CHAPTER 4 PROBLEMS

D. CHAPTER 4 PROBLEMS

(Unless otherwise stated assume T 0 = 25ºC and P 0 = 1 bar)

Problem D1 Is the relation s(T , p

HO 2 , ) R ln (p HO 2 , o /p HO 2 ,sat ) equivalent to s(T

Problem D2 In the condenser part of a power plant, is there an irreversibility due to Q o ?

Problem D3 Is it more practical to design for w opt than w s ?

Problem D4 Is the notion of availability based on an isentropic concept?

Problem D5 Is optimum work the same as reversible work?

Problem D6 When is g ≡ ψ?

Figure C. 84

Problem D7 Are ke and pe included in the definition of ψ?

Problem D8 Describe the concept of chemical availability.

Problem D9 Use an example to describe the availability for gasoline.

Problem D10 Differentiate between the absolute (availability-Europe) and the relative availability (exergy).

Problem D11 Explain the physical implications of the expression ψ= RT ln X k . Does this mean that

ψ chem < 0? Problem D12

Is chemical equilibrium satisfied when µ=µ o ? Problem D13

What is the typical range of COP? Problem D14

What is the difference between isentropic and optimum work? Problem D15

What is the absolute stream availability? Can it have negative values? Does the value depend upon the reference condition used for the properties, such as h, s, etc.?

Problem D16 What is the (relative) stream availability or exergy? Can it have negative values? Does the value depend upon the reference condition used for the properties, such as h, s, etc.?

Problem D17 What is the difference between closed system availability and open system availabil- ity ?

Problem D18 Can we assume that P o ∆v ≈ 0 for liquids?

Problem D19 What do we mean by useful and actual work?

Problem D20 Consider the universe. As S → ∞, does φ → 0?

Problem D21 What does a dead state imply?

Problem D22 How are irreversibilities avoided in practice?

Problem D23 For G to have a minimum value in a multicomponent system at specified values of T and P, what is the partial pressure of the species?

Problem D24 Can the availability be completely destroyed?

Problem D25 What are your thoughts regarding current oil consumption and availability?

Problem D26 What is the implication of W u,opt for compression work?

Problem D27 An irreversible expansion occurs in a piston–cylinder assembly with air as the me- dium. The initial and final specific volumes and temperatures are, respectively, 0.394

m 3 kg –1 and 1373 K, and 2.049 m 3 kg –1 and 813 K. Assume constant specific heats,

c v0 = 0.717 kJ kg –1 K –1 and c p0 = 1.0035 kJ kg –1 K –1 . a)

Determine the actual work delivered if the process is adiabatic and the adia- batic efficiency.

b) Assume that this is a reversible process between the two given states (not necessarily adiabatic for which Pv n = constant). What is the value of n? De- termine the reversible work delivered.

c) What is the maximum possible work if the only interactions are with the en- vironment, T amb = 300 K, and P amb = 100 kPa. What is the availability effi- ciency of this process? Is this the same as the adiabatic efficiency?

d)

What is the total entropy generated and the irreversibility?

Problem D28 Water flows through a 30 m long insulated hose at the rate of 2 kg min –1 at a pressure of 7 bar at its inlet (which is a faucet). The water hose is well insulated. Determine the entropy generation rate. What could have been the optimum work?

Problem D29 Steam enters a turbine at 5 bar and 240ºC (state 1). a)

Determine the absolute availability at state 1? What is the absolute availabil- ity at the dead state (considering thermomechanical equilibrium)?

b) What is the optimum work if the dead state is in mechanical and thermal equilibrium?

c) What is the chemical availability? d)

What is the optimum work if the steam eventually discharges at the dead state? The environmental conditions are 298 K, 1 bar, and air with a water vapor mole fraction of 0.0303.

Problem D30 Saturated liquid water (the mother phase) is contained in a piston–cylinder assembly at a pressure of 100 kPa. An infinitesimal amount of heat is added to form a single vapor bubble (the embryo phase). a)

If the embryo phase is assumed to be at the same temperature and pressure as the mother phase, determine the absolute availabilities ψ=h–T o s and Gibbs functions of the mother and embryo phases.

b) If the pressure of the embryo (vapor) phase is 20 bar at 100ºC, while the mother phase is at 1 bar, what are the values of the availability and Gibbs function of the vapor embryo? (Assume the properties for saturated vapor at 100ºC and that the vapor phase behaves as an ideal gas from its saturated va- por state at 1 bar and 100ºC to 20 bar and 100ºC to determine the properties.)

Problem D31 You’ve been engaged as a consultant for a manufacturing facility that uses steam. Their steam generator supplies high pressure steam at 800 psia, but they use the steam at 300 psia. How would you advise them to decrease the pressure such that they minimize irreversibilities? Be sure to explain your answer. If so, explain what and the mechanism responsible for the destruction. Show both the process and the throttling process on an h-s diagram and refer to it to illustrate your answer.

Problem D32 Consider the energy from the sun at T R,1 and the ocean water at T 0 . Derive expres- sions for W opt . Look at Figure Problem D.32 and interpret your results in terms of the figure.

Problem D33 Ice is to be heated at the North Pole where the ambient temperature is –30ºC to tem- perature of –25ºC, –20ºC, …, 90ºC. Determine the minimum work required. The heat of melting of ice is 334.7 kJ kg –1 , and c

ice is 1.925 kJ kg K .

Problem D34

A gas tank contains argon at T and P.

a) Obtain an expression for the maximum possible work if an open system is used when tank pressure is T and P. Assume that there is negligible change in T and P of the tank and constant specific heats for the ideal gas. The ambient temperature is T o and the ambient pressure is P o .

b) Suppose the gas is slowly transferred from the tank to a large piston–cylinder (PC) assembly in which the pressure and temperature decrease to the ambient values. Treat the tank and PC assembly as one closed system. What is the be- havior of φ/(RT o ) with respect to T/T o with P/P o as a parameter? Consider the

case when the gas state is at 350 K and 150 bar, and T o = 298 K and P o = 100 kPa.

Ocean water

Ambience at T 0

Figure Problem D.31 Relation between pressure and volume.

Problem D35 Natural gas (that can be assumed to be methane) is sometimes transported over thou- sands of miles in pipelines. The flow is normally turbulent with almost uniform ve- locity across the pipe cross sectional area. There is a large pressure loss in the pipe due to friction. The friction also generates heat that raises the gas temperature, which can result in an explosion hazard. Assume that the pipes are well insulated and the specific heats are constant. Assume that initially P = 10 bar and T = 300 K, and fi-

nally P = 8 bar for a mass flow rate of 90 kg s –1 1 2 1 m –2 . What is the entropy change per unit mass? What is the corresponding result if the velocity changes due to the pressure changes?

Problem D36 The adiabatic expansion of air takes place in a piston–cylinder assembly. The initial

and final volume and temperature are, respectively, 0.394 kg m –3 and 1100ºC, and 2.049 kg m –3 and 813 K. Assume constant specific heats c = 0.717 kJ kg –1 K –1

c p0 = 1.0035 kJ kg K . a)

What is the actual work? b)

What is the adiabatic efficiency of the process? c)

Assuming that a reversible path is followed between the same initial and fi- nal states according to the relation Pv n = constant, what is the work deliv- ered? Why is this different from the actual work?

d) Now assume isentropic expansion from the initial state 1 to a volume of 2.333 kg m –3 and isometric reversible heat addition until the final tempera- ture is achieved. What is the heat added in this case?

e) If the heat is first added isometrically and reversibly, and then isentropically expanded to achieve the final state, what is the value of the reversible work? e) If the heat is first added isometrically and reversibly, and then isentropically expanded to achieve the final state, what is the value of the reversible work?

Problem D37 Consider an ideal Rankine cycle nuclear power plant. The temperature of the heat source is 1400 K. The turbine inlet conditions are 6 MPa and 600ºC. The condenser pressure is 10 kPa. The ambient temperature is 25ºC. What is the irreversibility in KJ/kg and the maximum possible cycle work in KJ/kg?

Problem D38 Steam enters a non-adiabatic steady state steady flow turbine at 100 bar as saturated vapor and undergoes irreversible expansion to a quality of 0.9 at 1 bar. The heat loss from the turbine to the ambience is known to be 50 kJ/kg. Determine the a)

actual work, b)

optimum work, and c)

availability or exergetic or Second law efficiency for the turbine. Problem D39

Consider the generalized equation for work from a open system in terms of entropy generation. Using the Gauss divergence theorem, derive an expression for the work done per unit volume w ′′ by a device undergoing only heat interaction with its envi- ronment and show that w ′′ = –d/dt(e – T o s) – ∇(ρv(e T –T o s)) – T o σ. Obtain an ex-

pression for the steady state maximum work. Problem D40

Water is heated from the compressed liquid state of 40ºC and 60 bar (state 1) to satu- rated vapor at a pressure P 2 . Heat is supplied from a large reservoir of burnt gases at 1200 K. If the final pressure P 2 = 60 bar, calculate s 2 –s 1 and the value of the reversible heat transfer q 12 to the water. If P 2 = 58 bar due to frictional losses (state 2´) but h 2 ´=

h 1 , calculate s 2 ´–s 1 . Is this process internally reversible? Is there any entropy gener- ated and, if so, how much? If the value of Q H is identical for both cases (without and with frictional losses), what is the net entropy generated due to the irreversible heat transfer? Determine the changes in the availabilities ( ψ 2 – ψ 1 ) and ( ψ 2´ – ψ 1 ).

Problem D41

A water drop of radius a at a temperature T l is immersed in ambient air at a tempera- ture T ∞ and it vaporizes. The temperature and water vapor mole fraction profile can in terms of the radial spatial coordinate r be expressed through the following expression

under “slow evaporation” conditions

X v /X v,s = (T–T ∞ )/(T l –T ∞ ) = a/r, where r ≥a where X v denotes the mole fraction of the vapor and X v,s that at surface. Determine the difference between absolute availabilities at two locations r = a, and r = b. Plot the

variation of availability in kJ/kg of mix with a/r where r is the radius. Problem D42

Electrical work is employed to heat 2 kg of water from 25ºC to 100ºC. The specific heat of water is 4.184 kJ kg –1 K –1 . Determine the electrical work required, and the minimum work required (e.g., by using a heat pump instead).

Problem D43 Six pounds of air at 400ºF and 14.7 psia in a cylinder is placed in a piston-cylinder as- sembly and cooled isobarically until the temperature reaches 100ºF. Determine the optimum useful work, actual useful work, irreversibility and the availability or exer- getic or so called 2 nd law efficiency.

Problem D44 An adiabatic turbine receives 95,000 lbm of steam per hour at location 1. Steam is bled off (for processing use) at an intermediate location 2 at the rate of 18,000 lbm per hour. The balance of the steam leaves the turbine at location 3. The surroundings are at a pressure and temperature of 14.7 psia and 77ºF, respectively. Neglecting the

changes in the kinetic and potential energies and with the following information: P 1 = 400 psia, T = 600ºF, P = 50 psia, T = 290ºF, P = 2 psia, T = 127 ºF, v = 156.4

ft lbm , determine the maximum sssf work per hour, the actual work per hour, and the irreversibility.

Problem D45 In HiTAC (High temperature Air Combustion systems), preheating of air to 1000ºC is achieved using either a recuperator or a regenerator. The recuperator is a counterflow heat exchanger while the regenerator is based on a ceramic matrix mounted in a tank through which hot gases and cold air are alternately passed. The hot gas temperature or this particular application is 1000 K. Assume c p to be constant for the hot gas, and for it to be the same as that for the cold air. If the recuperator is used, cold air enters it

at 25ºC and the flowrate ratio of the hot to cold gases m ˙ H / m ˙ C = 0.5. The temperature differential between the air leaving the recuperator and the hot gases entering it is 50 K. Determine the availability efficiency for the recuperator. Will you recommend a regenerator instead? Why?

Problem D46 Large and uniformly sized rocks are to be lifted in a quarry from the ground to a higher level. The weight of a standard rock is such that the pressure exerted by it alone on the surrounding air is 2 bar. The rocks are moved by a piston–cylinder as- sembly that contains three pounds of air at 300ºF when it is at ground level. Heat is transferred from a reservoir at 1000ºF until the temperature of the air in the cylinder reaches 600ºF so that piston moves up, thereby lifting a rock. Assume that air is an ideal gas with a constant specific heat. If the surrounding temperature and pressure are 60ºF and 14.7 psia, determine: a)

The gas pressure. b)

The work performed by the gas. c)

The useful work (i.e., during the lifting of rocks) delivered by the gas. d)

The optimum work. e)

The optimum useful work. f)

The irreversibility and the availability efficiency (based on the useful work). Problem D47

A jar contains 1 kg of pure water at 25ºC. It is covered with a nonporous lid and placed in a rigid room which contains 0.4 kg of dry air at a temperature and pressure of 25ºC and 1 bar. The lid is suddenly removed. The specific heat of water is 4.184 kJ kg –1 K –1 , and that of air is 0.713 kJ kg –1 K –1 . a)

Determine the temperature and composition of the room, the atmosphere of which contains water vapor and dry air at equilibrium. Ignore the pressure change.

b) The change in the availability. Problem D48

Hot combustion products enter a boiler at 1 bar and 1500 K (state 1). The gases trans- fer heat to water and leave the stack at 1 bar and 450 K (state 2). Water enters the boiler at 100 bar and 20ºC (state 3) and leaves as saturated vapor at 100 bar (state 4). The saturated vapor enters a non-adiabatic turbine at 100 bar and undergoes irreversi- ble expansion to a quality of 0.9 at 1 bar (state 5). The combustion gases may be ap- proximated as air. And the total gas flow is 20 kg s –1 . Determine the: Hot combustion products enter a boiler at 1 bar and 1500 K (state 1). The gases trans- fer heat to water and leave the stack at 1 bar and 450 K (state 2). Water enters the boiler at 100 bar and 20ºC (state 3) and leaves as saturated vapor at 100 bar (state 4). The saturated vapor enters a non-adiabatic turbine at 100 bar and undergoes irreversi- ble expansion to a quality of 0.9 at 1 bar (state 5). The combustion gases may be ap- proximated as air. And the total gas flow is 20 kg s –1 . Determine the:

Absolute availability at the dead state for gas and water. c)

Relative availabilities at all states. d)

Optimum power for the gas loop, i.e., with the same inlet and exit conditions of the gas.

e) Optimum work for the entire plant including gas and water loops. f)

Irreversibilities in the heat exchanger and turbine. Problem D49

A nuclear reactor transfers heat at a 1727ºC temperature to water and produces steam at 60 bar and 1040ºC. The vapor enters the turbine at 60 bar and 1040ºC and expands isentropically to 0.1 bar. The vapor subsequently enters the condenser where it is condensed to a saturated liquid at 0.1 bar and then pumped to the boiler using an is- entropic pump. What are the values of η cyc , the optimum work and the availability ef-

ficiency, the overall cycle irreversibility, and the irreversibility in the boiler and con- denser? Perform an availability balance for the various states.

Problem D50

A house contains an air equivalent mass of 150 kg at 0ºC. It must be warmed to 25ºC. The only allowed interaction is with environment that is at a temperature T o = 273 K. What is the minimum work input? Assume that air leaves the house at a constant temperature of 12.5ºC and that the pressure in the house is near ambient. What is the minimum work input if outside air is circulated at the rate of 0.335 kg s –1 and the house must be warmed within 15 min?

Problem D51 Two efficiencies can be defined for heat exchangers. In a closed system Q s =Q used + Q loss , and η h =Q used /Q source = (end use) ÷(source energy). Since the end use and source

availabilities are respectively, Q used (1–T o /T used ), and Q source (1–T o /T source ), show that η avail = η h (1–T o /T used )/(1–T o /T source ). Discuss the two efficiencies.

Problem D52 During a cold wave the ambient air temperature is –20ºC. The temperature of a lake in the area is initially a uniform 25ºC, but, gradually, a thick layer of ice is formed. Under the ice layer there is water at 25ºC. The surface temperature of the ice layer is –10ºC, and the heat transfer from the warm water to the ice is 100 kJ kg –1 of ice. Determine the op- timum work. The heat of melting for ice is 335 kJ kg –1 , and the specific heats of ice and

water, respectively, are 1.925 kJ kg –1 K and 4.184 kJ kg K . Problem D53

Consider a non-adiabatic fire tube boiler. Hot gases at a temperature of 400ºC flow into the fire tube at a rate of 20 kg s –1 . The gas is used to heat water from a saturated liquid state to a saturated vapor condition at 150ºC. The heat loss from fire tube boiler is 50 kJ kg –1 of gas. If the gases exit the heat exchanger at 200ºC, determine the water flow required, the entropy generation if the control volume boundary is selected to be just inside the heat exchanger, entropy generation if control volume boundary is se- lected to be just outside the heat exchanger. and the optimum work. Assume that gases have the same properties as air (with c = 1 kJ kg –1 p K –1 ), and where T o = 298 K

and P 0 = 1 bar. Problem D54

A 10 m 3 tank contains air at 1 bar, 300 K. A compressor is used to evacuate the tank completely. The compressor exhausts to the ambience at 1 bar and 300 K. Assume

that the tank temperature remains constant through heat transfer from ambience at 300 K. You are asked to determine the minimum (optimum) work required. Select the

Steam 150ºC

Gas 400ºC Gas 200ºC

Problem D.53 Water 150ºC

control volume which includes the tank, compressor and the outlet from the compres- sor. a)

Does the tank mass remain constant? b)

Does the internal energy of unit mass within the tank remain constant if gas is assumed to be an ideal gas?

c) Does the absolute availability at the exit of the compressor change with time d)

Starting from mass conservation and generalized availability balance, then simplify the equation for the current problem., Indicate all the steps clearly and integrate over a period of time within which the tank is emptied.

e) Assuming that h= c p0 T, u = c v0 T, s = c p0 ln ( T/T ref ) - R ln ( P/P ref ), T ref =T 0 , P ref = 1 bar, determine the work in kJ.