Entropy. Chapter The Clausius Inequality and Entropy

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1 Chapter 7 Entrpy In the preceding chapter we btained a number f imprtant results by applying the secnd law t cyclic prcesses assciated with heat engines and reversed heat engines perating with ne and tw thermal reservirs. The cncept f a reversible prcess, thugh an idealizatin f real prcesses, prvided a means t find the upper limit f the efficiency f heat engines perating with tw heat reservirs. The maximum efficiency, called the Carnt-cycle efficiency, is a unique functin f the tw reservir temperatures. This lead t the develpment f the thermdynamic temperature scale. Finally, we generalized these results fr cyclic systems experiencing heat interactins with any number f thermal reservirs. In this chapter we shall extend the applicatin f the secnd law t clsed systems underging any prcess, reversible r therwise. The main utcme f this effrt is the emergence f a new system prperty called entrpy which has a brad significance and very wide applicatins. 7. The Clausius Inequality and Entrpy T apply the secnd law t a system that executes a cycle exchanging heat ver a range f temperatures we cnsider the arrangement shwn in Fig The cycle executed by the system S culd be internally reversible r irreversible but the heat interactins between the varius thermal reservirs and the system are reversible. This is the main difference between the present develpment and the analysis in Sec Fr external reversibility the temperature T i f the secndary 3

2 Entrpy 33 reservir must be equal t the temperature T f the system where heat is received during the prcess. Since we expect the temperature T t change during the prcess, we need t have a series f secndary reservirs with the assciated heat engines t achieve reversible heat transfer. Under these new cnditins we can rewrite Eq. (6.49) as Qsi 0 T i (7.) where Qsi is the heat flw frm the reservir whse temperature T i is equal t the temperature T f the system at the lcatin where heat is received. Cnsidering an infinitesimal heat transfer δ Q at temperature T, we can express Eq. (7.) in the cyclic integral frm δq 0 (7.) T cyclic Equatin (7.), which nw invlves nly the heat input t the system and the temperature f the system at the lcatin where heat is received, is called the Clausisus inequality. As we discussed in Sec the equality signs in Eqs. (7.) and (7.) apply when the cycle executed by the system is internally reversible and the inequality is fr irreversible cycles. Fr internally reversible prcesses the temperature T f the system is unifrm because the prcess ccurs quasi-statically. Hwever, fr internally irreversible prcesses there culd be temperature gradients within the system because the prcess is nt necessarily quasi-static. Fr this situatin, the term T in Eq. (7.) is the temperature at the lcatin n the bundary where the heat input δq enters the system as indicated in Fig Entrpy A thermdynamic prperty Cnsider a clsed system executing a reversible cycle -A--B- shwn in Fig. 7.. Applying Eq. (7.) with the equality sign we have A δq T + B δq 0 T (7.3)

3 34 Engineering Thermdynamics P B C D A Fig. 7. P-V diagram f prcess V We nw envisage an alternative reversible prcess, -C- that returns the system frm states t thus cmpleting the cycle. Applying Eq. (7.) t the cycle -A--C- we btain A δq T + Subtracting Eq. (7.3) frm Eq. (7.4) C δq T B δq 0 T C δq T (7.4) (7.5) Since the prcess C was selected arbitrarily, it fllw frm Eq. (7.5) that the quantity, ( δ Q / T ) is path-independent. It is, therefre, a prperty f the system which depends nly n state and state. Hence we have Q δ S S (7.6) T This new prperty, dented by S in Eq. (7.6), is called the entrpy the system. It is an extensive prperty that depends n the mass f the system. We recall that the internal energy f a system arse as a cnsequence f the first law. In a similar manner, the applicatin f the

4 Entrpy 35 secnd law has predicted the existence f this new prperty but ffers n additinal infrmatin n its physical meaning. Later in this chapter we shall briefly discuss the micrscpic interpretatin f entrpy. We nw select an irreversible prcess D, shwn in Fig. 7., t return the system frm state t state. This chice makes the cycle -A--D- internally irreversible. Applying Eq. (7.) with the inequality sign we have A δq T + D Frm Eqs. (7.4) and (7.7) it fllws that δq > T C Frm Eqs. (7.8) and (7.6) we have D δq < 0 T δq T (7.7) (7.8) δq S S > (7.9) T Fr an infinitesimal change f state we can write the differential frm f the abve relatins as D Q ds δ (7.0) T rev Q ds > δ (7.) T It is ntewrthy that in Eq. (7.0), T is the system temperature, which is unifrm fr internally reversible prcesses while in Eq. (7.), T is the temperature at the lcatin n the bundary where heat enters the system. The abve equatins cnstitute the mathematical frmulatin f the secnd law. Irr

5 36 Engineering Thermdynamics 7.. The temperature-entrpy diagram The P-V diagram is a useful graphical aid fr representing quasi-static prcesses because the area under the curve is prprtinal t the wrk dne during the prcess. In a similar manner we can use a temperatureentrpy diagram (T-S diagram) f a reversible prcess t btain the heat transfer. This fllws frm Eq. (7.0), which, fr a reversible prcess, may be written as Integratin f Eq. (7.) gives δ Q rev TdS (7.) f Q TdS (7.3) i where i and f represent the initial and final states f the system. A temperature-entrpy diagram fr a cyclic prcess cnsisting f three reversible prcesses is shwn in Fig. 7.. Frm Eq. (7.3) it fllws that the area under the curve - is the heat input t the system during the reversible prcess -. Fr the prcess -3 the area is traced in the negative S-directin when the system mves frm the initial state t the final state 3. We interpret this as a negative area which crrespnds t a heat utput fr the prcess -3. Fr the prcess 3-, ds 0. It fllws frm Eq. (7.) that δ Q rev 0, and therefre the prcess 3- is reversible and adiabatic. Such a prcess is called an isentrpic prcess Fig. 7. Temperature-entrpy (T-S) diagram

6 Entrpy 37 because the entrpy remains cnstant during the prcess and it can be represented by a straight line parallel t the T -axis as seen in Fig. 7.. Fr the cyclic prcess --3- the area enclsed by the three curves is the net heat input t the system during the prcess The T-ds equatin We shall nw derive a useful prperty relatin that invlves the zerth law, the first law and the secnd law fr a simple thermdynamic system. Cnsider an infinitesimal prcess in a clsed system. Applying the first law we have δ Q du + δw (7.4) Since the prcess is reversible δ W PdV (7.5) Applying the secnd law [Eq. (7.0)] t the reversible prcess Q ds δ (7.6) T Manipulating Eqs. (7.4), (7.5) and (7.6) we btain TdS du + PdV (7.7) Nte that Eq. (7.7) is a relatinship between thermdynamic prperties f the system and is therefre independent f the prcess. Mrever, it is independent f the substance cnstituting the system. This imprtant relatin, smetimes called the T-dS equatin, invlves the zerth law thrugh the cncept f temperature, the first law thrugh the internal energy functin and the secnd law thrugh the entrpy and thermdynamic temperature. rev 7..4 Entrpy f an ideal gas Since the entrpy is a prperty, it is pssible t express the entrpy as S S( V, T ), fllwing the tw-prperty rule fr a pure substance. We shall nw prceed t btain this prperty relatin fr an ideal gas.

7 38 Engineering Thermdynamics Cnsider a fixed mass m f an ideal gas subjected t a reversible prcess frm an initial state ( P, V, T ) t a final state ( P, V, T ). The equatin f state f the ideal gas is PV mrt (7.8) The internal energy f the ideal gas may be expressed as U mc T + (7.9) v U ref where U ref is the internal energy f the reference state. Differentiating Eq. (7.9) du mc dt (7.0) Substituting in the T-dS equatin [Eq. (7.7)] frm Eqs. (7.8) and (7.0) we btain v TdS mcv dt + mrtdv / V (7.) ds mcv dt / T + mrdv / V (7.) Integrating Eq. (7.) frm the initial state t the final state we have v ( T / T ) mr ln( V V ) S S + mc ln + / (7.3) Substituting fr V in Eq. (7.3) frm Eq. (7.8) we btain a prperty relatin f the type, S S( P, T ). ( T / T ) mr ln( TP T P) S S + mcv ln + / (7.4) v ( T / T ) mr ln( P P ) S S + m( c + R) ln / p ( T / T ) mr ln( P P ) S S + mc ln / (7.5) We nte that Eqs. (7.3) and (7.5) are relatinships between prperties f an ideal gas and therefre applicable t any equilibrium state. Mrever, the initial state dented by can be regarded as the reference state in a tabulatin f data fr the entrpy. The T-S diagram fr an ideal gas may be drawn by making use f the analytical expressins in Eqs. (7.3) and (7.5). Since these invlve three variables we need t draw families f curves keeping ne f the variables fixed. Fr example, fr an ideal gas, the cnstant-pressure lines n the

8 Entrpy 39 T-S diagram are lgarithmic curves accrding t Eq. (7.5). The relevant expressins fr these variatins are derived in wrked example Entrpy f a pure thermal system We shall nw btain a general expressin fr the entrpy change f a pure thermal system which is an idealized mdel applicable t slids and incmpressible fluids. Fr such systems the specific heat capacity is a cnstant. Cnsider an infinitesimal reversible heat transfer, δ Q t the system. Applying the secnd law we btain δ Q TdS (7.6) Applying the first law with zer wrk transfer δ Q du MCdT (7.7) where M and C are the mass and specific heat capacity f the pure thermal system. Frm Eqs. (7.6) and (7.7) we btain TdS MCdT (7.8) Integrating Eq. (7.8) frm the reference state t the general state ( T ) S S + MC ln / (7.9) ref T ref where the sub-script ref dentes the prperties at the reference state Entrpy f a pure substance Shwn in Fig. 7.3 is the T-S diagram fr a pure substance underging a reversible cnstant pressure heating prcess. At A, the fluid is a subcled liquid at a pressure P. As heat is supplied t the fluid at cnstant pressure, the prcess n the T-S diagram fllws the curve A-B up t B where evapratin just begins. The sectin B-C f the curve crrespnds t the evapratin prcess where S increases while T remains cnstant because f the cnstant pressure. At C evapratin is cmplete and alng the sectin C-D, the vapr becmes superheated.

9 30 Engineering Thermdynamics Fig. 7.3 T-S diagram fr a pure substance Fr a pure substance like steam, it is nt pssible t btain simple analytical expressins fr the entrpy as we did fr an ideal gas. In this case the data has t be btained frm tabulatins as fr the internal energy and enthalpy. Referring t the data tables in [6], we ntice that the entrpy per unit mass f a saturated liquid, s f is chsen t be zer at the triple pint f water. The saturated vapr entrpy is tabulated under s g. The change in entrpy when a unit mass underges phase change frm liquid t vapr is given by ( s g s f ) s fg. The specific entrpy f a wet vapr f quality x may be expressed as s x s + ( x) (7.30) g s f Fr superheated vapr, the entrpy is tabulated in [6] fr different pressures and temperatures. In general, t extract entrpy data frm the tables we fllw the same prcedure that was used earlier t btain the internal energy and the enthalpy f a pure substance. It fllws frm Eq. (7.3) that the ttal heat supplied during the evapratin prcess B-C in Fig. 7.3 is the area f the rectangle BCC B. Since the temperature, T s is cnstant during evapratin, we btain the fllwing relatin by applying Eq. (7.3) t a unit mass f steam s s q / T ( h h ) / T (7.3) g f fg s where q fg hg hf, is the heat supplied per unit mass during the evapratin prcess. g f s

10 7. Principle f Increase f Entrpy Entrpy 3 The principal f increase f entrpy can be deduced by cmbining Eqs. (7.0) and (7.) t the frm δq ds T (7.3) Cnsider the system A that interacts with a heat reservir R and a mechanical energy reservir MR as shwn in Fig MR culd be a simple pulley-weight arrangement cnnected t A with a frictinless shaft. During an infinitesimal change in the state f A its entrpy increases by ds while the crrespnding change in entrpy f the reservir is ds r. The heat and wrk interactins between the system and its surrundings, cnstituted by R and MR, during the prcess are δ Q and δw respectively. Cnsider an imaginary bundary C, indicated by the brken-lines in Fig. 7.4, enclsing A, R and MR such that nthing utside f it affects the inside significantly. We call the regin within C an islated system fr which the heat interactin δ Q c 0. Applying Eq. (7.3) t the islated system C we have ( ds ) 0 (7.33) Being an extensive prperty, we can add the entrpy changes f the different sub-regins that cnstitute C t write Eq. (7.33) in the frm c ( ds ) ds + ds + ds 0 (7.34) c r mr Fig. 7.4 Principle f increase f entrpy

11 3 Engineering Thermdynamics Since the entrpy change, zer, Eq. (7.34) becmes ds mr f the mechanical energy reservir MR is ds + ds r 0 (7.35) We cnclude frm the abve equatin that if all the prcesses ccurring within an islated system are reversible the entrpy f the system remains cnstant, therwise the entrpy must increase due the irreversible prcesses. This statement, whse mathematical frm is Eq. (7.35), is cmmnly called the principle f increase f entrpy. Althugh the principle f increase f entrpy stipulates that the entrpy f an islated system can nly increase r remain cnstant, the entrpy f sme f the individual sub-regins that cnstitute the islated system may decrease. Nw the heat transfer t the thermal reservir R is reversible because its temperature is cnstant. Therefre by applying Eq. (7.3) with the equality sign we btain the entrpy change f R as ds δq / (7.36) r T r Substituting frm Eq. (7.36) in (Eq. 7.35) we have ds δ Q / T 0 (7.37) Equatin (7.37) implies that the entrpy f the system A must increase t cmpensate fr the entrpy decrease f the reservir R s that the verall entrpy f the islated system C increases r remains cnstant. Otherwise the prcess wuld vilate the secnd law. We shall generalize the increase f entrpy principle and make it quantitative by intrducing a variable called the entrpy prductin. Cnsider any system and its interacting surrundings that are enclsed in an imaginary bundary and therefre islated (Fig 7.4). The entrpy prductin is given by r δσ ds system + ds surrundin gs 0 (7.38) The equality sign in Eq. (7.30) applies when all the prcesses within the islated system are reversible and the entrpy prductin, δσ is therefre zer. On the ther hand psitive values f δσ indicate the ccurrence f irreversible prcesses within the islated system and its magnitude prvides an indirect measure f the impact f the

12 Entrpy 33 irreversibilities n the wrk utput. Nte that the entrpy prductin σ is nt a prperty f the system like the entrpy S. Therefre σ depends n the type f irreversible prcess experienced by the system. 7.. Strage, prductin and transfer f entrpy When the principle f increase f entrpy, in the frm f Eq. (7.38), is applied t real systems we need t distinguish between three different quantities. These are called stred entrpy, entrpy transfer and entrpy prductin. We shall illustrate the difference between the abve terms using a practical situatin. Cnsider as a system the gas cntained in the rigid vessel shwn in Fig. 7.5 where a rtating paddle wheel supplies wrk t the system. The vessel is in thermal cmmunicatin with a single heat reservir at temperature T. The system underges a prcess where the paddle wheel is rtated fr fixed duratin f time and then stpped. Denting the initial and final equilibrium states by and we apply the first law t the prcess - executed by the system t btain U U W Q (7.39) where ( U U) is the increase in internal energy, Q heat utput frm the system t the reservir and W is the wrk input t the system thrugh the paddle wheel during the prcess. The applicatin f Eq. (7.3) in the integrated frm t the system gives ( S S) δq T (7.40) Fig. 7.5 Entrpy changes in system and reservir

13 34 Engineering Thermdynamics The negative sign in Eq. (7.40) signifies a heat utput frm the system as indicated in Fig We cnvert Eq. (7.40) t an equality by intrducing the entrpy prductin term σ. This gives ( S S) δq σ (7.4) T In Eq. (7.4), T is the temperature f the lcatin n the bundary at which heat is transferred frm the system t the reservir. In rder t evaluate the integral in the abve equatin we need t knw hw the temperature T varies during the prcess. Because the irreversible wrk input by the paddle wheel, shwn in Fig 7.5, causes cnsiderable turbulence, it is nt pssible t determine the variatin f T in a straightfrward manner. Mrever, T may nt be spatially unifrm within the system. Fr the purpse f the present discussin we reslve this difficulty by lcating the system bundary in clse cntact with the reservir s that T is equal t T, the cnstant reservir temperature. It is clear that with this arrangement f the system bundary any irreversibility due t heat transfer between the system and the reservir is nw attributed t the system. Hence upn integratin, Eq. (7.4) becmes Q ( S S σ (7.4) ) In sme respects Eq. (7.39), which is an energy balance, is analgus t Eq. (7.4) which culd be thught f as an entrpy balance equatin. In bth equatins the left hand side represents an increase in a stred prperty f the system. The term Q is the heat transfer which is a bundary interactin. Therefre we interpret the term ( Q / T ) as the entrpy transfer ut f the system t the reservir due t heat transfer at the bundary. Althugh the terms W and σ in the tw equatins are nt directly related we are aware that the irreversible frictinal dissipatin f the wrk input W cntributes t the entrpy prductin in the system. It is imprtant t nte that σ, is either psitive r zer but the entrpy change r strage, S ) in Eq. (7.4) culd be psitive r ( S T

14 Entrpy 35 negative depending n the magnitude f the entrpy transfer, ( Q / T ) cmpared t the entrpy prductin, σ. The entrpy balance equatin fr the reservir is given by Q ( S r Sr) σ r + (7.43) T The heat transfer prcess in the reservir is reversible because f its unifrm temperature, and therefre the entrpy prductin in the reservir, σ r 0. Cnsequently, the increase in the stred entrpy f the reservir is entirely due t the entrpy transfer assciated with heat transfer frm the system acrss the bundary. The entrpy balance equatin fr the cmpsite system cnsisting f the gas and the reservir is btained by adding Eqs. (7.4) and (7.43). This gives ( r S S ) + ( S r S ) σ (7.44) We culd have written Eq. (7.44) directly by applying the principle f increase f entrpy t the cmpsite system which is an islated system. In summary, the entrpy balance equatin fr a clsed system may be expressed in the general frm: Entrpy Entrpy + Entrpy (7.45) strage prductin transfer 7.. Entrpy transfer in a heat engine Cnsider the cyclic heat engine perating between tw heat reservirs as shwn in Fig. 7.6(a) where the varius heat and wrk interactins and temperatures are indicated. The engine experiences external irreversibilities due heat transfer acrss finite temperature differences ( T ) h T h and ( Tc Tc ) between the cyclic device and the reservirs. In additin, there are internal mechanical and thermal irreversibilities causing entrpy prductin within the cyclic device. The T-S diagram fr the irreversible heat engine cycle is shwn in Fig. 7.6(b). The cmpressin and expansin prcesses f this cycle are irreversible adiabatic prcesses. Hwever, we assume that during the

15 36 Engineering Thermdynamics heat interactins with the reservirs the temperature f the wrking fluid f the cyclic device remains cnstant. Applying the equatin f entrpy balance, expressed by Eq. (7.45), t the cyclic device we btain Q h Qc S + cycle σ (7.46) Th Tc It shuld be nted that in the abve equatin the entrpy prductin due t all irreversibilities, bth external and internal, are nw included in σ because we use the reservir temperatures t evaluate the entrpy transfers due t heat transfer and nt the temperatures at the bundary f the cyclic device. Since the heat engine perates in a cycle its entrpy change r strage, S 0. Therefre Eq. (7.46) becmes cycle Qh Qc σ (7.47) Th Tc Applying the first law t the heat engine, its wrk utput is btained as Substituting fr W act Q Q (7.48) h c Q c frm Eq. (7.47) in Eq. (7.48) Qh W + act Qh Tc σ (7.49) Th Th Th Qh Cyclic Device Wact Tc Qc Tc Fig. 7.6(a) Heat engine cycle Fig. 7.6(b) T-S diagram

16 Entrpy 37 Nw the wrk utput f a reversible engine perating between the same heat reservirs and receiving the same heat input Q h frm the ht reservir is Subtracting Eq. (7.49) frm Eq. (7.50) we have T c W rev Qh (7.50) Th W rev Wact Tcσ (7.5) We cnclude frm Eq. (7.5) that the entrpy prductin σ is a measure f the ptential lss in wrk utput due t internal and external irreversibilities f the heat engine Entrpy transfer in steady heat cnductin Cnsider the steady heat cnductin in a laterally insulated bar in thermal cntact with tw heat reservirs as shwn in Fig. 7.7 where the heat interactins and temperatures are indicated. Fr steady heat cnductin the temperature distributin in the bar is linear as indicated in the figure. Applying the entrpy balance equatin t the bar as the system we have Qɺ h Qɺ c Sɺ ɺ + bar σ (7.5) Th Tc Fig. 7.7 Steady heat cnductin in a bar

17 38 Engineering Thermdynamics where the dt ver a quantity stands fr differentiatin with respect t time. Since heat cnductin is a steady prcess the heat flws and the entrpy changes are expressed as rates in Eq. (7.5). The entrpy strage in the bar is zer because the prperties f the bar remain cnstant under steady cnditins. Therefre S ɺ 0 (7.53) bar Applying the first law t the bar we have Qɺ Qɺ Qɺ (7.54) h where Q ɺ is the cnstant heat flw rate thugh the bar. Substituting frm Eqs. (7.53) and (7.54) in Eq. (7.5) we btain Qɺ Qɺ ɺ σ > 0 (7.55) Tc T h We culd have btained Eq. (7.55) directly by applying the principle f increase f entrpy [Eq. (7.38)] t the cmpsite system cnsisting f the tw reservirs and the bar which is an islated system. Then the terms ( Qɺ / T c ) and ( Qɺ / T h ) are the rate f increase f entrpy in the cld reservir and the rate f decrease f entrpy f the ht reservir respectively. The entrpy prductin given by Eq. (7.55) is due t the internal thermal irreversibility in the bar which results frm the heat transfer acrss a finite temperature difference. If the same heat transfer was carried ut reversibly by perating a Carnt heat engine between the tw reservirs, then the rate f wrk utput f the engine is c T c Wɺ rev Q ɺ (7.56) T h Manipulating Eqs. (7.55) and (7.56) we have W ɺ σɺ (7.57) rev T c The abve relatin is similar t Eq. (7.5) fr the wrk utput f an irreversible engine, except that the ptential wrk available is entirely lst in the case f steady heat cnductin.

18 Entrpy Limitatins Impsed n Wrk Output by the Secnd Law In this sectin we shall derive expressins fr the upper limits f the heat transfer and wrk dne in a given prcess which are imprtant cnsequences f the secnd law. Denting the initial and final states f the clsed system by and we apply the first law t btain Q + (7.58) U U W Applicatin f the secnd law t an infinitesimal change f state gives δ Q TdS (7.59) Integrating Eq. (7.59) frm the initial t the final state we have Q TdS (7.60) The heat transfer Q is a maximum fr a reversible prcess fr which the equality sign in Eq. (7.60) applies. In rder t evaluate the integral n the right hand side f the Eq. (7.60) we need t knw the prcess path. Fr a reversible prcess, the path is usually well-defined and the heat transfer is als the area f the T-S diagram f the prcess. Hwever, fr an irreversible prcess, fr which the inequality sign applies, Eq. (7.60) establishes nly the upper limit f the heat transfer. Furthermre, the prcess path required t carry ut the integratin n the right hand side f Eq. (7.60) is ften difficult t determine fr an irreversible prcess. Eliminating Q between Eqs. (7.58) and (7.60) we btain U U + W TdS (7.6) ( U Hence W TdS U ) (7.6) The equality sign in Eq. (7.6) applies fr reversible prcesses fr which the prcess path is well defined and the equatin establishes the maximum wrk utput f the prcess. Hwever, fr irreversible prcesses the evaluatin f the integral n the right hand side f

19 330 Engineering Thermdynamics Eq. (7.6) culd pse a challenge because the prcess-path is ften nt readily available Helmhltz and Gibbs free energy We shall nw derive expressins fr the maximum wrk utput f several special prcesses that are f cnsiderable practical significance. First cnsider the applicatin f Eq. (7.6) t an isthermal prcess. The integral in the equatin is easily evaluated because the temperature is cnstant. This gives W W ( U T S S ) ( U ) (7.63) [( U TS ) ( U )] TS Hence, max [( U TS ) ( U TS)] (7.64) W is It is seen frm Eq. (7.64) that the maximum wrk utput fr an isthermal prcess is the change f the functin, ( U TS) frm state t state. This state functin is called the Helmhltz free energy and usually dented by F. It is a prperty f the system because U, T and S are prperties. The maximum wrk utput f an isthermal prcess may be written in terms f the Helmhltz free energy as W is, max F ( F ) (7.65) The physical interpretatin f Eq. (7.65) is that the maximum wrk utput f the system is the decrease in the prperty F. Althugh the system pssesses internal energy f magnitude U, a fractin TS f this internal energy is nt free t be used fr the prductin f useful wrk. Therefre fr an isthermal prcess the fractin f the internal energy that is free fr wrk prductin is the Helmhltz free energy F. It is imprtant t nte that Eq. (7.64) fr the maximum wrk utput is applicable t situatins invlving different frms f internal energy and wrk mdes. Fr example, we culd envisage an applicatin like a fuel cell where electrical wrk is prduced isthermally at the expense f chemical internal energy f the system. We shall nw btain the maximum wrk utput f an isthermal prcess that is carried ut under cnstant pressure. In ther wrds,

20 Entrpy 33 during the isthermal prcess, the system expands quasi-statically against a cnstant external pressure P. Of the maximum wrk utput f the system a fractin is nw used t vercme the external frce due t the pressure and therefre nt available fr useful applicatin. Separating the wrk utput int tw parts we have W W + P( V ) (7.66) is, max useful,max V where V and V are the initial and final vlumes f the system. Substituting fr W frm Eq. (7.66) in (7.64) W useful W useful is, max, max + P( V V ) [( U TS ) ( U TS)], max [( U + PV TS ) ( U + PV TS)] W useful, max [( H TS ) ( H TS)] (7.67) Frm Eq. (7.67) it is clear that the maximum wrk dne by a system under simultaneus isthermal and cnstant pressure cnditins is the change f a prperty defined as, G H TS where H is the enthalpy. The prperty G is called the Gibbs free energy. Therefre Eq. (7.67) can be written in terms f the Gibbs free energy as W useful, max G [ G ] (7.68) The functin G gives the fractin f the enthalpy f the system that culd be harnessed t prduce useful nn-expansin wrk. The fractin TS is nce again nt free fr wrk prductin Availability There are numerus engineering applicatins where a system prducing wrk exchanges heat with a single reservir like the atmsphere. The arrangement is similar t that shwn in Fig In this sectin we shall derive an expressin fr the maximum wrk utput f such a system. Let the initial and final states f the clsed system and the reservir be dented by and respectively. Applying the first law t the system Q + (7.69) U U W

21 33 Engineering Thermdynamics Cnsider the islated cmpsite system cnsisting f the given system and the reservir. We apply the principle f increase f entrpy t btain Q ( S S ) 0 (7.70) T r where T r is the reservir temperature. In Eq. (7.70) the first term is the increase in entrpy f the system while the secnd term is the decrease in entrpy f the reservir due t heat flw frm the reservir t the system. Eliminating Q between Eqs. (7.70) and (7.69) we have W [( U Tr S ) ( U Tr S)] (7.7) The maximum wrk utput is therefre the change in the functin ( U Tr S) frm the initial t the final state. Nw cnsider a system interacting with a single reservir as befre and underging a cnstant pressure prcess ding wrk against the atmsphere in a quasi-static manner. The useful wrk prduced is given by W useful W P ( V ) (7.7) atm V where P atm is the cnstant pressure f the atmsphere. Eliminating W between Eqs. (7.7) and (7.7) we btain W [( U + P V T S ) ( U + P V T S)] (7.73) useful atm r It is seen frm Eq. (7.73) that the maximum wrk utput is the change f the functin, Φ U + PatmV Tr S frm the initial t the final state. If the atmsphere is als the heat reservir, as it is the case in many engineering systems, then Φ U + PatmV Tatm S. The latter functin is clearly a prperty f the system-atmsphere cmbinatin. We can write Eq. (7.73) in terms f Φ as W [ Φ ] (7.74) useful Φ The minimum value f Φ ccurs when the system establishes equilibrium with the atmsphere and its temperature and pressure are therefre T atm and P atm respectively. If this minimum value f the prperty Φ is Φ min then we define the new prperty called the availability as atm r

22 Entrpy 333 Α Φ Φ min (7.75) In terms f the availability, the useful wrk utput given by Eq. (7.74) becmes W [ Α ] (7.76) useful Α The physical meaning f Eq. (7.76) is that the decrease in the availability f a system-atmsphere cmbinatin is the useful wrk utput f a reversible prcess f the system. Hwever, when the system underges an irreversible prcess with same initial and final states f the system, the wrk dne is less than the change in availability. 7.4 Maximum Wrk, Irreversibility and Entrpy Prductin In this sectin we shall btain a general relatinship between the entrpy prductin and the irreversibility fr a clsed system interacting with a series f thermal reservirs and prducing wrk as depicted in Fig The atmsphere, shwn separately, is chsen as the standard reservir that underges changes f state t accmmdate variatins f the wrk utput W f the system. The system underges a prcess whse initial and final states are dented by and respectively. The temperatures and the heat interactins between the system and the reservirs during the prcess are indicated in the Fig Applying the first law t the system Q + Qi U U + W (7.77) where Q is the heat flw frm the standard reservir (atmsphere). The secnd term n the left hand side f Eq. (7.77) is the ttal heat flw frm all the reservirs except the standard reservir. Apply the principle f increase f entrpy t the islated cmpsite system cnsisting f all the thermal reservirs and the given system. Hence Q Qi S S ) σ (7.78) T T ( i

23 334 Engineering Thermdynamics where σ is the entrpy prductin in the cmpsite system due t all irreversibilities. The first term in Eq. (7.78) is the change in entrpy f the system while the secnd and third terms are the changes in the entrpy f the atmsphere and the heat reservirs respectively. Eliminating Q between Eqs. (7.77) and (7.78) we have T T + S S) Qi ( U U ) Tσ + W (7.79) Ti ( T Q i [( U TS ) ( U TS)] Tσ + W (7.80) Ti Fig. 7.8 Prcess in a system interacting with multiple reservirs Nw cnsider a reversible prcess that wuld prduce the same change f state frm t in the system and all the reservirs except the atmsphere-reservir. The wrk utput fr this prcess, hwever, wuld be different frm the riginal prcess. It is seen frm Eq. (7.77) that a change f the wrk utput requires the heat interactin between the system and the atmsphere-reservir Q t adjust t satisfy the latter equatin, which is the first law. Nw fr the reversible prcess the entrpy prductin, σ 0. Therefre Eq. (7.80) becmes

24 Entrpy 335 T Q i [( U TS ) ( U TS)] Wrev (7.8) Ti Equatin (7.8) gives the maximum wrk utput f the system fr the given change f state frm t. Frm Eqs. (7.80) and (7.8) it fllws that W rev W Tσ (7.8) The difference between the actual wrk utput f the system W and the reversible wrk utput W rev is called the lst wrk r the irreversibility I. Substituting in Eq. (7.8) we have I T σ (7.83) We btained equatins similar t Eq. (7.8) fr the irreversibility f cyclic heat engines [Eq. (7.5)] and steady heat cnductin [Eq. (7.57)] in earlier sectins f this chapter. Equatin (7.83) is a general relatinship between the irreversibility and entrpy prductin that is applicable t any prcess in a clsed system. 7.5 Entrpy, Irreversibility and Natural Prcesses The new prperty called entrpy arse as a cnsequence f the secnd law and it was defined by Eq. (7.0). This equatin enables us t relate entrpy t ther macrscpic prperties like pressure, vlume and temperature (see Sec. 7..4) f a system and thereby btain its numerical value which is useful fr engineering analysis f systems. Hwever, t appreciate the brader significance f entrpy we still need t seek a mre satisfying physical interpretatin. Frm a micrscpic view pint entrpy can be thught f as a measure f the randmness r disrganizatin f the cnstituents f a system. Fr instance, when a gas is cmpressed the final state cnfines the mlecules f the gas t a smaller vlume thereby bringing mre rganizatin t the mlecules. Hwever, the higher temperature that may result frm the cmpressin will make the mlecules mre agitated and therefre their state mre randm. This view pint is reflected in the expressin fr the entrpy f an ideal gas in Eq. (7.3). The first term n the right hand side gives the level f disrganizatin due t the

25 336 Engineering Thermdynamics dispersin f energy amng the mlecules, which may be called the thermal entrpy and the secnd term gives the cnfiguratinal entrpy which is a measure f the disrganizatin due t the dispersin f the mlecules in the space ccupied by the gas. During the cmpressin prcess f an ideal gas, if the increase in thermal entrpy is exactly balanced by the decrease in cnfiguratinal entrpy, then the prcesses will be isentrpic as seen frm Eq. (7.3). In ur discussin n natural prcesses in Chapter 6 we nticed that these prcesses mved in a preferred directin. Fr instance, heat always flws unaided frm ht t cld regins making the latter regin mre disrganized. Furthermre, gases expand frm regins f high pressure t evacuated spaces increasing the level f disrganizatin f the mlecules f the gas. T reverse these prcesses we need t expend wrk which leaves permanent changes in the envirnment thus making the prcesses irreversible. In view f the abve bservatins we can cnclude that natural prcesses take systems frm mre rganized states t less rganized states thus increasing their entrpy. Often we maintain highly rganized states f systems by intrducing well designed cnstraints. Fr example, we need high quality thermal insulatin t prevent heat leaks between a ht bdy and a cld bdy. Similarly we use walls r membranes t cnfine high pressure gases frm expanding t lw pressure regins surrunding them. Once the cnstraints are remved the states f these systems evlve naturally r unaided t less rganized states. Since entrpy is a measure f the level f disrganizatin we are lead t the cnclusin that natural prcesses cause the entrpy f systems t increase and they are, therefre, irreversible. Nw cnsider a pure wrk interactin with n frictinal effects. The sliding f a metal blck dwn a smth inclined plane is a pssible example. The state f rganizatin f the mlecules f the blck remains unchanged during the sliding because the kinetic energy gained by the blck will affect all the mlecules f the blck equally in an rganized fashin. Therefre there is n change in the entrpy f the blck due t the mtin. Hwever, if sliding frictin is present sme f the wrk dne is cnverted t a heat interactin at the sliding surface resulting in a rise in temperature f the blck which, in turn, increases the disrganizatin

26 Entrpy 337 f its mlecules. Therefre the blck is at higher entrpy in the final state. An imprtant gal f engineering design t is t develp efficient systems t prduce wrk frm stred energy surces. Thermal pwer plants and cmbustin engines are examples f such systems. A questin that fllws naturally is hw t relate ur knwledge n irreversible prcesses, level f rganizatin f systems, and entrpy t prcesses in energy cnversin systems. The answer with regard t frictin is straight frward because frictin cnverts sme fractin f the wrk available r prduced directly t heat which cannt be recnverted t wrk cmpletely even using reversible devices. Nw cnsider the presence f cmpnents in an energy cnversin system where heat is transferred acrss finite temperature differences. Ideally we culd affect the same heat transfer using a reversible heat engine, thus prducing sme additinal wrk as was seen in Sec In ther wrds, heat transfer acrss a finite temperature difference is a lst pprtunity fr the prductin f wrk. Larger the temperature difference greater wuld be the ptential lss f wrk. Therefre by allwing the natural prcess, namely the flw f heat frm a higher t a lwer temperature in this case, t ccur we lst sme f the ptential wrk. The free expansin f a high-pressure fluid stream thrugh a cmpnent like a thrttle valve is a similar situatin. If the expansin is carried ut in a fully-resisted manner, perhaps with the aid f an expander, the wrk ptential in the high pressure stream culd have been realized. Mrever, in bth instances mentined abve the natural prcess, if allwed t ccur, increases the entrpy prductin f the system. In summary, differences f intensive prperties like the temperature and pressure, between systems are usually maintained using apprpriate cnstraints. These prperty gradients ffer us pprtunities t prduce useful wrk. If the cnstraints are simply remved, natural prcesses will level-ff these prperty gradients resulting in entrpy prductin. Hwever, if apprpriate devices are intrduced t equalize the prperty differences in a cntrlled manner sme f the wrk-ptential culd be realized. Equatins (7.8) and (7.83) are useful relatinships that enable

27 338 Engineering Thermdynamics us t evaluate the ptential lss f wrk by cmputing the entrpy prductin. 7.6 Wrked Examples Example 7. (i) An ideal gas underges a plytrpic prcess frm state t state accrding t the prcess law PV n C. Obtain an expressin fr the change in entrpy f the gas in terms f the initial and final pressures. (ii) A unit mass f methane is subjected t a quasi-static cmpressin. prcess which fllws the relatin PV C. The initial temperature and pressure are 8 C and 88 kpa respectively. The final pressure is 400 kpa. Fr methane is R 0.5 kjkg - K - and the rati f the specific capacities, γ. 3. Calculate the change in entrpy f the methane. Slutin We culd derive the required expressin fr the entrpy change fr a plytrpic prcess frm first principles as was dne in Sec Hwever, it is mre cnvenient t use the expressin already btained in Sec fr an ideal gas because the change in entrpy depends nly n the initial and final states f the ideal gas and nt n the prcess path. The general expressin fr the change in entrpy f an ideal gas is given by Eq. (7.5) as p ( T / T ) mr ln( P P ) S S + mc ln / (E7..) Manipulating the ideal gas equatin f state, plytrpic prcess relatin, PV n C we btain where C is a cnstant. Hence P ( n ) / n / T C ( n ) / n PV mrt and the (E7..) ( n ) / n P / T P / T (E7..3) where the sub-script dentes quantities at the initial state. Substituting fr T / T frm Eq. (E7..3) in Eq. (E7..) p ( P / P ) mr ln( P P ) S S + mc ( / n)ln / (E7..4)

28 Entrpy 339 Using the ideal gas relatins, ( c c ) R and c / c ) γ, p v ( p v Equatin (E7..4) can be expressed in the cmpact frm ( γ n) S S mr ln / n( γ ) ( P ) P (E7..5) Nte that if we substitute, n γ in Eq. (E7..5) the entrpy change becmes zer because the prcess then is isentrpic. (ii) The numerical data pertinent t the prblem are m kg, R 0. 5 kjkg - K -, γ. 3 and n.. Substituting in Eq. (E7..5) (.3.) S S 0.5 ln( 400 /88) 0. 9 kjk -.(.3 ) The entrpy f the gas has decreased because heat has been transferred ut f the gas during the prcess. Example 7. A fixed quantity f water f mass.3 kg at an initial temperature and pressure f 00 C and 30 bar respectively is cntained in a pistn-cylinder apparatus. The water underges a reversible isthermal expansin t a lwer pressure while receiving 3500 kj f heat. Calculate (i) the change in entrpy f the water and (ii) the final pressure. Slutin At the initial state, the water is a cmpressed liquid because the pressure is higher than the saturatin pressure at 00 C. The path f the isthermal heating prcess is indicated by the hrizntal line - in Fig. E7.. The area under the line - gives the heat supplied during the prcess because fr a reversible prcess Q TdS (E.7..) Since T is cnstant, Eq. (E7..) can be integrated directly t btain Q Tm( s ) (E.7..) s where s is the entrpy per unit mass.

29 340 Engineering Thermdynamics Fr cmpressed water we ignre the effect f pressure and find the saturated liquid entrpy at 00 C. Frm the tabulated data in [6] we btain by interplatin the liquid entrpy as, s. 33 kjk - kg -. Substituting the given numerical data in Eq. (E7..) we have 3500 ( ).3 ( s s) (E7.3.) Frm Eq. (7.3.) the entrpy change f water is given by ( s s) 5.69 kjk - kg - Fig. E7. T-S diagram fr water Hence s 8. 0 kjk - kg - In rder t determine the final state f the steam we need t btain the pressure f superheated steam at 00 C fr which s 8. 0 kjk - kg -. Frm the superheated steam data in [6] the pressure is btained by linear interplatin as 0.68 bar. Example 7.3 (a) Draw the temperature-entrpy diagram fr a Carnt refrigeratin cycle. Hence btain an expressin fr the COP f the cycle. (b) Obtain an expressin fr thermal efficiency f the reversible heat engine cycle whse T-S diagram is shwn in Fig. E7.3(b). Slutin (a) The temperature entrpy diagram fr a Carnt refrigeratin cycle is shwn in Fig. E7.3(a). Heat is extracted frm the cld reservir during the isthermal prcess -. The prcess -3 is an isentrpic cmpressin during which the temperature f the cycle becmes equal t the ht reservir temperature. Heat is rejected isthermally t the ht reservir during the prcess 3-4. Finally, the isentrpic expansin

30 Entrpy 34 prcess 4- cmpletes the cycle. Since all the prcesses are reversible, the area under the prcess path n the T-S diagram is the heat transfer. Using the quantities indicated in the figure we write the fllwing expressins fr the heat interactins Q Q Applying the first law t the cycle c h W T ( S ) S (E7.3.) c T S 3 S ) (E7.3.) in h ( 4 The COP f the refrigeratr is defined as COP Q Q (E7.3.3) Q h c Q c c ref (E7.3.4) Win Qh Qc Because f the rectangular shape f the T-S diagram ( S 3 S 4 S ) ( S ) (E7.3.5) Substituting frm Eqs. (E7.3.) and (E7.3.) in Eq. (E7.3.4) with the cnditin in Eq. (E7.3.5) we btain COP ref Tc T T h c Th T 4 3 Tc S S S Fig. E7.3(a) T-S diagram Fig. E7.3(b) T-S diagram

31 34 Engineering Thermdynamics (b) The T-S diagram fr the heat engine cycle, shwn in Fig. E7.3(b), is a triangle. The engine receives heat during the prcess -, underges an isentrpic expansin frm -3 and rejects heat during the prcess 3- t cmplete the cycle. The net heat transfer is the area enclsed by the triangle which by the first law is als equal t the net wrk utput. Therefre W net Q T T )( S ) / (E7.3.6) net ( S The ttal heat input t the cycle is the area f the trapezium --b-a, which can be written as Q in The thermal efficiency f the cycle is T + T )( S ) / (E7.3.7) ( S Wnet ( T T ) T η (E7.3.8) Q ( T + T ) ( T + T ) / in Equatin (E7.3.8) shws that the efficiency f the engine is equal t that f a Carnt engine perating with a heat surce temperature equal t the mean f the given cycle temperatures. Example 7.4 (a) Shw that fr an ideal gas the slpe f the cnstant vlume lines n the T-S diagram are larger than the slpe f the cnstant pressure lines. (b) Sketch the lines f cnstant pressure and cnstant vlume fr an ideal gas n the T-S diagram. (c) Draw the T-S diagram fr a reversible cycle executed by an ideal gas whse P-V diagram is a rectangle. Slutin (a) In Sec we btained the fllwing expressins fr the entrpy f an ideal gas v ( T / T ) mr ln( V V ) S S + mc ln + / (E7.4.) p ( T / T ) mr ln( P P ) S S mc ln / (E7.4.) The abve expressins are used t find the required gradients ( T S ) / V and ( T / S ) P f the cnstant vlume and cnstant pressure lines respectively n the T-S diagram.

32 Entrpy 343 Differentiate Eq. (E7.4.) with respect t T keeping V cnstant. Hence ( S / T ) mc T (E7.4.3) V v / Differentiate Eq. (E7.4.) with respect t T keeping P cnstant. Hence ( S / T ) mc T (E7.4.4) P p / Frm Eqs. (E7.4.3) and (E7.4.4) we have Nw fr an ideal gas, tw equatins abve that ( T / S ) T / mcv V ( T / S) P T / mc p c c + R > c. Therefre it fllws frm the p v ( T / S ) V > ( T / S) P (b) In rder t sketch the cnstant vlume and cnstant pressure lines we first btain functinal frms fr the variatin f S versus T under these cnditins. Rearranging Eq. (E7.4.) ( T / T ) exp( ( S S ) / mc R ln( V / V ) / c ) R / cv Hence ( T / T ) ( V / V ) exp [( S S ) / mc ] Similarly, by rearranging Eq. (E7.4.) v v (E7.4.5) R / c p ( T / T ) ( P / P ) exp [( S S ) / mc ] (E7.4.6) v p v Fig. E7.4(a) T-S diagrams f cnstant-p and cnstant-v prcesses

33 344 Engineering Thermdynamics The expnential curves given by Eqs. (E7.4.5) and (E7.4.6) are sketched in Fig. E7.4(a) where the pint O represents the reference state. Ntice that the T-V and T-P relatins fr an isentrpic prcess ( S S ) culd be deduced frm these equatins. (c) The P-V diagram shwn in Fig. 7.4(b) has a rectangular shape with tw cnstant-vlume prcesses - and 3-4 and tw cnstant-pressure prcesses -3 and 4-. We use the shapes btained in part (b) fr such prcesses t sketch the cycle n the T-S diagram shwn in Fig. 7.4(c). The cnstant pressure and cnstant vlume prcesses n the T-S diagram are represented by expnential functins accrding t Eqs. (E7.4.5) and (E7.4.6). Fig. E7.4(b) P-V diagram Fig. E7.4(c) T-S diagram Example 7.5 A fixed quantity f helium gas f mass 0.6 kg underges a reversible prcess that has a linear path n the T-S diagram. The temperature and vlume at the initial state are 38 C and 0.3 m 3 respectively. The crrespnding values are 60 C and 0.73 m 3 at the final state. Calculate the heat transfer during the prcess. Fr helium assume that c 3. kjkg - K - and R. 08 kjkg - K -. v Slutin Let the initial and final states f the prcess be dented by and respectively (see Fig. E7.5). Since helium is treated as an ideal gas the change in entrpy is btained frm the general ideal gas prperty relatin derived in Sec Hence S v + ( T / T ) mr ln( V V ) S mc ln /

34 Entrpy 345 Fig. E7.5 T-S diagram Substituting numerical values in the abve equatin S S ln ln S S.78 kjk -.78 Since the prcess is reversible, the heat transfer is equal t the area f the T-S diagram under the straight line -, which has the trapezidal shape as seen in Fig. E7.5. Therefre the heat transfer is Q 0.5 ( T + T )( S ) S Q 0.5 ( ) kj. Example 7.6 Tw separate quantities f water f masses m a and m b f the same cnstant specific heat capacity c are at temperatures T a and T b respectively. The tw masses f water are mixed in a well-insulated vessel f heat capacity c v and mass m v and left t attain equilibrium. (a) Obtain expressins fr (i) the entrpy changes f the water and the vessel after the final equilibrium state is reached and (ii) the entrpy prductin in the cmpsite system. (b) If the water is first mixed in a well-insulated vessel f negligible thermal capacity and then transferred t the riginal vessel, btain expressins fr the quantities listed in (a) abve.

35 346 Engineering Thermdynamics Slutin (a) Applying the first law t the mixing prcess, cnsidering the water and the vessel as the system we have Q + (E7.6.) U U W Fr the cmpsite system Q and W are bth zer. Therefre frm Eq. (E7.6.) where m c T + m c T + m c T ( m c + m c + m c ) T a a a b b b v v v T f is the final equilibrium temperature f the system. macata + mbcbtb + mvcvtv T f (E7.6.) m c + m c + m c a a The water and the vessel may be treated as pure thermal systems because f their lw cmpressibility. Since entrpy is a prperty we use the general expressin, Eq. (7.9), derived earlier in Sec fr a pure thermal system, t btain the entrpy changes as water a a b b a a ( T f / Ta ) mbcb ln( T f Tb ) m c ln( T T ) S m c ln + / (E7.6.3) vessel v v f v S / (E7.6.4) The water and the vessel cnstitute an islated system that des nt exchange heat with the envirnment. The entrpy prductin f the cmpsite system is therefre equal t the net change in entrpy f the water and the vessel. water S vessel v v σ S + (E7.6.5) Substituting frm Eqs. (E7.6.3) and (E7.6.4) in Eq. (E7.6.5) σ a a ( T / T ) + m c ln( T / T ) m c ln( T T ) m c ln + / f a b b (b) Applying the first law t the tw-step mixing prcess it is clear that the final equilibrium temperature f the water and the vessel is same as that given by Eq. (E7.6.). Since entrpy is a prperty and therefre independent f the prcess path, the entrpy changes and the entrpy prductin fr the tw-step mixing prcess are the same as thse fr part (a) abve. f b b v c v v f v f v