17, Thermodynamics For Processes In Finite Time - Ku
266Acc. Chem. Res. 1984, 17, 266-271Thermodynamics for Processes in Finite TimeBJARNEANDRESENPhysics Laboratory ZZ, University of Copenhagen, Universitatsparken 5, 2100 Copenhagen 0 , DenmarkR. STEPHENBERRY*Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Zllinois 60637MARYJo ONDRECHENDepartment of Chemistry, Northeastern University, Boston, Massachusetts 02115PETERSALAMONDepartment of Mathematical Sciences, San Diego State University, San Diego, California 92182Received September 1, 1983Thermodynamics serves several functions for chemistry and ita sister sciences and technologies. The firstof these historically, and still one of the most important,is the provision of limits on the performance of proceases and devices. The very origins of thermodynamicsdeveloped from the creative ways people addressed theproblem of how beat to pump water out of mines.’ Theprovision of limits of performance is the main concernof the work we describe here. Research during the past9 years has led to new ways of finding natural boundson performance under the constraint that the systemoperate at a nonzero rate, thereby giving more realisticbounds than those derived from reversible processes.Some approaches tell us only the limits; others tell usalso the process that would yield those limits. Frequently, we can infer how the lower bound on the energy we pay for operating at a nonzero rate depends onthat rate of operation. There even exist processes forwhich no zero-rate reversible counterpart exists.2The difference between the performance of a reversible process and a real-time process may be considerable, so great that relying on the bounds based on reversible models can be quite misleading. For example,a theoretical heat engine operating between 300 K and2700 K with friction, finite heat conductance betweenthe engine and its reservoirs, and a heat leak: all chosento correspond approximately to an automobile engine,has an ideal Carnot efficiency of (2700-300)/2700 0.89. The efficiency of the engine when operating togenerate maximum power is only 0.33.Another example is the frequent use of roasting withsubsequent reduction in ore processing, both processesBjarne Andresen was born In copenhagen in 1949. He studied at We&leyan Unhrerstty and at Cal Tech prlor to receMng hls Ph.D. from the University of Copenhagen in 1974. He is now a Lecturer in physics at the H. C.0rsted Institute, University of Copenhagen.R. Stephen Berry was born in Denver. Colorado. In 1931. He recehred hisA.B. (1952) and hk Ph.D.(1956) from W a r d Unlverslty. He was an Instructor at Haward and the Unhrerslty of Mchlgan and then moved to YaleUniversHy as an Assistant Professor. In 1964 he moved to the Unhrersity ofChicago and became a Professor thefe in 1967, and he remains there.Mary Jo W e c h e n was born In PMladelphia in 1953. She recehred herB.A. at Reed College In 1974 and her Ph.D. at Northwestern University in1978. She is now an Asslstanl Professor at Noctheastern University.Peter Salamon recehred his B.A. in 1971 from Llndenwood College, hisM.S. from Drexel Unhrersity in 1972. and his Ph.D. from the Unhrersity ofChicago in 1978. In 1981 he became Associate Professor at San DlegoState Universlty and remains there now.which usually are carried out with large expendituresof energy, even in cases when the metal has lower freeenergy content than the ore, and the refining thereforein principle could have supplied work. In short, theconstxaints attending operating in finite time reduce thelimits on performance so much that it is desirable tofind general methods to calculate more realistic limitsthat reflect those constraints.Extension to Finite-Time ThermodynamicsThe first natural pathway to those more realisticcriteria is the extension of the concepts of thermodynamic potential* and availability5 to incorporate theconstraints on time or rate with the otherconstraints-such as constant temperature, pressure,or volume-in the construction of the potential (seelater). Such extensions of thermodynamic potentials,traditionally functions only of the state variables of thesystem itself, to include time or rate, are justified bysome existence theorems that need not be reproducedhere.4 In many cases one wants more than just themaximum work that can be produced in a certain time.One may need to know the detailed time path thatyields this maximum work. The tool to obtain this isoptimal control theory. Applying this method is usuallymuch more involved than finding a potential and mustoften be done numerically, but in return the optimaltime path contains all the information about the optimized system, including the maximum work it canprovide-much as the wave function is the source of allinformation in quantum mechanics. The optimizationof the Otto cycle illustrates this procedure! In contrastto the usual parameter optimizations done in engi(1)D.S. L. Cardwell, “From Watt to Clausius”, Cornel1 UniversityPress, Ithaca, NY, 1971.(2)J. Wheatlev. T.Hofler. G. W. Swift. and A. Mieliori. Phvs. Reu.Lett., 50,499 (19g3);M. Mozurkewich and R.S. B e r g , J.Appi. Phys.,54,3651 (1983).(3) B. Andresen, P. Salamon, and R.S. Berry, J. Chem. Phys., 66,1571(1977).(4)P.Salamon, B. Andresen, and R.S. Berry, Phys. Reu. A , 15,2094(1977).(5)B. Andresen, M . H.Rubin, and R. S. Berry, J . Phys. Chem., 87,2704 (1983).(6)M. Mozurkewich and R. S . Berry, R o c . Nutl. Acad. Sci. U.S.A.,78, 1986 (1981);J.Appl. Phys., 53, 34 (1982).0001-4842/84/0117-0266 01.50/00 1984 American Chemical Society
Vol. 17, 1984Finite- Time Thermodynamicsneering design, in this calculation, the parameters ofperiod, heat transfer, fuel assumption, etc., were fixedat conventional values, and only the time path of thepiston was varied from its conventional sinusoidal formto optimize the power. Optimizing the piston motionalone could increase the power output of the engine by8-15%.The maximum work a system can do in a given process is one index of performance; it is not the only onewe could choose. Some others are as follows: the theoretical, or Carnot, efficiency qc 1- (TL/TH) (idealwork, W)/(heat to produce W); the thermal efficiencyq (work actually delivered, “,)/(heat to produce We);the effectiveness 6 “,/(change in availability); thepower; the dissipated availability (availability lost andnot transformed to work); the entropy produced; andthe net revenue. Some criteria of performance, such aspower, are not meaningful for reversible processes.Others, such as entropy generated, can be separatedinto loss terms that have no relation to the rate of theprocess and other loss terms that depend specificallyon time or rate. Entropy of mixing is of the first kind,and ohmic heating and other friction-based heat generation processes are clearly of the second kind. Asystem optimized by one criterion need not be optimalwith respect to others. A later section illustrates differences that follow from different choices of criterion.One goal of finite-time thermodynamics is the identification of generic models that can play roles of greatgenerality, in the way the Carnot engine is the archetypical heat engine in reversible thermodynamics.Assuredly, one uses many other idealized models suchas the air-standard cycles (Otto, Diesel, Stirling, Brayton, etc.) to describe classes of real engines approximately but more accurately than with the Carnot engine, even when retaining the conditions of reversibility.With the introduction of time constraints and thecorresponding time-dependent loss processes, the possibilities might seem to open without limit. It appearsnow, however, that one may need only a few idealizedparadigms for finite-time behavior.One generic class consists of mechanical processesdriven by heat, for which the irreversibilities occur entirely a t the boundaries between system and surroundings. These systems are called “endoreversible”,and two subclasses have been identified until now, thosedominated by constraints on heat transfer between thesystem and its reservoirs and those dominated byfriction and heat leaks. Systems with both kinds oflosses divide3 into sets dominated by one kind of irreversibility or the other.Mechanical Processes Driven by HeatWe will illustrate some of the consequences of finite-time operation by examining several generic processes. First, consider systems consisting of a workingfluid which may be connected to a range of heat reservoirs, operating either in a cycle (e.g., a reciprocatingsteam engine) or in continuous flow (e.g., a steam turbine). The most general approach to such heat enginesis provided by the black box model known as the tricycle f r m a l i s mwhich, only involves energy balanceand the second law, and keeps track of entropy flow andgeneration, without requiring any specific knowledgeof how the process works in detail. An early model ofa laser used this idea7 but without developing its gen-267ttt939,qfFigure 1. Tricycle decomposition of a general heat process intoits reversible and irreversible parts. Each side of the tricyclerepresents heat flow qi into a reservoir with temperature Ti. Theinput from the hot reservoir is q2; q b is set equal to q2 to makeall losses accountable to the work (ql) and waste heat (q3)flows.The first tricycle on the right, being reversible,has zero entropyproduction per cycle.described as flows between a working system and threereservoirs. In this way simple heat engines can berepresented by a triangle, standing for the system, withflows through each side of the triangle representingflows between the system and one of the (heat or work)reservoirs. This pictorial representation of a cyclicprocess is the “tricycle”. More complex systems mayrequire more than one triangle for their representation.The crux of the tricycle method is the decompositionof the real process into a reversible part and an irreversible part, as illustrated in Figure 1. Some processes,such as separation, are best decomposed into threeparts: a reversible part, a part containing the inherentlyirreversible contributions that occur even with infinitelyslow operation, and a third part that contains thetime-dependent irreversibilities. By choosing genericexpressions for the most important losses-friction,finite heat conductance, and heat leakage-in terms ofsystem parameters and the cycle time, it becomespossible to optimize the performance of the engine.The important finding3 is that the optimal operationof the general heat engine shows a bifurcation, as afunction of the ratio of friction to heat resistance, intotwo classes: those which are friction dominated andthose which are dominated by resistance to heat flow.Curzon and Ahlborn* considered a model of an enginewhich is a particularly simple yet rich elaboration of theCarnot engine. The Curzon-Ahlborn engine has a cycleof two adiabats and two isotherms and suffers losses dueonly to thermal resistance between the reservoirs attemperatures TH and TLand the working fluid. Weshall refer to such an engine as a CA engine. Its Carnotefficiency isoc 1 - TL/TH(1)obtained for infinitely slow operation. Curzon andAhlborn* asked not about maximum efficiency but instead about the efficiency when the engine producesmaximum power and found it to beow 1 - ( T L / T H ) l I 2independent of the thermal conductance K. This latterefficiency, Curzon and Ahlborn found, correspondsconsiderably closer to the actual performance of powerplants than does o,!In general, the CA engine can be optimized accordingto a variety of criteria, subject to the constraints of thereservoir temperatures and a fixed time T for the period(7) J. Geusic, E. 0. Schulz-duBois, and H. E. D. Scovil, Phys. Reo., 156.343 (1967).(8) F. L. Curzon and B. Ahlborn, Am. J . Phys., 43, 22 (1975).
Andresen et al.268LA-LII \(min. entropy productionLmin. loss of availability. max. revenuemax. efficiencymax. effectiveness(max. powerFigure 2. Distribution of the times an endoreversible engine isin contact with its hot reservoir (7J and its cold reservoir (72) whendifferent criteria of performance are optimized. The location ofsolutions corresponding to maximum revenue (see text) is situatedin the shaded region.of the cycle. Control to attain the optimum behavioris achieved by varying the portion of 7 allotted to eachbranch of the cycle. Adiabatic branches are ordinarilytaken to be instantaneous in analyses of this kind, sinceno dissipation is associated with them. Thus the optimization can be described by the durations of thehigh- and low-temperature isotherms 71 and 7 2 where71 7 2 7; a graph showing the optimag is presentedin Figure 2. The loci of solutions for the CA engine,determined by maximizing power, maximizing efficiency, minimizing entropy production, and minimizingloss of availability are straight lines. Note that for CAengines minimizing entropy production is equivalent tominimizing loss of availability.1 We can identify the rate of revenue R for the processas a linear combination of power and rate of loss ofavailability, R aP PA, because the power P is asaleable good-and the availability A represents inputresources, so A is negative; a and B represent the pricesof power and availability? Then, for any pair of positiveprices a and P optimal R defines a line between the lineof maximized power and the line of minimized loss ofavailability. Hence all economically optimal solutionsfor any pair of prices a,P lie inside the shaded regionof Figure 2. If resources are free, R is equivalent to P,and if power is given away, R is equivalent to A. Thisis an example of how one can bound the optimal solution with limits determined by nature, without knowingthe exact criterion of performance.A more general class of heat engines whose reservoirshave variable temperatures can be operated to minimizethe total entropy production.1 When this optimizationis applied, the rate of entropy production is a constant,given by(3)Accounts of Chemical Researchersible engine in which the working fluid receives generalized fluxes from reservoirs, provided such fluxes arefunctions, however nonlinear, of the generalized forcesat the boundary but are not functions of.time or of anytime derivatives of the forces. The maximum power ofsuch engines is given by1’Pmax K Variance (Tre8(t))1/2(4)where Variance refers to the time average of the squaret ) )1/2 from its mean value.deviation of (T,,,(These examples have either assumed a specific cycle(Carnot in the case of CA engines) or focused on thetemperature variations of the working fluid, sidestepping the volume variations necessary to achieve themand their possible restrictions. Rubin12made the firstcomplete optimal control calculations of the generalendoreversible engine, the only restrictions being limitson piston velocity, heat conductance between theworking fluid and the reservoir, and the range ofavailable reservoir temperatures. Fairen and Ross13have studied constraints due to inertial effects. Aqualitative summary of these findings is that theworking fluid should optimally accept and reject heatisothermally, at temperatures which minimize the lossesacross the thermal resistance, and “jump” from oneisotherm to the other as quickly as the constraints allow.If there are no limits on piston velocity, this meansinstantaneously, i.e., adiabatically, thus recovering theinterior Carnot cycle assumed in the Curzon-Ahlbornanalysis.8 In addition all endoreversible engines havethe same staging property as Carnot engines: If oneputs two or more engines of the same kind in sequence,then the whole system behaves as a single engine of thatkind.14 This makes the endoreversible engine a uniquebuilding block for analyzing larger finite-time thermodynamic systems.Much progress has also been made on the form of theoptimal trajectories for specific working fluids. Suchanalyses typically include considerations of friction 6,15,16inertia,13 and other losses and/or constraintsof operation and must eventually resort to numericalmethods.Processes Driven by Chemical ReactionsMany familiar power-producing systems are drivenby heat generated by an exothermic chemical reaction.Such engines have two important features. Firstly, theheat carrier is generally finite in size and, secondly,there is usually a limit on the rate of production of theheat used to drive the engine.The source of heat for a chemically driven engine isthe reaction product mixture. Because this has finiteheat capacity, although it is initially at temperature TH,heat from the mixture cannot be converted into workwith the Carnot efficiency qc, since the temperature ofthe high-temperature reservoir decreases as heat istransferred from the mixture. For a reversible engineworking between a heat source with finite, constant heatLwhere aiis the entropy change of the working fluidalong branch i. This result holds also for an endorev(9)P.Salmon and A. Nitzan, J. Chem. Phys., 74, 3546 (1981).(10)P.Salmon, A. Nitzan, B. Andresen, and R. S. Berry, Phys. Rev.A , 21, 2115 (1980).(11)P.Salmon, Y. B. Band, and 0. Kafri, J. Appl. Phys., 53, 197(1982).(12)M.H.Rubin, Phys. Rev. A, 19,1272 (1979);ibid., 19,1277(1979);ibid., 22, 1741 (1980).(13)V. Fairen and J. Ross, J. Chen. Phys., 75,5485 (1981).(14)M.H.Rubin and B. Andresen, J. Appl. Phys., 53,1 (1982).(15)Y.B. Band, 0. Kafii, and P. Salmon, J.Appl. Phys., 53,8(1982).(16)V. Fairen and J. Ross, J. Chem. Phys., 75,5490(1981).
269Finite- Time ThermodynamicsVol. 17, 1984t 2000//[peak power 1400/I200400600Distance Traversed ( c e l l s )Figure 3. Optimal temperature profiie along the reactor tubefor an exothermic reaction with a temperature-dependent rateconstant obeying an Arrhenius law. The dashed c w e with t lo00 corresponds to maximum power output.capacity initially at T H and an infinitely large coldreservoir at T L , the reversible efficiency is given by17for the case where the source is cooled all the way downto TL. Further discussion of the production of workfrom finite heat sources is given in ref 17 and 18.We have thus far discussed systems whose time dependence enters through friction, heat conductance, orheat loss. In other systems, performance is limited bythe rate at which heat is generated. Consider a continuous flow reactor that supplies heat to an engine-forexample, a combustion-heated, steam-driven electricgenerator. For a finite reaction tube and a nonzero flowrate, the reaction will not, in general, go to completion,and the temperature of the emerging product mixturewill depend upon the extent of the reaction. On theother hand the reactor must be run at a nonzero flowrate in order to produce power which clearly entailssome sacrifice of fuel efficiency.Conditions have been established for the achievementof maximum power from such aFor reactionsfollowing first-order, fmborder reversible, second-order,and second-order bilinear kinetics, the maximum poweris attained at a finite, positive flow rate; maximum fuelefficiency and minimum entropy production areachieved in the uninteresting limit of no flow. As onemight expect, power production approaches zero at veryslow flow rates (because heat is delivered to the enginevery slowly) and at very high flow rates (because thereaction hasn't time to go to completion). A temperature profile for the reactor (temperature as a functionof distance along the reactor tube) in Figure 3 showshow the temperature increases slowly at first and thenincreases quite suddenly because of the positive feedback due to the temperature dependence of the rateconstant. At the flow rate corresponding to maximumpower, the sudden rise in temperature-the(17)M. J. Ondrechen, B. Andresen, M. Monukewich, and R. S. Berry,Am. J. Phys., 49,681(1981);F.d'Isep and L. Sertorio, Nuouo CimentoB , 67,41 (1982).(18)M. J. Ondrechen, M. H. Rubin, and Y. B. Band, J.Chem. Phys.,78,4721(1983).(19)M. J. Ondrechen, R. S. Berry, and B. Andresen, J. Chem. Phys.,72,5118(1980);M.J. Ondrechen, B. Andresen, and R. S. Berry, J. Chem.Phys., 73,5838 (1980).qwJwFigure 4. Diagram of a distillation column with feed enteringat F, heavy component leaving at B, and light component leavingat D. Heat is supplied at temperature TBand withdrawn at TD.In the figure W for waste is used instead of B for bottoms."combustion zone"-occurs just before the end of thereactor tube. The maximum power obtainable fromsuch a process is a very sensitive and strictly decreasingfunction of the activation energy of the reaction whichdrives the system. Furthermore, the flow rate yieldingmaximum power is itself a decreasing function of theactivation energy.Systems Doing Chemical WorkThe theory of separation, and in particular isotopeseparation by distillation, got a tremendous boostduring and after World War II for obvious reasons. Thedevelopment was summed up in the 1950's by its maincontributors, Benedict and Pigford20 and Cohen andMurphy.21 Relatively few new ideas have appearedsince then.In conventional binary distillation feed of composition (mole fraction of light component) XF is separatedinto distillate of composition xD and bottoms of composition XB by passing heat Q through the distillationcolumn from temperature T B at the bottom to T D at thetop (see Figure 4). There is a trade-off, largely economic, between the number of plates in the column, i.e.,capital, and heat load required to perform a certainseparation, Le., operating cost. Even when heat requirements are pushed to their minimum, the effectiveness of separating a mixture of similar compoundsinto pure products by distillation, expressed as the reversible work of separation divided by the workequivalent of the heat used in the distillation process(heat used multiplied by qc(TB, TD)), is approximatelyonlyE - [ x F In X F (1- x F ) In (1- XF)] 1(6)This function has a maximum of 70% for an equimolarmixture, xF 0.5. The remaining 30% of the heat islost due to the inherent thermodynamics of the processand cannot be avoided withouth altering the distillationprocess itself, e.g., by adding extra boilers and condensers along the column.(20) M. Benedict and T. F. Pigford, 'Nuclear Chemical Engineering",McGraw-Hill, New York, 1957.(21)K.Cohen and G. M. Murphy, T h e The0 of Isotope Separationaa Applied to the Large-Scale Production of Uz", McGraw-Hill, NewYork, 1951.
Andresen et al.270Some work22has been spent in the past decade pinpointing the optimal position of such extra boilers andcondensers for complete separation of very similarcomponents. However, only very recently has the morerealistic case of dissimilar components and only partialseparation beenwith the goal of minimizingthe entropy production. It is found that the possiblesaving of entropy production per unit of feed by addition of intermediate heat exchangers is an almostconstant function of feed composition roughly in theinterval 0.25 xF 0.75. A very interesting result isthat this function has a (gentle) minimum at XF 0.5for xD 0.89 which turns into a (gentle) maximum forXD 0.89.Related to our own work on competing processess isthat of Robert T. Ross and co-workers" on the efficiency of solar energy converters. They analyze a modelin which solar energy is absorbed by a quantum systemunder less than reversible conditions and where someof the absorbed energy is lost by retransmission and bynonradiative recombination, much as in photosynthesis.The analysis yields the optimal spectral absorbance ofthe device for a range of decay rates and the associatedconversion efficiency. The most spectacular result isthat the efficiency can be made to exceed the idealthermal efficiency by rapid equilibration among theelectronically excited states which reduces reradiationand actually leaves the chemical potential of the excitedelectronic band below that of the ground band. A similar result for a lasing system has been discussed byBen-Shaul and L e i n e . The effects of an oscillatory reaction path have beenstudied by John Ross and co-workers, both for heatand for chemical systems.27 In both casesresonances are found where either dissipation is aminimum or power production is a maximum, frequently even exceeding the performance of the optimalmonotonic reaction path. This means that in somecases batch operation of a process may prove advantageous compared with steady-state peration. 't "Potential StructuresOn a more abstract level the availability (or exergy)of any chemical system has been generalized for finite-time processes.6 First let us recall that the availability, A , of a system is the maximum amount of workthat can be extracted reversibly while it comes toequilibrium with its surroundings. With subscript zerodenoting environment quantities, the traditionalavailability isA U - ToS PoV-CpoinTii(7)The decrease in availability of a system in going frominitial state i to final state f is then the reversible(22)Zs.Fonyo and P. Foldes, Acta Chin. Acad. Sei. Hung., 81,103(1974);F. Kayihan, AIChE Symp. Ser., 76, 1 (1980).(23)0.C. Mullins and R. S. Berry, J. Phys. Chem., 88, 723 (1984).(24)R.T. Roes and J. M. Collins, J. Appl. Phys., 51,4504(1980);R.T. Ross and A. J. Nozik, J. Appl. Phys., 53, 3813 (1982).(25)A. Ben-Shad and R. D. Levine, J. Non-Equil.Thennodyn., 4,363(1979).(26)P.H.Richter and J. Roes, J. Chem. Phys., 69, 5521 (1978).(27)Y.Termonia and J. Rosa, J. Chem. Phys., 74,2339 (1981);R o c .Natl. Acad. Sci. U.S.A., 78,3663 (1981);P.H.Richter, P. Rehmue, andJ. Ross, B o g . Theor. Phys., 66, 386 (1981).(28) J. S. Su and A. J. Engel, AZChE Symp. Ser., 76 (192),6 (1980).Accounts of Chemical Research(maximum) work that can be extracted in the processregardless of pathw,,, -AA(8)These two qualities are very desirable to preserve in thefinite-time generalization. This is accomplished byoptimizing the Tolman-Fine expression for the secondiaw29(9)where p is the (vector) set of generalized forces, dz' isthe (vector) set of generalize4 displacements corresponding to the components of F, Tois the temperatureof the surroundings into which all heat eventually goes,and S, is the total entropy of the system plus surroundings. The integrals are carried out along the pathtaken by the system. Then for an arbitrary process withno mass transferW -AA-TostfS,dt(10)t,which leads us to defiie the finite-time availability Aof a system undergoing any of a certain class of processes Bo in time 7 to be the maximum work extractableby any process within the class Bo:where the maximum is understood to be taken subjectto the constraints of the problem, including time or rateconstraints. We thus define the generalized (finitetime) availability A by changing eq 9 from a statementinto a recipe for computing that generalized availabilityfrom the dynamical equations for work, the constraints,and the equations governing the entropy production ofthe system.Another finite-time extension of thermodynamicswith immediate chemical applications is that of thermodynamic potentials! We are used to calculating thereversible work output of a process from accompanyingchanges in, e.g., the Gibbs or Helmholz free energies forconstant T, P and constant T, V processes, respectively.These potentials are obtained from the incompletedifferential dW P dV by adding integrating termsg dy, e.g., V dP for constant pressure and S d T forconstant temperature. However, this so-called Legendretransform is valid for any constraint dy 0, includingconstraints on time, complicated expressions beingconstant, or differential expressions vanishing, e.g.,arising from equations of motion expressed as differential equations. This complete generality opens up thepossibility of including the dynamics and nontrivialConstraints of a chemical process in its generalizedthermodynamic potential and thus being able to calculate its energetics without a complete kinetic analysis.Abstract geometry can be used to set a lower limit onthe amount of availability which must be lost in anyprocess leading from a specified initial state i to a finalequilibrium state f via states of local thermodynamice q i l i b r i u m . (This excludes turbulent processesduring which the local thermodynamic variables are not(29)R. C. Tolman and P. C. Fine, Rev. Mod. Phys., 20, 51 (1948).(30)P.Salamon and R. S. Berry, Phys. Rev. Lett., 51, 1127 (1983).
Acc. Chem. Res. 1984,17, 271-277well-defined.) The second derivatives d2U/ (dXi dXj)of the internal energy U as a function of all the extensive variables Xi have been shown by Weinhold31tohave the properties of d metric, which can be used toconstruct a distance D(f, i) (shortest length) betweenstates i and f,32 The dissipated availability is thenAA 1 D2(c/r)(12)where c is a mean relaxation time of the system, andr is the duration of the process. For endoreversiblesystems the bound is strengthened toAA 1 L2( /?)(13)where L is the length of the actual path traversed fromi to f. These expression give a direct measure of the costof finite process time, in terms of internal relaxationtimes and equilibrium properties.Concluding RemarksIn this Account we have pointed out the inadequacyof reversible thermodynamics to describe processeswhich proceed at nonvanishing rates and have pointed(31) F. Weinhold, J. Chem. Phys., 63, 2479 (1975).(32) P. Salamon, B. Andresen, P. D. Gait, and R. S. Berry, J . Chem.Phys., 73, 1001, 54073 (1980).271out a number of new methods to treat this situation.The primary goal has been to obtain bounds of performance which are more realistic than the reversibleones. Some of the finite-time procedures are generalizations of traditional quantities, like potentials andavailability; others are entirely new, like the thermodynamic length.Since the central ideas of reversible thermodynamicsare retained in finite-time thermodynamics, we arecontinuing our attempts to generalize traditional concepts to include time. Especially important are connections to statistical mechanics and irreversible thermodynamics, e.g., investigating the finite-time contentof Keizer's
its reversible and irreversible parts. Each side of the tricycle represents heat flow qi into a reservoir with temperature Ti. The input from the hot reservoir is q2; qb is set equal to q2 to make all losses accountable to the work (ql) and waste heat (q3) flows. The first tricycle on the right, being reversible, has zero entropy production per .
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1.4 Second Law of Thermodynamics 1.5 Ideal Gas Readings: M.J. Moran and H.N. Shapiro, Fundamentals of Engineering Thermodynamics,3rd ed., John Wiley & Sons, Inc., or Other thermodynamics texts 1.1 Introduction 1.1.1 Thermodynamics Thermodynamics is the science devoted to the study of energy, its transformations, and its
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Hotell För hotell anges de tre klasserna A/B, C och D. Det betyder att den "normala" standarden C är acceptabel men att motiven för en högre standard är starka. Ljudklass C motsvarar de tidigare normkraven för hotell, ljudklass A/B motsvarar kraven för moderna hotell med hög standard och ljudklass D kan användas vid
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Reversible and Irreversible processes First law of Thermodynamics Second law of Thermodynamics Entropy Thermodynamic Potential Third Law of Thermodynamics Phase diagram 84/120 Equivalent second law of thermodynamics W Q 1 1 for any heat engine. Heat cannot completely be converted to mechanical work. Second Law can be formulated as: (A .
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