Entropy. Objectives. MAE Chapter 7. Definition of Entropy. Definition of Entropy. Definition of Entropy. Definition of Entropy + Δ

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1 MAE Chapter 7 Entropy Objectives Defe a new property called entropy to quantify the second-law effects. Establish the crease of entropy prciple. Calculate the entropy changes that take place durg processes for pure substances, compressible substances, and ideal gases. Exame a special class of idealized processes, called isentropic processes, and develop the property relations for these processes. Derive the reversible steady-flow work relations. Develop the isentropic efficiencies for various steady-flow devices. Introduce and apply the entropy balance to various systems. he content and the pictures are from the text book: Çengel, Y. A. and Boles, M. A., hermodynamics: An Engeerg Approach, McGraw-Hill, New York, 6th Ed., 2008 Defition of Entropy For the reversible enge: H L + H L 1000kJ 300kJ = + = k 300K 1000kJ For the irreversible enge: H L + H L 1000kJ 550kJ = k 300K = 0.83 < kJ 1000kJ Defition of Entropy If we extend this to a thermodynamic cycle, there is a new statement as follows: Clausius equality: for any thermodynamic cycle, reversible or irreversible, the cyclic tegral of δ/ is always less than or equal to zero. he symbol (tegral symbol with a circle the middle) is used to dicate that the tegration is to be performed over the entire cycle. For the impossible enge: H L 1000kJ 200kJ + = k 300K H H H L + = 0.33 > 0 L L = δ δ H L ( + ) = H L 300kJ 550kJ 200kJ 300K 300k 300K δ <0 irreversible enge =0 reversible enge >0 impossible enge can be viewed as the sum of all the differential amount of heat transfer divided by the temperature at the boundary. For a device that undergoes an ternally reversible cycle: he equality the Clausius equality holds for totally or just ternally reversible cycles and the equality for the irreversible ones. Defition of Entropy A quantity whose cyclic tegral is zero (i.e., a property like volume) typically depends on the state and not the process path. hus such a quantity is a property. herefore (δ/), rev must represent a property the differential form. Clausius has realized this pot and discovered a thermodynamic property named entropy Defition of Entropy Entropy is a property, like all other properties, it has a fixed value at a fixed sate. he entropy change between two specified states is the same whether the process is reversible or irreversible. In a special case (an ternal reversible process), the entropy change between two specified states: Entropy (S) is an extensive property of a system. Entropy per unit mass, designated s, is an tensive property with the unit kj/kg.k. Note: C v, C p and R have the same unit kj/kg.k he net change volume (a property) durg a cycle is always zero. For a closed system that undergoes an irreversible process, the entropy change between two specified states: + Δ 1

2 Entropy A Special Case: Internally Reversible Isothermal Heat ransfer Processes Recall that isothermal heat transfer processes are ternally reversible: he Increase of Entropy Prciple Considerg a cycle that is made of two processes: Process 1-2, which is arbitrary (reversible or irreversible), and Process 2-1, which is ternally reversible. From the Clausius equality: his equation is particularly useful for determg the entropy changes of thermal energy reservoirs that can adsorb or supply heat defitely at a constant temperature. Heat transfer to a system creases the entropy of a system, whereas heat transfer from a system decreases it. In fact, losg heat is the only way the entropy of a system can be decreased. he term is equal to the entropy change S 1 -S 2 : A cycle composed of a reversible and an irreversible process. (1).he equality holds for an ternally reversible process. he entropy change becomes equal to, which represent entropy transfer with heat. (2). he equality for an irreversible process. he Increase of Entropy Prciple he Increase of Entropy Prciple he equality dicate that the entropy change of a closed system durg an irreversible process is greater than entropy transfer. In other words, some entropy is generated or created durg an irreversible process. he entropy generated is called entropy generation, S gen. Rewrite the above equation he entropy generation S gen is always a positive quantity or zero. Its value depends on the process path, and thus it is not a property of a system For an isolated system (an adiabatic closed system), the heat transfer is zero, thus his equation dicates that the entropy of an isolated system durg a process always creases. It never decreases. Increase of entropy prciple he entropy change of an isolated system is the sum of the entropy changes of its components, and is never less than zero. A system and its surroundgs form an isolated system. he Increase of Entropy Prciple Some entropy is generated or created durg an irreversible process, and this generation is due entirely to the presence of irreversibilities. he Increase of entropy prciple can summarized as follows: (1). he entropy generation, S gen, is not a property of a system. Its value depends on the process path. It is always a positive quantity or zero. (2). he entropy, S, is a property. It is dependent on the path. he entropy change (S 2 -S 1 ) can be zero, positive, or negative Some Remarks ab Entropy 1. Processes can occur a certa direction only, not any direction. A process must proceed the direction that complies with the crease of entropy prciple, that is, S gen 0. A process that violates this prciple is impossible. 2. Entropy is a nonconserved property, and there is no such thg as the conservation of entropy prciple. Entropy is conserved durg the idealized reversible processes only and creases durg all actual processes. 3. he performance of engeerg systems is degraded by the presence of irreversibilities, and entropy generation is a measure of the magnitudes of the irreversibilities durg that process. It is also used to establish criteria for the performance of engeerg devices. 4. Entropy can transferred by two ways: (i) heat transfer and (ii) mass transfer. Net work can not transfer entropy. 2

3 Energy balance & Entropy balance of closed system Conservation of energy is applied to a closed system that undergoes any processes regardless of reversible of irreversible, hence the energy balance equation: E E = ΔE + W sys W = ΔE system = ΔU + ΔPE + ΔKE he Increase of Entropy Prciple here is no conservation of entropy for a closed system that undergoes any processes regardless of reversible of irreversible, hence the entropy balance equation: + S gen = ΔS sys = S 2 S 1 In a special case, a closed system undergoes an ternal reversible processes, then the entropy balance equation becomes: = ΔS sys = S 2 S 1 he Increase of Entropy Prciple Example 7-2 he Increase of Entropy Prciple Example 7-2 Entropy Change of Pure Substances Entropy is a property, and thus the value of entropy of a system is fixed once the state of the system is fixed. he entropy of a pure substance is determed from the tables (like other properties). Especially, the -s diagram of water is shown Figure A-9 Pg. 926 Entropy Change of Pure Substances he value of entropy at a specific state is determed just like any other proper. In the saturated region: he entropy of a compressed liquid: P=constant -s diagram of water. 3

4 Entropy Change of Pure Substances Entropy Change of Pure Substances Example 7-3 Entropy Change of Pure Substances Example 7-3 able A-13 Superheated vapor Entropy Change of Pure Substances Example 7-3 able A-12 Isentropic Processes A process durg which the entropy remas constant is called an isentropic process. Isentropic Processes Durg an ternally reversible and adiabatic process, the entropy remas constant. he isentropic process appears as a vertical le segment on a -s diagram. 4

5 Isentropic Processes Example 7-5 Isentropic Processes Example 7-5 From able A-5, the sat = o C when P 1 = 5000 kpa. 1 =450 > sat = o C, herefore, it is superheated vapor under the State 1. So s 1 can be obtaed from able A-6, s 1 = KJ/kg K and h 1 = kj/kg k (page 922) Isentropic Processes Example 7-5 Property Diagrams Involvg Entropy For an ternal reversible process: From able A-5, the S g = KJ/kg K when P 1 = 1400 kpa. On a -S diagram, the area under the process curve represents the heat transfer for ternally reversible processes. But the area has no meang for an irreversible process s 2 =s 1 = > s g@p=1400 kpa = kj/kg K. herefore, it is superheated vapor under the State 2. So h 2 can be obtaed from able A-6, h 2 = kj/kg K through terpolation (page 921) Method (2), Figure A-9 Page 926 to get the h 2 emperature-entropy diagram For an ternal reversible isothermal process: or Property Diagrams Involvg Entropy Property Diagrams Involvg Entropy An isentropic process appears as a vertical le segment on a -s diagram. his expected sce an isentropic process volves no heat transfer, and therefore the area under the process path must be zero. P-v diagram of the Carnot Cycle he -s diagram of the Carnot Cycle he -s diagram serves as a valuable tool for visualizg the second law aspects of processes and cycles. An isentropic process appears as a vertical le segment on a -s diagram 5

6 Property Diagrams Involvg Entropy he h-s diagram is useful analysis of adiabatic steadyflow devices, such as turbes, compressors and nozzles: What is Entropy? Entropy is a measure of molecular disorder or molecular randomness (i). he vertical distance h (between the let and the exit states) on an h-s diagram is a measure of work. (ii). he horizontal distance s is a measure of irreversibilities associated with the process. Mollier diagram: he h-s diagram Fig. A-10 Appendix Page 927 he level of molecular disorder (entropy) of a substance creases as it melts or evaporates. A pure crystalle substance at absolute zero temperature is perfect order, and its entropy is zero (the third law of thermodynamics). he ds Relations For a closed system that undergog an ternal reversible process: he ds Relations For a closed system that undergog an ternal reversible process: ds PdV = du the first ds, or Gibbs equation ds PdV = du the first ds, or Gibbs equation the second ds equation the second ds equation he ds Relations he ds relations are derived from an ternal reversible process. However, it is valid for both reversible and irreversible processes and for both closed and open systems, sce entropy is a property and the change a property between two states is dependent on the type of process the system undergoes. Entropy Change of Liquids and Solids Sce for liquids and solids Liquids and solids can be approximated as compressible substances sce their specific volumes rema nearly constant durg a process. he ds relations are valid for both reversible and irreversible processes and for both closed and open systems. An isentropic process of an compressible substance is also an isothermal process: 6

7 he Entropy Change of Ideal Gases From the first ds relation From the second ds relation Constant Specific Heats (Approximate Analysis) Assumg constant specific heats for idea gases is a common approximation: for ideal gas Under the constant-specific-heat assumption, the specific heat is assumed to be constant at some average value. Variable Specific Heats (Exact Analysis) We choose absolute zero as the reference temperature and defe a function s as Variable Specific Heats On a unit mass basis On a unit mole basis he entropy of an ideal gas depends on both and P. he function s represents only the temperature-dependent part of entropy. Variable Specific Heats Example 7-9 Variable Specific Heats Example 7-9 7

8 Isentropic Processes of Ideal Gases Constant Specific Heats (Approximate Analysis) Isentropic Processes of Ideal Gases Variable Specific Heats (Exact Analysis) S 2 -S 1 = Settg this eq. equal to zero, we get Relative Pressure and Relative Specific Volume exp(s /R) is the relative pressure P r he isentropic relations of ideal gases are valid for the isentropic processes of ideal gases only. he use of P r data for calculatg the fal temperature durg an isentropic process. /P r is the relative specific volume v r he use of v r data for calculatg the fal temperature durg an isentropic process Isentropic Processes of Ideal Gases Variable Specific Heats (Exact Analysis) Isentropic Processes of Ideal Gases able A-17 able A-17 he use of P r data for calculatg the fal temperature durg an isentropic process. he use of v r data for calculatg the fal temperature durg an isentropic process Isentropic Processes of Ideal Gases Example 7-10 Isentropic Processes of Ideal Gases Example 7-10 If we use k=1.400 at the itial temperature 295 K 2 =(295K) (8) = K hus we can estimate the average temperature is 481 K At 481k, K=1.389 based on able A-2b 2 = K 8

9 Isentropic Efficiencies of Steady-flow Devices Isentropic process: ternal reversible, adiabatic he isentropic process volves no irreversibilities and serves as the ideal process for adiabatic devices. Isentropic Efficiency of urbes: Isentropic Efficiencies of Steady-flow Devices Isentropic Efficiency of urbes: Isentropic efficiency is also called the second law efficiency, which is a measure of the deviation of actual processes from the correspondg idealized ones. hermal efficiency is called the first law efficiency. ypically a turbe under the steady operation, the let state of the workg fluid and the exhaust pressure is fixed he h-s diagram for the actual and isentropic processes of an adiabatic turbe. Isentropic Efficiencies of Steady-flow Devices Isentropic Efficiencies of Steady-flow Devices 3. For a turbe under the steady operation, the let state of the workg fluid and the exhaust pressure is fixed. Isentropic Efficiencies of Steady-flow Devices Example 7-14 Isentropic Efficiencies of Steady-flow Devices Example 7-14, State 2s is the liquid-vapor region 9

10 Isentropic Efficiencies of Compressors Isentropic Efficiencies of Compressors Isentropic Efficiencies of Compressors Isentropic Efficiencies of Compressors When ketic and potential energies are negligible Can you use isentropic efficiency for a nonadiabatic compressor? Can you use isothermal efficiency for an adiabatic compressor? he h-s diagram of the actual and isentropic processes of an adiabatic compressor For a reversible isothermal process, then we can defe an isothermal efficiency: W t and W a are the required work puts to the compressor for the reversible isothermal and actual processes. Compressors are sometimes tentionally cooled to mimize the work put. Isentropic Efficiencies of Pumps Isentropic Efficiencies of Pumps: When ketic and potential energies of a liquid are negligible Isentropic Efficiencies of Pumps Isentropic Efficiencies of Pumps Example 7-15 Isentropic Efficiencies of Pumps Example 7-15 State 1: 10

11 Isentropic Efficiencies of Pumps Example 7-15 Isentropic Efficiency of Nozzles Isentropic Efficiency of Nozzles is defed as: E = E Sce = 0, W = 0, Δpe V 1 V2a m ( h1 + ) = m( h2a + ) 2 2 If the let velocity of the fluid is small relative to the exit velocity, the energy balance is hen he h-s diagram of the actual and isentropic processes of an adiabatic nozzle. Isentropic process of Nozzles Isentropic process of Nozzles Isentropic process of Nozzles Example 7-16 Isentropic process of Nozzles Example 7-16 V2 s = 2( h1 h2 ) = 2( ) = 666 ( m / s) Alternatively, use a constant C p 988 h2a 0.92 = h 2a = kj/kg h 1 = 988 kj/kg = 988,000 J/kg at 950K h 2s = 766 kj/kg = 766,000 J/kg at 748K V 1 = 0 Attention: unit conversion (able A-17) From table A-17: h=800.03kj/kg at 780K, and h= kj/kg at 760K Interpolation: a = a = 765K 11

12 Isentropic process of Nozzles Example 7-16 Alternatively, use a constant C p Entropy Balance Entropy Change of a System, S system When the properties of the system are not uniform Energy and entropy balances for a system, S gen Attention: the unit for followg three forms: Mechanisms of Entropy ransfer Entropy can be transferred to or from a system by two mechanisms: heat transfer and mass flow. Entropy transfer by heat: (1). For an irreversible process, the entropy generation (S gen ) is greater than zero (2). For a reversible process, the entropy generation (S gen ) is zero and thus the entropy change of a system is equal to the entropy transfer No entropy accompanies work as it crosses the system boundary. But entropy may be generated with the system as work is dissipated to a less useful form of energy. Entropy transfer by work: Heat transfer is always accompanied by entropy transfer the amount of /, where is the boundary temperature. Mechanisms of Entropy ransfer Entropy transfer by mass: Entropy Change of a System When the properties of the mass change durg the process he entropy change between the fal state and the itial state of system that undergoes a process: 1) he entropy transfer by heat 2) he entropy transfer by mass flow 3) he irreversibility extent of the process Mass contas entropy as well as energy, and thus mass flow to or of system is always accompanied by energy and entropy transfer. Mechanisms of entropy transfer for a general system. 12

13 Entropy Balance of Closed Systems A closed system volves no mass transfer across its boundary Entropy Balance of Closed Systems Notg that any closed system and its surroundgs can be treated as an adiabatic system, and then: For an adiabatic process, no heat transfer across the boundary of a closed system, the entropy change of the system only depends on the irreversibility of the process: Entropy generation side system boundaries can be accounted for by writg an entropy balance on an extended system that cludes the system and its immediate surroundgs. surr Balance Equations of Closed Systems A closed system volves no mass transfer across its boundary Energy balance: Entropy Balance of Control Volumes As compared with a closed system, a control volume volves mass transfer across its boundary. hus the entropy balance: ( ) + ( W W ) = ΔU + ΔKE + ΔPE For a closed system with change the KE and PE Entropy balance: net, Wnet, = ΔU he entropy change with a control volume durg a process is equal to the sum: (1) Entropy transfer by heat (2) the net entropy transfer by mass flow (3) the entropy generation as result from irreversibility Entropy Balance of Control Volumes For a general steady-flow process: k + mi si me se + S gen = 0 k For a sgle-stream, steady-flow device: k + m( si se ) + S gen = 0 k (kw/k) For an adiabatic, sgle-stream, steady-flow device: m( si se ) + S gen = 0 Balance Equations of Control Volumes For a general control volume: Mass balance : Energy balance : Entropy balance : For an adiabatic, sgle-stream, steady-flow device that undergoes a reversible process: s i = s e 13

14 Balance Equations of Steady-flow Control Volumes For a steady-flow control volume: Mass balance : Energy balance : Entropy balance : k + mi si me se + S gen = 0 k Example 7-18 Example 7-18 k + mi si me se + S gen = 0 k (able A-6) (able A-6) Interpolation Example

15 Example 7-19 Now we take the lake as the closed system. C avg = he heat transfer form the iron block to the lake while the lake keeps a constant temperature (285K). herefore, we can consider this isothermal process is an ternal reversible. hus S gen =0 For the iron block: net, Wnet, = ΔU Example 7-19 Example 7-20 Energy Balance: k + mi si me se + S gen = 0 k h 1 =? s 1 =? 15

16 Summary Entropy he Increase of entropy prciple Entropy change of pure substances Isentropic processes Property diagrams volvg entropy he ds relations Entropy change of liquids and solids he entropy change of ideal gases Isentropic efficiencies of steady-flow devices Entropy balance 16