General Properties of Approaches Maximizing Power Yield in Thermo-Chemical Systems

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1 Europan Assocaon or h Dvlopmn o Rnwabl Enrgs, Envronmn and Powr Qualy (EA4EPQ Inrnaonal Conrnc on Rnwabl Enrgs and Powr Qualy (ICREPQ Sanago d Composla (Span, 8h o 30h March, 0 Gnral Proprs o Approachs Maxmzng Powr Yld n hrmo-chmcal Sysms Sanslaw Snuycz Faculy o Chmcal and Procss Engnrng a Warsaw U, Waryńskgo Sr No, Warszawa, Poland (phon: ; ax: ; -mal: snuycz@ chp.pw.du.pl Absrac hs rsarch rprsns a hrmodynamc approach o modlng and powr opmzaon o nrgy convrrs, such lk hrmal, solar, chmcal and lcrochmcal ngns. hrmodynamcs lads o convrr s cncy and lmng gnrad powr. Ecncy quaons srv o solv problms o upgradng and downgradng o rsourcs. Ral work yld s a cumulav c oband n a sysm o a rsourc lud, ngns, and an nn bah. Whl opmzaon o sady sysms rqurs usng o drnal calculus and Lagrang mulplrs, dynamc opmzaon nds varaonal calculus and dynamc programmng. h prmary rsul o h sac opmzaon s h lmng powr, whras ha o dynamc opmzaon s a n-ra counrpar o h classcal ponal o rvrsbl work (xrgy. hs ponal dpnds on hrmal coordnas and a dsspaon ndx, h,.. h Hamlonan o h rlad problm o mnmum nropy producon. h gnralzd ponal mpls srongr bounds on work dlvrd or suppld han h rvrsbl ponal. In racng sysms h chmcal any consus a prvalng counrpar o h hrmal cncy. hror, n racng mxurs lux balancs ar appld o drv powr yld n rms o an acv par o chmcal any. Kywords: hrmal cncy, chmcal cncy, nropy producon, ngns. I. INRODUCION Applcaons o hrmodynamcs o n ras lad o soluons whch dscrb varous orms o bounds on powr and nrgy producon (consumpon ncludng n dynamcal cass n-ra gnralzaons o h sandard avalabls. In hs rsarch w ra powr lms n sac and dynamcal nrgy sysms drvn by nonlnar luds ha ar rsrcd n hr amoun or magnud o low, and, as such, play rol o rsourcs. A powr lm s an uppr (lowr bound on powr producd (consumd n h sysm. A rsourc s a valuabl subsanc or nrgy usd n a procss; s valu can b quand by spcyng s xrgy, a maxmum work ha can b oband whn h rsourc rlaxs o h qulbrum. Rvrsbl rlaxaon o h rsourc s assocad wh h classcal xrgy. Whn dsspav phnomna prval gnralzd xrgs ar ssnal. In ac, gnralzd xrgs quany dvaons o h sysm s cncy rom h Carno cncy. An xrgy s oband as h prncpal componn o soluon o h varaonal problm o xrmum work undr suabl boundary condons. Ohr componns o h soluon ar opmal rajcory and opmal conrol. In purly hrmal sysms (hos whou chmcal changs h rajcory s characrzd by mpraur o h rsourc lud, (, whras h conrol s Carno mpraur ( dnd n our prvous work [, ]. For chmcal sysms also chmcal ponal(s µ ( plays a rol. Whnvr ( and µ ( dr rom ( and µ( h rsourc rlaxs wh a n ra, and wh an cncy vcor drn rom h prc cncy. Only whn = and µ = µ h cncy s prc, bu hs corrsponds wh an nnly slow rlaxaon ra o h rsourc o h hrmodynamc qulbrum wh h nvronmnal lud. h srucur o hs papr s as ollows. Scon II dscusss varous aspcs powr opmzaon. Proprs o sady sysms ar oulnd n Sc. III, whras hos o dynamcal ons - n Sc. IV. Scon V dvlops quanav analyss o rsourc downgradng (n h rs rsrvor and oulns proprs o gnralzd ponals or n ras. Scons VI-VIII dscuss varous Hamlon-Jacob-Bllman quaons (HJB quaons or opmal work uncons, as soluons o powr yld problms. Exnsons or smpl chmcal sysms ar oulnd n Sc. IX. h sz lmaon o our papr dos no allow or ncluson o all drvaons o mak h papr sl-conand, hus h radr may nd o urn o som prvous works, [] - [5]. In vw o dculs n gng analycal soluons n complx sysms, drnc quaons and numrcal approachs ar rad n r. [3] whch also dscusss convrgnc o numrcal algorhms o soluons o HJB quaons and rol o Lagrang mulplrs n h dmnsonaly rducon. II. FINIE RESOURCES AND POWER OPIMIZAION Lmd amoun or low o a rsourc workng n an ngn causs a dcras o h rsourc ponal n m (chronologcal or spaal. hs s why suds o h rsourc downgradng apply h dynamcal opmzaon mhods. From h opmzaon vwpon, dynamcal procss s vry on wh squnc o sas, dvlopng hr n h chronologcal m or n (spaal holdup m. h rs group rrs o unsady procsss n non-saonary sysms, h scond group may nvolv sady sa sysms. In a procss o nrgy producon wo rsng rsrvors do nrac hrough an nrgy gnraor (ngn. In hs procss hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

2 powr low s sady only whn wo rsrvors ar nn. Whn on, say, uppr, rsrvor s n, s ponal mus dcras n m, whch s consqunc o h nrgy balanc. Any n rsrvor s hus a rsourc rsrvor. I s h rsourc propry ha lads o h dynamcal bhavor o h lud and s rlaxaon o h qulbrum wh an nn lowr rsrvor (usually h nvronmn. Alrnavly, lud a a sady low can rplac rsng uppr rsrvor. h rsourc downgradng s hn a sady-sa procss n whch h rsourc lud lows hrough a ppln or sags o a cascad and h lud s sa changs along a sady rajcory. As n h prvous cas h rajcory s a curv dscrbng h lud s rlaxaon owards h qulbrum bwn h lud and h lowr rsrvor (h nvronmn. hs s somms calld acv rlaxaon as s assocad wh h smulanous work producon. I should b conrasd wh dsspav rlaxaon, a wll-known, naural procss bwn a body or a lud and h nvronmn whou any powr producon. Rlaxaon (hr acv or dsspav lads o a dcras o h rsourc ponal (.. mpraur n m. An nvrs o h rlaxaon procss s h on n whch a body or a lud abandons h qulbrum. hs canno b sponanous; rahr h nvrs procss nds a supply o xrnal powr. hs s h procss rrrd o hrmal upgradng o h rsourc, whch can b accomplshd wh a ha pump. III. SEADY SYSEMS h gra dal o rsarch on powr lms publshd o da dals wh saonary sysms, n whch cas boh rsrvors ar nn. o hs cas rr sady-sa analyss o h Chambadal-Novkov-Curzon-Ahlborn ngn (CNCA ngn [6], n whch nrgy xchang s dscrbd by Nwonan law o coolng, or h San-Bolzmann ngn, a sysm wh h radaon luds and h nrgy xchang govrnd by h San-Bolzmann law [7]. Du o hr saonary (causd by h nnnss o boh rsrvors, conrols maxmzng powr ar lumpd o a xd pon n h sa spac. In ac, or h CNCA ngn, h maxmum powr pon may b rlad o h opmum valu o a r (unconsrand conrol varabl whch can b cncy η or Carno mpraur. In rms o rsrvors mpraurs and and nrnal loss acor Φ on nds / op = ( Φ [4]. Fgur. Maxmum powr rlaxaon curv or black radaon whou consran on h mpraur [8]. For h San-Bolzmann ngn xac xprsson or h opmal pon canno b drmnd analycally, y, hs mpraur can b ound graphcally rom h char P=(. Morovr, h mhod o Lagrang mulplrs can succssully b appld [8]. As hr lmnaon rom a s o rsulng quaons s qu asy, h problm s brokn down o h numrcal solvng o a nonlnar quaon or h opmal conrol. Fnally, h so-calld psudo-nwonan modl [4, 5], whch uss sa or mpraur dpndn ha xchang cocn, α( 3, oms, o a consdrabl xn, analycal dculs assocad wh h San-Bolzmann quaon. Applyng hs modl n h so-calld symmrc cas, whr boh rsrvors ar lld up wh radaon, on shows ha h opmal (powr maxmzng Carno mpraur o h radaon ngn s ha or h CNCA ngn,.. [4]. hs quaon s, n ac, a good approxmaon undr h assumpon o ransr cocns dpndn solly on bulk mpraurs o rsrvors. IV. DYNAMICAL SYSEMS h valuaon o dynamcal nrgy yld rqurs h knowldg o an xrmal curv rahr han an xrmum pon. hs s assocad wh applcaon o varaonal mods (o handl unconal xrma n plac o sac opmzaon mhods (o handl xrma o uncons. For xampl, h us o h psudo-nwonan modl o quany h dynamcal nrgy yld rom radaon, gvs rs o an xrmal curv dscrbng h radaon rlaxaon o h qulbrum. hs curv s non-xponal, h consqunc o h nonlnar proprs o h rlaxaon dynamcs. Non-xponal ar also ohr curvs dscrbng h radaon rlaxaon,.g. hos ollowng rom xac modls usng h San-Bolzmann quaon (symmrc and hybrd, [4,5]. Analycal dculs assocad wh dynamcal opmzaon o nonlnar sysms ar svr; hs s why dvrs modls o powr yld and dvrs numrcal approachs ar appld. Opmal (.g. powr-maxmzng rlaxaon curv ( s assocad wh h opmal conrol curv (; hy boh ar componns o h dynamc opmzaon soluon o a connuous problm. In h corrspondng dscr problm, ormulad or numrcal purposs, on sarchs or opmal mpraur squncs { n } and { n }. Varous dscr opmzaon mhods nvolv: drc sarch, dynamc programmng, dscr maxmum prncpl, and combnaons o hs mhods. Mnmum powr suppld o h sysm s dscrbd n a suabl way by uncon squncs R n ( n, n, whras maxmum powr producd by uncons V n ( n, n. Pro-yp prormanc uncon V and cos-yp prormanc uncon R smply dr by sgn,.. V n ( n, n = - R n ( n, n. h bgnnr may nd h chang rom symbol V o symbol R and back as unncssary and conusng. Y, ach uncon s posv n s own, naural rgm o workng (V - n h ngn rang and R - n h ha pump rang. Imporanly, nrgy lms o dynamcal procsss ar nhrnly conncd wh h xrgy uncons, h classcal xrgy and s ra-dpndn xnsons. o oban classcal xrgy rom powr uncons sucs o assum ha h hrmal cncy o h sysm s dncal wh h Carno cncy. On h ohr hand, non-carno cncs lad o hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

3 gnralzd xrgs. h lar dpnd no only on classcal hrmodynamc varabls bu also on hr ras. hs gnralzd xrgs rr o sa changs n a n m, and can b conrasd wh h classcal xrgs ha rr o rvrsbl quassac procsss volvng n m nnly slowly. h bn oband rom gnralzd xrgs s ha hy dn srongr nrgy lms han hos prdcd by classcal xrgs. Sysmac approach o xrgs (classcal or gnralzd basd on work unconals lads o svral orgnal rsuls n hrmodynamcs o nrgy sysms, n parcular allows o xplan unknown proprs o xrgy o black-body radaon or solar radaon, and o show ha h cncy o h solar nrgy lux ransormaon s qual o h Carno cncy. V. WO WORKS AND FINIE-IME EXERGY wo drn works, h rs assocad wh h rsourc downgradng durng s rlaxaon o h qulbrum and h scond wh h rvrs procss o rsourc upgradng, ar ssnal (Fg.. Durng h approach o h qulbrum ngn mod aks plac n whch work s rlasd, durng h dparur- ha-pump mod occurs n whch work s suppld. Work W dlvrd n h ngn mod s posv by assumpon ( ngn convnon. Squnc o rrvrsbl ngns (CNCA or San-Bolzmann srvs o drmn a ra-dpndn xrgy xndng h classcal xrgy or rrvrsbl, n ra procsss. Bor maxmzaon o a work ngral, procss cncy η has o b xprssd as a uncon o sa and a conrol,.. nrgy lux q or ra d/dτ, o assur h unconal propry (pah dpndnc o h work ngral. h ngraon mus b prcdd by maxmzaon o powr or work a low (h rao o powr and lux o drvng subsanc w o assur an opmal pah. h opmal work s sough n h orm o a ponal uncon ha dpnds on h nd sas and duraon. For appropra boundary condons, h prncpal uncon o h varaonal problm o xrmum work concds wh h noon o an xrgy, h uncon ha characrzs qualy o rsourcs. Fgur. wo works: Lmng work producd and lmng work consumd ar drn n an rrvrsbl procss. h da o an nn numbr o nnsmal CNCA sps, ncssary or xrgy calculaons, s llusrad n Fg.. Each sp s a work-producng (consumng sag wh h nrgy xchang bwn wo luds and h hrmal machn hrough n conducancs. For h radaon ngn ollows rom h San-Bolzmann law ha h cv ransr cocn α o h radaon lud s ncssarly 3 mpraur dpndn, α =. h scond or low- lud rprsns h usual nvronmn, as dnd n h xrgy hory. hs lud posssss s own boundary layr as a dsspav componn, and h corrspondng xchang cocn s α. In h physcal spac, h low drcon o h rsourc lud s along h horzonal coordna x. h opmzr s ask s o nd an opmal mpraur o h rsourc lud along h pah ha xrmzs h work consumd or dlvrd. oal powr oband rom an nn numbr o nnsmal ngns s drmnd as h Lagrang unconal o h ollowng srucur & [, ] = 0 (, d = Gc & ( η(, W d & ( whr 0 s powr gnraon nnsy, G & - rsourc lux, c(-spcc ha, η(, -cncy n rms o sa and conrol, urhr nlargd sa vcor comprsng sa and m, m varabl (rsdnc m or holdup m or h rsourc conacng wh ha ransr surac. Somms on uss a non-dmnsonal m τ, dncal wh h so-calld numbr o h ha ransr uns. No ha, or consan mass low o a rsourc, on can xrmz powr pr un mass lux,.. h quany o work dmnson calld work a low. In hs cas Eq. ( dscrbs a problm o xrmum work. Ingrand 0 s common or boh mods, y h numrcal rsuls gnras dr by sgn (posv or ngn mod; ngn convnon. Whn h rsourc lux s consan a work unconal dscrbng h hrmal xrgy lux pr un lux o rsourc can b oband rom Eq. ( w max d / d = = = c( d. ( d d (, / No ha h ndpndn varabl n hs quaon s,.. s drn han ha n Eq. (. h uncon 0 n Eq. ( conans hrmal cncy uncon, η, dscrbd by a praccal counrpar o h Carno ormula. Whn >, cncy η dcrass n h ngn mod abov η C and ncrass n h ha-pump mod blow η C. A h lm o vanshng ras, d/d = 0 and. hn work o ach mod smpls o h common ngral o h classcal xrgy. For h classcal hrmal xrgy = wmax = c( d = h h ( s s d / d 0. (3 = Nonlnars can hav boh hrmodynamc and knc orgns; h ormr rr, or xampl, o sa dpndn ha capacy, c(, h lar o nonlnar nrgy xchang. Problms wh lnar kncs (Nwonan ha ransr ar an mporan subclass. In problms wh lnar kncs, lud s spcc work a low, w, s dscrbd by an quaon = w[, ] W& / G& = c( d ( c( dτ (4 hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

4 whr τ x H U α avf α avfv = x = = (5 Gc & Gc & χ s non-dmnsonal m o h procss. Equaon (5 assums ha a rsourc lud lows wh vlocy v hrough cross-scon F and conacs wh h ha ransr xchang surac pr un volum a v []. Quany τ s dncal wh h so-calld numbr o h ha ransr uns. Soluons o work xrmum problms can b oband by: a varaonal mhods,.. va Eulr-Lagrang quaon o varaonal calculus L d L = 0. (6 d & In h xampl consdrd abov,.. or a hrmal sysm wh lnar kncs d d = 0 (7 dτ dτ whch corrsponds wh h opmal rajcory τ / τ (,,, = ( /. (8 τ τ (τ =0 s assumd n Eq. (8. Howvr, h soluon o Eulr-Lagrang quaon dos no conan any normaon abou h opmal work uncon. hs s assurd by solvng h Hamlon-Jacob-Bllman quaon (HJB quaon, [9]. b dynamc programmng va HJB quaon or h prncpal uncon (V or R, also calld xrmum work uncon. For h lnar kncs consdrd max ( c( ( = 0. (9 Obsrv ha all ras ( 0 and and drvavs o V ar valuad a h nal sa (h so-calld orward quaon. h xrmal work uncon V s a uncon o h nal sa and oal duraon. Ar valuaon o opmal conrol and s subsuon o Eq. (9 on obans a nonlnar quaon { ( + c / } = 0 c. (0 whch s h Hamlon-Jacob quaon o h problm. Is soluon can b ound by h ngraon o work nnsy along an opmal pah, bwn lms and. A rvrsbl (pah ndpndn par o V s h classcal xrgy A(,, 0. Modls o mulsag powr producon n squncs o nnsmal ngns []-[5] provd powr gnraon uncons 0 or hrmal Lagrangans l 0 = - 0 and dynamcal consrans. Numrcal mhods apply suabl dscr modls, or gvn ras 0 and. An mporan ssu s convrgnc o hs dscr modls o connuous ons [3]. VI. HJB EQUAIONS FOR NONLINEAR POWER SYSEMS W shall dsplay hr som Hamlon-Jacob-Bllman quaons or powr sysms dscrbd by nonlnar kncs. A suabl xampl s a radaon ngn whos powr ngral s approxmad by a psudo-nwonan modl o radav nrgy xchang assocad wh opmal uncon V (,,, max G& mcm ( Φ υ (, d ( '( whr υ =α( 3 ( -. An alrnav orm uss Carno mpraur xplc n υ [5]. Opmal powr ( can b rrrd o h ngral W& Gm chm cvm υd = 0 & ( ( G χυ υ m cvm Φ 0 & ( ( + ( d. ( ( + χυ + χυ hs procss s dscrbd by a psudolnar kncs d/d = (, conssn wh υ =α( 3 ( - and a gnral orm o HJB quaon or work uncon V s + max 0 (, (, = 0. (3 ( whr 0 s dnd as h ngrand o Eq. ( or (. A mor xac modl or radaon convrson rlaxs h assumpon o h psudo-nwonan ransr and appls h San-Bolzmann law. For a symmrc modl o radaon convrson (boh rsrvors composd o radaon Φ a a W G c β d & = & (. (4 a a ( Φ ( / + 0 h cocn s β σa c ( p s rlad o molar = v h m 0 consan o phoons dnsy p and San-Bolzmann m consan σ. In h physcal spac, powr xponn a=4 or radaon and a= or a lnar rsourc. Wh sa quaon d d β (5 a a = a ( ( / a Φ + [5] appld n gnral Eq. (9 w oban a HJB quaon a a G& ( c Φ (6 + max β = 0 ( a a ( ( / + / Φ + Dynamcs (5 s h characrsc quaon or Eq. (6. For a hybrd modl o radaon convrson (uppr rsrvor composd o h radaon and lowr rsrvor o a Nwonan lud, [5] h powr s Φ W& = G c( ud a a / a a u Φ ug g ( + β + β / τ (7 and h corrspondng Hamlon-Jacob-Bllman quaon s Gc ( ( + max ( a ( + β & (8 a Φ / a u + Φβ a u = 0 + ug / g VII. ANALYICAL ASPECS OF LINEAR AND PSEUDO-NEWONIAN KINEICS In all HJB quaons xrmzd xprssons ar som hamlonans. By applyng h dback conrol opmal drvng mpraur ' or ohr conrol s mplmnd as h hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

5 quany maxmzng h hamlonan wh rspc o a ach pon o h pah. h maxmzaon o H lads o wo quaons. h rs xprsss opmal conrol ' n rms o and z = - /. For h lnar kncs o Eq. (9 w oban (, 0 = + c( = 0 (9 whras h scond s h orgnal quaon (9 whou maxmzng opraon + ( + c( ( = 0 (0 o oban opmal conrol uncon '(z, on should solv h scond qualy n quaon (9 n rms o ', h rsul s Carno conrol ' n rms o and z = - /, / / = + c. ( hs s nx subsud no (0; h rsul s h nonlnar Hamlon-Jacob quaon + c ( + c / / = 0 ( whch conans h nrgylk (xrmum Hamlonan o h xrmal procss. V H (, = c ( + c / /. (3 For a posvly-dnd H, ach Hamlon-Jacob quaon or opmal work prsrvs h gnral orm o auonomous quaons known rom analycal mchancs and hory o opmal conrol. Exprssng xrmum Hamlonan (3 n rms o sa varabl and Carno conrol ' ylds an nrgy-lk uncon sasyng h ollowng rlaons E(, u 0 ( = 0 u = c (4 u E s h Lgndr ransorm o h work lagrangan l 0 = - 0 wh rspc o h ra u = d/dτ. Assumng a numrcal valu o h Hamlonan, say h, on can xplo h consancy o H o lmna /. Nx combnng quaon H=h wh opmal conrol (, or wh an quvaln rsul or nrgy low conrol u= - / u =. (5 c V + / ylds opmal ra u= & n rms o mpraur and h Hamlonan consan h & = { ± h / c ( ± h / c } (6 A mor gnral orm o hs rsul whch appls o sysms wh nrnal dsspaon (acor Φ and appls o h psudo-nwonan modl o radaon s hσ h & σ (7 = ± ± ξ ( hσ, Φ, Φcv ( Φcv ( whr ξ, dnd n h abov quaon, s an nnsy ndx and h σ =h/. hs rsul s oband by h applcaon o varaonal calculus o nonlnar radaon luds wh h mpraur dpndn ha capacy c v (=4a 0 3. Posv ξ rr o hang o h rsourc lud n h ha-pump mod, and h ngav - o coolng o hs lud n h ngn mod. hus psudo-nwonan rsourcs produc powr rlaxng wh h opmal ra & = ξ ( h σ,, Φ. (8 Equaons (7 and (8 dscrb h opmal rajcory n rms o sa varabl and consan h. h opmal (Carno conrol s = ( + ξ ( h, Φ, (9 σ h prsnc o rsourc mpraur n uncon ξ provs ha, n comparson wh h lnar sysms, h psudo-nwonan rlaxaon curv s no xponnal. VIII. OPIMAL WORK FUNCIONS FOR LINEAR AND PSEUDO-NEWONIAN KINEICS A soluon can now b ound o h problm o Hamlonan rprsnaon o xrmal work. L us bgn wh lnar sysms. Subsung mpraur conrol (9 wh a consan ξ no work unconal (4 and ngrang along an opmal pah ylds xrmal work uncon h V (,, h = c( c ln c ln (30 c hs xprsson s vald or vry procss mod. Ingraon o Eq (7 subjc o nd condons (τ = and (τ = allows o xprss Eq. (30 n rms o h procss duraon. For h radaon c v (=4a 0 3, whr a 0 s h radaon consan, an opmal rajcory solvng Eqs. (7 and (9 s / / -/ a Φ 3/ 3/ ± ( 4 / 3 0 hσ ln( / = τ τ (3 h ngraon lms rr o h nal sa ( and a currn sa o h radaon lud,.. mpraurs and corrspondng wh τ and τ. Opmal curv (3 rrs o h cas whn h radaon rlaxaon s subjc o a consran rsulng rom Eq. (8. Equaon (3 s assocad wh h nropy producon rm n Eq. (. h corrspondng xrmal work uncon pr un volum o lowng radaon s V hv hv ( sv sv (3 / / 3/ 3/ 3 3 (4 / 3 a h σ Φ ( + (4 / 3 a ( Φ( / 0 Also, h corrspondng xrgy uncon, oband rom (3 ar applyng xrgy boundary condons, has an xplc analycal orm. h classcal avalably o radaon a low rsds n h rsulng xrgy quaon n Jr s [0] orm A class v (, = h ( v 0,0 = hv hv ( sv sv. (33 4 / = (4/3 a0 ( / IX. WORK FUNCIONS FOR CHEMICAL SYSEMS h dvlopd mhodology can b xndd o chmcal and lcrochmcal ngns []. As opposd o hrmal machns, n chmcal ons gnralzd rsrvors ar prsn, capabl o provdng boh ha and subsanc. Whn nn rsrvors assur consancy o chmcal ponals, problms o xrmum powr (maxmum o powr producd and hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

6 mnmum o powr consumd ar sac opmzaon problms. For n rsrvors, howvr, amoun and chmcal ponal o an acv racan dcras n m, and consdrd problms ar hos o dynamc opmzaon and varaonal calculus. h smpls modl o powr producng chmcal ngn s ha wh an sohrmal and somrc racon, A +A =0 []. Powr xprsson and cncy ormula or h chmcal sysm ollow rom nropy consrvaon and nrgy balanc n h powr-producng zon. In ndorvrsbl ngns oal nropy lux s consan hrough h acv zon. Whn h consancy ormula s combnd wh nrgy balanc w nd n an sohrmal cas p = µ n ( ' µ ' (34 whr n s an nvaran molar lux o ragns. Procss cncy ζ s dnd as powr yld pr molar lux, n,.. ζ = p / n = µ µ (35 hs cncy s dncal wh h chmcal any o our racon n h chmcally acv par o h sysm. For a sady ngn h ollowng uncon dns h chmcal cncy n rms o n and mol racon x (Fg. 3 ' ' x ng ζ = ζ + 0 R ln (36 ng + x Fgur 3. Ecncy o powr producon ζ n rms o ul lux n n a chmcal ngn. Equaon (36 shows ha cv concnraon o racan n uppr rsrvor x = x g - n s dcrasd, whras cv concnraon o produc n lowr rsrvor x = x + g - n s ncrasd du o h n mass lux. hror cncy ζ dcrass nonlnarly wh n. Whn c o rssancs g s gnorabl or lux n s vry small, k rvrsbl cncy, ζ C, s aand. Powr uncon, dscrbd by h produc ζ(nn, xhbs a maxmum or a n ul lux, n. Exnsons o Eq. (36 ar avalabl []. Applcaon o Eq. (36 o an unsady sysm lads o a work uncon dscrbng h dynamcal lm o h sysm τ W = τ X /( + X + dx / dτ dx ζ 0 + R ln dτ x jdx / dτ dτ (37 (X=x/(-x. h pah opmaly condon may b xprssd n rms o h consancy o h ollowng Hamlonan x + jx( X + X j H ( X, X& = RX& = RX& +. (38 x( X x X x For low ras and larg concnraons X (mol racons x clos o h uny opmal rlaxaon ra s approxmaly consan. Y, n an arbrary suaon opmal ras ar sa dpndn so as o prsrv h consancy o H n Eq. (38. Ful cll xampls and hr hory ar analyzd n our prvous papr prsnd a h ICREPQ [3]. X. CONCLUSIONS Opmal powr daa show ha h daa dr or powr gnrad and consumd, and dpnd on paramrs o h sysm,.g.: lux nnsy, numbr o ransr uns, polarzaons, surac ara, avrag procss ra, rao o sram lows, sram drcons, c. h daa provd bounds or powr gnraors ha ar mor xac and srongr han rvrsbl bounds. As opposd o classcal hrmodynamcs, our bounds dpnd no only on changs o h hrmodynamc sa o parcpang rsourcs bu also on procss rrvrsbls, procss drcon and mchansm o ha and mass ransr. Only n hrmosacs h bound on h work producd concds wh ha on h work consumd. h gnralzd hrmo-knc bounds, oband hr, do no sasy h rvrsbly propry. Only or nnly long duraons or or procsss wh xclln ransr (an nn numbr o ransr uns h hrmoknc bounds rduc o classcal hrmosac bounds [4]. hus, wh rrvrsbl hrmodynamcs, w can conron and surmoun h lmaons o applyng classcal hrmodynamcs o ral procsss. hs s a drcon wh many opporuns. Rrncs [] S. Snuycz, Powr gnraon lms n hrmal, chmcal and lcrochmcal sysms, ICREPQ'0: Inrnaonal Conrnc on Rnwabl Enrgs and Powr Qualy, Granada, Span, 3-5 March, 00, hp:// [] S. Snuycz, Carno conrols o uny radonal and work-asssd opraons wh ha & mass ransr, Inrnaonal J. o Appld hrmodynamcs, 6 (003, [3] S. Snuycz, Dynamc programmng and Lagrang mulplrs or acv rlaxaon o luds n non-qulbrum sysms, Appld Mahmacal Modlng, 33 (009, [4] S. Snuycz and P. Kuran, Nonlnar modls or mchancal nrgy producon n mprc gnraors drvn by hrmal or solar nrgy, Inrn. J. Ha Mass ransr, 48 (005, [5] S. Snuycz and P. Kuran, Modlng hrmal bhavor and work lux n n-ra sysms wh radaon, Inrn. J. Ha and Mass ransr 49 (006, [6] F.L. Curzon and B. Ahlborn, Ecncy o Carno ngn a maxmum powr oupu, Amrcan J. Phys. 43 (975, -4. [7] A. d Vos, Endorvrsbl hrmodynamcs o Solar Enrgy Convrson, Oxord Unvrsy Prss, Oxord, 994, pp [8] P. Kuran, Nonlnar Modls o Producon o Mchancal Enrgy n Non-Idal Gnraors Drvn by hrmal or Solar Enrgy, PhD hss, Warsaw Unvrsy o chnology, 006. [9] R. E. Bllman, Adapv Conrol Procsss: a Gudd our, Prncon, Unvrsy Prss, 96, pp.-35. [0] J. Jr, Maxmum convrson cncy or h ulzaon o drc solar radaon, Solar Enrgy 6 (98, [] S. Snuycz, An analyss o powr and nropy gnraon n a chmcal ngn, Inrn. J. o Ha and Mass ransr 5 ( [] S. Snuycz, Complx chmcal sysms wh powr producon drvn by mass ransr, Inrn. J. o Ha and Mass ransr, 5 ( [3] P. Kuran and S. Snuycz, Modlng and smulaon o powr yld n chmcal and lcrochmcal sysms, ICREPO: Elvnh Inrnaonal Conrnc on Rnwabl Enrgs and Powr Qualy, Las Palmas GC, 3-5 Aprl, 0, hp:// hps://do.org/0.4084/rpqj RE&PQJ, Vol., No.0, Aprl 0

Maximization of Power Yield in Thermal and Chemical Systems

Maximization of Power Yield in Thermal and Chemical Systems Procdngs o h World Congrss on Engnrng 009 Vol II WCE 009, July 1-3, 009, London, U.K. Maxmzaon o Powr Yld n hrmal and Chmcal Sysms S. Snuycz Absrac hs rsarch nvolvs a hrmodynamc approach o modlng and powr