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2-Week ISTE Workshop on Engineering Thermodynamics

EXERCISES

U. N. Gaitonde U. V. Bhandarkar B. P. Puranik

Department of Mechanical Engineering Indian Institute of Technology Bombay December 1121, 2012
201212071520

Engineering Thermodynamics Exercises Texts/References There are a number of good books on Thermodynamics. An illustrative list is:

ii

M. Achuthan, Engineering Thermodynamics, Prentice-Hall of India, Second Edition, New Delhi, 2009. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, Fourth Edition, Wiley, New York, 2000. F. W. Sears and G. L. Salinger, Thermodynamics, Kinetic Theory and Statistical Thermodynamics, Addison-Wesley/Narosa, New Delhi, 1975. R. E. Sonntag, C. Borgnakke, and G. J. Van Wylen, Fundamentals of Thermodynamics, Sixth Edition, Wiley, Singapore, 2003. M. W. Zemansky, Heat and Thermodynamics, Fourth Edition, McGraw-Hill Kogakusha, New York/Tokyo, 1957. Steam Tables M. L. Mathur and F. S. Mehta, Steam and Other Tables (with Mollier Chart), Revised Edition, Jain Brothers, New Delhi, 2010.

Engineering Thermodynamics Exercises The Work Interaction

WI.1 Two kg of a substance undergoes the specied process in a cylinder-piston arrangement: p1 = 6 bar; V1 = 0.2 m3 ; p2 = 2 bar; V2 = 0.6 m3 . Determine the work done in each case. (a) p varies as a linear function of V . (b) pV = a constant. (c) p remains constant till the volume reaches 0.3 m3 ; pV n = a constant after that. WI.2 An electric heater has a resistance of 50. It is connected across a power supply of 240 V, for a period of 2 hr. Determine the work done by the power supply on the heater. How many units of electricity are consumed? WI.3 A system containing 5 kg of a substance is stirred with a torque of 0.3 kgf-m at a speed of 1000 RPM for 24 hr. The system meanwhile expands from 1 m3 to 2 m3 against a constant pressure of 4 kgf/cm2 . Determine the net work done in kJ. WI.4 Consider the situation shown in the gure below. The width of the channel
p o piston h water

is b, normal to the paper. The atmospheric pressure is p0 . (a) Show that the force exerted by the water on the piston is F = p0 + gh 2 hb,

where is the density of water. (b) Calculate the work done by the water on the piston and by the atmosphere on the water when the chamber length is increased slowly from x1 to x2 . Express your answers in terms of p0 , b, h1 , x1 , x2 , , and g. WI.5 The pressure on a 250 g block of metal in increased quasi-statically and isothermally from 0 to 1000 bar. Assume that the density and isothermal bulk modulus of the metal remain (almost) constant at 20.0 g/cm3 and 2 1012 dyne/cm2

Engineering Thermodynamics Exercises

respectively, determine the work done in J. Isothermal bulk modulus is dened as B = V (p/V )T . WI.6 The tension in a wire in increased quasi-statically and isothermally from Fi to Ff . If the length (L), cross-sectional are (A) and isothermal Youngs modulus (Y ) remain (almost) constant, show that the work done by the wire is
2 Ff Fi2 . W = L 2AY

The Youngs modulus Y = (L/A)(F/L)T . Determine the work done if L = 2 m, A = 0.0016 cm2 , Fi = 10 N, Ff = 100 N, and Y = 2.0 107 N/cm2 . WI.7 The equation of state of an ideal elastic substance is given by L2 L 0 , F = KT L0 L2 where K is a constant and L0 (the length at zero tension) depends only on T . Calculate the work necessary to compress the substance from L = L0 to L = L0 /2, quasi-statically and isothermally. WI.8 (a) Show, from rst principles, that when the surface area of a system, subject to a surface tension , changes by a small amount dA, the work done by the system is (dA). (b) A bubble rose from the bottom of a lake to near its surface. The bubble was initially 1 cm in diameter, at a pressure of 2 bar. The nal pressure experienced by the bubble was 1 bar. The initial as well as the nal shape of the bubble was spherical, and its volume varied inversely to the pressure. Determine the net work done by the bubble. For water, = 0.073 N/m. Which additional assumptions did you make? Please comment on your answer. Additional Exercises Achuthan, Chapter 3: Problems 6, 7, 9, 14-21 (pp 6873). Moran and Shapiro: Problems 2.18 to 2.45 (pp 7577). Some of these use non-metric units, you may convert these to metric/SI units. Sears and Salinger: Problems 3-1 to 3-3, and 3-6(p90). Sonntag et al.: Problems 4.20 to 4.94 (pp 106111).

Engineering Thermodynamics Exercises The First Law 1

F1.1 A system executes a quasi-static process from an initial state 1 to a nal state 2, absorbing 80 kJ of heat and expanding from 2.0 m3 to 2.25 m3 against a constant pressure of 1.5 bar. The system is brought back to its initial state by a non-quasi-static process, during which it rejects 100 kJ of heat. What is the work done during the second process? F1.2 Three kg of air in a rigid container changes its state from 5.0 bar and 75 C to 12 bar while it is stirred. The heat absorbed is 195 kJ. Assume air to be an ideal gas with cv = 0.714 kJ/kg K. Determine the nal temperature, change in internal energy and work done. F1.3 Two kg of air at 2.5 bar and 30 C forms a closed system. It undergoes a constant pressure process with heat addition of 450 kJ. Compute nal temperature, change in enthalpy, change in internal energy, and work done. Assume air to be an ideal gas, with MolWt of 29 kg/kmol and cv = 0.714 kJ/kg K. F1.4 A perfectly insulated system contains 0.1 m3 of hydrogen at 30 C, 5.0 bar. It is stirred at constant pressure till the temperature reaches 60 C. Determine heat transferred, change in internal energy, stirrer work, and net work. Treat hydrogen as an ideal gas with MolWt = 2 kg/kmol, = 1.4. F1.5 A closed system contains 2 kg of air at 3 bar, 150 C. It is stirred, and expands till its pressure reduces to 1 bar. During the process, the temperature of the system is maintained constant at 150 C. If the stirrer does 120 kJ of work, determine (a) expansion work done, and (b) heat transferred. Assume R = 287 J/kg K for air. F1.6 The heat transferred during the quasi-static process of an ideal gas with constant specic heats in which only expansion work is done can be represented by dq = c dT , where c is a constant. Show that the equation for this process can be represented as p v k = const. Determine k in terms of cp , cv and c. F1.7 Consider the action of an air gun. The gun consists of a chamber of volume VC connected to a long cylindrical barrel of volume VB . Initially, compressed air at pressure pC and temperature TC is lled in the chamber. The bullet is located at the chamber-end of the barrel, and is held in place by a stopper. When the bullet is relased, the air in the chamber expands into the barrel and accelerates the bullet. Assume that the bullet behaves like a leak-proof, frictionless piston; air expands adiabatically; and air behaves like an ideal gas with constant specic heats. (a) Obtain an expression for V0 , the muzzle

Engineering Thermodynamics Exercises

velocity of the bullet, in terms of pC , VB , VC , mb (mass of the bullet), and po (ambient pressure). (b) Is there an optimal length of the barrel? (c) In what way, in a real air-gun, will the situation dier? F1.8 Consider the electrolyte in a car battery as a system. It is charged over a period of 24[h]. During charging, the potential across the electrodes is 24[V] and the current drawn is 1.0[A]. During the process, 15.0[litre] of a gas is evolved. The evolution of gas may be modeled as the displacement of the atmosphere, which remains at 1.0[bar]. (a) Sketch the system diagram. (b) Sketch appropriate process diagrams. (c) Determine all components of work done by the system, and the total work. F1.9 Argon (MolWt 40 kg/kmol) is initially at 4 bar, 100 C in a closed system. During a process, it expands from an initial volume of 10 litre to a nal volume of 20 litre. The nal pressure is 2 bar, and the process may be represented by a straight line on a pV diagram. During the process, the system is stirred, and absorbs 3 kJ of stirrer work. Determine expansion work, total work, and heat absorbed by the system. F1.10 Two kg of Nitrogen gas, initially at 10 bar, 400 K, executes an adiabatic process to 5 bar, 200 K. The process is suspected to be non-quasi-static. During the process the system absorbs electric work of 1500 kJ. Determine (a) change in volume, (b) change in internal energy, (c) change in enthalpy, (d) net work done, and (e) expansion work done. For Nitrogen, molecular mass is 28 kg/kmol and = 1.4. F1.11 A closed contains 10 kg of an ideal gas. The initial state is 10 bar, 300 K, and the nal state is 5 bar, 300 K. The process is not necessarily quasi-static. During the process, electric work of 50 kJ is absorbed, and the heat loss is 30 kJ. Sketch the process on a pV diagram. Determine (a) change in energy, (b) expansion work, (c) total work. Additional Exercises Achuthan, Chapter 5: Problems 1(p103), 5, 6(p104), 7, 8, 11(p105), and 12, 13(p106). Moran and Shapiro: Problems 2.54 to 2.70 (pp 7880). Some of these use non-metric units, you may convert these to metric/SI units. Sears and Salinger (Read quasi-static where you see reversible.): Problems 2-4(p55)+3-4(p90) 2-5(p55)+3-5(p90), 3-7(p90), 3-13 to 3-18(pp91,92), 3-22 to 3-26(p93) Sonntag et al.: Problems 5.89 to 5.102(pp 152154).

Engineering Thermodynamics Exercises

Properties of Fluids, The First Law 2 F2.0 Plot, using your steam tables, the following diagrams for ordinary water substance. (a) pv diagram on linear scales. Plot a number of isotherms. (b) pv diagram on logarithmic scales. Plot a number of isotherms. (c) uT diagram on linear scales. Plot a number of isobars. (d) hT diagram on linear scales. Plot a number of isobars. (e) T v diagram, linear scale for T , logarithmic scale for v. (f) T s diagram on linear scales. Plot a number of isobars. F2.1 Find the state of a closed system containing 1 kg of water substance as mentioned below. Specify h, v, x, and s as appropriate: (a) saturated liquid at 5 bar, (b) saturated vapour at 10 bar, (c) wet steam of quality 0.8 and at 3 bar, (d) 5 bar, 200 C, and (e) 100 bar, 200 C. F2.2 Classify the following states of 1 kg of water substance as wet, dry saturated, superheated steam, subcooled liquid, etc.: (a) p = 1 bar, T = 150 C, (c) p = 2 bar, S = 6.2 kJ/K, (e) H = 2900 kJ, S = 6.2 kJ/K, (g) H = 2700 kJ, p = 0.5 bar, (i) T = 100 C, x = 0.8. (b) p = 2 bar, T (d) p = 2 bar, V (f) H = 2500 kJ, S (h) H = 2100 kJ, T = 200 C, = 0.1 m3 , = 8.8 kJ/K, = 50 C,

F2.3 For the following processes, nd the changes in h, s, v, T , and x, as appropriate. The initial state pressure is p1 = 5 bar. the nal state is 2. (a) constant volume : (b) constant entropy : (c) constant volume : (d) constant enthalpy : (e) constant volume : (f) constant pressure : v1 = 0.3 m3 /kg, s1 = 6.3 kJ/kg K, h1 = 2500 kJ/kg, s1 = 6.4 kJ/kg K, T1 = 200 C, x1 = 0.5, p2 = 3.0 bar; p2 = 1.5 bar; p2 = 2.0 bar; p2 = 2.0 bar; p2 = 2.0 bar; T2 = 400 C.

F2.4 For the Van der Waals equation of state, (p + a )(v b) = RT v2

Engineering Thermodynamics Exercises

show that pc = a/27b2 , vc = 3b, and Tc = 8a/27bR, where subscript c denotes critical conditions. F2.5 Derive the reduced equation of state for a Van der Waals gas: (pR + 3 )(3vR 1) = 8TR 2 vR

where the the reduced quantities are dened as follows: pR = p/pc , vR = v/vc , and TR = T /Tc . F2.6 Obtain an equation for the expansion work done by a Van der Waals gas during an isothermal process in terms of the reduced quantities. F2.7 Determine the region of pvT space in which a Van der Waals equation can be approximated by an ideal gas equation to an accuracy of 1% in pressure. (That is, |(pvg pig )/pig | 0.01, where pvg and pig are pressures calculated using the Van der Waals equation and the ideal gas equation, respectively, at the same v and T .) F2.8 From the steam tables, read o the critical conditions for water substance. Obtain the Van der Waals constants a and b. F2.9 Show that, for a Van der Waals gas, pc vc / R Tc = 3/8. Check this out for steam. How good is the Van der Waals equation for steam? Comment. F2.10 Calculate the work done per kg in compressing steam (a) from 1 bar to 25 bar, isothermally at 400 C, and (b) from 5 bar to 25 bar, isothermally at 600 C. Use steam tables and the trapezoidal rule for integration. Compare these values with those obtained by assuming steam to be a Van der Waals gas. F2.11 Two kg of steam at critical conditions is sealed in a rigid container. It is then placed inside an oven at 349.9 C, and allowed to reach thermal equilibrium. Determine the masses of water and steam in the nal state, and the heat transferred. F2.12 2 kg of saturated liquid water at 12 bar is mixed with 1 kg of superheated steam at 12 bar, 300 C. The mixing is adiabatic and at constant pressure. Determine the changes in volume and internal energy, the dryness fraction in the nal state, and work done. F2.13 Compute the changes in the internal energy and enthalpy, and the expansion work done when 1 kg of saturated liquid water at 100 C evaporates isothermally into dry saturated steam.

Engineering Thermodynamics Exercises

F2.14 Water at 100 C and 6 bar is supplied 1500 kJ/kg of heat at constant pressure. Determine the nal state of the system (enthalpy, dryness fraction, temperature). The system is a closed one. F2.15 A rigid metallic container is separated into two equal parts by a thin partition. One part contains 1 kg of saturated liquid water at 100 C. The other part is evacuated. The partition is broken, and equilibrium reestablished after some time. During the process, the temperature of the system is maintained by immersing it in an oil bath maintained at 100 C. Determine (a) nal state of steam, (b) work done, and (c) heat transferred. F2.16 Consider the system shown in the gure. The left chamber contains an ideal gas. The right chamber contains 2 kg of saturated liquid water at 1 bar. The gas is sirred, and the steam allowed to expand at constant pressure till it becomes dry saturated. Determine (a) change in internal energy of the ideal gas, (b) heat transferred across the diathermal wall, (c) stirrer work, and (d) work done by steam.
stirrer adiabatic cylinder

ideal gas

2 kg water

p=constant

adiabatic piston fixed diathermic partition

Rework the problem if the stirring is continued till the the steam reaches a temperature of 150 C. Assume that the ideal gas has a mass of 1 kg, and its specic heat at constant volume is 0.72 kJ/kg K. F2.17 A rigid, insulated vessel is divided into two chambers by an adiabatic partition wall. One chamber contains 1 kg of wet steam at 4 bar and dryness fraction 0.7, while the other chamber contains 0.5 kg of dry saturated steam at 2 bar. If the partition between the chambers is removed, and the uids on both sides allowed to mix, determine the specic volume, specic internal energy, pressure, and dryness fraction in the nal state. (You may have to use a trial-and-error procedure, or a graphical procedure, to determine the nal state.) F2.18 1.5 kg of steam is compressed from an initial state 1 (2 bar, dry saturated) to a nal state 2 (8 bar). The process may be modeled as pV = constant. Determine (a) nal temperature and nal state, (b) change in energy, (c) work done by steam, and (d) heat absorbed by steam.

Engineering Thermodynamics Exercises

The Second Law Whenever you work with the Second Law, keep the following questions in mind: Does the process satisfy the Second Law? Is it impossible, reversible, or irreversible? If irreversible, what are the causees of irreversibility? SL.1 A conductor connects two reservoirs at temperatures of 1200 K and 500 K. The steady ow rate of heat from the hot to the cold reservoir is 150 W. Determine the rate of entropy production by the conductor. SL.2 1 kg of water in a cylinder-piston arrangement is initially in the saturated liquid state at 8 bar. It absorbs heat from a reservoir at 250 C. During the process the piston moves in such a way that the pressure remains constant. At the end of the process, water has completely evaporated into steam. Determine (a) heat transferred, (b) change in entropy of the system, (c) change in entropy of the reservoir, and (d) change in entropy of the universe. SL.3 A thermally insulated cylinder, closed at both ends, is tted with a leakproof, frictionless, diathermic piston which divides the cylinder into two parts. Initially, the piston is clamped in the centre, with 1 litre of air at 300 K and 3 bar on one side and 1 litre of air at 300 K and 1 bar on the other side. The piston is released, and reaches equilibrium in pressure and temperature at a new position. Compute the nal pressure and temperature and the change in entropy. Which irreversible process has taken place? SL.4 Three kg of an ideal gas in a rigid insulated container changes its state from 600 kPa and 300 K to 2 MPa, while it is stirred. Assuming cp 1.0 kJ/kg K, and = 1.4, determine the change in entropy of the system. SL.5 An insulated chamber of volume 2V0 is divided by a thin, rigid partition into two parts of volume V0 each. Initially, one chamber contains an ideal gas at a pressure p0 and temperature T0 . The other chamber is evacuated. The partition is suddenly removed. Show that, when equilibrium is reestablished, the temperature is T0 . Determine the change in entropy. Which irreversible process has taken place? SL.6 Ten kg of dry saturated steam at 35 bar, contained in a cylinder-piston arrangement, is brought into thermal contact with a heat sink at 175 C. The steam rejects 16000 kJ of heat during a constant pressure process. Determine (a) nal state of steam, (b) change in entropy of steam, and (c) change in entropy of the universe.

Engineering Thermodynamics Exercises

SL.7 Two kg of saturated liquid water at 2 bar is mixed adiabatically with 5 kg of steam at 2 bar, 400 C. During the process, the pressure remains constant. Determine the nal state and the change in entropy. SL.8 One kg of a liquid at 300 K is mixed with 1 kg of the same liquid at 400 K in an adiabatic calorimeter. Assume that there is no change of phase and that the density and specic heat of the liquid remains constant at 1200 kg/m3 and 5 kJ/kg K. Calculate the change in entropy of the system. SL.9 One kg of an ideal gas at 12 bar and 500 K is mixed with 1 kg of the same gas at 5 bar and 300 K. The mixing takes place at constant volume. During the process, the system rejects 150 kJ of heat to the environment, which remains at 300 K. Gas properties: RM M = 32, = 1.33. Determine (a) nal state (temperature, volume, pressure), and (b) change in entropy of the universe. SL.10 A closed system undergoes a process between xed initial and nal states (1 and 2) such that the overall changes in energy, entropy, and volume are E2 E1 = E, S2 S1 = S, and V2 V1 = V respectively. The only input of energy to the system is from condensing steam at a temperature Ts . The only outputs of energy are the heat transfer to the environment and work done in displacing the environment. The amount of heat absorbed from steam by the system is Q. (a) Apply the rst law to the system. (b) Apply the second law to the system. (c) Show that any irreversibility in the process gives rise to a wastage of the thermal energy of steam by an amount Q Ts (E + po V To S) Ts To

where po and To are respectively the pressure and temperature of the environment. SL.11 An inventor has proposed a refrigeration system that works without any work input. The invention has the following characteristics: (a) It is a cyclic device, and does not have any work interaction with the surroundings. (b) It absorbs 35 kW of heat from the cold space at 10 C, and also absorbs 90 kW of heat from condensing steam at 100 C. (c) It rejects heat to the atmosphere, which is at 30 C. Apply the rst law and determine the rate of heat rejection to the atmosphere. Apply the second law and determine whether the proposed system is possible or not. SL.12 Water in a piston cylinder assembly is as shown in the gure below. There are two stops, a lower one at which the volume enclosed is 1 m3 and the upper one at which the volume enclosed is 3 m3 . The piston mass and the atmospheric

Engineering Thermodynamics Exercises

10

pressure are such that the piston oats at 500 kPa. The above described system has water initially at 1 MPa and 500 C. This is allowed to cool to 100 C, by rejecting heat to atmosphere at 30 C. Calculate the total entropy generated in the process?

Water
SL.13 Consider the system shown in the gure. The adiabatic cylinder(C)-piston(P) arrangement has a xed diathermic partition(D) inside it. The chamber away from the piston(A) contains M kg of an ideal gas, initially at a temperature T1 and pressure p1 . The other chamber(B) also contains M kg of the same gas, initially at a temperature T1 and pressure p1 . C P B D A

During a process, the piston is moved slowly, to compress the gas in B. The process continues till the temperature reaches T2 . Determine (a) nal pressure of gas in A, (b) heat transfer across the partition D, (c) work done by gas in B, (d) the entropy change of the universe, and (e) nal pressure of gas in B. (f) Show that the process for the gas in B can be represented by pv k = constant, and obtain an expression for k in terms of of the gas. Additional Exercises Achuthan, Chapter 7: 69, 11, 12, 17 (pp 165168). Moran and Shapiro: 5.1 to 5.23 (pp 231233). Sears and Salinger: 5-1, 5-4, 5-5, 5-7 to 5-9, 5-11, 5-13, 5-16, 5-20, 5-21, 5-27 (pp141145). Sonntag et al.: 7.33 to 7.38 (p 242).

Engineering Thermodynamics Exercises Property Relations PR.1 Consider s as a function of T and p. Show that T ds = cp dT T v T dp
p

11

PR.2 Now consider s as a function of T and v, and derive another expression for T ds, similar to that in Exercise PR.1. PR.3 Using the results of Exercises PR.1 and PR.2, show that cp cv = T v T p T

PR.4 Derive an expression for (cv /v)T in terms of p, v, and T . PR.5 Show that for a Van der Waals gas, cv is a function of T alone. PR.6 Determine cp cv for a Van der Waals gas. PR.7 Determine whether cp is a function of T alone for a Van der Waals gas. PR.8 Using the table below, estimate the values of cp , cv , , = 1 v v T , =
p

1 v

v p

, and =
T

1 v

v p

for steam at 7 bar and 500 C. p, bar 7.0 7.0 7.0 6.5 7.5 T , C 490 500 510 500 500 h, kJ/kg 3460 3482 3503 3482 3481 v, m3 /kg 0.5002 0.5069 0.5136 0.5461 0.4730 s, kJ/kg K 7.901 7.929 7.957 7.964 7.897

PR.9 The Joule-Kelvin coecient is dened as = (T /p)h . Show that = 1 T cp v T v

What is the utility of this equation? Show that the Joule-Kelvin coecient of an ideal gas is zero. PR.10 Prove that u p v T v p

= T
T

p
p

Engineering Thermodynamics Exercises PR.11 Prove that

12

p T

=
s

cp T v

Describe an experiment that will use this relation to determine cp without any energy measurements. PR.12 In a property table, three consecutive entries at constant pressure are as follows: 1 2 3 h, kJ/kg 3465 3486 3508 s, kJ/kg K 8.329 8.356 8.384 Estimate the value of temperature and specic Gibbs function at point 2. PR.13 Determine gf and gg for ordinary water substance at 1 bar, 10 bar, 100 bar, etc. Generalise the result. Explain it. PR.14 For a uid, the specic Gibbs function g(= hT s) is given by the expression g = RT ln(p/p0 ) b(T )p, where R and p0 are constants and b(T ) is a function only of T . For such a uid, (c) derive the equation of state, (d) an expression for s, and (e) an expression for cp . PR.15 Derive an expression for T v in terms of cv , and EoS information.
u

Check its value for an ideal gas. Explain the result. Now, consider the case of a Van-der-Waals gas. Comment on the result. PR.16 The velocity of sound c in a uid is related to the thermodynamic state by the relation p c2 = s Express c2 in terms of cp , cv , p, , and T . What does this expression reduce to for an ideal gas with constant specic heats? PR.17 Derive the Clausius-Clapeyron equation by applying Maxwells equation for (T /p)v or (T /p)s in the two-phase zone of a uid like water. From the textbooks you have used, nd out other methods to derive the C-C equation. Using Steam Tables, check whether the Clausius-Clapeyron equation is satised by steam-water equilibrium data at 100 bar. PR.18 Consider a Carnot cycle working between two temperatures T and T dT with a vapour as a working uid. The cycle lies entirely within the two-phase zone of the vapour. Sketch the cycle on T s coordinates. Write down an expression for its eciency, in terms of the work done and the heat absorbed.

Engineering Thermodynamics Exercises

13

Equate this to the standard expression for the eciency of the Carnot cycle. Hence, derive the Clausius-Clapeyron relation for (dT /dp)sat . PR.19 he following relation between the saturation pressure p and saturation temperature T of a uid has been obtained by using the Clausius-Clapeyron equation. hf g , ln p = A RT where A is a constant. One of the assumptions made for this derivation is that dry saturated vapour behaves like an ideal gas. Derive this equation. Which additional assumptions did you make? PR.20 If the specication for the properties of a substance is given as a(T, v) show that all its properties, including any change-of-phase (saturation) lines, can be derived. PR.21 Consider a Van-der-Waals gas, with a constant cv (note: you must have already shown that cv for such a gas is a function of T alone). Obtain an expression for a(T, v). It is recommended that you work with reduced properties. Use some reasonable values for cv and show, numerically/graphically, that there exists a critical point, and determine some saturation points and saturation states below critical temperature.

Open Thermodynamic Systems OS.1 Joule postulated that the temperature of water when it comes down a waterfall rises. Consider the waterfall to be an open thermodynamic system and determine the height through which the water should fall for its temperature to rise by 1 C. Assume water to be an incompressible uid with cp = 4.186 kJ/kg K. List all other assumptions made while solving the problem. OS.2 Steam is supplied to a turbine with hi = 3.2 MJ/kg and it leaves with he = 2.5 MJ/kg. The entrance and exit velocities are 170 m/s and 280 m/s respectively. If the heat loss is 50 kJ/kg, what is the work done? OS.3 A steady ow system receives 60 kg/min of gas at 2 bar, 90 C with negligible velocity, and discharges it at a point 20 m above the entrance section at a temperature of 300 C with a velocity of 2200 m/min. During this process, 2 kW of heat is supplied from external sources, and the increase in enthalpy is 7.8 kJ/kg. Determine the power output.

Engineering Thermodynamics Exercises

14

OS.4 A centrifugal air compressor compresses 5 m3 /min of gas from 10 N/cm2 to 70 N/cm2 . The inlet and exit specic volumes are 0.8 m3 /kg and 0.4 m3 /kg respectively. The duct diameter is 10 cm at the inlet and 5 cm at the exit. Determine (a) rate of ow work, (b) mass ow rate, and (c) change in velocity. OS.5 A steam turbine receives steam at a rate of 5400 kg/hr and develops power of 600 kW. Neglecting heat losses, determine the change in specic enthalpy of steam owing through the turbine if (a) the entrance and exit velocities and heights are negligible, and (b) if the entrance and exit velocities are 50 m/s and 320 m/s respectively, and the inlet is 4 m above the exhaust. OS.6 In a steady-state apparatus, the work done by the system is 80 kJ/kg of uid. The specic volume, pressure, and velocity at the inlet and exit are 0.5 m3 /kg, 8 bar, 12 m/s, and 0.7 m3 /kg, 1 bar, 220 m/s respectively. The inlet is 10 m above the exit and the total heat loss is 10 kJ/kg of uid. What is the change in specic internal energy? OS.7 A ship propulsion system incorporates a compressor which receives steam at 3.4 bar with 5 percent moisture. It delivers it dry and saturated at 8 bar. Steam ow rate is 5 kg/s. The compression is adiabatic. Diameters of the inlet and exit ducts are 20 cm. The mechanical eciency of the machine is 92%. (a) Determine the power required to drive the compressor. (b) Is the process possible or impossible? Why? What is the limiting exit state? Assume that the exit pressure is xed. OS.8 The inlet conditions for the nozzle of a steam turbine are 60 bar, 350 C. The exit conditions are 10 bar, 0.9 dry. (a) If the steam ow rate is 10,000 kg/hr, determine the exit velocity and exit area. (b) Is the process possible or impossible? Why? What is the limiting exit state and exit velocity? Assume that the exit pressure is xed. OS.9 The inlet conditions of a water pump are 1 bar, 25 C, and the exit pressure is 180 bar. The pump consumes 75 kW of power and pumps 12,000 litres of water per hour (at inlet conditions). Determine the temperature of water at the exit of the pump. If we dene the ideal pump as the one which does the pumping isothermally, what is the eciency of the pump? OS.10 Feedwater at 0.1 bar is pumped from a condenser into a boiler at 25 bar. Water at the exit of condenser is saturated and the compression is isentropic. Determine the work done per kg of water pumped, and the ow work. OS.11 Water at a rate of 120 kg/min enters a pump at 1 bar, 35 C. Thepump power is 110 kW, and the pump raises the pressure to 5 bar. The water then passes

Engineering Thermodynamics Exercises

15

through a boiler in which 1800 kJ/kg of heat is added. Assuming negligible pressure drop in the boiler, determine the state at the exit of the boiler and the velocity at that point if the diameter of the exit duct is 20 cm. OS.12 Water ows through a horizontal venturimeter at a steady rate of 600 kg/min. The inlet and throat diameters of the venturi are 6.0 cm and 3.0 cm respectively. If there is no transfer of heat or work, and no change in internal energy, and if the density remains constant at 1000 kg/m3 , what will be the pressure drop between the inlet and throat in bar? OS.13 Wet steam at 10 bar is passed through a throttling calorimeter. The state of steam after throttling is 0.75 bar, 100 C. What is the dryness fraction of steam before throttling? OS.14 A rigid, insulated bottle of volume V0 is perfectly evacuated. The stopper is opened and ambient air (at p0 , T0 ) is allowed to ow in. When the ow stops, the stopper is replaced. Determine the nal temperature of air in the bottle. OS.15 Steam enters the nozzle of a steam turbine with a velocity of 5 m/s at a pressure of 40 bar and 600 C. The pressure and temperature at nozzle exit were measured as 1 bar, 200 C. Determine (a) exit velocity, (b) entropy production rate if the ow rate of steam is 1.5 kg/s, and (c) isentropic eciency of the nozzle. OS.16 An adiabatic steam turbine handles 10 kg/s of steam. The inlet state is 10 bar, dry saturated. The exit pressure is 1 bar. The isentropic eciency of the turbine is 0.8. Determine (a) exit state, (b) power output, and (c) entropy production rate. OS.17 In a heat exchanger, air is heated from 30 C to 80 C by means of a second air stream which enters the heat exchanger at 150 C. Both streams have a ow rate of 2 kg/s and ow without any loss of pressure. Determine (a) heat transferred between the streams, and (b) entropy production rate. Assume cp = 1.0 kJ/kg K for air. OS.18 120 kg/hr of saturated water at 8 bar enters a heat exchanger and leaves at 4 bar, 200 C. Hot air enters at 600 C, 2 bar, and leaves at 240 C, 2 bar. Determine (a) ow rate of air, (b) heat transfer rate from air to water, (c) rate of entropy outow for each stream, and (d) entropy production rate. Use cp = 1.0 kJ/kg K for air. OS.19 An adiabatic cylinder C consists of two parts A and B, separated by a xed, diathermic, partition D as shown in the gure below:

Engineering Thermodynamics Exercises

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m
C D A B

The chamber at the left is continuously ushed with dry saturated steam at a pressure pA , and hence is initially full of it. The right chamber is now lled with an ideal gas (with constant specic heats at a temperature TB which is lower than TA (= Tsat (pA )). Heat is transferred across the diathermic partition from steam to air, and the temperature of air nally reaches TA . Any steam which condenses remains in chamber A (i.e. the exit state of A is always dry saturated steam at pA ). (a) Explain, with appropriate equations, how you will determine (1) the heat transferred across the partition, (2) the mass of condensed steam in chamber A, and (3) the change in entropy of air in B. (Nomenclature: mB : mass of air in B, cp , cv , : properties of air in B, m1 : initial mass of steam in A, mg : nal mass of steam in A, mf : nal mass of water (condensate) in A, hf , hg , hf g : properties of steam.) (b) Compute the quatities specied in (a) if pA = 40 bar, TB = 30 C, and mB = 10 kg. OS.20 The gure below shows a butane cylinder of diameter D = 0.4 m and height H = 0.8 m. The butane is at 300 K. Initially, the level of liquid in the cylinder in 0.7 m. The valve is opened, and some butane is taken out for consumption. During this process, (a) the temperature of butane is maintained at 300 K, (b) it absorbs heat from the environment, which is at 310 K, and (c) only dry saturated vapour leaves the cylinder. The nal level of butane is 0.3 m. Determine the mass of butane consumed, and the heat absorbed. Butane properties at 300 K are: psat = 2.607 bar, f = 570.5 kg/m3 , g = 6.559 kg/m3 , hf = 564.0 kJ/kg, hg = 924.7 kJ/kg.

Engineering Thermodynamics Exercises

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vap liq
H

D OS.21 Cold air at 0 C and 1 bar, and a ow rate of 1 kg/s is mixed with a hot stream of air at 80 C and 2 bar to provide warm air at 25 C and 1 bar. The mixer is an adiabatic open system with rigid walls. Determine (a) ow rate of hot air, and (b) rate of entropy production. (c) Which irreversible process has taken place? OS.22 A throttling and separating calorimeter is used to measure the dryness fraction of steam owing through a duct at 10 bar. wet steam, 10 bar Throttle Separator A, 1 bar, xA = 1, mA = 0.9 kg/s B, 1 bar, xB = 0, mB = 0.1 kg/s

A sample of steam is throttled to near ambient temperature, and then put through a separating chamber, where it is split into two streams. Stream A is dry saturated steam and stream B is saturated liquid. The ow rates of the two streams are measured as mA = 0.9 kg/s and mB = 0.1 kg/s. (a) Determine the dryness fraction of steam in the duct. (b) Can a simple throttling calorimeter be used for this measurement? Why? OS.23 An incompressible liquid (density , isobaric sp heat cp ) of mass M is lled in a tank, at an intial temperature T0 . The top surface is open to the atmosphere (at p0 ). For t 0, a stream of the same uid enters the tank at a temperature Ti , the inow rate is mi . At the same time, uid at a rate of me is withdrawn from the tank. Assume that the liquid in the tank is well mixed and neglect any heat transfer between the liquid and the surroundings. Derive expressions for (a) rate of change of temperature of the liquid in the tank at t = 0, and (b) rate of entropy production at t = 0.

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OS.24 Study the gure shown below. The pump P pumps water into the boiler B. The steam generated runs a turbine T. The exhaust of the turbine is split into two streams 5 and 6 in the separator S. The mixer M is used to mix some water from boiler inlet and some steam from boiler exit to provide a mixed steam 9. Data: State 1: 1 bar, sat.liq.; States 2, 8: 50 bar; States 3, 7: 50 bar, 400 C; State 4: 2 bar; State 5: 2 bar, sat.liq.; State 6: 2 bar, dry.sat.vap.; State 9: 50 bar, dry.sat.vap.; m4 = 10 kg/s; m9 = 1 kg/s; isentropic eciency of turbine 0.8. Determine (a) power produced by the turbine, (b) mass ows at 5, 6, 7, 8, and 1, (c) power comsumed by the pump, (d) rate of heat absorption in the boiler, and rate of production of entropy in (e) the turbine, (f) the separator, and (g) the mixer. Qb

3
Wt T

8 M

7 6 9

4 5
S

Combined First and Second Laws Assume that the environment is at p0 = 1 bar, T0 = 300 K, unless specied otherwise. CL.1 What is the maximum work that can be obtained from a perfectly evacuated space of volume Vvac ? CL.2 Determine the maximum useful work that can be obtained when the following systems undergo the specied change of state: (a) 1 kg of ordinary water substance at (i) 10 bar, saturated liquid, (ii) 10 bar, dry saturated vapour, (iii) 10 bar, 600 C. In each case, the nal state is (p0 , T0 ). (b) Initial state: Two identical systems of the same mass m, and specic heat capacity cp , temperatures TA and TB , and pressure p0 . Final state: equilibrium with each other. Process: no change of phase, adiabatic, constant pressure. CL.3 (a) Calculate the maximum useful power output from an adiabatic steam turbine if the inlet state is 10 bar, 350 C, exit pressure is 1 bar, and the ow

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rate 5 kg/s. Also determine the exit state. (b) Rework the problem if the steam loses 40 kJ/kg of heat to the atmosphere which is at 300 K. What is the exit state in this case? CL.4 Air at 1200 K and 8 bar expands in an adiabatic turbine to 1 bar, with an isentropic eciency of 0.85. The air ow rate is 2 kg/s. Determine (a) power output, (b) maximum possible power output for the same inlet and exit conditions, and (c) maximum power output if the turbine were to remain adiabatic. What is the exit state in this case? CL.5 A system containing 1 kg of water substance is compressed from 12 bar, 300 C to 20 bar in a quasi-static isothermal process, while being cooled by the environment. Determine (a) useful work done by the system, (b) maximum useful work, and (c) lost work. What is the cause of the lost work? CL.6 Just as your thermo prof was passing one of the new buildings (under construction), en-route to the Department to conduct an ME209 quiz, a 100 kg sack of sand (cp = 1000 kJ/kg K) smashed to the ground near him, just missing him. The zeroth, rst, second, and third thoughts to cross his mind were: (0) Who threw it from the top of the building? (1) what was the S for the bag of sand? S of the surroundings? (2) What was the amount of entropy produced? (3) What was the amount of lost work? Please help the prof by computing the quantities needed to answer (1), (2) and (3). You may assume that the height of the building is 30 m. Also, it may be a good idea to assume that the initial and nal temperature of the bag was 25 C, which was also the temperature of the environment. Even though frictional heating may have raised the temperature of sand as it fell and smashed to the ground, heat transfer to the environment would have cooled it to the ambient temperature.

Cycles CA.1 A Carnot engine uses 1 kg of air as the working uid. The temperature limits are 300 K and 900 K, the minimum pressure is 1 bar and the volumetric compression ratio is 20. Determine (a) eciency, (b) all state points, (c) work ratio, (d) work done per cycle, (e) heat supplied per cycle, and (f) mean eective pressure. CA.2 Repeat the previous exercise for a (a) Stirling cycle and (b) Ericsson cycle. Limit the maximum pressure, maximum temperature, and minimum volume to that in th previous exercise. Assume the regenerator to be an ideal one. Also determine the heat transfer in the regenerator per cycle.

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CA.3 Which is the more eective way to increase the eciency of a Carnot engine: (a) to increase the temperature of the hot reservoir, keeping that of the cold reservoir constant, OR (b) to decrease the temperature of the cold reservoir, keeping that of the hot reservoir constant? Discuss from the point of view of engineering. CA.4 What are the expressions for the work done, heat absorbed, and the eciency of the cycles shown? Assume the working uid to be an ideal gas with constant specic heats. (a) p
3

(b) s = const

V (c) p
3 4

CA.5 Undertake a standard analysis of the Brayton cycle. Derive expressions for the eciency, work ratio, and specic output in terms of the parameters rp (pressure ratio) and (temperature ratio). Determine the value of rp which maximises the specic output for a xed . Plot numerical values of , work ratio, and wsp /cp T1 against rp for dierent values of , for 3 rp 20 and 3 5. CA.6 A heat pump working on a reversed Carnot cycle is used to supply 80 kW of heat for maintaining the rooms of a building at 20 C when the outside temperature is 0 C. Determine (a) the COP of the system, (b) power consumed by the heat pump, and (c) heat absorbed from the atmosphere outside. Determine the power consumption if direct electric heating is used. CA.7 Determine the COP of a Joule-cycle refrigerator if the environment is at 300 K, the cold space is at 270 K, and (a) the compression ratio is 4, or (b) the compression ratio is 7. In either case, determine (c) power consumption in kW/tonne.

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CA.8 An Otto cycle with a compression ratio of 7.8 operates from the suction condition of 1 bar, 300 K. Find the pressure and temperature at the end of compression and the standard eciency. Can you determine the temperature at the end of expansion? Why? CA.9 In an Otto cycle with a compression ratio of 7, the suction temperature and pressure are 300 K and 1 bar. Heat supplied during the constant volume process is 700 kJ/kg. The air ow rate is 90 kg/hr. Determine (a) power output, (b) mean eective pressure, and (c) eciency. CA.10 Derive an expression for the eciency of a Diesel cycle in terms of the volumetric compression ratio (rv ) and the cut-o ratio (rc ). CA.11 Determine the eciency of a Diesel cycle having a compression ratio of 18, if the temperature at the beginning of compression is 300 K and that at the end of expansion is 1000 K. CA.12 The inlet state in a dual combustion cycle is 1 bar and 300 K. Its compression ratio is 10. The maximum pressure and temperature in the cycle are 45 bar and 1800 K. Determine the cycle eciency. CA.13 The inlet of a dual combustion cycle is 1 bar and 300 K. Its compression ratio is 8 and expansion ratio is 5.3. If the isobaric heat absorbed is twice the isochoric heat absorbed, determine (a) cycle eciency and (b) MEP. CA.14 The inlet conditions for an Otto cycle are 1 bar, 290 K. The pressures at beginning and end of combustion are 15 bar and 40 bar respectively. Determine (a) compression ratio, (b) standard eciency, and (c) MEP. If the relative eciency of an engine using this cycle is 50%, and the caloric value of the fuel burnt is 42 MJ/kg, determine (d) eciency, and (e) specic fuel consumption. CA.15 The fuel cut-o in a Diesel cycle takes place at 5% of the stroke. If the compression ratio is 20, determine the standard eciency. CA.16 Consider a Brayton cycle with reheat. Let rp and be the pressure ratio for the compressor and the temperature ratio for the cycle, respectively. Let rp1 and rp2 be the pressure ratios of the two turbines. Determine the values of rp1 and rp2 that maximise the specic output. For such an optimal-reheat cycle, what is the best value of rp for xed ? CA.17 Calculate the specic output and thermal eciency of the following ideal gas turbine plants. (a) Basic plant: rp = 8, compressor inlet at 300 K, turbine inlet at 1200 K.

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(b) Reheat plant: Equal pressure ratio for each turbine and the same inlet temperature for each turbine. Other details as in (a). (c) Intercooled plant: Equal pressure ratio for each compressor and the same inlet temperature for each compressor. Other details as in (a). (d) Regenerative plant: With an ideal regenerator. Other details as in (a). (e) A plant with reheat, intercooling, and regeneration, as specied in (a)(d). (f) Compare and comment on the performance of the cycles from (a) to (e). CA.18 Rework the previous exercise with compressor eciency of 0.80 and turbine eciency of 0.90. CA.19 A jet engine working on a clipped, standard Brayton cycle has compressor inlet state of 1 bar 300 K and turbine inlet state of 6 bar, 1500 K. The turbine produces just the right amount of power to drive the compressor. The turbine exhaust is expanded through a nozzle to a pressure of 1 bar. The nozzle exit area is 1 m2 . Determine (a) compressor exit temperature, (b) turbine exit temperature, (c) turbine exit pressure, (d) nozzle exit velocity, (e) mass ow rate of the working uid, (f) power output of the turbine, and (g) static thrust. CA.20 Consider a Rankine cycle with saturated steam. Undertake a study of such a plant with steam entering the turbine is dry and saturated at (a) 10 bar, (b) 30 bar, (c) 50 bar, (d) 70 bar, and (e) 90 bar. In each case, compute eciency, steam rate, and dryness fraction of steam at the exit of turbine. Compare and comment. Assume that the condenser is at 0.06 bar in all cases. CA.21 Consider a superheated steam plant working at 130 bar, with the condenser at 0.06 bar. Study the eect of superheating by determination of performance at 350 ( 50 ) 600 C. CA.22 A water heater system is used to provide heat input to a low-temperature Rankine cycle with ammonia as the working uid. Ammonia is superheated to 90 C at 19.62 bar and is condensed at 30 C after an isentropic expansion. The properties of ammonia are as follows: T C 30 p bar 11.6 hf kJ/kg 323.16 hg kJ/kg 1468.45 sf kJ/kg K 1.2035 sg vf 3 kJ/kg K m /kg 4.9820 0.00168

The properties of superheated ammonia are:

Engineering Thermodynamics Exercises p bar 11.66 19.62 T C 50 90 h kJ/kg 1529 1607 s kJ/kg K 5.1760 5.1760 v m /kg 122.46 81.03
3

23

(a) Determine the eciency of the cycle. (b) If an ideal heat exchanger is used to recover the superheat of the exhause ammonia, what will be the eciency? CA.23 A Rankine cycle plant using steam has the following parameters: boiler exit: 45 bar, 450 C; condensation temperature: 30 C. Determine: (a) heat absorbed, (b) heat rejected, (c) specic output, and (d) eciency. If the turbine has an isentropic eciency of 0.9, determine (e) specic output, and (f) eciency. CA.24 A steam plant works on the Rankine cycle with reheat. Steam enters the turbine at 35 bar, 350 C, and expands to 8 bar, where it passes through a reheater, emerging at 350 C. It then expands to the condenser temperature of 40 C. For the ideal cycle, compute (a) work done in HP and LP turbines, (b) heat added in the boiler, (c) heat added in the reheater, (d) pump work, and (e) eciency. CA.25 In an ideal regenerative cycle, steam is generated at 45 bar, 450 C. It then expands to 4 bar, where a fraction of the steam is extracted for feedwater heating. Condensation is at 30 C. Determine (a) fraction of steam extracted, (b) specic output, (c) total pump work, (d) enthalpy of feedwater entering the boiler, and (e) eciency. (f) Compare (b)(e) with those for a cycle without regeneration, and comment on the dierences. CA.26 In a nuclear plant, dry saturated steam is supplied to the turbine. The boiler is at 45 bar, and the condenser at 0.06 bar. During expansion, the steam is removed thrice from the turbine, the water separated in separators, and dry saturated steam fed back into the turbine. The pressures at which wet steam is taken to the separators are 20 bar, 5 bar, and 1 bar. The water separated in each separator is discharged into the condenser. Determine the state lines through the turbine, the steam rate, and the eciency of the plant, if (a) the turbine and separators are ideal, and (b) the turbine has an eciency of 0.8 and the separators have pressure drops of 1.0 bar, 0.5 bar, and 0.2 bar, respectively. CA.27 A power plant is to be designed to operate on a regenerative cycle with two contact heaters to heat the feed water to 198.3 C. The steam parameters are 60 bar, 400 C and 0.05 bar. If the plant produces 50 MW at generator terminals, determine the eciency of the plant. Compare it with the eciency of a plant without feed heating. You may assume equal heating in the two heaters.

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CA.28 A Joule-cycle refrigerator works with the environment at 300 K, the cold system at 250 K, and a compression ratio of 6. The working uid is air (assume ideal gas behaviour). For the ideal cycle, determine (a) COP, (b) specic refrigeration eect, and (c) power supply for a 12-tonne plant. (d) If the compressor and turbine eciencies are 90% and 80%, what will be the answers? CA.29 If, in the previous problem (with compressor and turbine eciencies as in (d), an ideal regenerator is used, determine the pressure ratio one can work with, assuming that the compressor exit temperature is xed at 400 K. (a) What are the answers now? (b) If the regenerator is not ideal, how will you account for its eectiveness? (c) Comculate the answers when the regnerator eectiveness is 0.90. (d) What is the minimum value of the regenerator eectiveness below which the refrigerator will not work? CA.30 A vapour compression cycle with ammonia has 10 C as the evaporation temperature and +35 C as the condensation temperature. Assume that the compressor inlet state is dry saturated vapour and the condenser exit state is saturated liquid. Determine (a) all state points, (b) COP, (c) power consumption in kW/tonne. (d) Repeat the calculations for a low-temperature cycle with evaporator and condenser temperatures as 20 C and +40 C. CA.31 Repeat the previous exercise with R-134a as the working uid. Compare the results and comment. CA.32 Repeat the preceding two exercises with a compressor isentropic eciency of 80%. CA.33 For air-conditioners, a common refrigerant is R22. Determine the COP of a vapour compression cycle with R22 that has evaporation at +10 C, and condensation at +40 C. Assume that the compressor inlet state is dry saturated vapour and the condenser exit state is saturated liquid. Determine (a) all state points, (b) COP, and (c) power consumption in kW/tonne. Study the eect of compressor eciency, by considering isentropic eciencies of 1.0 and 0.8.

Exercises on Psychrometry, Combustion, and Compressible Flow will be provided separately.