ME 201 Thermodynamics Homework

Summer 2004

(Note: Many of these problem sets were provided by the generosity of Prof. Craig Somerton)

Homework 1 - Due Friday 9 July Solution

1. If 500 gallons per minute of water is flowing in a 6 inch diameter pipe, what is the velocity if this flow in ft/min?

2. Consider filling a kitchen sink in preparation for washing dishes. Hot water flows out of the faucet at 0.05 kg/s. How long will it take to reach a water level of 0.13m? The bottom of the sink is 0.3 m by 0.4 m. The density of water may be taken to be 62.4 lb/ft3.

3. Now we leave the water running in the sink in problem 2 while we take out the drain plug. If water leaves through the drain at 0.15 kg/s, how long will it take the sink to empty?

Homework 2 - Due Friday 9 July Solution

1. If the weight of an object is 2 lb in a gravitational field where g = 10 ft/sec2, what is the mass of the object in slugs?  What is the mass in "lbm"?

2. What is the mass of 15 kg-moles of N2, CO2, and H2O?  

3.  If a 5 foot diameter sphere is full of air with a specific volume of 15 ft3/lb, what is its mass in kg?  What is its specific volume in SI units?

4.  If the specific volume of a substance at 200oF is 0.123 ft3/lb and at 220oF it rises to 0.144 ft3/lb, what do you estimate the specific volume at 205oF to be? (Note: Just assume specific volume is a linear function of temperature. This is called interpolation.)

Homework 3 – Due Monday 12 July Solution

1. Convert the following temperatures to °F, °C, K, R.

a. 10°F
b. 278 K
c. 1400 R
d. -57°C (Ans. 389 R)
2. Convert the following pressures to psia and kPa.
a. 38 in of Hg (Ans. 18.6 psia, 128.7 kPa)
b. 5 atm
c. 12.3 x 105 N/m2

(Note: One standard atmosphere is 29.92 in Hg)

3. Determine the pressure (in bars) when a 20 lb force is exerted over 5 cm2. (Ans 1.78 bar)

4. What is the kinetic energy (in Btu) associated with a 100 kg football player running 15 mph? (Ans. 21.3 Btu)

5. What is the potential energy (in kJ) of 15 grams at 50 feet above sea level?

6. What is the specific volume (ft3/lbm) of a material of mass 3 lb occupying a volume of dimensions 15 inches by 4 feet by 3 meters? 

Homework 4 - Due Wednesday 14 July Solution

1. Assuming ideal gas relationship calculate the specific volume in the appropriate units for:

a. CO at 300 kPa and 450 K
b. H2O at 2 psia and 300 R
c. Air at 100 kPa and 298 K (Ans: 0.855 m3/kg)

2. Calculate the specific enthalpy change for the following ideal gas processes by first assuming a temperature variable cp (i.e., integrating), then by using an average cp, and finally by using the ideal gas tables.  Since the ideal gas tables for H2 do not appear in the Moran and Shapiro textbook, a partial table is provided below.

a. H2 at 200 kPa goes from 300 K to 800 K (Ans: 7236, 7265 and 7266 kJ/kg)
b. H2 at 200 kPa goes from 300 K to 350 K
c. Air at 4 psia and 70°F goes to 15 psia and 70°F

Ideal Gas Properies for H2
T (K) h (kJ/kmole)
300 8522
340 9680
360 10262
800 23171

3. Carefully draw a graph of pressure versus specific volume showing lines of constant temperature and lines of constant specific internal energy using air as an ideal gas. Draw 3 lines on each graph, labeling the value of the constant property represented by that line.

Homework 5 – Due Wednesday 14 July Solution

1.  Calculate the entropy change for CO as it goes from 400 K and 100 kPa to 1000 K and 600 kPa. (Ans: 0.478 kJ/kgK)

2. For the two processes given below, determine the final temperature, pressure, and specific volume, as well as the change in enthalpy, internal energy, and entropy.

a.  Helium at 480 R and 6.6 psia goes isentropically to 17 psia.
b.  Air at 125°F and 7.5 atm goes isotropically to 3300°F. (Ans: 3300oF, 710.6 psia, 1.96 ft3/lb, 876.24 Btu/lb, 658.61 Btu/lb, 0.367Btu/lbR)

3. Calculate the final temperature and change in specific internal energy as copper at 30 psia and 500°F goes isentropically to 90 psia.

4. For the three processes given below, determine the changes in internal energy, enthalpy, and entropy.

a. Oil at 70 kPa and 300 K goes to 100 kPa and 430 K. (Ans: 234 kJ/kg, 234.03 kJ/kg, 0.648 kJ/kgK)
b. Wood with a density of 30 lbm/ft3 at 10 psia and 75°F goes isobarically to115°F.
c. Liquid water at 298 K and 95 kPa goes isothermally to 130 kPa

5. Carefully draw a graph of temperature versus specific entropy, showing lines of constant pressure and lines of constant specific enthalpy using air as an ideal gas. Draw 3 lines on each graph, labeling the value of the constant property represented by that line.

Homework 6 – Due Friday 16 July Solution

1.  Determine the internal energy change as saturated water vapor at 1.5 MPa goes to a two phase mixture of quality 0.25 at 1.5 MPa. (Ans: -1314 kJ/kg)

2.  What is the enthalpy, internal energy, entropy, and specific volume for steam at 331 psia and 546°F.

3. Give the behavior of entropy for steam when

a. quality increases
b. pressure decreases, constant enthalpy
c. temperature increases, constant pressure
d. pressure increases, constant temperature

4.  What is the quality and enthaply of a two-phase mixture of R-134a at 15 psia which has an entropy of 0.1 Btu/(lb R)?  (Ans: x = 0.40, h = 44.3 Btu/lb)

Homework 7 – Due Friday 16 July Solution

1.  Saturated liquid water at 400 K undergoes an isenthalpic expansion to 200 kPa. Determine the final fluid state and entropy change for the process. (Ans: x = 0.013, s1 = s2 = 1.60 kJ/kg)

2.  Consider a system of steam undergoing a constant enthalpy process from 200 kPa and 160°C to 200°C.  Determine the final fluid phase and its pressure and entropy.

3. A large pump in a power plant takes saturated liquid water at 0.01 MPa and increases its pressure to 50 MPa in an isentropic process. Determine the final state temperature and the change in enthaply for the process.  Below is the table information you'll need for 50 MPa water: (Ans: T2 = 47.7°C, h2 = h1 = 50.8 kJ/kg)

T(oC) h(kJ/kg) s(kJ/kgK)
40 211.21 0.5527
60 292.79 0.8052

4.  In a jet engine, air at 120 kPa and 50°C is compressed isentropically to a pressure of 800 kPa. Determine the temperature and specific volume in the final state and the change in enthalply for the process. 

Homework 8 – Due Monday 19 July Solution

1.  Superheated steam at 300 psia and 600°F is contained in a piston/cylinder system. It undergoes a constant pressure process that results in a final state of saturated liquid. What is the boundary work for this process in Btu/lbm? (Ans: -110.2 Btu/lb)

2.  What is the boundary work in kJ involved in blowing up a balloon to a volume of 5 x 10-4m3? You may take the initial pressure and volume to be 100 kPa and 2x10-5m3. The process may be considered to have a polytropic exponent of -0.4 or PV-0.4= constant. (Ans: 0.128 kJ)

3.  If the time to blow up the balloon is 5 seconds, determine the boundary power.(Ans: 0.0256 kW)

Homework 9 – Due Wednesday 21 July Solution

 1.  A container containing oxygen at 19 psia and 47°F is combined with a container containing oxygen at 35 psia and 125°F in an adiabatic process. Determine the final pressure and temperature. Each of the containers is 0.6 ft3. (Ans: P = 26 psia, T = 555R)

2.  An ideal gas at 4 MPa and 460°C is expanded in a polytropic process with n = 0.8 to three times its initial volume.  Determine the work and heat transfer in kJ/kg associated with this process.  You are also given that R = 0.444 kJ/kg K and cp = 1.108 kJ/kg for this substance. (Ans: q = 517 kJ/kg, w = 398 kJ/kg)

3.  Determine how much heat is required to take a pan of water at 22°C up to its boiling point in a pressure of 100 kPa. How much additional heat is required to boil off all of the water? (Ans: 325 kJ/kg, 2089 kJ/kg)
 

Exam 1 – Wednesday 21 July

Homework 10 – Due Monday 26 July Solution

A piston cylinder device initially contains 0.3 liters of air at 30°C and 105 kPa. The air undergoes an isentropic compression to 0.05 liters. Energy is then added by heat transfer while the air is at constant volume until the temperature reaches 1000°C. Finally the air is expanded isentropically back to its initial pressure. For each of the three processes the air undergoes find the work and heat transfer in kJ. What is the total work and heat transfer for all three processes together? (Ans: Q = 0.2042 kJ and W = 0.1368 kJ)
 

Homework 11 – Due Wednesday 28 July Solution

 1.  Steam enters an ideal turbine at 400 psia and 800°F. If the exhaust pressure is 10 psia, determine the work output. (Ans: 340.7 Btu / lbm)

2.  Water exits a faucet at the top of a 10 story building (100 ft high) at 25°C and 1.2 atm pressure. If the water enters at the bottom of the building at 25°C, what pressure (in psia) is required? (Ans: 60.7 psia)

3.  Air at 0.8 atm pressure, 60°F, and velocity 60 mph enters a diffuser. If the air leaving the diffuser has negligible kinetic energy, determine the pressure and temperature at the exit of the diffuser. Clearly state any assumptions. (Ans: 11.81 psia)

Homework 12 – Due Wednesday 28 July Solution

 1.  A 0.3 m3 rigid tank is filled with saturated liquid water at 200°C. A valve at the bottom of the tank is opened, and liquid is withdrawn from the tank. Heat is transferred to the water such that the temperature in the tank remains constant. Determine the amount of heat that must be transferred by the time one-half of the total mass has been withdrawn. (Ans: Q = 2314.98 kJ)

2.  An adiabatic tank of compressed air with volume 100 m3 at 600 kPa and 500 K is vented to the atmosphere until the pressure in the tank becomes 350 kPa. What would be the maximum amount of work that could be generated by the air released from this system? (Ans: 22,308 kJ)

Homework 13 – Due Friday 30 July Solution

1. A reversible process has been defined as a process, which having taken place, can be reversed and in so doing leaves no change in either the system or the surroundings. Six restrictions were imposed:

a. no friction
b. heat transfer occurs only for infinitesimal temperature differences
c. unrestrained expansion does not occur
d. no mixing
e. no turbulence
f. no combustion

Choose a process for which one of these restrictions is relaxed and discuss how this process is not reversible.

2.  Water enters an ice machine at 55°F and leaves as ice at 25°F. If the COP of the ice machine is 2.4 during this operation, determine the required power input for an ice production rate of 20 lbm/hr. Assume this process takes place at atmospheric pressure. (Ans: 1415.8 Btu/hr)

Homework 14 – Due Friday 30 July Solution

Consider the Carnot cycle occurring in a piston-cylinder device with operating conditions given below:

Process A: Isothermal heat addition at TH = 300°C to convert saturated liquid to saturated vapor
Process B: Isentropic and adiabatic expansion to TL = 15°C
Process C: Isothermal heat removal at 15°C
Process D: Isentropic and adiabatic compression back to the initial state.
a. If steam is the working fluid for this cycle calculate the thermal efficiency using the actual heat and work addition and removal, using the steam tables. (Ans: 0.5)
b. Compare this to the ideal Carnot cycle efficiency, using the temperature information alone. (Ans: 0.5)

Homework 15 – Due Monday 2 August Solution

1.  A solar collector receives heat from the sun, at a rate of 0.315 kW per m2 of collector area and provides energy to a storage unit whose temperature remains constant at 500 K. A heat engine interacts with the storage unit as its high temperature reservoir, generates 1000 kW of net work output, and interacts with the surroundings at 20°C as its low temperature reservoir. Determine the minimum theoretical collector area required. (Ans: 7668 m2

2.  The surroundings of a freezer are at a temperature of 27°C. To maintain the freezer compartment at -40°C, it is necessary to remove energy by heat transfer at a rate of 1.25 kW. Determine

a. the maximum possible COP (Ans: 3.48)
b. the minimum theoretical power input required (Ans: 0.36 kW)

3. In the month of January, a certain dwelling requires 2.4 x 106 kJ per day to maintain its temperature at 20°C. As a means of proving the required heating, a heat pump cycle is under consideration that would operate between the dwelling an a nearby pond at 10oC. The manufacturer claims that the net work required to operate the heat pump is 1.6 x 105 kJ per day. Evaluate this claim. (Ans: possible)

Homework 16 – Due Wednesday 4 August Solution

1.  One kilogram of saturated liquid water in a closed system at 160°C expands to a final pressure of 150 kPa in an isothermal reversible process. Determine the work and heat transfer. (Ans: W = 471.5 kJ, Q = 2392 kJ)

2.  One tenth kilogram of water in a closed container undergoes a process from 300 kPa and 200°C to 1.5 MPa and 280°C during which the work input is measured to be 17.5 kJ. Determine whether it is possible for the work measurement to be correct if the ambient temperature is 295K. (Ans: impossible)

3.  Steam at 14.7 psia and 250°F enters a non-ideal compressor and exits at 160 psia and 400°F. The ambient temperature is 70°F. The work input is claimed to be 120 Btu/lbm. Determine whether this claim can be correct. (Ans: impossible)

Homework 17 – Due Wednesday 4 August Solution

1. Steam enters a turbine at 3000 psia and 1000°F and the outlet pressure is 1.0 psia with a quality of 83%. What is the turbine mechanical efficiency? (Ans: 85%)

2. A compressor with an isentropic efficiency of 90% intakes saturated vapor refrigerant-134a at 100 kPa and exhausts at 1.2 MPa. Determine

a. Ideal work (Ans: -51.76 kJ/kg)
b. Actual work (Ans: -57.51 kJ/kg)

Exam 2 – Wednesday 4 August


Homework 18 – Due Monday 9 August Solution

1. Consider a steam power plant operating on a Rankine cycle with single reheating and a single open feedwater heater receiving steam from the high pressure turbine.  Steam leaves the boiler at 15 MPa, 500°C, and 100 kg/s. The feedwater heater operates at 3 MPa. The condenser operates at 0.01 MPa. All devices in the system may be assumed to be ideal. The plant design calls for steam leaving the high pressure turbine to be saturated vapor. The temperature at the exit of the reheater is 450°C. Determine

a. pressure of the reheat leg (Ans: 1.9825 MPa)
b. power output (Ans: 119.82 MW)
c. plant thermal efficiency (Ans: 0.448)


Homework 19 – Due Wednesday 11 August Solution

An engine operates on the ideal Dual cycle with air as its working fluid. The cycle begins with air at 200 kPa and 320 K, due to supercharging and intercooling. The engine has the following operating conditions:

Bottom Dead Center Volume: 0.5 liters
Compression Ratio: 7
Cut-off Ratio: 1.5
Maximum Temperature: 2300 K

Determine

a. Net work output (Ans: 0.833 kJ)
b. Overall thermal efficiency (Ans: 0.452)

Homework 20 – Due Wednesday 11 August Solution

Consider a jet propulsion cycle with air entering the engine at 500 mph, 95 kPa and 285 K. Neglect air velocity inside the engine, i.e. after the diffuser and before the nozzle. The compressor operates with a pressure ratio of 12 and the exit temperature of the combustion chamber is 1700 K. The compressor and turbine both have an isentropic efficiency of 0.85 and you may assume that the remaining devices are ideal. 

a. Determine the exit air velocity and the total heat added.(Ans: 1192.5 kJ/kg, 1066 m/s)
b. Determine the exit air velocity and the total heat if an afterburner is added raising the exit temperature back up to 1700 K. (Ans: 1505.1 kJ/kg 1157 m/s)

Homework 21 – Due Friday 13 August Solution

Considering the same engine as that in Homework 19, we will now delete the supercharger  and see what the difference in performance is. We will now have air entering the cylinder at atmospheric pressure and 285K with the same amount of heat added per kg of air (1693 kJ/kg). Determine:

a. Net work output (Ans: 875 kJ/kg)
b. Overall thermal efficiency (Ans: 51.7%)

Homework 22 – Due Friday 13 August Solution

Considering the jet engine in Homework 20, we will now add a propulsion turbine of 85% mechanical efficiency to drive a helicopter rotor. If the inlet and outlet velocities of the engine are negligible, and the atmospheric pressure is again 95 kPa, find the net work output of the helicopter rotor and compare this to the power of the jet engine. (Recall that power is thrust times speed. For the jet engine, thrust is mass flow rate times change in velocity.) (Ans: 408 kJ/kg, 188 kJ/kg)
 

Homework 23 – Due Monday 16 August Solution

A refrigeration system using R-22 has a compressor suction temperature of 40°F and a condenser saturation temperature of 110°F. Find:

a. COP (Ans: 5.8)
b. Carnot COP for the same temperatures (Ans: 7.14)
c. Input power (in HP) required for 3 tons of cooling (i.e. 36,000 Btu/hr) (Ans: 2.44 Hp)

Homework 24 – Due Wednesday 18 August Solution

1. Complete the following table for the properties of an air/ water vapor mixture.

2. Air enters a cooling coil at 30oC and 50% relative humidity and exits at 12oC. Determine

a. Exit relative humidity (Ans: 100%)
b. Heat transfer required (Ans: -30.5 kJ/ kg of dry air)
c. Amount of water condensed (Ans: -.0048 kg/ kg of dry air)

Final Exam – Wednesday 18 August