© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
Problems for Chapter 2: Thermodynamic Properties
A. Property Data from Tables
Use the property information from the tables in the appendix to do the problems in this section.
2.A-1 Pure water is held in a container. The temperature of the water is T1 = 520°C and the pressure is
P1 = 800 kPa.
a.) Sketch a T-v diagram and locate the state of the water on the diagram. Your sketch should be
qualitatively correct, but it does not have to be to scale.
b.) Determine the specific volume of the water (m3/kg) and the density of the water (kg/m3).
c.) If the mass of water in the container is m = 7.2 kg then what is the volume of the container?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-2 Determine the boiling temperature (°F) of water:
a.) at normal atmospheric pressure (Patm = 14.7 psi, which corresponds to sea-level),
b.) in Denver, where the pressure is P = 24.58 inch Hg, and
c.) at the summit of Mount Everest, where the pressure is P = 30 kPa.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-3 A rigid tank with volume V = 8000 cm3 is filled with water with quality x1 = 0.05 and temperature
T1 = 140°C.
a.) What is the specific volume (m3/kg) and the pressure (kPa) of the water.
b.) What is the total mass of water in the tank (kg)? What is the mass of liquid (kg) and the mass
of vapor (kg) in the tank?
c.) What are the volumes of liquid and vapor in the tank (m3)?
The water in the tank is heated to T2 = 200°C. The tank is rigid (i.e., its volume doesn't change)
and leak tight.
d.) What is the pressure (kPa) and quality of the water in the tank at state 2?
e.) What is the mass of liquid (kg) in the tank at state 2?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-4
a.) Sketch a T-v diagram and locate the states listed in Table 2.A-4 (for water) on the diagram.
The sketch can be approximate but it should clearly show the conditions that define each state
(that is, each state is determined by an intersection of two property lines - show this
intersection).
Table 2.A-4: Some states of water.
T
P (kPa)
(°C)
State 1: saturated liquid at T = 200°C
State 2: saturated vapor at P = 0.50 bar
State 3: P = 0.8 bar, T = 220°C
State 4: T = 400°C, u = 2850 kJ/kg
State 5: T = 100°C, P = 100 bar
State
v (m3/kg)
u (kJ/kg)
Table
b.) Fill in Table P2.A-4. Show your work where interpolation is required and indicate which
table you used for each calculation.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-5 Figure 2.A-5 illustrates Mount Everest, a mountain with an elevation of H = 29,029 ft above sealevel
Figure 2.A-5: A photograph of Mount Everest.
a.) Estimate the atmospheric pressure at the top of Mount Everest (kPa) assuming that the
density of air is constant at all elevations and equal to its value at sea level, ρa = 1 kg/m3. Is
this an upper or a lower bound on the pressure? Why?
The actual atmospheric pressure at the top of Mount Everest is PEverest = 33.7 kPa.
b.) If you put a pot of water on a cook stove at the top of Mount Everest in order to make coffee,
what temperature would the water boil at (ºF)?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-6 You are interested in locating the state of water at P = 300 kPa and T = 165ºC.
a.) Sketch the isobar P = 300 kPa and the isotherm T = 165ºC on a T-v diagram. Your sketch
need not be quantitatively correct. However, the shape of the lines should be qualitatively
right and they should intersect in the correct region.
b.) Determine the specific volume of water at P = 300 kPa and T = 165ºC. If you need to do an
interpolation to accomplish this please show your work clearly.
You are interested in locating the state of water with properties T = 200ºC and v = 1.0 m3/kg.
c.) On a T-v diagram, sketch the isotherm T = 200ºC and the isochor (line of constant specific
volume) v = 1.0 m3/kg. Your sketch need not be quantitatively correct. However, the shape
of the lines should be qualitatively right and they should intersect in the correct region.
d.) Determine the pressure of water at T = 200ºC and v = 1.0 m3/kg. If you need to do an
interpolation to accomplish this please show your work clearly.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-7 Figure 2.A-7 illustrates a pressure cooker.
Patm = 101.3 kPa
relief disk
Drd = 0.88 inch
mrd = 0.05 kg
spring
x = 0.25 inch
K = 200 lbf/inch
m = 0.25 kg
V = 2.5 liter
Figure 2.A-7: A pressure cooker.
The pressure cooker has an internal volume of V = 2.5 liter and contains m = 0.25 kg of pure
water. The pressure relief valve consists of a spring loaded disk that is positioned over a hole in
the top of the cooker. The disk has diameter Drd = 0.88 inch and mass mrd = 0.05 kg. The spring
is compressed x = 0.25 inch and has a spring constant of K = 200 lbf/inch. The atmospheric
pressure is Patm = 101.3 kPa.
a.) Determine the internal pressure in the pressure cooker that is required to open the pressure
relief valve.
b.) The pressure relief valve allows vapor to escape in order to maintain the pressure that you
calculated in (a). What is the temperature of the water remaining in the pressure cooker after
the relief valve opens assuming that some liquid remains?
You have been asked to examine the possibility that the relief valve fails to open. In this case, no
water can escape and therefore the temperature and pressure of the contents of the pressure
cooker will continue to rise until the device fails. Assume that the pressure cooker is rigid (i.e.,
the volume does not change).
c.) What is the temperature in the pressure cooker when the pressure reaches P2 = 24 MPa?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-8 A rigid tank with volume V = 8000 cm3 is filled with water at state 1, which has a quality x1 =
0.05 at temperature T1 = 140°C.
a.) What are the specific volume (m3/kg) and the pressure (kPa) of the water at state 1?
b.) What is the total mass of water in the tank (kg) at state 1? What is the mass of liquid and the
mass of vapor (kg) in the tank at state 1?
c.) What are the volume of liquid and the volume of vapor in the tank (m3) at state 1?
The water in the tank is heated to T2 = 200°C. The tank is rigid (its volume doesn't change) and
leak tight.
d.) What is the pressure (kPa) and quality of the water in the tank at state 2?
e.) What is the mass of liquid (kg) at state 2?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-9 Refrigerant R134a is contained in a tank with volume V = 0.25 m3 at pressure P = 4 bar and
temperature T = 65°C.
a.) Sketch a T-v diagram and locate the state of the R134a on your sketch. The sketch can be
approximate but it should clearly show the isobar and isotherm that define the state.
b.) Determine the mass of R134a in the tank (kg).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-10 Refrigerant R134a is contained in a small recharge tank with volume V = 2 liter. Initially, the
tank is filled with m1 = 2.0 kg of R134a at T1 = 15°C.
a.) Sketch a T-v diagram and locate the state of the R134a on your sketch. The sketch can be
approximate, but it should clearly show the intersection of the two property lines that define
the state.
b.) What is the pressure in the tank at state 1?
c.) What is the quality of the R134a in the tank?
d.) What is the mass of liquid R134a in the tank? What is the mass of vapor R134a in the tank?
e.) What is the volume of liquid R134a in the tank? What is the volume of vapor R134a in the
tank?
The recharge tank is equipped with a pressure relief valve that allows refrigerant to escape when
the pressure within the tank reaches P2 = 9.0 bar. The tank is accidentally transported in an unconditioned truck where the temperature is Ttruck = 40ºC. As a result, the temperature of the
R134a in the tank slowly starts to rise causing the pressure to rise. State 2 is defined to be the
state of the refrigerant where the pressure relief valve just opens.
f.) On your T-v diagram from (a) overlay state 2. Indicate on your diagram what two properties
define state 2.
g.) What is the temperature of the R134a at the time that it reaches state 2?
The refrigerant continues to increase in temperature until finally it reaches T3 = Ttruck. During this
time, the pressure relief valve vents refrigerant in order to maintain the pressure in the tank
always at P3 = P2 = 9.0 bar.
h.) On your T-v diagram from (a) overlay state 3. Indicate on your diagram what two properties
define state 3.
i.) What mass of refrigerant is vented from the tank during the time that it goes from state 2 to
state 3?
Eventually, the tank is unloaded from the truck and cooled to T4 = 20°C.
j.) On your T-v diagram from (a) overlay state 4. It should be clear on your diagram what two
properties define state 4.
k.) What is the volume of liquid refrigerant in the tank at state 4?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-11 A rigid tank with volume Vtank = 5.0 gallon is maintained at T1 = 300ºC and initially contains m1 =
0.08 kg of water, as shown in Figure 2.A-11. At some time, a valve is opened allowing Vin = 0.5
gallons of water at Tin = 20ºC and Pin = 20.0 MPa to enter the tank. The valve is shut and
eventually all of the water in the tank comes to T2 = 300ºC.
rigid tank
Vtank = 5.0 gal
m1 = 0.08 kg
T1 = T2 = 300°C
Vin = 0.5 gal
Tin = 20°C
Pin = 20 MPa
Figure 2.A-11: Rigid tank of water.
a.) Locate state 1 and state in, the states of the water initially in the tank and the water added to
the tank, respectively, on a sketch of a T-v diagram.
b.) What is the initial pressure in the tank (MPa)?
c.) What is the mass of water added to the tank (kg)?
d.) Locate state 2, the final state of the water in the tank, on the T-v diagram from (a).
e.) What is the final pressure in the tank (MPa)?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-12 Up until very recently, steam catapults were used to launch aircraft from aircraft carriers. Figure
2.A-12(a) illustrates an aircraft connected to a shuttle that is connected to a steam catapult located
under the deck.
Figure 2.A-12(a): Aircraft connected to a steam catapult.
A simplified schematic of one tube of a steam catapult is shown in Figure 2.1-12(b).
inlet valve
P1 = 300 kPa
T1 = 200°C
Lmin = 5 ft
D = 18 inch
Figure 2.A-12(b): Simple schematic of a steam catapult.
The diameter of the piston is D = 18 inch and its initial position is Lmin = 5 ft. The cylinder is
initially filled with water at P1 = 300 kPa and T1 = 200ºC.
a.) Locate state 1 on a sketch of a T-v diagram (the sketch need not be quantitatively correct, but
it should clearly show the intersection of the two constant property lines that fix state 1).
b.) Determine the specific volume at state 1. Use this value to determine the initial mass of
water in the cylinder.
c.) The inlet valve is opened allowing high pressure steam into the cylinder which causes the
temperature and pressure to rise to T2 = 700ºC and P2 = 10 MPa, respectively). The cylinder
does not move during this process. Locate state 2 on a T-v diagram. Again - show clearly the
two constant property lines that fix state 2.
d.) Determine the specific volume at state 2. What is the mass of steam added to the cylinder
during this process?
e.) The inlet valve is closed and the catapult is activated, causing the position of the piston to
move from Lmin to Lmax = 100 ft. The final pressure in the cylinder is P3 = 500 kPa.
Determine the specific volume of the water at the conclusion of the launch process.
f.) Locate state (3) on a T-v diagram. Again - show clearly the two constant property lines that
fix state (3).
g.) Determine the temperature at state 3.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-13 Figure 2.A13(a) illustrates a pressure cooker with the pressure relief valve removed.
Patm = 100 kPa
V = 2 liter
bottom 5%
is filled with
liquid water
Figure 2.A-13(a): A pressure cooker with the pressure relief valve removed.
The pressure cooker has an internal volume of V = 2 liter. Because the relief valve is removed,
the contents are initially at atmospheric pressure, Patm = 100 kPa. The pressure cooker contains
water in a two-phase state. The bottom 5% of the volume of the vessel is filled with liquid water
while the remainder of the vessel contains water vapor.
a.) Determine the initial temperature of the water in the cooker.
b.) Determine the quality of the water initially in the cooker.
c.) Locate the initial state of the water (state 1) on a sketch of a T-v diagram.
The pressure relief valve is installed on the pressure cooker, as shown in Figure 2.A-13(b).
Patm = 100 kPa
relief disk
Drv = 0.50 inch
mrv = 0.1 kg
spring
cs = 0.1694 inch
K = 150 lbf/inch
Figure 2.A-13(b): A pressure cooker with the pressure relief valve installed.
The pressure relief valve consists of a spring loaded disk that is positioned over a hole in the top
of the cooker. The disk has diameter Drv = 0.5 inch and mass mrv = 0.1 kg. The spring is
compressed cs = 0.1694 inch and has a spring constant of K = 150 lbf/inch.
d.) Determine the internal pressure that is required to open the pressure relief valve.
e.) Heat is added to the pressure cooker with the relief valve in place. The pressure relief valve
opens when the pressure reaches the value calculated in (d) and allows vapor to escape in
order to maintain the pressure at this value. What are the temperature and quality of the
water in the pressure cooker at the instant that the pressure relief valve opens at state 2? Add
state 2 to your T-v sketch from (c).
f.) What is the fraction of the volume of the pressure cooker that is filled with liquid at the
instant that the pressure relief valve opens (at state 2).
g.) Heat continues to be added to the pressure cooker until all of the liquid disappears at state 3.
What is the mass of water that has passed through the pressure relief valve at this instant?
Locate state 3 on your sketch from (c).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
h.) Heat continues to be added to the pressure cooker until the temperature reaches T4 = 400ºC.
Locate state 4 on your sketch from (c). What is the mass of water that passes through the
pressure relief valve between the time that the pressure cooker is at state 3 and at state 4?
i.) The pressure cooker is removed from the source of heat and begins to cool. The pressure
begins to drop and therefore the pressure relief valve closes. At what temperature does liquid
water begin to form in the vessel? Locate this state 5 in your sketch from part (c).
j.) The pressure cooker is cooled until the temperature reaches T6 = 20ºC. Determine the
pressure and quality of the water at state 6 and locate state 6 in your sketch from part (c).
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-14
Refrigerant R134a is contained at its critical point in a piston-cylinder device.
a.) What is the temperature and the pressure of the R134a?
b.) If the fluid is cooled slightly at constant volume, how many phases will be present in the
cylinder? Provide a sketch to show your result.
c.) Repeat part (b) assuming that pressure rather than the volume remains constant as the fluid is
cooled.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.A-15 At state 1, superheated water vapor is contained in a sealed glass vial at T1 = 200°C. You would
like to know the pressure at this state, but have no means of measuring it directly. However,
when the vial is slowly cooled to state 2, T2 = 120°C, you notice that droplets of liquid begin to
form on the glass walls.
a.) Use this information to determine the pressure at state 1.
b.) Sketch the process on a temperature-volume diagram.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
B. Property Data from EES
Use the property functions in EES to do the problems in this section.
2.B-1
A piston-cylinder device with volume V1 = 40 cm3 is filled with saturated vapor R134a at
temperature T1 = -20°C, as shown in Figure 2.B-1(a).
R134a
V1 = 40 cm3
sat. vapor
T1 = -20°C
Figure 2.B-1(a): Piston-cylinder device at state 1.
a.) Use EES' internal property routines to determine the pressure (kPa) of the R134a at state 1.
b.) Use the tables in the appendix of your book to determine the pressure (kPa) of the R134a at
state 1 and compare your answer with part (a).
c.) What is the mass of R134a (kg) in the piston?
The piston is moved so that the volume decreases to V2 = 10 cm3, as shown Figure 2.B-1(b). At
the conclusion of this process, the pressure in the cylinder is P2 = 950 kPa.
R134a
V1 = 40 cm3
sat. vapor
T1 = -20°C
V2 = 10 cm3
P2 = 950 kPa
Figure 2.B-1(b): Piston-cylinder device at state 2.
d.) What is the temperature of the R134a (°C) at state 2?
The piston is locked in place and the contents are cooled until the R134a reaches T3 = -20°C.
e.) What is the pressure of the R134a (kPa) at state 3?
f.) Sketch the locations of states 1, 2, and 3 on a T-v diagram that includes the vapor dome.
Make sure that you determine whether these states are sub-cooled liquid, two-phase, or
superheated vapor and place them accordingly.
g.) Using EES, prepare a plot showing the pressure at state 3 as a function of T3 for -40ºC < T3 <
100ºC; assume all other parameters do not change. Your plot should have pressure in kPa as
the y-axis and temperature in ºC as the x-axis. Explain the shape of your plot.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.B-2
One problem facing people who build large-scale refrigeration systems that include a lot of piping
and components is that it is not easy to estimate the total volume contained in the system. This is
important because you must order refrigerant in a quantity that is sufficient to charge the system.
If you order too much then you'll have to ship the excess back (if the company will take it back)
and if you order too little then your project faces delays as you end up re-ordering. You have
come up with a technique that you think will allow the volume enclosed in an installed
refrigeration system to be measured. The idea is to attach a high pressure air bottle to the system,
as shown in Figure 2.B-2.
refrigeration system
Vsystem = 15,000 liter
Tamb = 20°C
Patm = 1 atm
high pressure air bottle
Vbottle = 20 liter
Tamb = 20°C
Pbottle,g = 2000 psig
Figure 2.B-2: High pressure air bottle connected to the system.
The air in the system is initially at ambient temperature and atmospheric pressure, Tamb = 20ºC
and Patm = 1 atm, respectively. Assume for now that the system volume is Vsystem = 15,000 liter.
The air in the bottle is initially at ambient temperature and a high pressure, Pbottle,g = 2000 psig
(note that the gage mounted on the bottle measures the gage pressure of the air). The volume of
the bottle is Vbottle = 20 liter. The valve connecting the bottle to the system is opened, allowing
high pressure air from the bottle to flow into the system. Eventually the air in the bottle and the
system come to the same final pressure, P2. The final temperature of the air in the bottle and in
the system is T2 = Tamb. You would like to use the measurement of the final pressure, P2, in order
to determine the volume of the system. Use the substance 'Air_ha' in EES to model the air for this
problem (i.e., do not model the air as an ideal gas).
a.) Determine the final pressure (in psig, the gage pressure) that exists in the system and in the
bottle.
b.) Generate a graph showing the system volume (the quantity you are interested in) as a function
of the final pressure (the quantity that you can measure). Does this technique work? Can you
determine the system volume if you measure the pressure?
c.) Your EES program uses system volume as an input and computes the final pressure as an
output. However, you can comment out the system volume that you input and instead specify
the measured output pressure (i.e., you can turn an input into an output). If you measure a
final pressure of P2 = 4.5 psig (again, a gage pressure) then what is the system volume?
d.) During the commissioning process you want to charge your system (after first evacuating it to
remove all of the air) with refrigerant R134a at pressure Pcharge = 450 kPa and temperature
Tcharge = Tamb. Based on your calculation from (c), how much refrigerant (in lbm) do you need
to order?
You are in the process of specifying the pressure and temperature instrumentation that are
required to carry out the procedure described above.
e.) If you can measure the bottle pressure to within δPbottle = 25 psi then what is the related
uncertainty in your measurement of the system volume? One easy way to evaluate this is to
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
change the specified value of Pbottle by the uncertainty δPbottle and see how much the
calculated system volume changes.
f.) If you can measure the final pressure to within δP2 = 0.25 psi then what is the related
uncertainty in your measurement of the system volume?
g.) If you can measure the ambient temperature to within δTamb = 2.5ºC then what is the related
uncertainty in your measurement of the system volume?
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.B-3
A sealed rigid vessel has a volume of V = 0.0425 m3. The vessel contains m = 0.34 kg of water at
T1 = 93°C. A pressure relief valve is attached to the vessel to ensure that it is not subjected to
destructives pressures when it is heated.
a.) If the maximum temperature of the vessel contents is expected to be T2 = 200°C, determine
the pressure at which the relief valve should be set.
b.) The mass of water in the vessel is adjusted so that the water will pass through the critical
point when it is heated (assume that the safety valve fails to open). Determine the mass of
water in the vessel.
c.) The vessel contains the amount of mass you determined in part (b). What is the quality of
water when the temperature is equal to T3 = T1 = 93°C?
d.) Plot the heating processes with the two different amounts of mass in the vessel on a
temperature-volume plot.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
C. The Ideal Gas and Incompressible Fluid Models
2.C-1
A piston-cylinder device has volume V1 = 1.0 m3 and contains air at T1 = 20ºC. The piston is
frictionless and has no mass. The piston-cylinder device is in an environment with atmospheric
pressure Patm = 1 atm. Therefore, as long as the piston is free to move, the pressure of the air in
the cylinder is also at P = 1 atm. Assume that the air in the tank can be modeled as an ideal gas.
a.) What is the mass of air in the cylinder (kg)?
b.) How many moles of air are in the cylinder (kmol)? How many air molecules (molecules) are
in the cylinder?
The air is heated to T2 = 100ºC with the piston free to move. No air leaks out of the piston during
this process.
c.) What is the volume of the cylinder at state 2 (m3)?
The piston is locked in place so that the volume cannot change. The air is cooled to T3 = 20ºC;
again, no air leaks out during this process.
d.) What is the final pressure in the tank, P3 (kPa)?
e.) Sketch states 1, 2, and 3 on a T-v diagram for the air. Your diagram should clearly show the
intersecting lines that define each state.
© S.A. Klein and G.F. Nellis
Cambridge University Press, 2011
2.C-2
Refrigerant R134a (MW = 102 kg/kmol) is contained in a tank with volume V = 0.25 m3 and
temperature T = 105°C.
a.) On the same plot show the mass of R134a contained in the tank as a function of pressure for
pressures ranging between 0.5 bar and 40 bar calculated using the ideal gas law and using the
EES property data base for R134a.
b.) Under what conditions could you use the ideal gas law to provide an approximate answer for
the mass in the tank?