Dissipated energy and Entropy Production for an Unconventional Heat Engine: The Stepwise Circular Cycle

Transcription

1 sspated energy and Entropy roducton for an nconventonal Heat Engne: he Stepwse rcular ycle Francesco d Lberto, Raffaele astore and Fulvo erugg partmento Scenze Fsche - nverstà d apol Federco II - SI-R, ISM and IF, Sezon d apol partmento Scenze Fsche - nverstà d apol Federco II dlberto@na.nfn.t bstract hen some entropy s transferred, by means of a reversble engne, from a hot heat source to a colder one, we have the maxmum of effcency,.e. we obtan the maxmum avalable work. Smlarly the reversble heat pumps transfer entropy from a cold heat source to a hotter one wth the mnmum expense of energy. On the contrary f we are faced wth non reversble devces there s some Lost ork for heat engnes, and some Extra ork for heat pumps. hese quanttes are both related to the Entropy producton. he Lost ork,.e., s also called degraded energy or Energy unavalable to do work. he Extra ork,.e. Extra Lost Irrev Irrev, s the excess of work performed on the system n the rreversble process wth respect to the reversble one (or the excess of heat gven to the hotter source n the rreversble process). In ths paper, whch follows two prevous ones on the Lost ork [hl. Mag. 87, 569 (7), hl. Mag (8)] both quanttes are analyzed n deep and are evaluated for a process wth complexty,.e. the stepwse rcular ycle whch s smlar to the stepwse arnot cycle [hysca 34, 33 ()]. he stepwse rcular ycle s a cycle performed by means of small weghts dw whch are frst added and then removed from the pston of the vessel contanng the gas or vceversa. he work performed by the gas can be found as ncrease of the potental energy of the dw s. e dentfy each sngle dw and thus evaluate ts rsng.e. ts ncrease n potental energy. In such a way we fnd how the energy put of the cycle s dstrbuted among the dw s. he sze of the dw s affects the Entropy producton and therefore the Lost and Extra work. he rsng dstrbuton depends on the removng process we choose. - Introducton s ponted n a prevous paper [], entropy producton and ts relaton to the avalable energy are fascnatng subjects whch n last years have attracted many physcs researches [5-]. It s well known [-] that for some elementary rreversble process, lke the rreversble sothermal expanson of a gas n contact wth a heat source at temperature, the work done by the gas Irrev s related to the reversble sothermal work (.e. the work performed by the gas n the correspondng reversble process) by the relaton S () where s the total entropy change of the unverse (system + envronment). he degraded energy S S s usually called the Lost work Lost () Lost he latter can be nterpreted as the mssng work:.e. the addtonal work that could have been done n the related reversble process (here the reversble sothermal expanson); t s also called energy unavalable to do work. On another hand n the rreversble sothermal compresson S s called Extra.e. the excess of work performed on the system n the rreversble process wth respect to the reversble one. (3) Extra n where now Irrev n. ue to the energy balance, the same relaton holds for the amounts of heat gven to the source,.e. we have S (4)

2 herefore S s also called the Excess of heat ( Extra gven to the source [8,9]. he total varaton of Entropy, S. he second Law clams that ),.e. the addtonal heat that has been, s usually called Entropy producton ; we shall call the latter and the entropy s an ensve quantty whch n the transfers between systems can only ncrease or stay unchanged.. - Entropy producton and Lost ork and Extra ork n sothermal rreversble processes. Let us frst consder the sothermal rreversble expanson ( ) of an deal gas n contact wth a heat source at temperature where and wth,. In such a smple process some heat n goes from the heat source to the deal gas. here s an ncrease of entropy of the deal gas, Sgas and a decrease of the entropy of the heat source n, where d R ln R ln R therefore the entropy producton s n S (5) Snce we fnd n the deal gas an amount of entropy greater than that taken from the heat n 3 source. If, for example, 3 we have R ln 4 R. 636R. On the other hand for the 4 sothermal rreversble compresson of the deal gas ( ) some heat goes from the gas to the source at temperature. e have a decrease of the gas entropy, where Sgas and an ncrease of the source entropy, n. herefore the entropy producton n the compresson s S (snce ), (6) whch, for 3 gves R ln 3R R ln 4. 64R. Observe that n the compresson the entropy producton s much bgger than n the expanson; here we wll show how ths s related to the wastng of energy. In order to fnd how the prevous entropy productons affects the dsspaton of energy, we have to remark that the rreversblty of a generc process ( ) s due, n general, to nternal and ernal rreversblty, therefore, as shown n [,,5] the related entropy producton can be expressed as a sum of two terms: the nternal entropy producton, nt and the ernal entropy producton,.e. nt (7) he quanttes n,, are postve.

3 hs result s not trval snce S ; there are n fact many processes for whch S and nt sys nt. he system entropy producton nt s defned [,,5] by the relaton S S S (8) sys n where S n and S are respectvely the quantty of entropy whch respectvely comes nto and comes of the system n the rreversble process; S sys nt sys sys 3 s the entropy varaton of the system from to and does not depend on the partcular process. Smlarly the ernal Entropy producton, s gven by the relaton S S S (9) n or by relaton (7). It s easy to verfy that for both prevous rreversble sothermal processes and therefore that for both the expanson and the compresson nt. In the ppendx we gve the relatons for the Lost ork and Extra ork for sothermal processes wth nternal and ernal rreversblty ( ). From relatons (4) and (6) t follows that the Lost ork for an sothermal expanson at temperature and wth ernal rreversblty ( ) s () Lost and that the Extra ork for an sothermal compresson at (wth ) s Extra n nt () herefore we understand that n the rreversble compresson much more energy s wasted than n the rreversble expanson. nt. - Entropy producton, Lost ork and Extra ork n sobarc rreversble processes Let us consder the rreversble sobarc expanson at pressure (heatng) of one mole of monatomc deal gas from the state to the state for whch, for example,. he deal gas, ntally at temperature, s brought n thermal contact wth the source, then an Fgure - rreversble sobarc expanson at pressure takes place and the deal gas reaches the fnal state. Let be the ntal volume and the fnal volume. In the expanson the gas has performed the work (= E ) ( ) R( ) () rrev

4 and has racted from the source we understand that there has been an ncrease of nternal energy E E ( ) the heat (= E ) ( ). From the energy balance n where and are the molar specfc heats respectvely at constant pressure and at constant volume. For ths process rrev Ssys S ln (3) s n the prevous case we want to fnd nt and,.e. the Entropy producton due to the nternal rreversblty and the Entropy producton due to the ernal rreversblty. he path we follow s to analyse the related ernally reversble process (Eso-reversble process); for ths we evaluate the Entropy producton, whch s therefore due only to the nternal rreversblty. hs wll be. From ths we can have (the Entropy producton due to the ernal rreversblty) by subtractng nt from,.e. n nt rrev p (4) o perform the Eso-reversble process we need a sequence of heat sources rangng from to, from whch the gas takes, at each nfntesmal step, the heat to perform the rreversble sobarc Eso expanson, and an auxlary reversble heat engne whch takes the heat from the source at temperature and gves the heat d to the source at temperature of the sequence. Such an engne performs the work Eso at each step. Obvously Eso Eso d ln. In ths Externally reversble process (Eso-reversble) the Entropy producton due to the Internal rreversblty at each step s due to the nfntesmal varaton of Entropy of the gas (.e. ds Eso syst d nt ) and to the nfntesmal varaton of Entropy of the heat source of the sequence whch s actve n the step (.e. Eso Eso d d nt dssyst ds, whch means that there s no nternal Entropy producton n ths Eso- e fnd therefore that nt reversble process; therefore ds Eso d ), hence rrev (5) ln Remark that the global Entropy change s related the local Entropy productons by means of the followng relaton S S S sys For the rreversble sobarc expanson at pressure (heatng) of one mole of monatomc deal gas 5 from the state to the state for whch, for example, and R the rreversble work done by the gas and the heat taken from the source are rrev ( ) R( ) R and ( ) ; therefore rrev nt 4 nt

5 . (6) rrev ln ln. 93 o fnd the Lost ork we need the avalable total ersble ork. he total ersble work s the reversble work made by the gas (whch s dentcal to the rreversble work) and the work made by the auxlary reversble engne workng between and the varable temperature of the sequence of sources whch we use to perform the reversble sobarc expanson. otal gas engne Eso ( ) where Eso d herefore otal Lost rrev ln ( ) On the other hand by relaton (4) Lost ln ( ) (7).e Lost For the rreversble sobarc compresson at pressure (coolng) of one mole of monatomc deal 5 gas from the state to the state for whch as before,, the rreversble work s rrev ( ) R( ). Followng the same steps as for the expanson we fnd rrev ln and Extra u rrev ln (8).e. for ( ln ). 37 and Extra u. 37 Observe that here, as opposed to the prevous sothermal process, we have Lost Extra In the n secton by means of relatons (,) and (7,8) we study the Lost ork and the Extra ork for the Stepwse rcular ycle. a 3 - he step-wse deal gas rcular cycle and dsspated energy In order to perform an deal gas stepwse cycle we need a lot of heat sources () and heat snks (), a vessel wth a free pston and a large number () of small drvng weghts to ncrease or decrease slowly, step by step, the ernal pressure. If the steps are nfntesmally small the cycle s reversble. In order to evaluate the work performed by the deal gas durng the cycle, the dsplacements of the small drvng weghts (dw) must be done carefully. e let them move on and off the pston only horzontally. o ths end we assume that the handle of the pston s endowed wth

6 so many shelves that we can move each dw horzontally (and wth frcton) from (or to) the correspondng fxed shelf whch belongs to the dw s Reservor. (he dw s Reservor s a vertcal sequence of horzontal shelves on whch the dw s are ntally located). Such an deal devce s shown schematcally n Fgure. 6 Fgure a) he adabatc vessel wth some dw s on the pston. b) ross secton vew of the vessel showng two supports for the dw s (the dw s Reservor). he rcular cycle can be performed through Z= steps. In each of the frst steps one dw s added on the pston (and removed from the Reservor at ts ntal heght h ); n each of the followng steps one dw s removed from the pston (and brought back to the Reservor at ts fnal heght, say h,f ). he k-th dw s the dw whch has been added on the pston at the k-th step n the compresson. hen the dw s added on the pston of the vessel n thermal contact wth the snk,the gas performs an sothermal compresson; when t s removed the process s the sothermal expanson. Each sothermal process s followed by an sobarc process: ths s a compresson (volume reducton) n the frst / steps and an expanson n the followng / steps. he reverse happens n the removng steps. herefore at the end of the cycle the overall rasng, on the dw s Reservor, of the k-th dw from ts ntal heght (h k ) to the fnal one h ) s h k kf k ( kf h h (6) Snce a frcton-less process s assumed, the vertcal moton of the dw s s only due to the gas and the total work () performed by the deal gas can be found as ncrease of potental energy of the dw s on the Reservor,.e. mg where s the ernal pressure at step (after the addton or removal of the -th dw), s the volume varaton from step ( ) to step, and mg s the weght of the generc dw. Relaton (7) has been proved elsewhere [3]. In the n secton the rasngs of the sngle dw on the reservor are evaluated. k h k (7) 3. - he rasngs of the dw s for a step-wse rcular cycle he cycle we consder s reported n Fgure 3. he chosen values of and are easly avalable n ordnary condtons. In the frst steps the dw s are added on the pston to perform frst the process ( ). In the remanng steps the dw s are removed from the pston n order to return to the ntal state ( ). he workng flud s the deal gas and we assume the free pston has no mass.

7 he vertcal vessel s walls are heat nsulatng and the vessel s dathermal floor s made adabatc when needed. he chosen crcular cycle s descrbed n the plane by the relaton (8) r r where = 5 atm, = 7.4 l, r = 5 l and r = 5 atm. Snce mn s at mn 3/ 4 and Max s at Max 7 / 4, t follows that the steps from to /4 are coolng steps. hese are followed by heatng steps and 3 / 4 coolng steps. 7 Fgure 3 - he step-wse rcular cycle wth very small steps and (θ) the temperature along the clock-wse cycle. Let us call atm and atm, respectvely, the values of the pressure at bottom and at the top of the cycle. e have consdered here = 33 dw s and therefore = 66 steps. he mass of each dw s m =. Kg. he surface of the pston s S = cm, so that at each step n the compresson the pressure ncrease s / 33,.e., and (9) for nd for each step n the expanson the pressure decreases by.e. l ( l) l for l, () otce moreover that and,.e. the volume at step s the ntal volume and the,,, by means of volume at step s the volume at the top of the cycle. Of course for each relaton (8), there are ( ) exp and ( ) comp whch are the volume at the pressure taken respectvely n the expanson ( ) and n the compresson ( ), and also two temperatures ( ) exp and ( ) comp. ll that can also be wrtten n the followng way: for each,, there s a volume ( ) and a heat (source or snk) at temperature ( ). Keepng n mnd how we perform the rcular cycle, let us take a closer look at the last dw: when ths small weght leaves the Reservor and s added on the pston, t (together wth the pston and the prevous dw s) moves downward, performng the sothermal compresson step ( ) at temperature ; then the gas s heated by the heat source at pressure performng the expanson ( ); afterwards, at step +, the small weght s removed and goes to rest on the fxed shelf of the Reservor n front of t. It wll stay on the pston for one step only!.e. h ( ) S. p For ths last step let us call,, ; from whch p we have. In ths last step the snk at temperature takes the n entropy, and the source at temperature gves the entropy to the system, where and n p.

8 Snce R, t follows that R n. Smlarly for the last but one dw: h ( ) ( ) () S S herefore for the k-th dw h k k k, S and h. S For a very large number we can wrte h( ) ( ) exp ( ) comp S ere () exp and () comp are the volume at the pressure n the expanson and n the compresson, and h() s the rasng of the dw whch, added on the pston, gves rse to the pressure. From ths: r r r and ) comp r r / r ( ) exp / For r r one has ellptc cycles. ( For a reversble rcular ycle that starts from =7.4 l and = = atm, the rasngs h() are easly obtaned from relatons (8) and (): h( ) r r / r S, () whose values are shown n Fgure 4. 8 and Fgure 4 - Overall rasng on the reservor of each dw Lost work and ra work step by step and the total dsspated energy One may observe that n the cycle there has been an Entropy producton: n fact when the last dw s added on the pston (the -th step) we had an sothermal compresson at temperature and an

9 9 sobarc expanson from to. y means of relaton (6) and (9), for the sothermal compresson (, ) we have nt ln R R. For the expanson and sobarc heatng we have n Ext ln. Observe that the same result holds for sobarc coolng. herefore n the -th step we have nt ra and Ext Lost. Fnally we can conclude that the sspated energy.e. s Ext nt ow we gve some upper bound to ) (. Let / n be the frst / addng steps for whch,, mn / n be the second / addng steps for whch,, 3 / n be the thrd / removng steps for whch,, 4 / n be the fourth / removng steps for whch,, mn mn Ext nt mn nd, snce, Ext nt ) ( 4 Snce 5 4, 3, choosng 5 R 5, we have Ext nt, 3 Ext nt. From ths upper bound we see that, for, ) (. 4 - Summary

10 In ths paper we have ntroduced the Extra ork whch, together wth the Lost ork, gves the sspated energy n the rreversble processes. he analyss s very accurate for rreversble sothermal process and for sobarc processes. he new and prevous results are used to evaluate the sspated energy for a stepwse deal gas rcular ycle, a system wth complexty. cknowledgements e thankfully acknowledge G. Monroy,. Ruggero and M. Zannett for helpful dscussons. hs paper s dedcated to the memory of Gulano Martnell for hs poneerng works on energy preservaton. ppendx - Lost ork and Extra ork for sothermal processes wth ernal rreversblty Here we evaluate the Lost ork for the expanson and the Extra work for the compresson when there s ernal rreversblty. In Sec.. and n Sec. 3 of paper [] we have shown that, f the rreversble sothermal expanson s performed by means of a (shorter) contact wth a heat source at, we have,.e. n n () and for the Endo reversble process,.e. the process n whch the gas performs the reversble sothermal expanson, Endo () Smlarly, f the rreversble sothermal compresson s performed by means of a (shorter) contact wth an heat source at, we have,.e. Endo and. (3) o evaluate the Lost ork for the expanson wth we calculate the work avalable n the related ersble process and subtract from t, the effectve work done n the rreversble process. hs dfference gves the Lost ork. he ersble ork s the ersble work of the gas plus the work of an auxlary reversble gas engne workng between and. For the gas. he auxlary reversble engne, whch brngs the heat to the system (the deal gas at temperature ) and takes from the heat source engne at temperature the heat, performs the work. herefore the total reversble work s otal gas engne he ork performed by the gas n the rreversble expanson s, therefore On the other hand for the compresson wth a heat source at otal Endo Lost n n (4) Extra n (5) but f one uses a heat source at, we have to subtract the work of the reversble engne from the ersble work necessary to perform the sothermal compresson at temperature, whch nt n

11 Mn subtracts from the heat source at temperature (the gas) and gves the heat to the source at temperature,.e. therefore the Extra work s Extra n mn ; nt. (6) References [] F. d Lberto, hl. Mag. 87, 569 (7), hl. Mag. 88, (8)] [] F. d Lberto, Gornale d Fsca 49, -4 (8) [3] F. d Lberto, hysca 34, () [4] F. d Lberto, [5] G. Job and R. Ruffler hyscal hemstry Job Foundaton (Hamburg 7) [6].. onevey, ature 333, 49 (988) [7] H.S. Leff, m. J. hys (978) [8].. Marcella, m. J. hys (99) [9] R.E. Reynolds, m. J. hys. 6 9 (994) [] H.. Fuchs, he dynamcs of heat (Sprnger, ew York 996) [] L.G. hen,. u and F.R. Sun,., J. on- Equl. hermodyn (999) and references theren.

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