 2?
e.) What is the specific volume of the fluid at state 2?
f.) What is the change in the potential energy of the piston that occurs as the system goes from
state 1 to state 2? If your answer to part (f) is not the same as part (c) then explain?
In going from state 2 to state 3, a total heat transfer of Q2,3 = 0.5 kJ is transferred to the fluid.
During this process, the piston is pinned in place and not allowed to move.
g.) What is the work done by the fluid during this process?
h.) What is the specific internal energy of the fluid at state 3?
i.) What is the specific volume of the fluid at state 3?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.A-6 Water is contained in a piston-cylinder device as shown in Figure 3.A-6. You may neglect
friction between the piston and the wall and assume that the piston does not leak.
spring
K = 5000 N/cm
m = 0.05 kg
T1 = 125°C
x1 = 0.90
T2 = 250°C
Ac = 0.20 m2
z1
z2
state 2
state 1
Figure 3.A-6: Water in a spring-loaded piston-cylinder device at state 1 and state 2.
The mass of the water is m = 0.05 kg and the area of the piston face is Ac = 0.2 m2. Initially, the
water is at T1 = 125ºC with a quality of x1 = 0.90. The water is heated and the piston begins to
rise; as this occurs, the spring is compressed and so the pressure in the cylinder begins to rise.
The pressure rise is proportional to the amount that the spring is compressed, according to:
(z − z )
P2 = P1 + K 2 1
Ac
where K = 5000 N/cm is the spring constant and s2 and s1 are the final and initial positions of the
piston, respectively, as shown in Figure 3.A-2. The heating stops when the water temperature
reaches T2 = 250ºC.
a.) What is the initial pressure in the piston? What is the initial position of the piston, z1?
b.) What is the pressure in the piston at the end of the heating process?
c.) Using EES, prepare a T-v diagram for water using the Property Plot option from the Plots
menu and overlay states 1 and 2 on this plot.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
B. Closed System Energy Balances
3.B-1
A well-insulated rigid container with an internal volume of V = 0.01 m3 holds m = 2 kg of R134a,
as shown in Figure 3.B-1.
relief valve,
opens at
Popen = 1 MPa
R134a
m = 2 kg
V = 0.01 m3
T1 = 20°C
insulation
electrical
resistance
heater
Welec = 250 W
Figure 3.B-1: Container holding R134a.
The container is initially at room temperature (T1 = 20°C). The container is fitted with an
electrical resistance heater as shown in the figure. The heater draws Welec = 250 W when it is
switched on. The pressure relief valve opens when the internal pressure in the container reaches
Popen = 1 MPa. Assume that the mass of the container is negligible.
a.) Determine the initial pressure in the container.
b.) Determine the time that the electrical heater can operate before the pressure relief valve
opens.
c.) Determine the temperature of the R134a at the time that the pressure relief just opens.
You have carried out some experiments and measured the time required to open the relief valve
for the container analyzed in Figure 3.B-1. You have found that the measured time required is
much longer than what you predicted in (b). It is suspected that the discrepancy is related to the
energy required to change the temperature of the container walls. The container is made of AISI
304 stainless steel and has a total mass of mcyl = 7.25 kg. The specific heat of this material is ccyl
= 478.2 J/kg-K. (Note that this property is also available in the EES Solid/Liquid Property
library.) Assume that the container wall temperature is uniform and that the container is in
thermal equilibrium with the R134a (i.e., they are at the same temperature).
d.) Determine the time required for the pressure relief valve to open if the container wall is
included in the analysis.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-2
A cylinder containing air has a piston held by a lock, as shown in Figure 3.B-2. The diameter of
the piston is 0.25 m and its mass is 5.0 kg. Initially, the air in the cylinder is at 25°C, 110 kPa
and the bottom of the piston is 0.65 m from the bottom of the cylinder. The pressure and
temperature of the air surrounding the cylinder are 100 kPa and 25°C, respectively. A block of
unknown mass is resting on the piston. The lock is released, allowing the piston to move with
friction to a new location that is 0.55 m from the bottom of the cylinder. After some time, the
entire apparatus returns to a temperature of 25°C and the process is concluded.
25°C and 100 kPa
mass
lock
piston mass = 5 kg
piston diameter = 0.25 m
air
0.65 m
Figure 3.B-2: Piston-cylinder device with air.
a.) Determine the mass of air in the cylinder.
b.) Determine the mass of the block.
c.) Taking the cylinder, the air in the cylinder and the piston as your system, determine the
change in internal energy (ΔU), the change in kinetic energy (ΔKE), the change in potential
energy (ΔPE), the work done by the system to the surroundings (Wout), and the heat transfer
from the surroundings to the system (Qin).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-3
You want to build a device that can be used to lift weights. The device a piston cylinder
apparatus filled with the fluid R22 (R22 is a fluid that is built into EES), as shown in Figure
P3.B-3(a).
g
weight
mw = 50 kg
R22
sat. liquid
piston
Ac = 2 cm2
mp = 15 kg
s1 = 10 cm
Figure 3.B-3(a): Piston cylinder device filled with R22.
The mass of the piston is mp = 15 kg and the mass of the weight that you want to lift is mw = 50
kg. Atmospheric pressure is Patm = 100 kPa. The cross-sectional area of the piston is Ac = 2 cm2
and the initial position of the piston is s1 = 10 cm. The R22 is initially saturated liquid. At state
1, the weight is placed on the piston as shown in Figure P3.B-3(a).
a.) Determine the pressure experienced by the R22 when the weight is placed on the piston (Pwt)
and the pressure that would be experienced if the weight is removed (Pnwt).
b.) Sketch a T-v diagram and locate state 1 on the diagram.
c.) What is the mass of R22 in the piston? What is the initial temperature of the R22?
With the weight on the piston, heat is added to the R22, Figure 3.B-3(b). This process is
continued until the R22 completely evaporates, at state 2.
heat
addition
R22
sat. liquid
R22
sat. vapor
state 1
state 2
Figure 3.B-3(b): Process 1-2, heat addition at constant pressure.
d.) Locate and draw state 2 on your T-v sketch from part (b).
e.) Determine the volume of the R22 at state 2. What is the distance that the weight been lifted?
f.) What is the heat transfer required to accomplish this process (Q12)? What is the work transfer
from the R22 during the process (W12)?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
The piston is locked into place at state 2 so that it can't move and the weight is removed. Heat is
removed from the R22 until the pressure in the cylinder is reduced from Pwt to Pnwt as shown in
Figure 3.B-3(c).
piston is locked in place
heat
removal
R22
sat. vapor
R22
P = Pnwt
state 2
state 3
Figure 3.B-3(c): Process 2-3, heat removal at constant volume.
g.) Locate and draw state 3 on your T-v sketch from part (b).
h.) What are the temperature and quality of the R22 at state 3?
i.) What are the mass of liquid and mass of vapor at state 3? What are the volume of liquid and
the volume of vapor at state 3?
j.) What is the heat transfer that must be removed in order to accomplish this process (Q23)?
What is the work transfer from the R22 during this process (W23)?
The piston is unlocked at state 3 and heat is removed until the volume of the R22 returns to its
initial value (i.e., the value that it had at state 1), as shown in Figure 3.B-3(d). The state of the
R22 at the conclusion of this process is state 4.
piston is unlocked
R22
P = Pnwt
heat
removal
V = V1
state 3
state 4
Figure 3.B-3(d): Process 3-4, heat removal at constant pressure.
k.) Locate and draw state 4 on your T-v sketch from part (b).
l.) What are the temperature and quality of the R22 at state 4?
m.) What is the heat transfer that must be removed to accomplish this process (Q34)? What is the
work transfer from the R22 (W34)?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
Finally, the piston is locked in place at state 4 and heat is added until the pressure in the R22
returns to Pwt as shown in Figure 3.B-3(e). Note that at this point we have returned our device to
state 1.
piston is locked in place
heat
addition
P = Pnwt
P = Pwt
state 4
state 4
Figure 3.B-3(e): Process 4-1, heat addition at constant volume.
n.) What is the heat transfer that must be added in order to accomplish this process (Q41)? What
is the work transfer from the R22 during this process (W41)?
Your device has just undergone a complete cycle. It started at state 1 and, after doing some
things, ended back at state 1. We analyzed each process that made up the cycle using an energy
balance. However, it is worth checking our answers by doing an energy balance on the system
for the entire cycle (i.e., the time required to go from state 1 to 2, 2 to 3, 3 to 4, and finally 4 back
to 1). The storage of energy for this energy balance must be zero because the system begins and
ends at the same state.
o.) Check your answers by verifying that the net energy transfer into the system equals the net
energy transfer out for the entire cycle.
Your device is a very simple engine - it takes in heat (during processes 1-2 and 4-1) and produces
a net amount of work (the work produced during process 1-2 less the work required by process 34). The heat addition is presumably obtained by burning something that costs money - therefore
you pay for the heat addition. The work produced is valuable to you - somehow you get paid for
it. Therefore, the efficiency of a power cycle like this is defined as:
η=
W
what you want
= net
what you had to pay for Qin
(1)
p.) Compute the efficiency of your power cycle.
q.) Plot the efficiency of your engine as a function of the mass of weight that is lifted (mwt) for a
range of mwt from 0 kg to 84 kg. You should see an optimal value of mwt - can you think of
any reason for this?
r.) If the engine operates with a frequency of f = 5 cycles/s then what is the average rate at which
work is produced; that is, what is the power produced by the engine (W)?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-4
A piston-cylinder device is used to fill a balloon with helium, as shown in Figure 3.B-4.
Dp = 3 mm
P1 = 103 kPa
T1 = 22°C
z1 = 5 cm
Db,1 = 3.5 mm
z2
Db,2 = 7 mm
Figure 3.B-4: Piston-cylinder device used to fill a balloon.
The piston diameter is Dp = 3 mm and the piston position is initially z1 = 5 cm from the bottom of
the cylinder. At this point, the balloon diameter is Db,1 = 3.5 mm and the pressure and
temperature of the helium in the piston and the balloon are P1 = 103 kPa and T1 = 22°C,
respectively. The balloon stretches as it is filled and so the pressure in the balloon increases. The
relationship between the pressure within the balloon and the volume of the balloon is given by:
P = Patm + K b Vb
where Patm is atmospheric pressure (1 atm) and Kb is a constant. The final diameter of the balloon
is Db,2 = 7 mm. The temperature of the helium in the piston is maintained at Tin = T1 during the
process (i.e., all of the helium leaving the piston and entering the balloon is at Tin). The final
temperature of the helium in the balloon is T2 = 30°C. Model helium as an ideal gas with R =
2076.9 J/kg-K. Assume that the piston and balloon are both leak-tight. You may ignore the
volume of the tube connecting the balloon to the piston. The pressure in the piston and the
pressure in the balloon are always the same.
a.) What is the work done by the helium on the balloon during the filling process?
b.) What is the mass of helium that was added to the balloon in order to inflate it?
c.) What is the final position of the piston (z2 in Figure 3.B-4)?
d.) What is the heat transfer from the balloon to the surroundings?
e.) Plot the diameter of the inflated balloon as a function of the final position of the piston.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-5
Figure 3.B-5 illustrates the operation of a very small, reciprocating compressor that operates at a
low frequency. This "syringe-type" compressor is being designed to compress air for a medical
application. Assume that air behaves as an ideal gas (R = 287 J/kg-K) with a constant specific
heat capacity at constant volume (cv = 718 J/kg-K). Note that there is no substance in EES that
assumes ideal gas behavior with constant specific heat capacity - you cannot use the substance
'Air' or 'Air_ha' or the property routines in EES in order to solve this problem.
state 1
Vmax = 1 cc
Tin = 20°C
Pin = 1 atm
compression
state 2
T=Tin
P = Pin +ΔP
exhaust
valve
intake
valve
`
intake
exhaust
out flow
in flow
`
state 4
Tin = 20°C
P = Pin
expansion
state 3
T=Tin
P = Pin + ΔP
Vcl = 0.1 cc
Figure 3.B-5: Syringe compressor operation.
The compressor undergoes the cycle shown in Figure P3.B-5. At state 1 the compressor piston is
drawn all the way up so that the volume in the piston is at a maximum, Vmax = 1 cm3. The piston
is filled with air that has been pulled through the inlet check valve. The air is at Tin = 20ºC and
Pin = 1 atm.
a.) Determine the mass of air in the compressor at state 1 (mg).
During the compression stroke from state 1 to state 2, the piston moves downwards causing the
pressure in the air to rise. In a typical compressor, this process occurs nearly adiabatically and
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
therefore the temperature of the air rises. However, in this very small and low frequency
compressor it is appropriate to model the process as being isothermal due to heat transfer from the
air to the walls of the piston. That is, the temperature of the air in the piston can be assumed to be
constant during the compression stroke. The compression stroke continues until the pressure of
the air increases to Pin + ΔP where ΔP = 50 psi is the pressure rise produced by the compressor.
The piston does not leak.
b.) Determine the volume in the cylinder at the conclusion of the compression stroke, V2 (cm3).
c.) Determine the work transfer from the piston to the air during the compression process, W1-2
(J).
d.) What is the heat transfer from the air during the compression stroke, Q1-2 (J)?
During the exhaust stroke from state 2 to state 3 the exhaust valve is opened and the piston is
pushed down until the clearance volume in the device is reached, V3 = Vcl = 0.1 cm3. Mass is
pushed out of the exhaust valve at high pressure. The pressure in the piston is constant during
this process (we are assuming that the exhaust valve is perfect and so no pressure drop is
associated with the flow). Again, heat transfer keeps the temperature always at Tin during the
process.
e.) Determine the work done to the air by the piston during the exhaust stroke, W2-3 (J)?
f.) Determine the amount of high pressure air pushed out of the compressor during the exhaust
stroke, mout (mg).
During the expansion stroke from state 3 to state 4 the exhaust valve is closed and therefore the
mass of air in the piston remains constant. The piston is retracted until the pressure reaches the
inlet pressure, P4 = Pin. Again, heat transfer keeps the temperature always at Tin during the
process.
g.) Determine the volume of the piston at the conclusion of the expansion stroke, V4 (cm3).
h.) Determine the work done by the air to the piston during the expansion stroke, W3-4 (J).
During the intake stroke, the piston is moved back to its original position, V1 = Vmax. The intake
valve is opened and air at pressure Pin and Tin is drawn through the valve. The temperature
remains constant at Tin and the pressure remains constant at Pin during this process.
i.) What is the work transfer from the air to the piston during the intake stroke, W4-1 (J)?
j.) What is the net work required to run the compressor over a cycle, Wnet (J)? The net work is
the sum of the work into the system less the work out of the system for each of the processes.
k.) The frequency of the compressor is f = 1 Hz; that is, one cycle is executed per second.
Determine the average power required to run the pump (mW).
l.) What is the average mass flow rate produced by the pump (mg/s)?
m.) Prepare a pump curve for this pump. A pump curve shows the pressure rise (psi) as a
function of mass flow rate (mg/s). Your pump curve should go from dead-head (i.e., zero
mass flow rate) to open circuit (i.e., zero pressure rise).
n.) Overlay on your pump curve from (m) the power required by the pump (in mW) as a function
of the mass flow rate.
p.) On a single plot, overlay a family of pump curves, each with different values of the clearance
volume.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-6
At the end of a manufacturing process, you are left with small pieces of scrap metal that are
relatively hot, TH = 700 K, and have mass mb = 2.0 kg. Rather than just throw the metal pieces
away, you have come up with an idea for a simple machine that you hope can produce useful
work. The machine is a piston-cylinder device that is filled with air and it goes through four
processes in order to lift a mass (i.e., produce useful work). The processes take the air in the
piston from state 1 to state 2 to state 3 to state 4 and then back to state 1. The cumulative effect
of the four processes is referred to as a cycle. Model the metal pieces as incompressible with a
constant specific heat capacity, cb = 900 J/kg-K. Model the air as an ideal gas with R = 287.1 Nm/kg-K and cv = 717.6 J/kg-K.
The piston-cylinder device is initially filled with air at atmospheric pressure, P1 = Patm = 1 atm
and ambient temperature T1 = Tamb = 300 K. The piston area is Ac = 0.05 m2 and the distance
between the piston and the bottom of the cylinder is initially z1 = 0.5 m. The piston is massless,
frictionless, and leak-tight. A mass, mp = 100 kg, is slowly placed on the piston so that the piston
moves downward. During this process, the piston is transfers heat to the ambient atmosphere and
therefore the air in the piston is compressed isothermally (i.e., the temperature of the air in the
piston during this process is always T = Tamb). This process of going from state 1 to state 2 is
shown in Figure 3.B-6(a).
Process 1
(1) to (2)
Po = 1 atm
air
state (1)
T1 = Tamb
P1 = Po
z1 = 0.5 m
mp = 100 kg
massless,
frictionless
piston
piston
A = 0.05 m2
state (2)
T2 = T1
z2
Figure 3.B-6(a): Process of going from state 1 to state 2.
a.) What is the final pressure of the air, P2? What is the final position of the piston, z2?
b.) Determine the work done by the air, W12; note that this quantity may be negative if work is
transferred to the air.
c.) Determine the heat transfer to the air, Q12; note that this quantity may be negative if heat is
transferred from the air.
Next, one of the pieces of hot scrap metal is brought into thermal contact with the piston; this
causes a heat transfer from the metal to the air in the piston. The air temperature rises while the
metal temperature drops and this continues until the air and metal come to the same temperature,
T3. The increase in the air temperature causes the piston to rise, lifting the weight. Assume that
there is no heat transfer with ambient atmosphere during this process (i.e., the only heat transfer is
from the metal piece to the air in the cylinder). This process of going from state 2 to state 3 is
shown in Figure 3.B-6(b).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
Process 2
state 2 to state 3
state 3
T3 = metal
temperature
state 2
scrap metal
piece at T = TH
z2
z3
scrap metal
piece at T = T3
Figure 3.B-6(b): Process of going from state 2 to state 3.
d.)
e.)
f.)
g.)
What is the final temperature of the air and the scrap metal piece, T3?
What is the work done by the air, W23?
What is the heat transfer from the scrap metal piece to the air, Q23?
What is the final position of the piston, z3?
Next, the piston is locked in place and the mass is removed from the piston. The scrap metal
piece is taken out of thermal contact with the piston. With the piston locked in place, the air in
the piston is allowed to cool by transferring heat to the ambient atmosphere until the pressure in
the piston reaches atmospheric pressure, P4 = Patm. This process of going from state 3 to state 4 is
shown in Figure 3.B-6(c).
Process 3
state 3 to state 4
state 4
P4 = Patm
state 3
z4 = z3
z3
Figure 3.B-6(c): Process of going from state 3 to state 4.
h.) What is the final temperature of the air, T4?
i.) What is the work done by the air, W34?
j.) What is the heat transfer to the air, Q34?
Finally, the piston is unlocked and allowed to float freely (note that the mass has been removed)
and the air in the piston is allowed to cool by transferring heat to the ambient environment until it
returns to state 1, P1 = Patm and T1 = Tamb. This process of going from state 4 to state 1 is shown
in Figure 3.B-6(d).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
Process 4
(4) to (1)
state (4)
state (1)
P1 = Po
T1 = Tamb
z4
z1
Figure 3.B-6(d): Process of going from state 4 to state 1.
k.) What is the work done by the air, W41?
l.) What is the heat transfer to the air, Q41?
Note that your machine has gone through a cycle - the final state of the air is equal to its initial
state and so it is ready to go through the cycle again.
m.) Sketch states 1 through 4 on a T-v diagram for air (or use EES to draw one for you).
n.) What is the net work done by your machine (i.e., what is the sum of the work terms done by
the air that you calculated for each process)?
o.) If you can run your system at a frequency of f = 5 Hz (i.e., the piston goes back and forth 5x
per second) then what is the average power produced by the machine?
p.) One way to double-check your answer is to calculate the net heat transfer to the air (the sum
of the heat transfers to air that you calculated for each process) and compare it with the net
work done by the air (the sum of the work done by the air that you calculated for each
process). These two quantities ought to be the same - are they? Why should these quantities
be equal?
q.) The efficiency of your machine is defined as the ratio of what you get out, the net work
calculated in (n), to what you put in, the heat transfer from the metal calculated in (f). What
is the efficiency of your machine?
r.) Prepare a plot showing the efficiency of your machine as a function of the initial temperature
of the scrap metal, TH.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-7
The refrigerant R134a is held in the piston/cylinder apparatus shown in Figure 3.B-7.
spring
saturated liquid R134a
T1 = 20°C, V1 = 1 m3
Figure 3.B-7: Piston/cylinder apparatus.
The piston is connected to a spring, as shown. As the piston is pushed up, the spring exerts an
increasing force on the piston. As a result, the pressure within the cylinder is a linear function of
the volume of the R134a:
P = KV
where K is a constant associated with the spring. Initially, the cylinder contains saturated liquid
R134a at T1 = 20°C. The initial volume of the R134a is V1 = 1 m3.
a.) Locate state 1 on a T-v sketch. This qualitative sketch does not need to be accurate, but it
should clearly show the two properties that define the state.
b.) Determine the initial pressure in the cylinder and the constant K.
c.) Determine the mass of R134a in the cylinder.
Heat is added to the R134a until the pressure in the cylinder increases to P2 = 20 bar.
d.) Determine the final volume of the R134a, V2.
e.) Sketch state 2 on the T-v diagram from part (a). This qualitative sketch does not need to be
accurate, but it should clearly show the two properties that define the state. Determine the
final temperature of the R134a, T2.
f.) Determine the heat transfer required to accomplish the process, Q12.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-8
Water is held in a piston/cylinder apparatus. The pressure in the cylinder is initially P1 = 1.5 bar
and the temperature is T1 = 150°C. The volume of water contained in the cylinder is V1 = 1 m3.
a.) On a T-v sketch, locate state 1. This qualitative sketch does not need to be accurate, but it
should clearly show the two properties that define the state.
b.) Determine the mass of water in the cylinder (kg).
The piston is pushed in so that the volume of the water is reduced until liquid water droplets just
start to form in the cylinder at state 2. During this process, the contents of the cylinder are
maintained at a constant temperature (T2 = 150°C) by heat transfer with the surroundings.
c.) On the T-v sketch from (a), locate state 2. This qualitative sketch does not need to be
accurate, but it should clearly show the two properties that define the state.
d.) What is the volume in the cylinder (m3) at the instant that liquid water begins to form?
The piston is pushed in further, reducing the volume to V3 = 0.05 m3 at state 3. During this
process, the contents of the cylinder are maintained at a constant temperature (T3 = 150°C) by
heat transfer with the surroundings.
e.) On the T-v sketch from (a), locate state 3. This qualitative sketch does not need to be
accurate, but it should clearly show the two properties that define the state.
f.) What is the pressure in the cylinder at state 3?
g.) What is the volume of liquid in the cylinder at state 3?
h.) Determine the heat transfer required to maintain the contents of the cylinder at 150ºC as the
piston is pushed in from state 2 to state 3 (kJ). Be sure to indicate clearly whether the heat
transfer that you calculate is into or out of the water.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-9
Consider the piston-cylinder device shown in Figure 3.B-9. The diameter of the piston is Dp =
0.10 m. Initially, the piston is resting on a set of stops and the distance from the bottom of the
cylinder is z = 0.1 m. The entire apparatus is at T1 = 25°C and in thermal equilibrium with its
surroundings. The cylinder is evacuated except for a spherical capsule having an inner diameter
of Dc = 2.5 cm containing carbon dioxide at T1 and a known high pressure, P1. The piston has a
mass of mp = 75 kg. The capsule then ruptures and the carbon dioxide is rapidly released into
the cylinder.
Tamb = 25°C
Patm = 1 atm
mp = 75 kg
Dp = 0.1 m
carbon dioxide
at T1 and P1
z = 0.1 m
Dc = 2.5 cm
Figure 3.B-9: Piston-cylinder apparatus with capsule containing CO2
a.) Prepare a plot indicating the work done by the piston-cylinder apparatus during this process
as a function of the initial pressure of the carbon dioxide in the capsule for pressures ranging
1 MPa < P1 < 10 MPa. Clearly indicate your system and state any assumptions you employ.
Note that carbon dioxide does not obey the ideal gas law at the conditions it exists in the
capsule.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B.10 A quantity of air is contained in a cylinder by a movable piston as shown in Figure 3.B-10. After
the latch holding the piston in place is removed, the air expands slowly (because of friction
between the piston and cylinder wall) from a volume of V = 0.025 m3 and a pressure of P = 6
1
1
bar to a volume of V2 = 0.050 m3 at which point the piston encounters another latch. There is a
complete vacuum on the left side of the piston, as shown in Figure 3.B-10. The cylinder is wellinsulated on its outer surface. Thermal energy is freely exchanged between the piston, the
cylinder and the air. The mass and specific heat of the piston and cylinder are not known.
Latch
Evacuated space
Air
Piston
Figure 3.B-10: Piston-cylinder apparatus.
a.) Calculate the temperature change and final pressure of the air in the cylinder and estimate the
heat and work effects.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-11 Figure P3.B-11 shows m = 0.5 g of hydrogen gas contained within a piston-cylinder
assembly that is fitted with an electrical heater.
Patm = 1 atm
Tamb = 25°C
spring
K = 9000 N/m
x2 = 0.06 m
x1 = 0 m
mcyl = 0.25 kg
ρs = 8000 kg/m3
hydrogen
m = 0.5 g
T1 = Tamb
state 1
mp = 10 kg
Ac = 0.0078 m2
state 2
cs = 480 J/kg-K
Figure P3.B-11: Piston-cylinder apparatus with electrical heater.
The cylinder and piston are made of steel (ρs = 8000 kg/m3 and cs = 0.48 kJ/kg-K). The entire
apparatus is well-insulated on its outside surfaces from the surroundings which are at Patm = 1 bar
and Tamb = 25°C throughout this process. The apparatus and gas are initially at T1 = Tamb. The
cross-sectional area of the piston is Ac = 0.0078 m2 and its mass is mp = 10 kg. The mass of the
cylinder is mcyl = 84 kg. Initially, the piston face is at position x1 = 0 and the spring exerts no
force on the piston. The spring constant is K = 9,000 N/m. The electrical heater is engaged and
the gas then expands raising the piston until the position of the piston is x2 = 0.06 m. At this
point, the electrical heater is turned off. You may assume frictional effects are negligible.
Assume that hydrogen obeys the ideal gas law but does not have a constant specific heat capacity
at constant volume.
a.) What is the initial pressure and volume of the hydrogen before the heating is started.
b.) Determine the work done by the hydrogen and the amount of electrical energy required if the
process occurs quickly so that there is negligible time to allow heat transfer between the
hydrogen gas and the surrounding metal piston and cylinder walls.
c.) Determine the work done by the hydrogen and the amount of electrical energy required if the
heating process is done slowly such that, at the final state, the hydrogen gas is in thermal
equilibrium with the surrounding metal piston and cylinder walls.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-12 Immediately after a high pressure tank of air is filled, the air in the tank is hot, Ta,1 = 200°C, and
the tank material itself remains at room temperature, Tt,1 = 20°C. The valve on the tank is shut
and the volume of air in the tank is Va = 5 liter. The initial pressure of the air is P1 = 350 psi. The
tank material has a mass of mt = 0.45 kg and a specific heat capacity of ct = 471.9 J/kg-K. Model
the air as an ideal gas, R = 287 J/kg-K, with constant specific heat capacity at constant specific
volume, cv = 726 J/kg-K. There is a transfer of heat between the tank material and the air that
continues until they come to the same temperature, T2. Assume that there is no heat transfer from
the tank to the surroundings during this process.
a.) Determine the final temperature of the air and the tank and the final pressure of the air.
b.) What is the heat transfer to the tank that occurs during this process?
Eventually, the air and the tank both return to room temperature, T3 = 20°C due to heat transfer
with the surroundings.
c.) Determine the final pressure of the air.
d.) What is the heat transfer to the surroundings that occurs during this process?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-13 An emergency flotation device is made by attaching a high pressure canister of air to a balloon
via a valve. Initially, the balloon is deflated (i.e., its volume is zero) and the canister is
pressurized to Pc,1 = 6000 psi. The canister has inner radius Rc = 2.5 cm and length Lc = 20 cm.
In order to activate the flotation device, the valve is opened allowing air to flow into the balloon
causing it to inflate. The internal pressure in the balloon is higher than atmospheric pressure due
to the tension in the balloon material. The internal pressure is given by:
Pb = Patm + K b Vb
where Patm = 100 kPa is the atmospheric pressure, Vb is the balloon volume, and Kb = 1x106 N/m5.
The inflation process is complete when the pressure within the canister and the pressure within
the balloon are the same. Assume that the air within the canister and the balloon is maintained at
Tatm = 15ºC by heat transfer with the surroundings. Model air as an ideal gas.
a.) Determine the final radius of the balloon.
b.) What is the work done by the air on the balloon? What is the heat transfer from the
surroundings to the air?
c.) Determine the buoyancy force associated with the flotation device once it is activated.
Assume that the density of water is ρw = 1000 kg/m3.
d.) Plot the buoyancy force produced by the flotation device as a function of the initial pressure
in the canister.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-14 Figure 3.B-14 illustrates a piston cylinder device that contains water that is initially at T1 = 400°C
and P1 = 1000 kPa (10 bar).
g = 9.81 m/s2
Patm = 100 kPa
stops
water
T1 = 400°C
P1 = 1000 kPa
piston
Ac = 0.01 m2
mp = 102 kg
L1 = 1 m
Figure 3.B-14: Piston/cylinder apparatus.
Initially, the piston is pressed up against a set of stops by the pressure within the cylinder, as
shown. The atmospheric pressure is Patm = 100 kPa. The cross sectional area of the piston is Ac =
0.01 m2 and the mass of the piston is mp = 102 kg. The acceleration of gravity is g = 9.81 m/s2.
The initial position of the piston is L1 = 1 m.
a.) On a T-v sketch, locate state 1. This qualitative sketch does not need to be accurate, but it
should clearly show the two properties that define the state.
b.) What is the mass of water in the cylinder?
The cylinder is cooled (heat is removed) and therefore the pressure in the cylinder drops. At state
2, the piston just starts to move away from the stops.
c.) Determine the pressure in the cylinder at state 2.
d.) Locate state 2 on the T-v diagram from (a).
e.) Determine the temperature in the cylinder at state 2.
f.) At what temperature during the cooling process does liquid water start to form in the
cylinder?
g.) Determine the heat transfer from the water that occurs between states 1 and 2.
Additional heat is removed from the piston until the system reaches state 3 where the position of
the piston has been reduced to L3 = 0.2 m.
h.) Locate state 3 on the T-v diagram from (a).
i.) Determine the heat transfer from the water that occurs between states 2 and 3.
j.) What is the volume of liquid water at state 3?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-15 A flexible bag contains steam at T1 = 500°F and P1 = 15 psia. The pressure within the bag is
independent of its volume. The initial volume of the bag is V1 = 19 ft3. The bag and its contents
are cooled at constant pressure until the volume is reduced to V2 = 6.7 ft3.
a.) What is the mass of steam in the bag?
b.) What is the state (temperature, pressure, quality) of the steam after cooling?
c.) Determine the heat transfer from the steam.
d.) Plot the initial and final states on a temperature-volume plot. Label the states.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-16
The piston-cylinder apparatus shown in Figure 3.B-16 contains air that is initially at T1 = 25°C
and P1 = 105 kPa. The piston is initially z1 = 0.64 m above the cylinder bottom and is held in
place by a lock. The cross-sectional area of the piston is Ac = 0.05 m2. A weight is placed on the
piston. The lock is released and the piston falls. After a period of time, heat transfer between the
air in the cylinder and the surroundings (which are at Tamb = 25°C and Patm= 101.3 kPa) restores
the air temperature to T2 = 25°C. At this point, the piston is z2 = 0.58 m above the cylinder
bottom, as shown in Figure 3.B-16.
Tamb = 25°C,
Pamb = 101.3 kPa
weight
piston, Ac = 0.05 m2
lock
air
T1 = 25°C
P1 = 105 kPa
z1 = 0.64 m
Figure 3.B-16: Piston-cylinder apparatus.
a.) Determine the combined mass of the piston and weight.
b.) Determine the work done on the air during this process.
c.) Determine the heat transfer to the air during this process.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-17 An elevator is shown in Figure 3.B-17. The air in the V = 30 m3 tank is initially at T1 = 25°C and
P1 = 100 kPa. The piston has mass mp = 230 kg and diameter Dp = 0.75 m. A casting with mass
mc = 1000 kg is slid onto the platform at its lower level (level 1). Then saturated steam at Ts =
150°C is provided to raise the temperature of the air in the tank by heat exchange. The heating
continues until the platform reaches its upper level (level 2) that is z = 6 m above the lower level,
at which point the platform hits a stop and heating is stopped. Here, the casting is slid off of the
platform. Then, cooling water is provided to lower the air temperature until the platform returns
to the initial level where the piston rests on a stop. Cooling continues until the air in the tank is
returned to T1. Please answer the following questions and state any assumptions that you employ.
z=6m
casting
mc = 1000 kg
piston
mp = 230 kg
Dp = 0.75 m
saturated steam
at Ts = 150°C
cooling water
air
V = 30 m3
condensate
Figure 3.B-17: Heat powered elevator.
a.) Determine the temperature and pressure of the air in the tank when the platform just reaches
level 2.
b.) Determine the work done by the air in raising the platform and casting from level 1 to level 2.
c.) What is the heat transfer to the air from the steam while raising the platform and casting from
level 1 to level 2?
d.) What is the heat transfer between the air and the cooling water during the process in which
the platform returns from level 2 to level 1?
e.) What is the overall efficiency of the elevator for the completion of one cycle, lifting the
casting to level 2 and returning to level 1?
f.) What is the maximum casting mass that can be lifted by this elevator?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.B-18 A cheese plant has a number of large storage tanks that are used to store milk and other liquids.
The volume of these tanks must be accurately known. For one of the tanks, however, company
records show two significantly different volumes. Obviously one (or both) of these figures are
wrong and so it is necessary to determine the volume of this tank. An estimate of the tank
volume could be obtained by measuring the external dimensions of the tank, but the tank is buried
in insulation which the company would rather not remove. One of the engineers at the plant has
implemented the following plan. She used a vacuum pump to evacuate this tank to a very low
pressure. Then, she used existing piping and a valve to connect the tank to a second tank of
volume VA = 83.4 m3 containing air at PA,1 = 100 kPa and TA,1 = 300 K. She opened the valve and
found that, shortly after the pressure in the two tanks equilibrated, the pressure was PA,2 = 72.3
kPa.
a.) Based on this information, what is your estimate of the tank volume. (Hint: Assume a
reasonable value for the temperature of the air in one of the two tanks and see how your
results are affected by the assumption. State any assumptions you employ.)
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
C: Advanced Problems
3.C-1
A piston, latched into place within a well-insulated cylinder (shown in Figure 3.C-1) encloses
helium initially at a pressure of PHe,1 = 400 kPa and a temperature of T1 = 290 K. The inside
diameter of the cylinder is d = 0.2 m. A compressed spring is located in the right compartment of
the cylinder which contains air, initially at Pair,1 = 100 kPa and T1. The piston and cylinder are
made of th = 5 mm stainless steel having a density of ρ = 8,000 kg/m3 and a specific heat
ss
capacity of css = 480 J/kg-K. The spring exerts a force towards the left of a magnitude F = K x
where the spring constant K = 4,500 N/m and x is the distance from the far left side of the
cylinder. The piston is initially latched in place at location x1 = 0.30 m. The total length of the
cylinder is L = 1.2 m. The latch is released, allowing the piston to move. You may assume that
the air and the helium both obey the ideal gas law and have constant specific heat capacity values.
The volume occupied by the spring is negligible. State and justify any other assumptions that you
employ.
helium
PHe,1 = 400 kPa
T1 = 290 K
air
Pair,1 = 100 kPa
T1
x1 = 0.3 m
L = 1.2 m
Figure 3.C-1 Piston and spring in cylinder with helium.
a.) Describe what an observer would see when the piston latch is released. Explain these
observations using words and equations. You do not need to solve these equations.
b.) Assuming that the helium and air have no heat interaction with the piston or cylinder,
estimate the equilibrium value of x and the temperature and pressure of the helium and the
air.
c.) Repeat the calculations for part (b) assuming that there is a heat interaction between the
cylinder and piston walls with the helium and air.
d.) What role does friction play in your answers to questions (b) and (c)? Are frictional effects
considered in your analysis? How would your answers to (b) and (c) change if the friction
were increased or decreased?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-2
An aluminum cylinder has an inner diameter of D = 12 cm. The cylinder initially contains Vini =
4 liters of air under a piston that is held in place by a pin, as shown in Figure 3.C-2. The cylinder
and its contents are initially at Tini = 25°C and Pini = 100 kPa. Note that the thermal capacitance
of the cylinder is much higher than that of the contained air. Consequently, you may neglect the
small temperature variation of the tank wall. A small vent in the cylinder connects the space
above the piston to the atmosphere, which is at Tatm = 25°C and Patm = 100 kPa. The piston has a
mass of mpw = 10 kg. The pin is now removed and the piston moves in an oscillatory manner.
The heat transfer coefficient between the air and the cylinder surfaces is estimated to be hconv = 20
W/m2-K acting on the inside surface area of the cylinder.
Tatm = 25°C
Patm = 100 kPa
air
Vini = 4 liter
Tini = 25°C
Pini = 100 kPa
piston
mpw = 10 kg
D = 12 cm
Figure 3.C-2: Cylinder with air and locked piston and weight.
a.) Using the above information and your engineering expertise, prepare plots showing the
elevation of the piston above the bottom of the cylinder, the temperature of the air, and the
pressure of the air as a function of time for a 3 second period following the release of the
piston pin.
b.) Repeat part (a) assuming that there is no heat transfer between the air and inside cylinder
surfaces.
c.) Repeat part (a) assuming that there is a frictional force of 0.5 N between the piston and the
cylinder acting in a direction that resists the motion of the piston. One way to simulate the
direction of the frictional force in EES is to use the Sign function applied to the velocity, e.g.,
Friction = 0.5 [N]*sign(Vel).
d.) What is the period of oscillation for part (a)?
e.) What is the observed effect of heat transfer? How is the period affected by the thermal
interaction with the cylinder wall?
f.) What is the observed effect of friction? How is the period affected by friction?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-3
The purpose of this problem is to analyze the dynamic behavior of a vertical piston-cylinder
device containing a gas. The gas is placed in the cylinder at an initial pressure that is above
atmospheric and the piston is locked in place. In a particular case, the cylinder contains carbon
dioxide gas initially at Tini = 25°C and Pini = 1.5 bar. Assume that the carbon dioxide obeys the
ideal gas law (i.e., if you are using EES, then the fluid name should be 'CO2' rather than
'CarbonDioxide'). The internal radius of the cylinder is R = 0.05 m. The piston, which has a mass
of mp = 20 kg, is initially locked at a position Hini = 0.15 m above the bottom of the cylinder. The
piston and cylinder are made of metal and initially this metal is also at Tini. When the piston locks
are released, the piston moves in an oscillatory manner that is damped by frictional effects.
Prepare an analysis of this experiment. Use your analysis to calculate and plot the position of the
piston above the bottom of the cylinder for a simulation time of tsim = 2 s for the following cases:
a.) No friction and no heat exchange between gas and metal. Determine the frequency of the
oscillations.
b.) Include frictional effects (without heat transfer) by setting the frictional force resisting piston
motion to a value of Ff = 10 [N-s/m] Vp where Vp is the instantaneous piston velocity.
Indicate how the amplitude and frequency are affected by frictional effects based on
examination of your plot.
c..) Include heat transfer between the gas and cylinder walls with a convection heat transfer
coefficient of hconv = 100 W/m2-K. Ignore friction for this simulation. You may assume that
the wall temperature remains constant at Tini due to its large thermal mass. Indicate how the
amplitude and frequency are affected by heat transfer based on examination of your plot.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-4
As part of team that is building a new compressor, you have been asked to analyze the
dynamic behavior of a vertical piston-cylinder device containing refrigerant R134a. The
internal radius of the cylinder is R = 0.025 m. A weight is attached to the top of the piston to
give it an equivalent mass of mp = 2.5 kg. The piston is initially locked at a position where its
lower edge is Lo = 0.1 m above the bottom of the cylinder. The piston and cylinder are made
of stainless steel and are initially at To = 25°C. The refrigerant is charged into the cylinder so
that it initially is at To and Po = 2.5 bar. When the piston lock is released, the piston moves in
an oscillatory manner. Develop a model that can predict (i) the position of the piston above
the bottom of the cylinder; (ii) the R134a pressure, and (iii) the R134a temperature for a
period of two seconds. Plot the piston position, pressure and temperature versus time on
separate plots for the following cases. Also plot the R134a temperature versus pressure.
Overlay the plots for each case to facilitate comparisons:
a.) No friction, no heat exchange and metal. Determine the frequency of the oscillations.
b.) Include frictional effects (without heat transfer) by setting the frictional force resisting piston
motion to a value of 2.5 [N/m-s]*(Piston velocity).
c.) Include heat transfer between the gas and cylinder walls with a convection coefficient of h =
200 W/m2-K (without friction). You may assume that the wall temperature remains constant
at its initial temperature due to its large thermal mass.
d.) Include the frictional effects from part (b) and the heat transfer from part (c).
e.) Explain the results observed in the plots. Indicate how the amplitude and frequency are
affected by heat transfer and friction. Explain the observed behavior of the temperature
versus pressure plot for the different cases.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-5
A glass jar having a volume of V = 8 liters is fitted with a stopper through which a glass tube
having an internal radius of R = 0.8 cm is inserted, as shown in Figure 3.C-5.
Patm = 758.7 mm Hg
tube
R = 0.8 cm
ball
mball = 15 g
Argon
Vo = 8 liter
To = 24°C
Figure 3.C-5: Apparatus.
The glass tube is open to the atmosphere, which is at To = 24°C and Patm = 758.7 mm Hg. Within
the glass tube is a tightly fitted ball having a mass of mball = 15 g. Initially, the jar contains argon
gas at To and an equilibrium pressure that just supports the ball. However, this equilibrium
condition is disturbed by forcing the ball down Lo = -1 cm (i.e., the ball is moved into the tube
towards the jar). When the ball is released, it oscillates up and down in the tube until frictional
effects eventually bring it to rest. This process occurs over a short period of time with very small
temperature differences so that it is reasonable to assume the process is adiabatic. You may
assume ideal gas behavior and constant properties. State any other assumptions you employ.
a.) What is the equilibrium pressure in the jar?
b.) Assuming friction to be negligible, calculate and plot the position of the ball as a function of
time for 5 seconds. Determine the period of oscillation from your plot.
c.) There will of course be some friction between the ball and the tube. What effect will friction
have on this experiment? In particular, how will it effect the period of oscillation?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-6
An insulated, cylindrical steel tank having a diameter of D = 0.30 m and a height of H = 1.0 m
is to be used to transport liquid nitrogen to a manufacturing facility. The wall thickness of the
tank is tht = 0.75 cm. The tank material is composed of steel with properties ρt = 7850 kg/m3,
ct = 0.45 kJ/kg-K, and kt = 64 W/m-K. The tank is insulated on the top, bottom and sides with
a rigid polystyrene insulation. The properties of the insulation are ρins = 56 kg/m3, kins = 0.025
W/m-K, and cins = 0.75 kJ/kg-K. The tank is equipped with a pressure relief valve that opens
at Prv = 200 kPa. The tank is filled to f = 80% by volume with liquid nitrogen at P1 = 1 bar
with the remainder of the tank volume containing gaseous nitrogen in equilibrium with the
liquid. The tank is to be shipped in to the facility with an expected delivery time of tship = 4
hours. The outdoor temperature is Tamb = 25°C. The convection heat transfer coefficient
between the liquid nitrogen and the tank wall is hconv,in = 750 W/m2-K and the convection heat
transfer coefficient between the surroundings and the insulation is hconv,out = 15 W/m2-K.
a.) Determine the minimum amount of insulation needed to prevent the pressure relief valve
from opening during the transport period. State and justify any assumptions you employ.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-7
A stainless-steel cylinder with a mass of mcylinder = 95 kg is fitted with a mpiston = 50 kg
stainless steel piston, as shown in Figure 3.C-7.
Patm = 1 atm
Tamb = 20°C
mass
air
V1 = 0.3 m3
P1 = 7 atm
T1 = 20°C
piston
Ac = 0.15 m2
mpiston = 50 kg
cylinder
mcylinder = 95 kg
Figure 3.C-7: Piston cylinder apparatus.
The cylinder contains V1 = 0.30 m3 of air initially at P1 = 7 atm and T1 = 20ºC. The exterior of
the cylinder and piston are heavily insulated. The piston, which is locked in place, has a
cross-sectional area of Ac = 0.15 m2. A mass is placed on the piston, as shown in the figure.
The surroundings are at Patm = 1 atm and Tamb = 20ºC. When the lock is removed, the piston
rises to a new equilibrium position, lifting the mass and thereby doing some useful work.
(Useful work refers to the elevation of the mass, excluding the piston.) If the mass is small,
relatively little useful work will be done during the piston motion. On the other hand, if the
mass is large, the piston will rise only a small amount, or perhaps not at all, and therefore little
or no useful work will be done. You may assume air to behave as an ideal gas. The specific
heat of stainless steel is cs = 0.48 kJ/kg-K.
a.) Determine the mass that will result in the maximum amount of useful work and the useful
work and final temperature and pressure of the air in the insulated cylinder corresponding
to this mass. Neglect any heat interaction between the cylinder and piston surfaces with
the air for this calculation.
b.) Repeat part (a) but consider thermal equilibrium between the cylinder and piston surfaces
with the air.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-8
The cylinder of a reciprocating refrigeration compressor initially contains mR134a = 0.042 kg
of refrigerant R134a at P1 = 290 kPa. The initial cylinder volume is V1 = 3.229 liter. The
refrigerant is compressed to P2 = 1 MPa. The final cylinder volume is V2 = 0.9613 liter. The
compressor is turning at N = 1520 revolutions per minute. Simultaneous measurements of
the pressure in the cylinder and the cylinder volume are provided in Table 3.C-8. The
compressor is tested in a room containing air at Tamb = 25°C.
Table 3.C-8. Pressure volume data for a refrigeration compressor.
Pressure
Volume
(kPa)
(liter)
290
3.229
375
2.552
441
2.122
533
1.816
608
1.584
694
1.405
754
1.262
842
1.144
918
1.045
1000
0.9613
a.) What are the temperatures of the R134a at the start and end of the compression process?
b.) Compression processes can often be represented by assuming PVn = constant where n is
called the polytropic index and C is a constant. Estimate the polytropic index for this
compression process.
c.) Calculate the work provided to the refrigerant during the compression process assuming that
the relation between pressure and volume during the process is represented by PVn = C.
d.) Calculate the work provided to the refrigerant by integration the experimental measurements
of pressure and volume. Compare the answer with the result in part (c).
e.) Determine the heat transfer between the refrigerant and its surroundings during the process.
f.) Calculate the temperature of the R134a as a function of specific volume during the
compression process. Overlay your calculated results on a temperature-specific volume
property plot for R134a.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-9
A two-phase liquid-vapor mixture of water with an initial quality of x1 = 0.25 is contained in a
piston-cylinder assembly as shown in Figure 3.C-9. The cylinder is well-insulated on its outside
surface. The mass of the piston is mpiston = 40 kg and its diameter is D = 10 cm. The atmospheric
pressure is Patm = 1 bar. The initial and final positions of the piston are z1 = 1 cm and z2 = 4 cm,
respectively, above the bottom of the cylinder. The piston rises as the water is electrically heated
until it hits the piston stops. Heating continues until the pressure of the water is P2 = 3 bar. The
total mass of the cylinder is mcylinder = 60 kg and it is made out of AISI302 stainless steel. The
piston is made from the same material. Neglect friction between the piston and cylinder.
Patm = 1 atm
z2 = 4 cm
piston
D = 10 cm
mpiston = 40 kg
water
x1 = 0.25
z1 = 1 cm
cylinder
mcylinder = 60 kg
electrical heater
Figure 3.C-9: Piston-cylinder apparatus with electrical heater containing water.
a.) Determine the final temperature of the water and all energy transfers, assuming that the
piston and cylinder material remain in thermal equilibrium with the water.
b.) Show the process on T-v and P-v diagrams.
c.) Repeat part (a) assuming that the there is no heat transfer between the water and the piston or
cylinder.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-10 The experiment shown in Figure 3.C-10 attempts to determine the constant relating mechanical
and thermal energy transfer. Air, initially at T1 = 75°F and P1 = 15 psi, is contained in a wellinsulated cylinder that has diameter D = 2 ft and length L = 4 ft. The cylinder is fitted with a fan
blade (that occupies a negligibly small volume) connected by a shaft to a spool of thin flexible
wire. The fan blade rotates as an mw = 50 lbm weight descends a total vertical distance of H = 30
ft in te = 3.5 seconds as it unwinds the wire and spins the spool. The air within the cylinder
experiences convection with internal cylinder walls, which remain at T1 = 75°F during this
experiment. The convection coefficient is estimated to be hconv = 23 W/m2-K.
fan
pulley
air
T1 = 75°F
P1 = 15 psi
D = 2 ft
L = 4 ft
weight
mw = 50 lbm
Figure 3.C-10: Experiment to relate mechanical and thermal energy
a.) What is the final temperature of the air in cylinder that you would expect if heat transfer
between the air and cylinder walls did not occur?
b.) What is the accepted value of the constant relating ft-lbf of work to Btu of thermal energy?
c.) Prepare a plot of the temperature of the air in the cylinder as a function of time for the
duration of the experiment; include heat transfer with the cylinder walls in your analysis.
d.) If you did not know that heat transfer was occurring between the air and the internal cylinder
walls, what value would you have reported for the constant relating ft-lbf of work to Btu of
thermal energy based on this experiment?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-11 Shown in Figure 3.C-11 are test data for a small air compressor in the form of gage pressure
versus cylinder volume. These data are provided in file Compressor_3C11.lkt, which can be read
directly into an EES Lookup table with the Open Lookup Table command in the Tables menu.
The intake valve closes at point 1 and the exhaust valve opens at point 2. The exhaust valve is a
reed valve that flutters during the expulsion of compressed air, but it finally closes at point 3. The
intake reed valve opens at point 4. State 1, 2, 3, and 4 are on lines 954, 1630, 1, and 307,
respectively of the data file. The inlet air is at Tin = 25°C, Pin = 101.3 kPa and the compressor
operates at N = 1200 cycles per minute. The air in the cylinder at state 1, which has mixed with
residual air at state 4, is at T1 = 32°C.
600
2
500
Gage pressure (kPa)
3
400
300
200
100
0
1
4
-100
0
50
100
150
200
250
300
350
Volume (cm3)
Figure 3.C-11: Plot of data from file Compressor_3C11.lkt.
a.) What are the compressor clearance volume and displacement?
b.) The volumetric efficiency of a compressor is the ratio of the volume of the air that is actually
passed through the compressor at the start of compression (state 1) to the compressor
displacement. What is the volumetric efficiency of the compressor at these operating
conditions?
c.) What is your estimate for the temperature of the air at point 2?
d.) What is the minimum power required to operate this air compressor under steady conditions
at N = 1200 cycles per minute? What motor power would you recommend to drive the
compressor?
e.) The compression process from state 1 to state 2 (which occurs with all valves closed) can be
represented as a polytropic process for which P V n = Constant. Curve-fit the experimental
data to determine the value of n that best represents this process. Is your value physically
reasonable?
f.) Compare the work per cycle between states 1 and 2 calculated using your polytropic
equation from part (e) with the actual work determined by integration of the experimental
data. Comment on the agreement.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-12 A spherical balloon containing air is located with its top surface at the air-water interface in a
pool of water at Tw = 25ºC. The diameter of the balloon at this location is Db,1 = 0.35 m. The
balloon is made of an elastic material that expands or contracts such that the difference in
pressure between the air in the balloon and its surroundings is directly proportional to the surface
area of the balloon. The mass of the elastic material is mb = 0.015 kg. The atmospheric pressure
at the surface of the pool is Patm = 101.3 kPa and the pressure of the air in the balloon at this
location is P1 = 104.8 kPa. The balloon is now slowly lowered a distance of H = 10 m into the
pool of water. You may assume ideal gas behavior for air. Water may be assumed to be
incompressible with constant density. The entire process may be assumed to be isothermal at Tw.
State any other assumptions you employ.
a.) Prepare a plot of the diameter of the balloon and the air pressure in the balloon as a function
of depth between 0 (the surface) and 10 m.
b.) The balloon is buoyant in water. Calculate the work required to move the balloon from the
surface to a depth of 10 m.
c.) The volume of the air changes as it descends into the water. Calculate the work done on the
air in the balloon during this process.
d.) The balloon material contracts as the balloon descends into the pool. What is the work done
on the balloon material during this process?
e.) Is the process adiabatic? If not, estimate the heat transfer to the balloon.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-13 A balloon containing helium has a volume of Vo = 1.45 m3 and pressure of Po = 104.2 kPa at
ground level where the atmospheric pressure and temperature are Patm,o = 101.3 kPa and To =
15°C, respectively. The balloon is made of an elastic material with a mass of mmaterial = 0.80 kg
that stretches such that the difference in pressure between the helium and the atmosphere is
inversely proportional to the radius of the balloon.
a.) Estimate the work required to fill the balloon with helium from a near-zero initial volume to
its initial volume and pressure, Vo and Po, at ground level.
b.) The balloon is now released and it slowly rises to a height of 3 km. The temperature of the
balloon and the contained helium may be assumed to be the same as the atmospheric
temperature because of the adequate time and surface area for heat exchange. The
atmospheric pressure and temperature vary with elevation as indicated in the Table 3.C-13.
Taking the balloon as the system, what is the total work done in the process in which the
balloon rises 3 km? Be sure to indicate the direction of the work transfer. Is work done on
the balloon or is the balloon doing work on its surroundings? You may assume ideal gas
behavior for air and helium. State any other assumptions you employ. (The data in Table
3.C-13 are provided in EES lookup table format in file altitude_3C13.LKT.)
Table 3.C-13: Atmospheric pressure and temperature as a function of elevation
Elevation,
Pressure,
Temperature,
z (m)
P (kPa)
T (ºC)
0
152.4
304.8
457.2
609.6
762.0
914.4
1067
1219
1372
1524
1829
2134
2438
2743
3048
101.3
99.49
97.63
95.91
94.19
92.46
90.81
89.15
87.49
85.91
84.33
81.22
78.19
75.22
72.4
69.64
15
14
13
12
11
10
9
8
7
6
5
3
1
-1
-3
-5
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-14 One type of household humidifier operates by expelling water droplets into the air. The droplet
radii are assumed to be normally distributed with a mean radius of r = 1 μm and a standard
deviation of σ = 0.1 μm. These small droplets then evaporate into T = 25°C air to provide
humidification. If the droplets follow a normal distribution, the probability density function of a
droplet having radius less than or equal to r is given by:
⎡ ( r − r )2 ⎤
1
f =
exp ⎢ −
⎥
2
σ 2π
⎢⎣ 2σ ⎥⎦
The integral
∫
rmax
rmin
f dr =1; the limits of integration can be approximated as r ± 4σ .
a.) What are the total mass, volume, and surface area of a sample of N = 1x1014 droplets?
b.) Assuming that water is supplied to the humidifier at temperature T and atmospheric pressure,
estimate the mechanical work required to produce these droplets.
c.) What would the energy requirement be if the same mass of water were directly evaporated in
the humidifier by adding heat (as many types of humidifiers do)?
d.) Comment on the energy use and the advantages and disadvantages of these two different
designs. If it is relevant, assume that the humidifier is in use in a home on a winter day when
the outdoor temperature is -5°C.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
3.C-15 The performance of a diesel engine is very much influenced by the efficiency of the combustion
process which in turn is affected by the quality of the atomization process that occurs when high
pressure diesel fuel is injected into the cylinder. In one set of experiments on a 6-cylinder 4stroke diesel engine operating steadily at 3200 rpm, diesel fuel was injected at P = 75 bar to
produce droplets that were, approximately, normally distributed with a mean diameter of dm =
48.9µm and a standard deviation of σ = 19.1µm. The mass flowrate of diesel fuel was m = 8.976
g/s. For the purposes of this problem, diesel can be represented by n-dodecane (C12H26). If the
droplets follow a normal distribution, the probability density function of a droplet having
diameter less than or equal to the mean diameter is:
⎡ ( d − d m )2 ⎤
1
exp ⎢ −
f =
⎥
2σ 2 ⎥⎦
σ 2π
⎢⎣
The integral of the distribution is 1. The limits of integration where the integration limits can be
approximated as dm ± 4 σ. However, negative diameters are obviously not possible.
a.) Estimate the mass of fuel and the corresponding number of droplets produced during one
injection and their total surface area.
b.) Calculate the total mechanical power for all cylinders required to produce the droplets.
c.) Compare your answer to part b to an estimate of the steady-state engine power. The engine is
reported to operate at η = 32% efficiency based on the lower heating value of the fuel, which
is LHV = 44,100 kJ/kg.