5/6/ :41 PM. Chapter 6. Using Entropy. Dr. Mohammad Abuhaiba, PE

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1 Chapter 6 Using Entropy 1

2 2 Chapter Objective Means are introduced for analyzing systems from the 2 nd law perspective as they undergo processes that are not necessarily cycles. Objective: introduce entropy and show its use for thermodynamic analysis.

3 3 6.1 Introducing Entropy Clausius inequality The Clausius inequality provides the basis for introducing two ideas instrumental for analyses of both closed systems and control volumes from a 2 nd law perspective.

4 4 6.1 Introducing Entropy Clausius inequality Clausius inequality states that for any thermodynamic cycle Q = heat transfer at a part of the system boundary during a portion of the cycle T = absolute temperature at that part of the boundary. Subscript b = a reminder that the integrand is evaluated at the boundary of the system executing the cycle. The integral is to be performed over all parts of the boundary and over the entire cycle. Equality applies when there are no internal irreversibilities as the system executes the cycle, and inequality applies when internal irreversibilities are present.

5 5 6.1 Introducing Entropy Clausius inequality Equation 6.1 can be expressed equivalently as is a measure of the effect of irreversibilities present within the system executing the cycle.

6 6 Defining Entropy Change A quantity is a property if, and only if, its change in value between two states is independent of the process.

7 7 Defining Entropy Change Integral of Q/T has same value for any internally reversible process between the two states. Value of integral depends on end states only. The integral represents the change in some property of the system, which is called entropy, its change is given by

8 8 Retrieving Entropy Data

9 9 Retrieving Entropy Data The TdS equations are developed by considering a pure, simple compressible system undergoing an internally reversible process. In the absence of overall system motion and the effects of gravity, an energy balance in differential form is

10 10 Retrieving Entropy Data Entropy Change of an Ideal Gas

11 11 Retrieving Entropy Data Entropy Change of an Ideal Gas USING IDEAL GAS TABLES Value of specific entropy is set to zero at state temperature of 0 K & pressure of 1 atmosphere. Using Eq. 6.19, specific entropy at a state where temperature is T and pressure is 1 atm is determined relative to this reference state and reference value as Tables A-22

12 12 Retrieving Entropy Data Entropy Change of an Ideal Gas

13 13 Retrieving Entropy Data Entropy Change of an Ideal Gas ASSUMING CONSTANT SPECIFIC HEATS

14 14 Retrieving Entropy Data Entropy Change of an Incompressible Substance

15 15 Entropy Change in Internally Reversible Processes When a closed system undergoing an internally reversible process receives energy by heat transfer, the system experiences an increase in entropy. Conversely, when energy is removed from the system by heat transfer, entropy of the system decreases. An entropy transfer accompanies heat transfer. Direction of entropy transfer is same as that of heat transfer. Isentropic process: A constant-entropy process, adiabatic internally reversible process.

16 16 Entropy Change in Internally Reversible Processes

17 17 Entropy Change in Internally Reversible Processes Carnot cycle

18 18 EXAMPLE 6.1 Internally Reversible Process of Water Water, initially a saturated liquid at 100 C, is contained in a piston cylinder assembly. The water undergoes a process to the corresponding saturated vapor state, during which the piston moves freely in the cylinder. If the change of state is brought about by heating the water as it undergoes an internally reversible process at constant pressure and temperature, determine the work and heat transfer per unit of mass, each in kj/kg.

19 19 EXAMPLE 6.1 Internally Reversible Process of Water

20 20 Entropy Balance for Closed Systems - Developing the Entropy Balance The cycle consists of process I, during which internal irreversibilities are present, followed by internally reversible process R. For this cycle, Eq. 6.2 takes the form Subscript b in 1 st integral serves as a reminder that the integrand is evaluated at the system boundary.

21 21 Entropy Balance for Closed Systems - Developing the Entropy Balance closed system entropy balance

22 22 Entropy Balance for Closed Systems - Developing the Entropy Balance If end states are fixed, entropy change on LS of Eq can be evaluated independently of the details of process. The two terms on RS depend explicitly on nature of process and cannot be determined solely from knowledge of end states. 1 st term is associated with heat transfer to or from system during the process. Entropy transfer accompanying heat transfer. Direction of entropy transfer is same as direction of heat transfer, and same sign convention applies as for heat transfer: A positive value: entropy is transferred into system A negative value : entropy is transferred out When there is no heat transfer, there is no entropy transfer.

23 23 Entropy Balance for Closed Systems - Developing the Entropy Balance Entropy change of a system is not accounted for solely by entropy transfer, but is due in part to 2 nd term on RS of Eq denoted by s. s is positive when internal irreversibilities are present during the process and vanishes when no internal irreversibilities are present. Entropy is produced within the system by the action of irreversibilities. 2 nd law of thermodynamics: entropy is produced by irreversibilities and conserved only in the limit as irreversibilities are reduced to zero. Since s measures effect of irreversibilities present within the system during a process, its value depends on the nature of the process and not solely on the end states. It is not a property.

24 24 Entropy Balance for Closed Systems - Developing the Entropy Balance 2 nd law requires that entropy production be positive, or zero, in value The value of the entropy production cannot be negative. Change in entropy of system may be positive, negative, or zero: Entropy change can be determined without knowledge of the details of the process.

25 Entropy Balance for Closed Systems - Developing the Entropy Balance Temperature at the portion of boundary where heat transfer occurs is the same as the constant temperature of the

25 25 Entropy Balance for Closed Systems - Developing the Entropy Balance Temperature at the portion of boundary where heat transfer occurs is the same as the constant temperature of the reservoir, T b. Reservoir is free of irreversibilities The system is not without irreversibilities, for fluid friction is evidently present, and there may be other irreversibilities within the system.

26 26 Entropy Balance for Closed Systems - Developing the Entropy Balance Apply entropy balance to the system and to the reservoir. Since T b is constant, integral in Eq is readily evaluated, and entropy balance for the system reduces to Q/T b accounts for entropy transfer into the system accompanying heat transfer Q. Entropy balance for reservoir takes the form

27 27 Entropy Balance for Closed Systems - Developing the Entropy Balance Entropy of reservoir decreases by an amount equal to entropy transferred from it to the system. As shown by Eq. 6.30, entropy change of system exceeds amount of entropy transferred to it because of entropy production within the system. If heat transfer were passing instead from the system to reservoir, magnitude of entropy transfer would remain the same, but its direction would be reversed. In such a case, entropy of system would decrease if amount of entropy transferred from system to reservoir exceeded amount of entropy produced within the system due to irreversibilities.

28 28 Entropy Balance for Closed Systems - Other Forms of Entropy Balance If heat transfer takes place at several locations on boundary of a system where temperatures do not vary with position or time, Closed system entropy rate balance

29 29 Entropy Balance for Closed Systems - Evaluating Entropy Production and Transfer Value of entropy production for a given process of a system often does not have much significance by itself. Significance is normally determined through comparison. For example, entropy production within a given component might be compared to entropy production values of the other components included in an overall system formed by these components. By comparing entropy production values, components where appreciable irreversibilities occur can be identified and rank ordered. This allows attention to be focused on components that contribute most to inefficient operation of overall system.

30 30 EXAMPLE 6.2 Irreversible Process of Water Water initially a saturated liquid at 100 C is contained within a piston cylinder assembly. The water undergoes a process to the corresponding saturated vapor state, during which the piston moves freely in the cylinder. There is no heat transfer with the surroundings. If the change of state is brought about by the action of a paddle wheel, determine net work per unit mass, in kj/kg, and amount of entropy produced per unit mass, in kj/kg.k

31 31 EXAMPLE 6.2 Irreversible Process of Water

32 32 EXAMPLE 6.3 Evaluating Minimum Theoretical Compression Work Refrigerant 134a is compressed adiabatically in a piston cylinder assembly from saturated vapor at 0C to a final pressure of 0.7 MPa. Determine the minimum theoretical work input required per unit mass of refrigerant, in kj/kg.

33 33 EXAMPLE 6.3 Evaluating Minimum Theoretical Compression Work

34 34 EXAMPLE 6.4 Pinpointing Irreversibilities Referring to Example 2.4, evaluate the rate of entropy production in kw/k, for a. the gearbox as the system b. enlarged system consisting of the gearbox and enough of its surroundings that heat transfer occurs at the temperature of the surroundings away from the immediate vicinity of the gearbox, T f = 293 K (20C).

35 35 EXAMPLE 6.4 Pinpointing Irreversibilities

36 36 Entropy Balance for Closed Systems - Increase of Entropy Principle An enlarged system comprising a system and portion of surroundings affected by system as it undergoes a process. Since all energy and mass transfers taking place are included within boundary of the system, enlarged system can be regarded as an isolated system. An energy balance for the isolated system reduces to because no energy transfers take place across its boundary. Thus, energy of isolated system remains constant. Since energy is an extensive property, its value for the isolated system is sum of its values for the system and surroundings, respectively, so Eq. 6.34a can be written as

37 37 Entropy Balance for Closed Systems - Increase of Entropy Principle Conservation of energy principle places a constraint on the processes that can occur. For a process to take place, it is necessary for the energy of the system plus the surroundings to remain constant. However, not all processes for which this constraint is satisfied can actually occur. Processes also must satisfy the 2 nd law. An entropy balance for the isolated system reduces to s isol is total amount of entropy produced within system and its surroundings. Since entropy is produced in all actual processes, only processes that can occur are those for which the entropy of the isolated system increases. This is known as increase of entropy principle. Alternative statement of 2 nd law.

38 38 Entropy Balance for Closed Systems - Increase of Entropy Principle Since entropy is an extensive property, its value for the isolated system is the sum of its values for the system and surroundings, respectively, so Eq. 6.35a can be written as This equation does not require the entropy change to be positive for both system and surroundings but only that sum of the changes is positive. Increase of entropy principle dictates the direction in which any process can proceed: direction that causes total entropy of system plus surroundings to increase. The tendency of systems left to themselves to undergo processes until a condition of equilibrium is attained. The increase of entropy principle suggests that entropy of an isolated system increases as the state of equilibrium is approached, with the equilibrium state being attained when the entropy reaches a maximum.

39 39 EXAMPLE 6.5 Quenching a Hot Metal Bar A 0.3 kg metal bar initially at 1200K is removed from an oven and quenched by immersing it in a closed tank containing 9 kg of water initially at 300K. Each substance can be modeled as incompressible. An appropriate constant specific heat value for the water is c w = 4.2 kj/kg K, and an appropriate value for the metal is cm 0.42 kj/kg K. Heat transfer from the tank contents can be neglected. Determine a. final equilibrium temperature of the metal bar and the water, in K b. amount of entropy produced, in kj/

40 40 EXAMPLE 6.5 Quenching a Hot Metal Bar

41 41 Entropy Balance for Closed Systems - Statistical Interpretation of Entropy Entropy is associated with the notion of disorder and the 2 nd law statement that entropy of an isolated system undergoing a spontaneous process tends to increase is equivalent to saying that the disorder of the isolated system tends to increase.

42 42 Entropy Rate Balance for Control Volumes Like mass and energy, entropy is an extensive property, so it too can be transferred into or out of a control volume by streams of matter. control volume entropy rate balance

43 43 Entropy Rate Balance for Control Volumes INTEGRAL FORM S cv (t) = total entropy associated with control volume at time t,

44 44 Entropy Rate Balance for Control Volumes INTEGRAL FORM = heat flux, time rate of heat transfer per unit of surface area, through the location on the boundary where the instantaneous temperature is T. V n = normal component in the direction of flow of the velocity relative to the flow area. For a closed system the sums accounting for entropy transfer at inlets and exits drop out, and Eq reduces to give a more general form of Eq

45 45 Entropy Rate Balance for Control Volumes - Steady State conservation of mass energy rate balance steady-state form of the entropy rate balance Mass and energy are conserved quantities, but entropy is not conserved Eq requires that rate at which entropy is transferred out must exceed rate at which entropy enters, the difference being rate of entropy production within the control volume owing to irreversibilities.

46 46 Entropy Rate Balance for Control Volumes - Steady State, One-inlet, One-exit Control Volumes From Eq. 6.40, entropy of a unit of mass passing from inlet to exit can increase, decrease, or remain the same. Because value of 2 nd term on RS can never be negative, a decrease in specific entropy from inlet to exit can be realized only when more entropy is transferred out of the control volume accompanying heat transfer than is produced by irreversibilities within the control volume. When value of this entropy transfer term is positive, specific entropy at exit is greater than specific entropy at inlet whether internal irreversibilities are present or not. In the special case where there is no entropy transfer accompanying heat transfer, Eq reduces to

47 47 EXAMPLE 6.6 Entropy Production in a Steam Turbine Steam enters a turbine with a pressure of 30 bar, a temperature of 400 C, and a velocity of 160 m/s. Saturated vapor at 100 C exits with a velocity of 100 m/s. At steady state, the turbine develops work equal to 540 kj/kg of steam flowing through the turbine. Heat transfer between the turbine and its surroundings occurs at an average outer surface temperature of 350 K. Determine the rate at which entropy is produced within the turbine per kg of steam flowing, in Neglect the change in potential energy between inlet and exit.

48 48 EXAMPLE 6.6 Entropy Production in a Steam Turbine

49 49 EXAMPLE 6.7 Evaluating a Performance Claim An inventor claims to have developed a device requiring no energy transfer by work or heat transfer, yet able to produce hot and cold streams of air from a single stream of air at an intermediate temperature. The inventor provides steady-state test data indicating that when air enters at a temperature of 39 C and a pressure of 5.0 bars, separate streams of air exit at temperatures of 18 C and 79 C, respectively, and each at a pressure of 1 bar. 60% of the mass entering the device exits at the lower temperature. Evaluate the inventor s claim, employing the ideal gas model for air and ignoring changes in the kinetic and potential energies of the streams from inlet to exit.

50 50 EXAMPLE 6.7 Evaluating a Performance Claim

51 51 EXAMPLE 6.8 Entropy Production in Heat Pump Components Components of a heat pump for supplying heated air to a dwelling are shown in the schematic below. At steady state, Refrigerant 22 enters the compressor at 5 C, 3.5 bar and is compressed adiabatically to 75 C, 14 bar. From the compressor, the refrigerant passes through the condenser, where it condenses to liquid at 28 C, 14 bar. The refrigerant then expands through a throttling valve to 3.5 bar. The states of the refrigerant are shown on the accompanying T s diagram. Return air from the dwelling enters the condenser at 20 C, 1 bar with a volumetric flow rate of 0.42 m 3 /s and exits at 50 C with a negligible change in pressure. Using the ideal gas model for the air and neglecting kinetic and potential energy effects, a. determine the rates of entropy production, in kw/k, for control volumes enclosing the condenser, compressor, and expansion valve, respectively. b. Discuss the sources of irreversibility in the components considered in part (a).

52 52 EXAMPLE 6.8 Entropy Production in Heat Pump Components

53 53 Isentropic Processes General Considerations

54 54 Isentropic Processes Using the Ideal Gas Model - Ideal Gas Tables For two states having same specific entropy, Eq. 6.21a reduces to

55 55 Isentropic Processes Using the Ideal Gas Model Assuming Constant Specific Heats Eqs and 6.23 reduce to the equations

56 56 Isentropic Processes Using the Ideal Gas Model Assuming Constant Specific Heats

57 57 EXAMPLE 6.9 Isentropic Process of Air Air undergoes an isentropic process from p 1 = 1 bar, T 1 = 300K to a final state where the temperature is T 2 = 650K. Employing the ideal gas model, determine the final pressure p 2, in atm. Solve using a. Table A-22 b. a constant specific heat ratio k evaluated at the mean temperature, 475K, from Table A-20.

58 58 EXAMPLE 6.10 Air Leaking from a Tank A rigid, well-insulated tank is filled initially with 5 kg of air at a pressure of 5 bar and a temperature of 500 K. A leak develops, and air slowly escapes until the pressure of the air remaining in the tank is 1 bar. Employing the ideal gas model, determine the amount of mass remaining in the tank and its temperature.

59 59 EXAMPLE 6.10 Air Leaking from a Tank

60 60 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps Isentropic efficiencies involve a comparison between the actual performance of a device and the performance that would be achieved under idealized circumstances for the same inlet state and the same exit pressure.

61 61 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps Isentropic Turbine Efficiency Fig shows a turbine expansion on a Mollier diagram. State of matter entering the turbine and exit pressure are fixed. Heat transfer between the turbine and its surroundings is ignored, as are kinetic and potential energy effects. Mass and energy rate balances reduce, at steady state, to give the work developed per unit of mass flowing through the turbine

62 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps Isentropic Turbine Efficiency Since there is no heat transfer, the allowed exit states are constrained by Eq. 6.

62 62 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps Isentropic Turbine Efficiency Since there is no heat transfer, the allowed exit states are constrained by Eq Because entropy production cannot be negative, states with s 2 < s 1 are not accessible in an adiabatic expansion. The only states that actually can be attained adiabatically are those with s 2 > s 1.

63 63 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps ISENTROPIC NOZZLE EFFICIENCY The isentropic nozzle efficiency = ratio of actual specific kinetic energy of gas leaving the nozzle, to the kinetic energy at the exit that would be achieved in an isentropic expansion between the same inlet state and the same exhaust pressure, Nozzle efficiencies of 95% or more are common,

64 64 Isentropic Efficiencies of Turbines, Nozzles, Compressors, and Pumps Isentropic Compressor and Pump Efficiencies The state of matter entering the compressor and exit pressure are fixed. Negligible heat transfer with surroundings and no appreciable kinetic and potential energy effects.

65 65 EXAMPLE 6.11 Evaluating Turbine Work Using the Isentropic Efficiency A steam turbine operates at steady state with inlet conditions of p 1 = 5 bar, T 1 = 320 C. Steam leaves the turbine at a pressure of 1 bar. There is no significant heat transfer between the turbine and its surroundings, and kinetic and potential energy changes between inlet and exit are negligible. If the isentropic turbine efficiency is 75%, determine the work developed per unit mass of steam flowing through the turbine, in kj/kg.

66 66 EXAMPLE 6.11 Evaluating Turbine Work Using the Isentropic Efficiency

67 67 EXAMPLE 6.12 Evaluating the Isentropic Turbine Efficiency A turbine operating at steady state receives air at a pressure of p 1 = 3.0 bar and a temperature of T 1 = 390 K. Air exits the turbine at a pressure of p 2 = 1.0 bar. The work developed is measured as 74 kj per kg of air flowing through the turbine. The turbine operates adiabatically, and changes in kinetic and potential energy between inlet and exit can be neglected. Using the ideal gas model for air, determine the turbine efficiency.

68 68 EXAMPLE 6.12 Evaluating the Isentropic Turbine Efficiency

69 69 EXAMPLE 6.13 Evaluating the Isentropic Nozzle Efficiency Steam enters a nozzle operating at steady state at p 1 = 1.0 MPa and T 1 = 320 C with a velocity of 30 m/s. The pressure and temperature at the exit are p 2 = 0.3 MPa and T 2 = 180 C. There is no significant heat transfer between the nozzle and its surroundings, and changes in potential energy between inlet and exit can be neglected. Determine the nozzle efficiency.

70 70 EXAMPLE 6.13 Evaluating the Isentropic Nozzle Efficiency

71 71 EXAMPLE 6.14 Evaluating the Isentropic Compressor Efficiency For the compressor of the heat pump system in Example 6.8, determine the power, in kw, and the isentropic efficiency using data from property tables,

72 72 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes Heat Transfer For a control volume at steady state in which flow is both isothermal and internally reversible, the appropriate form of the entropy rate balance is

73 73 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes Work Since internal irreversibilities are absent, a unit of mass traverses a sequence of equilibrium states as it passes from inlet to exit. Entropy, enthalpy, and pressure changes are therefore related by Eq. 6.12b

74 74 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes Work in Polytropic Processes

75 75 Heat Transfer and Work in Internally Reversible, Steady-State Flow Processes Work in Polytropic Processes Ideal Gas Case

76 EXAMPLE 6.15 Polytropic Compression of Air An air compressor operates at steady state with air entering at p 1 = 1 bar, T 1 = 20 C, and exiting at p 2 = 5 bar.

76 76 EXAMPLE 6.15 Polytropic Compression of Air An air compressor operates at steady state with air entering at p 1 = 1 bar, T 1 = 20 C, and exiting at p 2 = 5 bar. Determine the work and heat transfer per unit of mass passing through the device, in kj/kg, if the air undergoes a polytropic process with n = 1.3. Neglect changes in kinetic and potential energy between the inlet and the exit. Use the ideal gas model for air.

77 77 Home Work Assignment of Ch6 1, 10, 17, 18, 25, 28, 36, 38, 46, 53, 63, 70, 77,84, 96, 104, 112, 120 Due Wednesday 9/5/2012

I. (20%) Answer the following True (T) or False (F). If false, explain why for full credit.

I. (20%) Answer the following True (T) or False (F). If false, explain why for full credit. Both the Kelvin and Fahrenheit scales are absolute temperature scales. Specific volume, v, is an intensive property,