CHAPTER 3 PROBLEMS

C. CHAPTER 3 PROBLEMS

Problem C1 An Otto cycle involves reversible processes. Otto cycle A uses Ar while Otto cycle B involves N 2 . Then using Clausius theorem and T-s diagram for Otto cycle one of the following must be true for the same initial (T 1 ) and peak temperatures (T 3 ) a) η A = η B ,

b) η A > η B , c) η A < η B . Which one is it?

Problem C2 Innovations, Inc. claims that they have developed a heart pump driven by a heat en- gine which uses the warm reservoir of a human body 98.6ºF and the cold reservoir of ambience 60ºF. It is claimed that for every 100 BTU of heat absorbed from the body, the engine could deliver work of 10.5 BTU. Can you verify the claim?

Problem C3 The S vs. T a during the cooling of a coffee cup in room air (T a ) shows a maximum at equilibrium. Can the same curve yield the time scale required for equilibrium? What is a spontaneous process?

Problem C4 How can changing heat transfer into work reduce entropy production?

Problem C5 Can we define equilibrium in terms of E, V, m?

Problem C6 Will the equilibrium condition within a tank containing N 2 change if the tank walls are porous?

Problem C7 Will the work added to a system always change entropy? Specifically, consider QE adiabatic work and NQE adiabatic work.

Problem C8 If the concept related to σ is important, why do not engineers use it in common prac- tice?

Problem C9 Consider a cup full of coffee placed in room air. If the pressure and entropy are maintained constant within the rigid room, in practice how can there be a heat loss?

Problem C10 Why is there a negative sign associated with the equality ∂S/∂N 1 =- µ 1 /T?

Problem C11 Consider problem B.10. The entropy change after compression process is a) zero, or

b) positive? Problem C12

Explain entropy using physical principles. What is an "endoreversible" engine?

Problem C13

If the entropy increases, do you believe S=0 at the beginning of the universe?

Problem C14

What prevents O 2 being on one side within a room?

Problem C15 Is entropy generated as a result of electrical and gravitational fields?

Problem C16 Can you predict time to reach equilibrium be using thermodynamics?

Problem C17 The temperature of air in a rigid container initially at 300K (T1) and 100Kpa (P1) has to be increased to a final temperature of 600K. The student found that the gas tem- perature was not uniform inside the tank during the heating process with a gas burner Problem C17 The temperature of air in a rigid container initially at 300K (T1) and 100Kpa (P1) has to be increased to a final temperature of 600K. The student found that the gas tem- perature was not uniform inside the tank during the heating process with a gas burner

The specific entropy change (s 2 -s 1 ) in kJ/kg-K is given as 0.49 kJ/kg-K. b)

The specific entropy change cannot be determined since the process is inter- nally irreversible.

c) The entropy change is zero. Problem C18

What is the physical significance of A and G reaching a minimum value? Problem C19

How does the entropy change if a partition between two fluids is porous? Problem C20

What is the physical meaning of the entropy flux? (Hint: heat transfer through fins.) Problem C21

Consider the cooling of coffee and air for which S 1 =S coffee + air = 2 kJ/K, S 2 =S max = S M = 5 kJ/K after reaching a maximum (at state M). Can this entropy be lowered to S 3 <S 2 through an adiabatic process?

Problem C22 Why should σ be different if energy is supplied via heat or via electrical work cross-

ing a system boundary? Problem C23

If microscopic fluctuations occur at equilibrium, why does it not violate the Second Law?

Problem C24 What are the ways of increasing the temperature of a closed system?

Problem C25 Are there ways of determining irreversibility without evaluating σ?

Problem C26 If the entropy never decreases we obtain greater disorder. If a room is messy we can clean it up . Why can’t we do the same for a thermodynamic system?

Problem C27 What is the difference between compressible and incompressible materials?

Problem C28 When can you assume constant specific heats and when can you assume variable spe- cific heats?

Problem C29

What is the difference between s 2 -s 1 and σ 12 ?

Problem C30

Physically, what is the difference between chemical and deformation work ?

Problem C31 Recall that we can use the expression dS = ( δQ/T) rev = dU/T + P dV/T for a fixed

mass system to develop tables for s = s(T,P). Can we use this expression to determine the entropy of a fluid which enters a turbine or for any other open system?

Problem C32 If the human body is assumed to have the same properties as liquid water, then during fever, the entropy of the human body should be higher, since a) the volume of the human body slightly decreases, and b) its energy increases during fever. State which of the two is the correct answer.

Problem C33 When a violinist performs, only 2% of the work done is used to produce musical sound, and the remainder is dissipated in the form of heat. Assume that violin strings are made of steel. As a string is heated, the damping coefficient changes, which alters the sound. How can you employ the Second Law to relate the damping and heating processes in order to produce a better violin design?

Problem C34 Consider an insulated piston cylinder assembly. The piston mass is negligible and is fabricated of good insulation material. A ambient pressure P 0 = 100 kPa acts on the piston. The temperature of the gas (which is air) is 300K (T a ). A mass of 100 kg (m E ) is suddenly placed on the piston and the surface area of the piston is 1 cm 2 (area A). The initial volume is 0.01m 3 (V 1 ). The local gravitational acceleration is 10m/s 2 (g). Make any reasonable approximation and indicate the formulations required to obtain answers to the following questions.

a) The final pressure (P 2 ),

b) The final temperature (T 2 ) and volume (V 2 ), and

c) Entropy change between states 1 and 2. Problem C35

Consider the reversible polytropic process pv n = constant between the states i (initial) and f (final). We must devise an equivalent process that consists of an adiabatic proc- ess ig, an isothermal process gh, and an adiabatic process hf, such that work per- formed by the equivalent processes is the same as the work done by the reversible polytropic process. a)

Show that for the isothermal process,

g = (T i –T f ) (((k–1)/(n–1))– 1)/(ln ((P f v f ) (P i v i ))). b)

What is the value of n for constant P process? Using the answer to part (a) obtain an expression for a constant pressure process.

c) What is the value of n for a constant volume process? Using the answer to part (a) obtain an expression for a constant volume process.

d) Plot the processes within the range P i = 60 bar to P f = 48.2 bar, with v i = 0.1m 3 /kg and v f = 0.12 m 3 kg –1 . Assume the medium to be air and treat it as an ideal gas.

Problem C36 Derive an expression for the efficiency of the reversible cycles illustrated in the figure in terms of T L and T H , and for a Carnot cycle operating between the same tempera- tures.

Problem C37 Consider an adiabatic reversible process for an incompressible liquid flowing through an expanding duct. a)

What is the entropy change? b)

Can the process be considered isothermal? Problem C38 Consider an adiabatic reversible process for an incompressible liquid flowing through

an expanding duct. Using the basic energy and mass conservation relations for an

Figure 1: Chapter 3, Problem C37.

open system obtain a relation between the pressure and the area of the duct. Show that this reduces to Bernoulli energy equation. (Recall that for liquids du = cdT.)

Problem C39 Heat (Q H ) is rejected from the condenser to the ambient in a refrigeration cycle for which the temperature T H (ambience) is 10ºC below the condenser temperature. Similarly, heat is added to the evaporator from a cold space at a temperature T L . The evaporator coil is at a temperature that is 10ºC below T L . Is it possible to use the heat

transfer Q H to reduce the work input to the compressor? Problem C40

a) Determine the entropy generation rate in a bar of cross sectional area A which is maintained at T o (x = 0) and at T L (at x = L). The peripheral area is insulated. Derive expressions for the: i) entropy generation per unit volume in terms of local temperature, ii) total entropy generation rate per unit cross sectional area in the bar, and iii) entropy generation rate per unit heat loss rate.

b) What is the: i) entropy generation per unit volume at x = 0.1 m for aluminum if T o = 500 K, T L = 300K, L = 0.3m, ii) total entropy generated within the whole bar, iii) entropy generated per unit heat loss, iv) lost work rate per unit mass loss rate, and v) lost work rate per unit mass loss rate?

c) Determine the difference between the specific entropies at x = L and x = 0. d)

The heat flux Q ″ enters at x= 0 and leaves at x=L. Calculate the entropy flux in and out? Why is the exit entropy flux different?

Problem C41 As a patent officer you receive a patent application for a cyclic device which consists of irreversible adiabatic compression from (P 1 ,V 1 ) to (P 2 ,V 2 ), quasi-static isothermal heat addition from (P 2 ,V 2 ) to (P 3 ,V 3 ) (V 2 <V 3 <V 1 ) and then finally adiabatic and quasi-static expansion from (P 3 ,V 3 ) to (P 1 ,V 1 ). Will you issue the patent? Justify.

Problem C42 One kmol of CO is contained in a piston–cylinder assembly at 3000 K and 1 bar. What is the system entropy? Now, one kmol of O 2 at 1 bar and 3000 K is introduced, but constant pressure is maintained in the system. What is the entropy change for the

Problem C43 An ideal gas can be heated in a closed system using a) an isobaric process or b) an isometric process through a similar temperature rise. What is the ratio of isobaric en- tropy change to isometric entropy change? Discuss the results briefly.

Problem C44 An ideal Otto cycle which uses the same gas for all four processes (adiabatic com- pression, isometric heat addition, adiabatic expansion, isometric heat rejection) has the following expression for efficiency

η = 1 – 1/(r (k–1) v )

where r v denotes the compression ratio v 1 /v 2 ,v 1 the initial volume, and v 2 the volume after compression. a)

Determine η for with k = 1.4, r v = 8. b)

Determine η for a monatomic gas for which k = 1.6667, and r v = 8. c)

Discuss the results for a) and b). Problem C45

Consider the transient three–dimensional heat conduction equation and the entropy balance equations for solids. Show that the entropy generation rate per unit volume is

given by the expression σ ˙ = –(q ″/T 2 ) ∇T.

a) What is the result for σ ˙ if q ″ = – λ ∇T, b)

What is the result for σ ˙ if q ″ = λ ∇T, c)

What is your conclusion? Problem C46

Ten kg of hot air is stored at 1000K and 1 bar in compartment A of a rigid insulated tank, and 5 kg of cold air at 500 K is stored in compartment B of the same rigid insu- lated tank at 1 bar. a)

If you open the partition between compartments A and B i)

What is the final temperature T 2 ?

ii)

What is the final pressure P 2 ?

b) What is the entropy generated? c)

If we connect a Carnot engine between reservoirs A and B (i.e., by extracting heat from compartment A and rejecting it to compartment B to produce work), i)

What is the final temperature in A and B?

ii)

What is the maximum possible work?

iii)

What is the entropy generated?

Problem C47 Assume that you have a “finned head” on your body. “Finned heads” argue that they are the “coolest” people on the earth, since each strand of hair behaves as a fin to keep their heads cool. However “bald heads” argue that the radiation heat loss from their heads is a more important form of cooling. Looking at the problem from a thermody- namic point of view, which “head” generates more entropy for the same ambient tem- perature?. The temperature of a “head” can be assumed to be 98.6ºF for both groups. State your assumptions.

Problem C48 Consider the piston–cylinder assemblies A and B, where the head of cylinder A is in contact with head of cylinder B. Equal masses of ideal gases exist initially at P =10 bar and T = 500 K in both cylinders. Cylinder A is quasi-statically compressed to 15 bar while the pressure in B is reduced to 5 bar. Each cylinder is insulated except at the Problem C48 Consider the piston–cylinder assemblies A and B, where the head of cylinder A is in contact with head of cylinder B. Equal masses of ideal gases exist initially at P =10 bar and T = 500 K in both cylinders. Cylinder A is quasi-statically compressed to 15 bar while the pressure in B is reduced to 5 bar. Each cylinder is insulated except at the

Problem C49 The following problem will illustrate the Clausius inequality. A professor asked his research assistant (student A) to compress 0.002 kg of air in an insulated cylinder which was initially at 100 kPa and 300 K (state 1). He wanted the student to do this as

slowly as possible so that when the volume was reduced to 1/10 th of the original vol- ume, he obtained a pressure of 25120 kPa and a temperature of 753 K. He advised the

student that it might take many hours to do the job. As soon as the professor left the room, the student got anxious and in a few milliseconds he/she moved the piston to reach 1/10 th of the original volume and left. The professor returned after a few hours. He found to his dismay that the temperature reading was 900 K and correspondingly the pressure was also high (state 2). He figured out what had happened and he imme- diately fired the student. He hired a new research assistant (student B) who he thought would act more responsibly. He told the research assistant to move the piston back as slowly as possible to the original volume to the initial state. The cylinder was still in- sulated. After a few hours the student moved the piston back to the original position. But the temperature and pressure after expansion (state 3) were not the same as the values at state 1. He reported the results to the professor. The professor told the stu- dent to remove the insulation and to isometrically cool the cylinder to a temperature of 300 K. The student did that and found the pressure to be almost the same as the pressure at state 1. a)

If the compression process was adiabatic and reversible what would the tem- perature (T 2s ) and pressure (P 2s ) have been after compression? Assume con- stant specific heats (evaluated at 300 K). Why are these values different from the values measured by student A? What would be a best determine for the actual pressure at state 2 for the measured temperature?

b) Why is state 3 different from state 1? c)

How much heat must be removed between states 3 and 1?

d) Do you believe the pressure measurement after the cooling process? e)

Determine the cyclic integral of δQ/T. What is the sign of this quantity? Ex- plain the significance of the sign.

f) Assuming that the entropy at 300 K, 100 kPa is 2.515 kJ kg –1 K –1 , evaluate the entropies at states (2) and (3).

g) What is the cyclic integral of dS? Why is this integral different from the an- swer to part (e)?

h) If the atmospheric temperature is 300 K, what is the entropy change of an isolated system during a single cyclical process?

Problem C50 Consider a system A of mass 2 kg at 8 bar at 500 K, a system B of mass 1 kg at 10 bar at 300 K. a)

The two systems are adiabatic and divided by an insulated partition and by a pin. The pin is released. What is the final pressure? What are T A and T B ? As- sume that the process is quasi-steady. Assume an adiabatic expansion proc- ess for one cylinder.

b) The two systems are divided by a diathermal wall. Except at the partition, there is no heat transfer. What is the final pressure? What are the final tem- peratures?

Problem C51 Obtain an expression for the entropy generated over a time period t when a pressur- ized gas at the state (T i , P i ) enters an adiabatic piston–cylinder–weight assembly of cross sectional area A and the weight W is just lifted. The ambient temperature is T o .

Problem C52 The generalized entropy relation for any simple compressible substance following the state equation v = v(T,P) is ds = c v dT/T + ( ∂P/∂T) v dv. Assume that a solid substance

undergoes adiabatic reversible compression or expansion. a)

Obtain an expression for ( ∂T/∂v) s in terms of c v , β P , β T and T where β P = (1/v) ( ∂v/∂T) P and β T = –(1/v) ( ∂v/∂P) T .

b) Discuss the results qualitatively for a substance that expands or contracts upon heating.

c)

Determine ( ∂T/∂v) s for copper assuming its properties at 25ºC.

d) Irreversibilities exist in systems (e.g., temperature gradients involved in the bending of the copper beam) and as a result entropy δσ is generated. Since

ds – δq/T = δσ., qualitatively compare (∂T/∂v) obtained under adiabatic irre- versible conditions with those obtained for part (a).

Problem C53

A family returns from vacation to find their house at a temperature of 15 °C, while the outside temperature is 5 °C. The house has a volume of 2000 cubic meters and the ef- fective heat capacity of the house, furniture, and fixtures (exclusive of the air) is

3 ×10 5 kcal K –1 , while the heat capacity of air (c p ) may be assumed to be constant and its value can be fixed with respect to some average temperature. As the house is

heated, air is expelled (through the sides of windows, the chimney, etc.) to maintain the pressure at one atmosphere. a)

How much heat is required to raise the temperature of the house and its con- tents to 25 °C (assuming negligible heat losses, except through the expelled air)?

b) How much electrical energy would it take to run a heat pump to achieve the same objective? Assume that the heat pump and motor combined run at 35% of the theoretical efficiency (independent of temperature) and that the expan- sion coils are outside the house and are maintained at 0 °C, while the com-

pressor and condenser and motor are inside the house at a temperature equal to that of the house.

Problem C54 Suppose that saturated liquid data for h, v and s are available for water from 10 to 180ºC. Produce a compressed liquid table for h, u (= h–Pv) and s at P = 10 bar for temperatures from 10ºC to 180ºC using the saturated liquid data. Assume that the liq- uid specific volume does not change with pressure at a given temperature.

Problem C55 Consider a high intensity discharge lamp in which the electric discharge occurs with an energy U in a narrow ultra violet wavelength range band ∆λ around λ. The energy

intensity of the photons is expressed by Planck’s law. The entropy of the photons S = 4(U/3)T. Mercury vapor under high pressure absorbs this energy but emits in the visi- ble range of wavelength. assuming that the vapor absorbs all of the energy in the visi- ble wave length range.

a) determine the temperature of vapor (T v ) assuming it to be a black body (U =

4 σT 4 v ), and b)

determine the entropy generated. Problem C56

Consider a fin of arbitrary cross sectional area with its base maintained at the tem- perature T w (x = 0) that loses heat to its ambient at the temperature T ∞ . Show that the

expression for the work loss rate per unit heat loss rate is the same as the Carnot effi- ciency. (Hint: use a control volume which includes the base area with other bounda- ries extending far away from the fin.)

Problem C57 Hot water (W) at the temperature T W,0 is kept in an adiabatic classroom which con- tains air (A) initially at a temperature T A,0 . Initially, there is a constraint on the cup in the form of an insulation around the cup. Once we remove the insulation, there is ir- reversible heat transfer that leads to an increase in the entropy of the combined sys- tem. Concerned with this, a graduate student connects a Carnot engine between the water and room air (A) and delivers work to the outside of the classroom to run an elevator. a)

What is the energy of the combined system (A+W)? b)

What are the energy of the subsystems, A and W? c)

What is the entropy of the combined system? d)

Compare the final temperatures of the subsystems, A and W.

e) Determine the work done. f)

Discuss the internal energy variation of A+W with time as work is delivered. Problem C58

Consider an Otto cycle (which is also a reversible cycle) operating with compression ratio of 5 (=V 1 -V 2 ). It can be shown that the temperature ratio (T 4 /T 1 ) = (T 3 /T 2 ). If T 1 =300K, what is T 2 ? Assume T 3 = 800K. What is the Carnot efficiency for the same maximum and minimum temperatures? Do you believe that the Otto efficiency is greater, smaller or equal to the Carnot efficiency? Provide reasons for your answer in five or six lines.

Problem C59 Consider a reversible Otto cycle operating with a compression ratio (=v 1 /v 2 ) of 5. The temperature ratio (T 4 /T 1 ) = (T 3 /T 2 ). If T 1 = 300 K and T 3 = 800 K, determine T 2 ? What is the Carnot efficiency for the same maximum and minimum temperatures? What is the Otto efficiency and the corresponding Carnot efficiencies?

Problem C60

A cooker A of 30 cm diameter and a 30 L volume is filled with 4 kg of water. The cooker operates at a pressure of 5 bar. A metal tube of 10 mm 2 cross sectional area is contained inside the cooker from a position slightly removed from its bottom surface and attached to a valve at the top of the cooker. When the valve is opened, the re- maining water in the cooker is injected into another open adiabatic cooker B of equal dimension in order to conserve energy. However we would like to monitor the pres- sure in cooker A with respect to time until there is no liquid left in it. Assume that water vapor behaves as an ideal gas with a specific heat of 1.65 kJ kg –1 K –1 . Liquid water is incompressible with a specific heat of 4.184 kJ kg –1 K –1 and a specific vol-

ume v = 0.001 m 3 kg –1 . The area of the metal tube is 10 mm 2 . Assume that the vessels are insulated, and that there is no heat transfer when the valve is opened and that the cooker free space is occupied by vapor alone. a)

What is the water quality when the valve in the metal tube is opened?

Figure Prob. C.59

b) What is the water quality when all water has exited from cooker A? c)

Illustrate the process for cooker A on a P–v diagram. d)

Write the energy balance equation for the process occurring in the metal tube and obtain an expression for the gas velocity through the tube for a steady state process.

e) Obtain an expression for mass flow through the tube. f)

Write the mass conservation equation for liquid phase (assume phase equi- librium between vapor and liquid).

g)

Write the entropy balance equation for the combined system.

h) Derive the expression for P(t) in terms of the vapor volume in the cooker? i)

What is the pressure in cooker A when all of the water has evaporated?

6 sat 0 h) sat If P (bar) = 1.8 exp (5199/T ( K)) compare the T (t) behavior with T(t). Check the assumption. Problem C61

sat

Consider a Carnot cycle in which the air is adiabatically and reversibly compressed say from V

1 = 0.1 m ,P 1 = 100 kPa, T 1 = 300 K to V 2 = 0.06 m 3 , and P 2 = 205 kPa. Heat is then isothermally added (i.e., T H during heat addition) where Q in =Q H = 14.75

kJ, and the air expanded to state 3. The gases are adiabatically and quasistatically ex- panded to a temperature T 4 = 300 K. Finally heat is isothermally rejected so that the

volume returns to its original value. Assume ideal gas behavior, c –1

= 0.714 kJ kg K –1 , and constant specific heats. a)

vo

Determine Q out (= Q L ). b)

Determine Q L /Q H .

c) What is W cycle ? d)

Is Q L /Q H =T H /T L ?

Problem C62 Determine the entropy of N 2 (g) at 373 K and 1 bar. If N 2 is a solid at 0 K, and h sf =h fg = 0, what is the entropy s of N 2 (g) at 0 K? Assume ideal gas behavior between 0 and 373 K (undoubtedly, a drastic assumption).

Problem C63 Consider a gas turbine that is 2 m long, with a net power output of 100 kW, and oper-

ating with a monatomic gas for which c –1 po = 20.79 kJ kmol K . Let P i = 10 bar and P e = 1 bar. Since the gas is monatomic, the specific heat is not a function of T. As- sume steady state steady flow and that the turbine walls are well insulated. The inlet and exit velocities are very low. In order to obtain additional work from the turbine it

T=T i – (T i –T e ) x/L,

where L denotes the length of turbine, and x the distance from inlet. The turbine walls are insulated, and the blades are cooled as long as the gas temperature exceeds the turbine blade temperature. We also assume that when the gas temperature falls below 10 K, there is no appreciable heat loss to the blades. The heat loss rate per unit length and per unit mass flow is given by the relation h´(T– T bl ) where h´ denotes the heat loss per unit length of turbine and is 0.5 kW/m K. The exit temperature is 600 K. a)

Start from generalized mass and energy conservation and entropy balance equations

dN cv /dt = N i –N e ,

dE cv /dt = Q cv –W cv +N i (h + ke + pe) i –N e (h + ke + pe) e , and

dS cv /dt = ∫δQ/T b +N i s i –N e s e + σ.

Using assumptions stated in the problem and any additional assumption, pre- sent the mass and energy conservation and entropy balance equations in a simplified form.

b) What is the work under steady state operation? c)

Determine the entropy generated per unit mass flow in the turbine. d)

If the turbine runs at an inlet temperature of 10 K, but with no cooling and no heat loss, what is the work done for the same exit conditions?

Problem C64 Assume that there is a secondary system, which is a reservoir at a fixed pressure, in- side a spacecraft at the state T s,o and P s,o ,. Determine the optimum work done if an ideal gas initially at T p,o ,P p,o in a primary system that undergoes change of state to T pf ,P pf due to interaction with the secondary system. Assume that T po = 1000 K, P po =

20 bar, m p = 4 kg, the gas in primary system is Ar, P pf = 10 bar, T pf = 600 K, T so = 300 K, P so = 1 bar, and m s = 8 kg. Assume that the gas in the secondary system is He.

Problem C65 In a conventional Carnot cycle T L =T 4 =T o. In an unconventional cycle an ideal gas is expanded to the temperature T 4 ´ (< T o ) and volume v 4 ´. Show that the additional work during the expansion process w add <P o (v 4 ´– v 4 ). Assume constant specific heats.

Problem C66 Consider a cycle consisting of reversible adiabatic compression from state 1 to 2, isothermal heat addition at T H , and reversible adiabatic expansion to state 1, which is at 0 K. Draw a T–P diagram for the cycle. Is this cycle possible?

Problem C67 Is it possible to obtain an efficiency η = 1 if compressed gas at room temperature is

available at the state (P 1 , T o ), then adiabatically and reversibly compressed to state 2 (P 2 ,T 2 ) heat is added at a constant high temperature T H to state 3, and the gas is fi- nally expanded to state 4 (P 0 ,T o ) and then discharged to the atmosphere?

Problem C68 Consider saturated water in an insulated blender at 100 O

C, P =–101kPa (state 1). A weightless piston is kept above the water. The ambient pressure is 101kPa. As the motor is turned on, the water just starts evaporating and reaches saturated vapor state (state 2). Sketch the process on P-v and T-s diagrams.

a) What is the boundary work?

b) What is the work input through the blender shaft? b) What is the work input through the blender shaft?

d) Comment on the areas under process 1-2 in the P-v and T-s diagrams. Problem C69

10 kg of Ar is contained in the piston–cylinder section A of a system at the state (1.0135 bar, 100ºC). The gas is in contact through a rigidly fixed diathermal wall with

a piston–cylinder section B of the system that contains a wet mixture of water with a quality x = 0.5 that is constrained by a weight. As the gas in section A is compressed the temperature in A remains at 100ºC using QE process due to contact with section

B. Assume that the quality in section B increases to 90%. Both systems are well in- sulated except at the diathermal wall. Determine:

a) the initial pressure in Chamber B,

b) the heat transfer Q in kJ to Chamber B during compression of Ar in Chamber 12, B A,

c) the work for sections A and B in kJ,

d) the change in the entropies of Ar and H 2 O (both liquid and vapor), and

e) the volume V 2 in Chamber A

f) Is the process for the composite system (A+B combined together) isothermal and isentropic?

Problem C70

A piston–cylinder assembly contains Ar(g) at 60 bar and 1543 K (state 1).

a) Determine the work done if the gas undergoes isothermal expansion to 1 bar (state 2). What is the heat transfer? Does this work process violate the Sec- ond Law?

b) Determine the work done if the gas undergoes quasistatic adiabatic expan- sion to 1 bar (state 3). Can we continue the expansion to v 3 → ∞ by remov- ing the insulation and adding heat?

Problem C71

A rigid container of volume V is divided into two rigid subsystems A and B by a rigid partition covered with insulation. Both subsystems are at the same initial pressure P o . Subsystem B contains 4 kg of air at 350 K, while subsystem A contains 0.4 kg of air at 290 K. The insulation is suddenly removed and A and B are allowed to reach ther- mal equilibrium. a)

What is the behavior of the overall entropy with respect to the temperature in subsystem A. What is the equilibrium temperature?

b) As heat is transferred, the entropy of subsystem A increases while that of subsytem B decreases. The entropy in the combined system A and B is held constant by removing heat from subsystem A. Plot the behavior of the over- all internal energy with respect to the temperature in subsystem A. What is the equilibrium temperature?

c) Both subsystems are allowed to move mechanically in order to maintain the same pressure as the initial pressure P o . The entropy is held constant by al- lowing for heat transfer. Plot the behavior of the overall enthalpy with re- spect to the temperature in subsystem A. What is the equilibrium tempera- ture?

Problem C72

A piston–cylinder–weight assembly is divided into two insulated subsystems A and B separated by a copper plate. The plate is initially locked and covered with insulation. The subsystem A contains 0.4 kg of N 2 while subsystem B contains 0.2 kg of N 2 .

a) B The insulation is removed, but the plate is

kept locked in locked positions. Both subsys- tems are at the same initial pressure P 1A =P 1B = 1.5 bar with temperatures T 1A = 350 K, and T 1B = 290 K. Both A and B reach thermal equilibrium slowly. Assuming that internal equilibrium exists within each subsystem,

A plot (S = S A + S B ) with respect to T B for specified values of U, V, and m. What is the

Figure Problem C.72 value of T B at equilibrium?

b) The plate insulation is maintained, but the lock is removed. Assume P 1B = 2.48 bar and P 1A = 1.29 bar and equal temperatures T A,1 = T B,1 = 335 K. Assume quasiequilibrium expansion in subsystem B and plot S with respect to P A for specified values of U, V, and m

b) The insulation is removed, but heat transfer to outside ambience is allowed with the restraint that the entropy of the combined system A+B is constant.

Plot U with respect to T B . What is the value of T B at equilibrium? Problem C73

An adiabatic rigid tank is divided into two sections A (one part by volume) and B (two parts by volume) by an insulated movable piston. Section B contains air at 400 K and 1 bar, while section A contains air at 300 K and 3 bar. Assume ideal gas behavior. The insulation is suddenly removed. Determine:

a)

The final system temperatures.

b)

The final volumes in sections A and B.

c)

The final pressures in sections A and B.

d)

The entropy generated per unit volume.

Problem C74 Steam enters a turbine at 40 bar and 400ºC, at a velocity of 200 m s –1 and exits at

36.2ºC as saturated vapor, at a velocity of 100 m/s. If the turbine work output is 600 kJ kg –1 , determine:

a) The heat loss. b)

The entropy generation assuming that the control surface temperature T b is the average temperature of the steam considering both inlet and exit. c)

The entropy generation if the control surface temperature T b = T o = 298 K, which is the ambient temperature

Problem C75 Determine entropy generated during the process of adding ice to tap water. A 5 kg glass jar (c = 0.84 kJ kg –1 K –1 ) contains 15 kg of liquid water (c = 4.184 kJ kg –1 K –1 ) at 24ºC. Two kg of ice (c = 2 kJ kg –1 K –1 ) at –25ºC wrapped in a thin insulating foil of negligible mass is added to water. The ambient temperature T o = 25ºC. The insulation is suddenly removed. What is the equilibrium temperature assuming that no ice is left

(the heat of fusion is 335 kJ kg –1 ), and what is the entropy generated? Problem C76

Consider the isentropic compression process in an automobile engine. The compres- sion ratio r v = (V 1 /V 2 ) = 8 and T 1 = 300 K. Assuming constant specific heats, deter- mine the final temperature and T 2 and the work done if the fluid is air and Ar respec- tively. Explain your answers.

Problem C77 The fuel element of a pool–type nuclear reactor is composed of a core which is a ver- tical plate of thickness 2L and a cladding material of thickness t on both sides of the

plate. It generates uniform energy q ′′′ , and there is heat loss h H (T s – T ∞ ) from the plate surface, where T s denotes the surface temperature of the cladding material. The

temperature profiles are as follows: In the core,

(T – T ∞ )/( q ′′′ L 2 core /2k core ) = 1 – (x/L) 2 – B, where

B= 2(k core /k clad ) + 2 (L clad /L core ) (k core /k clad ) (1 + k clad /(h H L clad )). For the cladding material (T – T )/(

L 2 /2k ) = –(x/L) ∞ 2 q ′′′ core clad + c, where

C = (L clad /L core )(1 + k clad /(h H L clad )) and L clad =L core + t. Here L denotes length, k the thermal conductivity, h H the convective heat transfer co- efficient, and t thickness. a)

Obtain expressions for the entropy generated per unit volume for the core and clad.

b) Simplify the expression for the entropy generated per unit volume at the center of core?

c) Determine the entropy generated per unit surface area for the core and clad. Problem C78

The energy form of the fundamental equation for photon gas is U = (3/4) 4/3 (c/(4 σ)) 1/3 S 4/3 V –1/3 where c denotes speed of light, σ Stefan Boltzmann constant, and V volume. a)

Obtain an expression for T(S,V). b)

Obtain an expression for (P/T) in terms of S and V. c)

Using the results for parts (a) and (b) determine P(T,V). Problem C79

A heat engine cycle involves a closed system containing an unknown fluid (that is not an ideal gas). The cycle involves heat addition at constant volume from state 1, which is saturated liquid, to state 2, adiabatic reversible expansion from state 2 to state 3 which is a saturated vapor, and isobaric and isothermal heat rejection from state 3 to state 1 (that involves condensation from saturated vapor to saturated liquid). The cy- cle data are contained in the table below. The heat addition takes place from a thermal energy reservoir at 113ºC to the system. Heat rejection occurs from the system to the ambient at 5ºC. Determine the heat added and rejected, the cycle efficiency, the asso- ciated Carnot efficiency, and the entropy generated during the cyclical process

860 Problem C80

An ideal gas available at state (P 1 ,T 1 ) is to be isentropically expanded to a pressure P 2 . Given the choice that you can either use a turbine or a piston–cylinder assembly, which one do you recommend? Are the isentropic efficiencies the same for both de- vices if the final states are the same?

Problem C81 Show that the reversible work for an isothermal process undergoing expansion from a pressure of P 1 to P 2 in a closed system is same as the work in an open system (neglect Problem C81 Show that the reversible work for an isothermal process undergoing expansion from a pressure of P 1 to P 2 in a closed system is same as the work in an open system (neglect

Problem C82 Show that the expression

dU = T dS - P dV + µdN (A) reduces to the expression du = Tds – Pdv.

Problem C83 Assume that we have 2 kmol of N 2 at 400 K and 1 bar in a rigid tank, and S 1 = 200.1 ×2 = 400.2 kJ/K. We add 0.1 kmols of N 2 and transfer heat from the system

such that S 2 =S 1 .

a) Determine U at states 1 and 2. b)

Determine the temperature at state 2. b)

Determine the chemical potential µ(= ∂U/∂N) S,V

Problem C84 Consider a counter-flow heat exchanger in which two streams H and C of specific heats c pH and c pC flow counter to each other. The inlet is denoted as i and the exit as e. If T H,i and T H,e are the inlet and exit temperatures of stream H, and T C,i is the inlet of stream C., then obtain an expression for the maximum most temperature T C,e . Assume

that C p,H m H < C pC m C and T H,e = T C,i . Determine the entropy generated per kg of smaller heat capacity fluid

Problem C85 Consider an adiabatic reversible compression from 1 to 2 via path A from volume v 1 to v 2 followed by irreversible adiabatic expansion from 2-3 and cooling from 3-1 (path B: 2-3 and 3-1). Apply Clausius in-equality for such a cycle and discuss the re- sult.