ENTROPY ‘Thermal reservoir 1 Reversible, Ieyelic Idevice ' ' ‘Combined system (system and cyclic device) The system considered in the development of the Clausius inequality. SW, { 80 — 9| 8We = 50x — dE; 7 50, 80 . _. 8Q Te oF) BWe = Teg ~ dBc . 60 We = Ty T f 6Q _ » Clasius r inequality 5Q =o 50 Formal dS = ( T ) (\J/8) definition ‘ of entropy ases-s=[(®) The equality in the Clausius inequality holds for totally or just internally reversible cycles and the inequality for the irreversible ones.AS=S,-5,=04 KK +(%2) =0 T A quantity whose cyclic integral is zero (i.e., a property like volume) Irreversible Entropy is an extensive 3m property of a system. Reversible 03 07 SKK The entropy change between two The net change specified states is the same whether in volume (a the process is reversible or irreversible. property) suring 7 a cycle is $ A Special Case: Internally Reversible always zero. Isothermal Heat Transfer Processes : : . as= [(%)_ = | (2), : This equation is particularly useful for determining the entropy changes of thermal energy reservoirs. dV = AV, feycte = 0 ip A | (8Q)in rev ASTHE INCREASE OF ENTROPY PRINCIPLE Process 1-2 > f 8, [' 82 [' a) r p (F 1 6Q The equality holds for an internally Process 2-1 dS = reversible process and the inequality (internally I for an irreversible process. reversible) Acycle composed of a AS S s s reversible and an " F - irreversible process. Sg = MSvni = ASm, + SSuz ZO Some entropy is generated or created during an irreversible process, and this generation is due entirely to the presence of irreversibilities. The entropy generation S,,, is always a positive quantity or zero. Can the entropy of a system during a process decrease?(Isolated) Subsystem 1 Subsystem \ 2 Subsystem The entropy change of an isolated system is the sum of the entropy changes of its components, and is never less than zero. AS cineca = 0) Spa = ASwui = ASu, + ASumr = 0 gee N ASworai = LAS;> 0 is Subsystem a N Isolated system boundary, Surroundings Asystem and its surroundings form an isolated system. > 0 Irreversible process The increase Seon) = 0 Reversible process of entropy < 0 Impossible process _ principleSome Remarks about Entropy Surroundings Sou = 3 KK Sgon = AS vrai = ASeys + ASgure= 1 RIK The entropy change of a system can be negative, but the entropy generation cannot. 4 Processes can occur in a certain direction only, not in any direction. A process must proceed in the direction that complies with the increase of entropy principle, that is, Scen 2 0. A process that violates this principle is impossible. Entropy is a nonconserved property, and there is no such thing as the conservation of entropy principle. Entropy is conserved during the idealized reversible processes only and increases during ai/ actual processes. The performance of engineering systems is degraded by the presence of irreversibilities, and entropy generation is a measure of the magnitudes of the irreversibilities during that process. It is also used to establish criteria for the performance of engineering devicesENTROPY CHANGE OF PURE SUBSTANCES Entropy is a property, and thus the x. value of entropy of a system is fixed once the state of the system is fixed. T; Pps x00 Superheated vapor Compressed liquid ® Saturated liquid—vapor mixture Tr), = xy 2 508 5 fe ‘678 sing K Schematic of the 7-s diagram for water. The entropy of a pure substance Entropy change is determined from the tables AS = mAs = m(s; — 5;) (kJ/K) (like other properties). 8ISENTROPIC PROCESSES Aprocess during which the entropy remains constant is called an isentropic process As=0 ors, =, (kJ/kg-K) Isentropic fs process No heat transfer (adiabatic) During an internally reversible, adiabatic x 4 (isentropic) process, the The isentropic process appears as a entropy remains constant. vertical line segment on a T-s diagram. 52 = 5) sPROPERTY DIAGRAMS INVOLVING ENTROPY h Internally reversible Ona TS process diagram, the dA =TdS area under the process curve represents the heat transfer for internally reversible ‘5 processes. For adiabatic steady-flow 8Qinrev = TAS) Qin ey = | T dS devices, the vertical distance 1 Ahon an h-s diagram is a -2 measure of work, and the Sint reve = T dS int rev | T ds horizontal distance Asis a t measure of irreversibilities. Qin re Ty AS | dien Ty As Mollier diagram : The h-s diagramWHAT IS ENTROPY? Entropy, Se, kiikg + K Boltzmann Vpbrcecpecl ge relation S=kinp K k= 1.3806 * 10-2 JK Apure crystalline substance at absolute zero temperature is in perfect order, and its entropy is zero (the third law of thermodynamics). The level of molecular disorder (entropy) of a substance increases as Disorganized energy does not create much it melts or evaporates. useful effect, no matter how large it is.uring this process. During a heat shaft does not increases. (The create any disorder increase in the (entropy), and thus entropy of the cold energy is not body more than degraded during this Offsets the decrease process. in the entropy of the hot body.) 42THE Tds RELATIONS 5. 5Q vot rev T ds éW Pdv int ev — OWoes revo = AL TdS=dU+PdV (kJ) T ds = du + P dv (kJ/kg) the first T ds, or Gibbs equation h=u + Pv =du+P ; ; dh = du + P dv +v ms Ta =an= var The T ds relations are valid for T ds = du + P dv both reversible and irreversible the second T ds equation processes and for both closed To} te : ds Pd and open systems. 4 - Differential changes T r in entropy in terms yy ~ 2h — © 2? of other properties T T ds =ENTROPY CHANGE OF LIQUIDS AND SOLIDS _ du T Pdv a, Since dv=0 du ds = T since ¢ c dT Z Liquids, solids: for liquids and solids cand du cdT Liquids and solids can be approximated as incompressible substances since their specific volumes remain nearly constant during a process. T, eg Ae 1 (kJ/kg + K) For and isentropic process of an incompressible substance Isentropic T,=T,THE ENTROPY CHANGE OF IDEAL GASES From the first T ds relation From the second T ds relation ‘i du | P dv du = c,dT js = th _ vaP “TT |P=RIW = oo Is =< a. R dv dh =c, dT vy = RTIP T v aT v dT P. 5 — 5 (Tp + Rin $ — 5 } of) Ring A broadcast from channel IG.Constant Specific Heats (Approximate Analysis) : aT vy t, V2 s-s= | cf(T)= + Rin aS, — S; = Cyn, In — + Rin — , T YW 2 Rn Y 2 aT P, 1; . S, — 8, = © (1) > ~ Rin Pa 2 Cig On T, — Rin P, 1 (kJ/kg + K) Entropy change of an ideal gas on a unit-mole basis B+ Rein (kd/kmol-K) By — Hy = Cay IM 5 Actual ¢, I P 5p — 5) = Cpang in Rin (ki/kmol « K) T P, Under the constant-specific- heat assumption, the specific heat is assumed to be constant T, Tayg ‘Tr T at some average value. ' a mice saeIsentropic Processes of Ideal Gases Constant Specific Heats (Approximate Analysis) rt, Vy % 3 Sp — 5) = Cumg in 7, * Rin a Setting this eq. equal to zero, we get T, Vy In==- z In— ideal gas T, cy Y VALID FOR tropic process constant specific heats h vy, \* in Bon (4) ‘ ; 1 R=c, — k= ec The isentropic relations of ideal and thus R/c k-1 gases are valid for the isentropic processes of ideal gases only. (F)_.. 2) T, 7 smcona. vy Tv‘! = constant C2) = (F) UF) (3) Pu‘ = constantWhen the properties of the system are not uniform s sm = | sp d\ Energy and entropy balances for a system.Mechanisms of Entropy Transfer, S, and S,, 1 Heat Transfer LO Entropy transfer by heat transfer: A Surroundings \ ) Sve g (T= constant) \ 6Q : Sreat = é 2 =>% ; Q=500KI Entropy transfer by work: } Swot = 0 2 wor ‘i qT, - = 1.25 KK Entropy is not Wit transferred Heat transfer is always with work i Entropy accompanied by entropy transfer generation, in the amount of Q/T, where Tis via friction the boundary temperature Eperines place . No entropy accompanies work as it crosses the system boundary. But entropy may be generated within the system as work is dissipated into a less useful form of energy. 28Mechanisms of Entropy Transfer, S$, and Sout 2 Mass Flow i = = Entropy transfer by mass: ! Control volume h. Sinass = ms s m i mh | ms When the properties of the mass | change during the process il | : J, spV,dA Mass contains entropy as well as energy, and thus mass flow into or s | sdm S wun lt out of system is always ¥ accompanied by energy and entropy transfer.Entropy Generation, S,., Si, — S + § AS coco (kJ/K) Sin— Som + $8 Ss stom/ dt (kW/K Mechanisms of entropy transfer for a general system. Entropy generation outside system boundaries can be accounted for by writing an entropy balance on an extended system that includes the system and its immediate surroundings.Closed Systems =e Closed system: 3s Os = AS... = S,- 5, (kJ/K) hi gen System The entropy change of a closed system during a process is equal to the sum of the net entropy transferred through the system boundary by heat transfer and the entropy generated within the system boundaries. Adiabatic closed system: Sogou = AS csicnatic syne System + Surroundings: Seq = AS = AS.yrcm + AS eyo arvoundiogs| AS, = mS, — SAS ue = Qeurr!T surr surr! # sure?, SS! + San = Dents Dt Do Diss + Sa De ae + Deis, - Deis, + § Steady-flow Steady-flow, single-stream. Steady-flow, single-stream, adiabatic (S2 — Si)ev (kJ/K) dSey/dt_— (KW/K) > Or Syn = ts, - 5) - > gon = Hin 1-25 Sen = tt(8, ~ 8,) The entropy of a substance always increases (or remains constant in the case of a reversible process) as it flows through a single-stream, adiabatic, steady- flow device. Control Volumes Surroundings ASey= 2 + m5; meso Spon Entropy Entropy transfer transfer by heat by mass The entropy of a control volume changes as a result of mass flow as well as heat transfer. 32A large condenser in a steam power plant dumps 15 MW at 45°C with an ambient at 25°C. ‘What is the entropy generation rate? A 50-kg copper block initially at 80°C is dropped into an insulated tank that contains 120 L of water at 25°C. Determine the entropy change of a copper block. A Carnot engine operates between 327°C and 27°C. If the engine produces 300 kJ of work, what is the entropy change during heat addition? An electric current of one amp flows through a resistor of 300 ohm, which is in contact with a reservoir at 300K. at steady state, what is the rate of entropy generation of the ‘universe?Air initially occupying Im* at 1Sbar and 20°C undergoes an internally reversible compression for which PV"=C to a final state where the pressure is 6 bar and the temperature is 120°C. determine i) the value of n, ii)the work and heat transfer iii) the change in entropy take Cv=0.718 kI/kgK A 50-kg copper block initially at 80°C is dropped into an insulated tank that contains 120 L of water at 25°C. Determine the entropy change of a water. Two flows of air both at 200 kPa: one has 1 ke/s at 400 K and the other has 2 ke/s at 290 K. The two flows age, mixed in an insulated box to produce a single exit flow at 200 kPa Write a procedure to find the total zate of entropy generation for this process.Two flow streams of water, one at 0.6 MPa, saturated vapor, and the other at 0.6 MPa, 600°C, mix adiabatically in a steady flow process to produce a single flow out at 0.6 MPa, 400°C. Develop an equation for the total entropy generation for this process. A geothermal supply of hot water at 500 kPa, 150°C ig fed to an insulated flash evaporator at the sate of 1.5 kg/s. A stream of saturated liquid at 200 kPa is drained from the bottom of the chamber and a stream of saturated vapor at 200 kPa is drawn from the top and fed to a turbine. Develop an equation for finding the rate of entropy generation in the flash evaporator. Heat is lost through a plane wall steadily at arate of 600 W. If the inner and outer surface temperatures of the wall are 20°C and 5°C, respectively. evaluate the rate of entropy generation within the wall.Steam enters an adiabatic turbine at 8 MPa and 500°C at a rate of 18 kg/s, and exits at 0.2 MPa and 300°C. Calculate the rate of entropy generation in the turbine. Liquid water enters an adiabatic piping system at 15°C at a rate of $ kg/s. If the water temperature rises by 0.2°C during flow due to friction, Estimate the rate of entropy generation in the pipe An apple with an average mass of 0.15 kg and average specific heat of 3.65 ki/kg - °C is cooled from 20°C to 5°C. Find the entropy change of the apple. Steam is condensed at a constant temperature of 30°C as it flows through the condenser of a power plant by rejecting heat at a rate of 35 MW. Find the rate of entropy change of steam as it flows through the condenser.A well-insulated, thin-walled, double-pipe, counter-flow heat exchanger is to be used to cool oil (cm =2.20 kI/ke. °C) from 150°C to 40°C at a rate of 2 ke/s by water (cp =4.18 kI/ke. °C) that enters at 22°C at arate of 1.5 ke/s. Determine (i) the rate of heat transfer and (ii) the rate of entropy generation in the heat exchanger. Air is compressed steadily by a 5-kW compressor from 100 kPa and 17°C to 600 kPa and 167°C at arate of 1.6kg/min. During this process, some heat transfer takes place between the compressor and the surrounding medium at 17°C. Determine the rate of entropy change of air during this process.A frictionless piston-cylinder device contains a saturated liquid-vapor mixture of water at 100°C. During a constant-pressure process, 600 kI of heat is transferred to the surrounding air at 25°C. As a result, part of the water vapor contained in the cylinder condenses. Determine ({) the entropy change of the water and (ji) the total entropy generation during this heat tranefer process. 1.5 kg of air at | bar, 300K is contained in a rigid insulated tank. During the process, 18kJ of work ig.done on the gas through a paddle-wheel mechanism. Determine the final temperature, final pressure of air in the tank and change in entropy. An iron cube at 400°C is dropped into an insulated bath having 10 kg water at 25°C. Final temperature of water is 50°C. Assume the process as reversible and find the change in entropy of iron and water. Take Cyy=4.186 kW/keK.