Irreversible Work of Separation and Heat-Driven Separation

Transcription

1 J. Phys. Che. B 004, 08, Irreversble Wor of Separaton and Heat-Drven Separaton Anatoly M. Tsrln and Vladr Kazaov*, Progra Syste Insttute, Russan Acadey of Scence, set. Botc, PerejaslaVl-Zalesy, Russa 500, and School of Fnance & Econocs, UnVersty of Technology, Sydney, NSW 007 Australa ReceVed: NoVeber 4, 003; In Fnal For: February, 004 Estates of the nal necessary wor of separaton for echancal separaton processes and of the nal necessary heat consupton for heat-drven separaton processes wth gven productvty are derved n ths paper. It s shown that for heat-drven processes the productvty s lted, and ths ltng productvty s estated.. Introducton The nal aount of energy needed for separaton a xture wth a gven coposton can be estated usng reversble therodynacs. These estates turn out to be very loose and unrealstc. They also do not tae nto account netc factors (laws and coeffcents of heat and ass transfer, productvty of the syste, etc.). In ths paper we derve rreversble estates of the wor of separaton that tae nto account all these factors. The ajorty of separaton systes are open systes that exchange ass and energy wth the envronent. If ass and heat transfer coeffcents (deterned by the sze and constructon of the apparatus) are fnte and f the productvty of the syste s fnte then the processes n such systes are reversble. The energy flows, the copostons of the ass flows, and the productvty of the syste are lned va the balance equatons of energy, ass, and entropy. The latter also ncludes entropy producton n the syste. Mnal energy used for separaton corresponds to nal entropy producton n the syste subject to varous constrants. Ths allows us to estate ths nal energy. There s a qualtatve as well as a quanttatve dfference between the reversble and rreversble estates obtaned n ths paper. For exaple, the rreversble estate of the wor of separaton for poor xtures (where the concentraton of one of the coponents s close to one) tends to a fnte nonzero lt, whch depends on the netcs factors. The reversble wor of separaton for such xtures tends to zero. The reversble estate dffers fro the aount of energy needed n practce for separaton of poor xtures by a factor of 0 5. For heat-drven separaton processes the novel results obtaned n ths paper nclude the estate of the nal heat consupton as a functon of netc factors and the therodynac lt on the productvty of a heat-drven separaton. Fgure. Splfed scheatc of therodynac balances for separaton processes. pressure P 0 s separated nto two flows wth the correspondng paraeters g, x, T, P ( ), ). The flow of heat q + wth the teperature T + can be suppled, and the flow of heat q - wth the teperature T - can be reoved. The echancal wor wth the rate (power) p can be suppled. In centrfugng, ebrane separaton, and adsorptondesorpton cycles that are drven by pressure varatons, no heat s suppled/reoved and only echancal wor s spent. In absorpton-desorpton cycles, dstllaton, and so forth, no echancal wor s spent, only heat s consued (heat-drven separaton). In soe cases the nuber of nput and output flows can be larger. As a rule one can stll represent the syste as an assebly of separate blocs, whose structure s shown n Fgure... Heat-Drven Separaton. Consder a heat-drven separaton (p ) 0) and assue that each of the vectors x ) (x,..., x j,..., x ), ) 0,,, conssts of coponents whch denote the olar fracton of the jth substance n the th flow. The therodynac balance equatons of ass, energy, and entropy here tae the followng for g 0 x 0j - g x j - g x j ) 0, j ),..., () x j ), ) 0,, (). Therodynac Balances of Separaton Processes and the Ln between Energy Consupton and Entropy Producton Consder the syste, shown n Fgure, where the flow of xture wth rate g 0, coposton x 0, teperature T 0, and * Correspondng author. E-al:Vladr.Kazaov@uts.edu.au. Russan Acadey of Scence. E-al: tsrln@sarc.bot.ru. Unversty of Technology, Sydney. q + - q - + g 0 h 0 - g h - g h ) 0 (3) where h s the enthalpy of the th flow; q + T + - q - T - + g 0 s 0 - g s - g s + σ ) 0 (4) σ denotes entropy producton. Fro eq, eq follows that g 0 ) g + g. After elnaton of g 0 fro eqs 3 and 4 and 0.0/jp CCC: $ Aercan Checal Socety Publshed on Web 04/0/004

2 6036 J. Phys. Che. B, Vol. 08, No. 9, 004 Tsrln and Kazaov ntroducton of enthalpy ncreents h and entropy ncreent s we get Here, h 0 ) h 0 - h, s 0 ) s 0 - s ( ), ). Elnaton of q - usng eq 5 and ts substtuton nto eq 6 yelds g ( h 0 s 0 - T - ) + q + ( - T + T -) + σ ) 0 and the flow of used heat for heat-drven separaton s q + ) q + - q - + g h 0 + g h 0 ) 0 (5) g s 0 + g s 0 + q + T + - q - T - + σ ) 0 (6) T + The frst ter n the square bracets depends only on the paraeters of the nput and output flows and represents the reversble wor of separaton per unt of te (reversble power of separaton). The second ter there represents the process netcs and correspondng energy dsspaton. For xtures that are close to deal gases and deal solutons, olar enthalpes and entropes h and s n the eqs 3 and 4 can be expressed n ters of copostons and specfc enthalpes and entropes of the pure substances. We obtan for each of the flows where R s the unversal gas constant. The reversble energy consupton here s We denote here the Carnot effcency of the deal cycle of the heat engne as Condton 7 can be rewrtten as T + - T -[ g ( s 0 T - - h 0 ) + σt -] (7) h 0 ) [x 0j h j (T 0, P 0 ) - x j h j (T, P )] s 0 ) [x 0j s 0 j (T 0, P 0 ) - x j s 0 j (T, P ) - R(x 0j ln x 0j - x j ln x j )], ), (8) q 0 + ) η K g [[x 0j s j 0 (T 0, P 0 ) - x j s j 0 (T, P ) - R(x 0j ln x 0j - x j ln x j )]T - + x j h j (T, P ) - x 0j h j (T 0, P 0 )] (9) η C ) T + - T - T + q + ) η C (p 0 + σt - ) (0) Here, p 0 s the reversble power of separaton that s equal to the reversble flow of heat gven by eq 9 ultpled by the Carnot effcency. When eq 0 was derved we too nto account only the rreversblty σ of the separaton process (the rreversblty of the heat transfer was not taen nto account). In realty heat can be suppled/reoved wth a fnte rate only rreversbly. Any transforaton of heat nto wor wth fnte heat transfer coeffcents and fnte power s rreversble. Ths leads to a lower effcency than the Carnot effcency. The closed for expresson for ths effcency was obtaned n ref 7. It depends on the power p and on heat transfer coeffcents for heat supply and heat reoval R + and R -. For the Newton (lnear) law of heat transfer t has the for η p ) ax p q + ) - T + (T + + T - - 4p R - (T + - T - ) + ( 4p R ) - 8 p R (T + + T - ) ) () where t s assued that there s constant contact of the worng body wth the heat reservors and It s easy to show that f p f 0 then η p tends to the Carnot effcency. Substtuton of η p nstead of η C n eq 0 allows us to derve a tghter estate for the heat consupton n heat-drven separaton processes by fndng the nal possble entropy producton σ subject to varous constrants where R) 4R + R - R + +R - () p n q + g q n + ) η p (p n, R, T +, T - ) (3) p n ) p 0 + σ n T - (4) Condtons -4 sngle out the area of therodynacally feasble heat-drven separaton systes. Expressons 9 and 0 and eq 7 can be further specfed by assung the constancy of heat capactes, that the xture s bnary, and so forth... Mechancal Separaton. Consder a separaton syste that uses echancal wor wth rate p. Assue that no heat s suppled/reoved (q + ) q - ) 0) and that nput and output flows have the sae teperature T and the sae pressure. Multplcaton of eq 6 by T and subtracton of the result fro the energy balance (eq 5), where (q + - q - ) s replaced wth the suppled power p, yelds p ) Tσ + g 0 γ (T s 0 - h 0 ) (5) Here γ ) g /g 0. After tang nto account (eq 9) that the enthalpy ncreent h 0 n a echancal separaton s zero, we get p ) g 0 RT[ γ x j ln x j - x 0j ln x 0j] + Tσ ) p0 + Tσ (6) The frst ter n ths expresson represents the nal power for separaton that corresponds to the reversble process (σ ) 0). Ths power p 0 s equal to the dfference between the reversble power for coplete separaton of the nput flow p 0 0 )-g 0 RT j x 0j ln x 0j and the cobned reversble power of separaton of the output flows p 0 and p 0.

3 Irreversble Wor of Separaton J. Phys. Che. B, Vol. 08, No. 9, Fgure. reservor. Here Coputatonal structure of the separaton syste wth p 0 (x ) )-RTg 0 γ x j ln x j, ) 0,, (7) s the reversble power of separaton of the th flow nto pure substances. 3. Mnal Wor of Separaton n Irreversble Processes 3.. Assuptons and Proble Forulaton. Assue that the coponents of the nput xture are close to deal gases or deal solutons. The checal potental of the th coponent can then be wrtten n the followng for µ (T, P) ) µ 0 (T, P) + RT ln x, ),..., (8) where x s the concentraton of the th coponent. Frst we consder a syste that ncludes three eleents (see Fgure ), a reservor wth the te ndependent teperature T, pressure P, and vector of concentratons x 0 ) {x 0,..., x 0 } (therefore ts checal potental µ 0 s also te ndependent), the fnte capacty output subsyste wth checal potental µ that depends on the current copostons of the xture and of the worng body that has controllable values of checal potental µ 0 w and µ w, at the ponts of contact wth reservor and output subsyste. At the te the ntensve varables of the output subsyste concde wth the values of the reservor s ntensve varables, and the nuber of oles n t s gven and equal to N 0. At te the nuber of oles N() and the coposton x() n the output subsyste are gven. The ass transfer coeffcents between the reservor and the worng body and the worng body and the output subsyste are fnte and fxed. The nal necessary wor requred for the separaton s sought. We do not consder here how to pleent the derved optal dependence of the checal potental of the worng body because of two reasons. Frst, our an objectve s to derve a lower bound on the wor of separaton. However, posng constrants on feasble varatons of checal potental would lead to an ncrease n energy consupton. Second, we wll deonstrate that for the ajorty of ass transfer laws the optal ass transfer flow s te ndependent, and ts pleentaton s straghtforward. The wor of separaton n an sotheral process for an adabatcally nsulated syste can be found usng the Stodola forula n ters of the reversble wor A 0 and the entropy ncreent S A ) A 0 + T S (9) The reversble wor s equal to the ncreent of the syste s nternal energy. Snce as a result of the process (N() - N(0)) oles of xture wth the coposton x 0 s reoved fro the reservor, and the energy of the output subsyste rses because of the ncrease of the aount of oles n t fro N(0) to N() and ts coposton fro x 0 to x, the total change of the Fgure 3. Dependence of the entropy producton on the rate for the * constant (a) and swtchng (b) solutons (g * and g are the basc values of the rate). syste s nternal energy s A 0 ) N() µ ) N()RT [x () lnx () - x 0 ln x 0 ] (0) and t s ndependent of N(0). Because A 0 s deterned by N, x(), x(0), the nu of A corresponds to the nu of the entropy ncreent S ) T 0 ) T 0 [g 0 (µ 0 - µ w ) + g (µ w - µ )] dt (g 0 µ 0 + g µ )dt () Because the worng body s paraeters have the sae values at the begnnng and at the end of a cycle 0 g0 dt ) 0 g dt () N()x () - N(0)x (0) ) (Nx ), 3.. Optal Soluton. The proble of nzaton of S subject to constrants (eq ) on g 0 g 0, g g 0 becoes spler n a coon case where the checal potentals ncreents µ 0, µ are unque functons of flows g 0 and g, correspondngly. If processes are close to equlbru then ths dependence s lnear. Assue µ 0 ) φ 0 (g 0 ), µ ) φ (g ) ),,..., then the probles and can be decoposed nto probles S j ) 0 σj (g j )dt f n / 0 gj d ) (Nx ) j ) 0,, ),,..., (3) where σ j ) g j φ j (g j ) s the functon that deternes dsspaton. Probles (eq 3) are averaged nonlnear prograng * probles. Ther optal solutons g 3 j are ether constants and equal to g * j ) g * ) (Nx ) (4) or swtches between two so-called basc values on the nterval (0, ), the soluton (eq 4) correspondng to the case where the convex envelope of the functon σ j (g j ) s lower than the value of ths functon at g * j. Characterstc fors of the functon σ j (g j ) for the constant and swtchng reges are shown n Fgure 3.

4 6038 J. Phys. Che. B, Vol. 08, No. 9, 004 Tsrln and Kazaov If the functon σ j s concave then the optal rate g j s always constant. Let us calculate the second dervatve of σ on g (we ot subscrpts for splcty). If t s postve then the constancy of the rate n the optal process s guaranteed. σ (g) ) φ (g) + gφ (g) g 0 (5) The frst ter n ths expresson s always postve because the checal potentals dfference s the drvng force of ass transfer and onotoncally depends on the flow. For the ajorty of laws of ass transfer the nequalty (eq 5) holds. In partcular, t holds f the flow of ass transfer s proportonal to the dfference of checal potentals n any postve degree. Consder ass transfer flow that depends lnearly on the checal potental dfference for all, j. Then Fgure 4. Reversble (A 0 ) and rreversble (A r) estates of the nal wor of separaton of bnary xture as functons of ey coponent s concentratons. g j )R j µ j w φ j ) g j R j (6) It s clear that the condtons (eq 5) hold and the optal rates of flows obey equaltes 4. Equaltes 4 hold for any nonswtchng soluton. The nal ncreent of the entropy producton for such soluton s S n ),j and the nal wor of separaton s The optal rates are deterned by the ntal and fnal states whch allows us to specfy the estate (eq 8). Near equlbru the flows obey Onsanger s netcs (eq 6), and fro eq 8 t follows that + (Nx ) R 0 A n ) A 0 + g ( S j n ),j σ j( (Nx ) s the equvalent ass transfer coeffcent on the th coponent and the nal entropy producton s The lower bound for the average power of separaton s p 0 ) A 0 / s the reversble power of separaton. If ) (7) A n ) A 0 + T σ,j j( (Nx ) ) (8) R ) ) A 0 + R (9) R ) R 0 R (30) R 0 +R σ n ) T (Nx ) R (3) A n A 0 p n ) ) + (Nx ) (3) R N(0) ) 0, (Nx ) ) Nx () Fgure 5. Separaton of the syste wth fnte capacty on subsystes. then expressons 9 and 3 tae the for where A n ) A 0 + N x () R (33) x () p n ) p 0 + g (34) R A 0 ) NRT [x () lnx () - x ln x ] (35) Note that the rreversble estate of the wor of separaton (eq 33) does not tend to zero for poor xtures when the concentraton of one of the coponents tends to one (Fgure 4). If syste ncludes not one but a nuber of output subsystes then t s clear that the estate for the nal wor of separaton s equal to the su of the estates for each subsyste. A n ) j A n, p n ) j p n (36) The superscrpt j here denotes the subsystes Separaton of a Syste wth Fnte Capacty nto Subsystes. Consder a syste that s shown n Fgure 5. Its ntal state s descrbed by the vector of concentratons x 0, the nuber of oles of the xture N 0, and ts fnal state by the nuber of oles Nh j, j ),..., n each of the subsystes and ther concentratons, x j. The ass balances yelds Nh j xj j ) N 0 x 0, Nh j ) N 0 (37) ),,...,

5 Irreversble Wor of Separaton J. Phys. Che. B, Vol. 08, No. 9, The wor n the reversble separaton process here s A 0 r (x 0,xj) ) RT[ Nh j xj j ln xj j - N 0 ) A 0 r0 (x 0, N 0 ) - A 0 rj (xj j, Nh j ) x 0 ln x 0] (38) The reversble wor of separaton s equal to the dfference of the reversble wor of separaton of the ntal xture nto pure coponents and the reversble wor of separaton for xtures n each of the subsystes. We agan assue that flows g j have coponents g j proportonal to the dfference of the checal potental of the subsyste and the worng body wth the coeffcent R j. Here, the condton of nal wor of separaton corresponds to the condton of flow constancy g j ) Nh j xj j, ),,...,, j ),..., (39) µ j ) g j Rj j, j ) 0,,..., (40) Here, Rj j s the equvalent ass transfer coeffcent calculated usng eq 30 for the flow nto the jth output subsyste of the th coponent. Slarly as was done above for the syste wth the reservor and one fnte capacty output subsyste and flows proportonal to the fnal concentratons (eq 39), these concentratons n the output subsystes are te ndependent and equal to xj j, correspondngly, and the nuber of oles Nh j (t) depends lnearly on te. The power p here s constant xj j p ) RT Nh j xj j ln + N j xj j /Rj j (4) x 0 The nal wor of separaton for the xture wth concentratons x 0 nto subsystes wth concentratons xj over the te s xj j N 0 A r ) RTN 0 γ j xj j ln + γ j xj j /Rj j (4) x 0 Here, γ j ) N j /N 0, Rj j )R j R 0 /(R 0 +R j ). The frst ter here concdes wth the reversble wor of separaton A r 0 of the xture of N 0 oles wth concentraton x 0 nto subsystes wth nuber of oles Nh j and concentratons xj j. The second ter taes nto account rreversblty of the process. A r decreases onotoncally and tends to A r 0 when process duraton and ass transfer coeffcent Rj j ncreases Exaple. Consder separaton of the bnary xture nto pure coponents n te. In ths case N ) x 0 N 0, N ) ( - x 0 )N 0, where x 0 s the concentraton of the ey coponent, xj ) xj ). Fro the forula 4 we get A r )-RTN 0 (x 0 ln x 0 + ( - x 0 )ln(- x 0 )) + N 0 ( x 0 + ( - x 0 ) Rj Rj ) ) A 0 r(x 0 ) + N 0 ( x 0 + ( - x 0 ) Rj Rj ) (43) The estate (eq 43) was derved n ref 8 by solvng the proble of optal separaton of the bnary xture n the gven te n Van t Hoff s thought experent wth ovable pstons and Fgure 6. Scheatc of a contnuous separaton syste. setransparent ebrane where Rj and Rj are the pereablty coeffcents on the frst and second coponent. If flows do not depend explctly on the checal potentals dfferentals, for exaple, are proportonal to the concentratons dfferental, then an estate slar to the one obtaned above can be constructed by solvng the followng auxlary nonlnear prograng proble Here, (P 0, P ) are partal pressures of the coponents n contactng subsystes that depend on the checal potentals dfferentals µ. The flow g depends on the sae dfferentals. Mnus n these probles are sought for dfferent values of constant g > 0 and nonpostve P 0 and P. We denote the nal values of the objectve n each of these probles µ n (g ) as µ * (g ). Ths dependence can be used n the estate (eq 3) of the rreversble wor of separaton Exaple. Assue µ ) RT ln(p 0 /P), g(p 0, P) ) (P 0 - P)/R, and 0 < P < P ax. Let us express P 0 n ters of g and P: µ ) RT ln(rg/p + ) attans ts nu at P ) P ax g. Therefore, µ * (g ) ) RT ln(r g /P ax + ). 4. Potental Applcaton of Obtaned Estates We wll llustrate the possbltes of the applcaton of the derved estates. 4.. Estate of the Power of Separaton n a Contnuous Separaton Syste. Consder a contnuous separaton syste wth the nput flow g 0 wth concentraton x 0 and output flows g j (j ),..., ) wth concentratons x j ) {x j0, x j,..., x j }, (Fgure 6. Here, the teperatures on the nput and output flows are close to each other. Equaton 4 allows us to estate the nal power requred for contnuous separaton n such syste where µ (P 0, P ) f n P 0, P /g (P 0, P ) ) g, ),,... (44) P 0 )R g + P, ), p n ) g j γ j ) g 0, g 0 p 0j + g 0 γ j R j (45) γ j ) (46) p 0j ) g 0 γ j RT [ ln - x 0 ln x 0 ] ) γ j M j (g 0, x j ) (47)

6 6040 J. Phys. Che. B, Vol. 08, No. 9, 004 Tsrln and Kazaov Mass balance equatons yeld The nuber of condtons (eq 48) s -, because the concentraton of one of the coponents s deterned by the condtons (eq 46). If the nuber of flows >, and ther copostons are gven, then the reoval fractons can be chosen n such a way that the power of separaton s nal subject to constrants (eqs 46 and 48). The Lagrange functon of ths proble s here L s the concave functon on γ j, and ts condtons of statonarty deterne the flows that nze the power for separaton for a gven flow s copostons We have lnear equatons for λ 0 and λ [ γ j ) x 0, ),..., -, ), j ) 0,..., (48) { - L ) γ j M j + γ j r j - λ 0 γ j - λ γ j x j} (49) γ j * ) ( λ 0 - M j r j r j (g 0, x j ) ) g 0 λ 0 - M j + λ [ λ 0 - M j r j - R j 4.. Exaple. Assue ) 3, ), g 0 ) ol/s, T ) 300 K, and the copostons and transfer coeffcents are Fro eq 47 we obtan M ) 90, M ) 97, M 3 ) 90, and r ) 05, r ) 580, r 3 ) 37. r j, j ),..., (50) - + λ j] r ) (5) + - λ x r j j)] ) x 0 ),..., - (5) x 0 ) x 0 ) 0.5 x ) 0.9; x ) 0.; Rj )Rj ) ol /(J s) x ) 0.3; x ) 0.7; Rj )Rj ) 0.0 ol /(J s) x 3 ) 0.; x 3 ) 0.9; Rj 3 )Rj 3 ) 0.06 ol /(J s) Equatons 5 and 5 for λ-ultplers tae the for [ λ 0 - M + λ 0 - M + λ 0 - M 3 + r r r 3 [ ( x λ 0 - M λ ( x + x + x 3 r r r 3 )] ) + λ x r r ) + + λ x r r ) + x ( λ 0 - M x 3( λ 0 - M 3 + λ x 3 r ) x 0 3 We obtan λ 0 ) 894, λ ) 83. Ther substtuton n eq 50 yelds γ * ) 0.36, γ * ) 0.64, γ 3 * ) 0 and the correspondng estate for the nal rreversble power of separaton (eq 45) s p n ) 78 J/S 4.3. The Selecton of the Separaton Sequence for a Multcoponent Mxture. In practce, separaton of ultcoponent xtures s often realzed va a sequence of bnary separatons. So, a three-coponent xture s frst separated nto two flows, one of whch does not contan one of the coponents. The second flow s then separated nto two uncoponent flows. The reversble wor of separaton (that corresponds to the power p 0 ) does not depend on the sequence of separaton, because p 0 s deterned by the rates and copostons of the nput and output flows of the syste as a whole. The rreversble coponent of the power p n eq 45 depends on the sequence of separaton and can be used to fnd the optal one. Consder a three-coponent xture wth concentraton x 0 ) (x 0, x 0, x 03 ), and rate g 0 we set to one. We denote the ass transfer coeffcents at the frst and second stages of separaton as R and R. They depend on the constructon of the apparatus. Frst, we assue for splcty that these coeffcents do not depend on the xture s coposton (n the general case they do depend on t). We consder rreversble power consupton for two cases: (a) The frst coponent s frst separated, then the second and the thrd are separated. (b) The second coponent s separated, and then the frst and the thrd are separated. We assue that the separaton at each stage s coplete. We get up to the constant ultpler p a ) p a + p a ) x 0 /R + (x 0 + x 03 ) + (x R 0 + x 03 ) + (x 0 /R + x 03 /R 3 ) (53) The frst two ters n ths su represent the loss of rreversblty durng the frst stage of separaton. For g 0 ) and coplete separaton the output rates of ths stage g and g are x 0 and (x 0 + x 03 ), correspondngly. Consder the frst stage of case a for g 0 ) and coplete separaton and vew the second and thrd coponent as the sae substance wth the output rate x 0 + x 03 ) - x 0. The rreversble expenses (eq 45) are p a ) x 0 + ( - x 0 ) ) x x 0 (54) R R R r 3 )]

7 Irreversble Wor of Separaton J. Phys. Che. B, Vol. 08, No. 9, When the second flow s separated nto two flows ther rates are and the rreversble power s The cobned rreversble power s Slarly n case b we get g ) x 0 ( - x 0 ), g 3 ) x 03 ( - x 0 ) p a ) R ( - x 0 ) (x 0 + x 03 ) p a (x 0, x 0 ) ) x 0 - x x 0 + ( - x 0 - x 0 ) R R ( - x 0 ) p b (x 0, x 0 ) ) x 0 - x x 0 + ( - x 0 - x 0 ) R R ( - x 0 ) The dfferental between these two values s p ab ) p a - p b ) [(x R 0 - x 0 ) - (x 0 - x 0 )] + R ( - x 0 )( - x 0 ) [( - x 0 ) (x 0 + x 03 ) - ( - x 0 ) (x 0 + x 3 03 )] (55) If p ab > 0, then sequence b s preferable. Note that t s not possble to forulate the general rule to choose the optal separaton sequence for a ultcoponent xture, n partcular, on the bass of the reversble wor of separaton. It s necessary here to copare rreversble losses for each sequence Exaple. Assue that the coposton of the nput threecoponent xture s x 0 ) 0.6, x 0 ) 0.3, x 03 ) - x 0 - x 0 ; the ass transfer coeffcents are R ) 0.0 ol /(J s), R ) 0.0 ol /(J s). Fro eq 55 we fnd that the dfference n power between sequences a and b s p ab ) p a - p b )-7.8 J The coparson of the cobned nal rreversble power for the sae ntal data shows that the power for separaton of a xture usng sequence b s hgher than the power used for sequence a, that s, p ab < 0. Thus, sequence a s preferable, and t s better to perfor the coplete separaton by separatng the frst coponent. 5. Ltng Productvty and Mnal Heat Consupton for a Heat-Drven Separaton In any separaton processes a heat engne s used to create the dfferental of the checal potental between the worng body and the reservors (the drvng force of ass transfer). Here, the worng body s heated durng contact wth one reservor and s cooled durng contact wth the other reservor. One can represent the heat-drven separaton syste as a transforer of heat nto the wor of separaton that generates power p, consues heat flow fro hot reservor g +, and rejects flow g - to the cold reservor. Heat transfer coeffcents for contacts wth the hot and cold reservor R + and R - are fxed. It was shown n refs 5 and 6 that the potental of the drect transforaton of heat to wor s lted and the axal generated power for the worng body wth the dstrbuted paraeters s In ths expresson Rj )(R + R - )/(R + +R - ) s the equvalent heat transfer coeffcent for contnuous contact wth the reservors; Rj ) (R + R - )/ ( R + + R -) s the equvalent heat transfer coeffcent for sequental contact. The axal power deternes the heat flow consued fro the hot reservor. Further ncrease of heat consupton for gven values of heat transfer coeffcents requres an ncrease of the teperature dfferental between the reservors and the worng body and reduces the power. The dependence of the used power on the productvty of rreversble separaton processes s onotonc (eq 45). Therefore, the ltng productvty of heat-drven separaton processes corresponds to the axal possble power produced by transforaton of heat nto wor. Further ncrease of heat consupton q + reduces power and therefore reduces the productvty of separaton process. For the Newton (lnear) law of ass transfer and heat-wor transforer the dependence of the power on the heat used 7 s q + (p) ) p η p ) Here, η C ) (T + - T - )/T + s the Carnot effcency, T + and T - are the hot and cold reservor s teperatures, and Rj )(R + R - )/ (R + +R - ) s the equvalent heat transfer coeffcent. The nal heat consupton q + as a functon of productvty g 0 for a heat-drven separaton can be obtaned by substtutng expresson 57 nstead of p n the rght-hand sde of eq 45. The result holds for p e p ax and therefore for g 0 e g 0ax. The duraton here ust not exceed the axal possble duraton. Substtuton of the rght-hand sde of eq 56 nstead of p n eq 45 yelds the axal possble productvty of the syste (where Rj s chosen accordng to the type of contact between the transforer and reservor). We denote We obtan B ) RT j and the ltng productvty s p ax )Rj ( T + - T -) (56) Forulas 58 and 59 allow us to estate the ltng productvty of a heat-drven separaton process for Newton s laws of heat transfer between the worng body and reservors and ass transfer proportonal to the dfferentals n checal potentals (ass transfer s close to sotheral wth the teperature T). 5.. Exaple. Consder heat-drven onoethanade gas cleansng. One of the coponents s absorbed by the cold soluton fro the nput gas xture. Ths soluton s then heated p ( p RjT + + η ) + ( p RjT + + η C ) - 4p RjT + γ j ln, x 0 D ) T j γ j p ax )Rj( T + - T - ) ) Bg 0ax + Dg 0ax -B + g 0ax ) B + 4RjD ( T + - T -) D (57) R j (58) (59)

8 604 J. Phys. Che. B, Vol. 08, No. 9, 004 Tsrln and Kazaov and ths coponent s vaporzed. The nput xture s paraeters are Th )350 K, the ey coponent s olar concentraton x ) 0.5, the rate of xture g 0 ) 5 ol/s. The teperatures of heat suppled/reoved are correspondngly T h ) 400 K, T c ) 300 K, and the heat transfer coeffcents are R + ) J/(s K) and R - ) J/(s K). The concentratons of the ey coponents n the output flows are x ) 0.9, x ) 0.; the ass transfer coeffcents for each of the coponents (ntegral values over the whole contact surface) for the hot and cold reservor s contacts are R ) 0.07 ol /(g s), R ) 0.03 ol /(g s). Because the soluton crculates and s heated and cooled n turns, the ltng power for transforaton of heat nto wor s gven by the expresson 56 wth the correspondng Rj The power for separaton s gven by eq 45. We have p 0 ) RTg 0 The nal wor requred for a syste wth Onsanger s equatons are (see eq 45) p ) g 0 p ax ) 0.7 J s γ j γ j ln I Thus, p ) p 0 + p ).636 J/s < p ax. The wor needed for separaton does not exceed the axal possble value for gven heat transfer coeffcents. Let us estate the nal heat consupton. Fro eq 57 we get If the teperatures of the nput and output flows are not the q + ) 3.46 J s ) J x 0 s ) 7.38 J R j s sae then the nal energy requred for separaton can be estated usng the therodynac balance equatons 3 and 4 and the expresson for σ n (eq 3). 6. Concluson New rreversble estates of the n-prncple ltng possbltes of separaton processes are derved n ths paper. They tae nto account the unavodable rreversblty caused by the fnte rate of flows and heat and ass transfer coeffcents. They also allow us to estate the ltng productvty of a heatdrven separaton and to fnd the ost energy effcent separaton sequence/rege of separaton for a ultcoponent xture. Acnowledgent. Ths wor s drectly supported by a grant fro RFFI (Grant ). References and Notes () Orlov, V. N.; Rozonoer, L. I. Estates of Effcency of Controlled Therodynac Processes Based on the Balance Equatons of Mass, Energy and Entropy. Abstracts of the 0th All-Unon Conference on Control Probles,Moscow, AN SSSR, IPU, 75-76, 986. () Shabadal, P. DeVelopent and Applcaton of the Concept of Entropy; Naua: Moscow, 967. (3) Tsrln, A. M. Optal Processes n Therodynacs and Mcroeconocs; Naua: Moscow, 003. (4) Tsrln, A. M.; Mronova, V. A.; Aeln, S. A.; Kazaov, V. A. Fnte-Te Therodynacs: Condtons of Mnal Dsspaton for Therodynac Process wth Gven Rate. Phys. ReV. E998, 580,. (5) Novov, I. I. The Effcency of Atoc Power Statons. J. Nucl. Energy II (USSR) 958, 7, 5-8. (6) Curzon, F. L.; Ahlborn, B. Effcency of a Carnot Engne at Maxu Power Output. A. J. Phys. 975, 43, -4. (7) Rozonoer, L. I.; Tsrln, A. M. Optal Control of Therodynac Processes. Auto. Reote Control 983, 70-79, 88-0, (8) Aeln, S. A.; Butcer, I. M.; Hoffan, K. H.; Tsrln, A. M. Estates of Ltng Possbltes of Separaton Processes. Theor. Found. Che. Tech. 983, 35, 3, (9) Bosnajaovc, F. Techncal Therodynacs; Holt, Rnehart and Wnston: New Yor, 965; Vol.. (0) Berry, R. S.; Kazaov, V.; Senutycz, S.; Szwast, Z.; Tsrln, A. M. Therodynac Optzaton of Fnte Te Processes, Wley: Chchester, U.K., 999.