EMERGY SYNTHESIS 2: Theory and Applications of the Emergy Methodology. The Center for Environmental Policy

Transcription

1 EMERGY SYNTHESIS 2: Theory an Alications of the Emergy Methoology Proceeings from the Secon Biennial Emergy Analysis Research Conference, Gainesville, Floria, Setember, 2001 Eite by Mark T Brown niversity of Floria Gainesville, Floria Associate Eitors Howar T Oum niversity of Floria Gainesville, Floria Davi Tilley niversity of Marylan College Park, Marylan Sergio lgiati niversity of Siena Siena, Italy December 2003 The Center for Environmental Policy Deartment of Environmental Engineering Sciences niversity of Floria Gainesville, FL The Center for Environmental Policy PO Box vi

2 2 Mathematical Formulation of the Maximum Em-Power Princile Corrao Giannantoni ABSTRACT After having briefly recalle the main results shown in the revious Conference (a rigorous efinition of Emergy in mathematical terms an a general formulation of the Emergy Balance equation) the aer resents an extremely general mathematical formulation of the Maximum Em-Power Princile Such a formulation allows us to analyze in articular (but not exclusively): i) in what sense the maximum flow has to be unerstoo, not only in steay state conitions but also in stationary an variable conitions; ii) what recirocal role Transformity an Exergy lay in maximizing such a flow; iii) in what sense such an extremely General Princile can be interrete (but only reuctively) as a traitional Thermoynamic Princile The last asect enables us to oint out the ifferent an wier resuositions of the M Em-P Princile (as a more general Thermoynamic Princile) in comarison with the other Thermoynamic Princiles (esecially the First an the Secon ones) an furnishes a clear answer to the (aarently) contraictory assertions between the Maximum Em-Power Princile an the Minimum Action Princile In aition, such a mathematical formulation throws new light on the thermoynamic concets of orer an isorer, an aves the way to a better unerstaning of the so-calle Fifth Princile 1 INTRODCTION This aer exressly eals with the mathematical formulation of the Maximum Em-Power Princile To this urose it is worth starting from the basic results alreay achieve an resente at the First Emergy Research Conference (Giannantoni, 2000a), which constitute the most correct resuositions obtaine for such a ossible formulation Let us recall two of them very synthetically: a general mathematical efinition of Emergy an the structure of the global Emergy Balance Equation i) A rigorous an general efinition of Emergy, vali in whatever variable conitions, can be given by the following exression where Ex eq ( τ ) is efine as t Em = Exeq( τ) τ (11) Exeq( τ) = c( x, y, z, τ) ρ( x, y, z, τ) ex( x, y, z, τ) 3 V (12) D ( τ )

3 an is the instantaneous equivalent Exergy Power use u uring the rocess of generating a secific rouct Eqs (11) an (12) translate, in mathematical terms, the general efinition of Emergy given by H T Oum as the total solar equivalentavailable Energy irectly an inirectly use u to generate a secific form of Energy (or rouct)ª In fact, the coefficient which aears in Eq (12) is a imensional structural factor (whose imensions are sej/j, that is solar emergy joules er Joule) an eens, among other things, on co-injection or co-rouction factors It is thus efine in such a way as to summarize all the rules of Emergetic Algebra (this is the reason for the term equivalent ); D ( τ ) is the Domain of integration which efines the quantity of the consiere matter, is the mass ensity, is the secific Exergy, while Newton s ot notation in Eq (11) stans for the total erivative with resect to time Therefore, when we assume a Lagrangian ersective an steay state conitions, Eqs (11) an (12) efine the traitional an usual concet of Emergy, while uner whatever variable conitions they efine Emergy in its wiest an most general concetion ii) A general Global Balance Equation for any System mae u of n sub-systems can be written as follows m n α j α j Em uj γ k γ k k u u um D βl βl l t A t Em ( ) Φ ( )= y 1, 2,, ( ) s ( ) (13) j = 1 k = 1 sej where sec Φ ( k u u u ) 1, 2,, m is the equivalent Source Term1 relative to the k- sub-system, which is exresse as a linear combination of all the real sub-system contributions Φ n k 1 2 m kr r = 1 ( u, u,, u )= λ Φ (14), r D s while A (t) is the Global Accumulation Term ue to all the istinct contributions of the n subsystems, thus it is given by an aroriate sum of equivalent accumulation terms, each one (in turn) exresse as a linear combination of all the real sub-system contributions n A = δi δi AD ( AD, AD,, AD )= δi δi εij A n 1 2 D (15) i= 1 i= 1 j = 1 s i n j where, are the co-injection an co-rouction coefficients for each sub-system are their associate re-normalization factors (referre to the Whole System), δ i are the ertinent weights in the corresoning sub-system balance equation, δ are the associate re-normalization factors i λ, kr ε are the secific incience coefficients ij Moreover, it is worth recalling that the consiere Global Balance Equation (13), in aition to accounting for inut an outut quantities as a global result of the ifferent co-injective or co-rouctive sub-system structures, contemorarily accounts for three istinct aitional contributions: - Accumulation Terms, amlifie by both the rouctive an connective structure of the System; - Source Terms, characterize by similar (rouctive an connective) amlification effects; n l = 1

4 - Circulating Emergy Flow, which quantifies the Flow of Information through fee-back athways The latter quantity, if Eq (13) is alreay structure in its stanar form2, may be efine as = ( ) ( l ) Em β β Em y Em y circ l l l l = 1 l = 1 an corresons to the Flow of Information through the fee-back athways of the System On the basis of such resuositions we will firstly try to answer a funamental question 2 WHAT EXACTLY IS MAXIMM? This is certainly not a trivial question In fact the general Global Balance Equation (13), which is vali for any Comlex System, may be written in several forms by organizing the ifferent terms in such a way as they coul aear, accoring to secific exigencies, either on the first or on the secon sie of the equation Let us consier some of these forms In aition to Eq (13) which, restructure as follows through Eq (16), may be enominate as form A A) m n α j α j Em( uj) γ k γ k Φ k( u1, u2,, um)= AD ( t) Emcirc Em y s ( l ) j = 1 k = 1 t l = 1 (22) we coul also consier other main forms, for instance the following ones: (16) B) n m γ k γ k Φ k u1, u2,, um AD ( t) Emcirc Em yl α j α j Em u s t k = 1 C) n k = 1 ( )= ( ) ( j) l = 1 j = 1 m γ k γ k k( u u um) α j α j Em( uj) Em( yl )= D t A t Em Φ 1, 2,, ( ) s D) n k = 1 j = 1 l = 1 m γ k γ k Φ k( u u um) α j α j Em( uj) Em( yl ) Emcirc = D t A t 1, 2,, ( ) s j = 1 l = 1 circ (23) (24) (25) which o not eviently exhaust all the various other ossibilities If we now consier the form A as the reference structure in orer to formulate the M Em-P P, this assumtion is equivalent to saying that the System tens to maximize the sum of inut an autogenerate Emergy flows (first sie) in orer to maximize the rate of accumulate Emergy together with circulating an outut Emergy flows (secon sie) Analogously, if we take the other consiere forms as reference structure, the ertaining formulation of the M Em-P P woul resectively assert that:

5 B) The System tens to maximize the autogenerate Emergy flow (first sie) in orer to maximize the rate of accumulate Emergy, together with circulating an outut Emergy flows, at net of the equivalent inut one (secon sie) ; C) The System tens to maximize the autogenerate Emergy lus the net inut/outut Emergy flows (first sie) in orer to maximize the rate of accumulate an circulating Emergy in the system (secon sie) ; D) The System tens to maximize the autogenerate Emergy lus the net inut/outut Emergy flows at net of circulating Emergy flow (first sie), in orer to maximize the rate of accumulate Emergy in the system (secon sie) Such ifferent ossible statements allow us to reformulate the initial question in a more correct form: what is the mathematical structure that is otentially more suitable to best translate Oum s formulation of the Princile uner consieration? We will now try to show that form B is the most suitable starting structure in orer to formulate such a general reference rincile for self-organizing systems which is known as Maximum Em-Power Princile 3 SEFL EMERGY AND PROCESSED EMERGY In many ublications HT Oum reeately asserts that Em-ower has to be unerstoo as useful Emergy ower This coul lea us to think (erroneously) that the Emergy Balance in form A is the one which is more suitable to reresent such a concet, esecially if we consier the terms which aear on the secon sie of Eq (22) But it is also evient that, if the System has all the Source Terms Φ ( k u u u ) 1, 2,, m equal to zero, it is intrinsically unable to organize inut resources, so that it simly transforms the equivalent inut Emergy flow into other forms of flow, without any incremental Emergy contribution that might be really useful to imrove either its internal structure or its external Emergy roucts It is not a self-organizing system at all, but it is only a mechanical transucer In fact, uner such conitions, all the ertinent co-rouction coefficients ( β l ) are all equal to 1, as well as the corresoning re-normalization factors ( β ) Thus Eq (16) imlies that = ( ) ( l )= Em β β Em y Em y circ l l l l = 1 l = 1 l 0 (31) Consequently: a) In steay state conitions such a system merely transfers inut Emergy flow irectly into outut; b) In variable conitions inut Emergy flow is simly transforme into the rate of Accumulate Emergy an outut Emergy flow, but without showing any organizing henomenon (it acts like a mass anchore to a sring an contemorarily subjecte to an external forcing energy source) In aition, if the outut flow is ersistently equal to zero, the system reuces to an Emergy storage (which eviently cannot be efine as a self-organizing system) We may thus conclue that a self-organizing System (in its most meaningful sense) is only the one that manages inut resources by increasing them by an aitional ositive an net contribution which is the only really useful one to increase both its rate of Accumulate an Circulating Emergy flow an its outut Emergy flows In other wors, what is really useful is not simly an inut/outut Emergy transfer, but the net contribution given by the internal generation rocess of Emergy which, through an aroriate organization of inut resources, finalizes everything to a net increase of Emergy both in terms of rate of Accumulation, Circulation an outut Prouction It shoul now be clear that the Balance Equation in form B is the one which is secifically inicate to exress Oum s finings At the same time, if we take into consieration that, aart from the

6 Emergy Source Term, all the other forms of Emergy are involve in such a (re)organization rocess, we may consequently name them by a comrehensive term such as rocesse Emergy flows (obviously each flow is unerstoo as accounte for with its own secific algebraic sign) We can now roose the researche formulation of the M Em-P Princile 4 MATHEMATICAL FORMLATION OF THE MAXIMM EM-POWER PRINCIPLE Eq (23) can be at first re-organize in a more synthetic an comact form In fact such characteristics, as alreay ointe out in Giannantoni (2000a), are articularly necessary in orer to clearly answer the question as to whether the Maximum Em-Power Princile is really a Thermoynamic Princile or not At the same time it is also esirable for such a formulation to be structure in this way for the following aitional reasons: to facilitate its comarison with the structure of the other Thermoynamic Princiles (esecially the First an the Secon ones); to inclue the most general ossible conitions ertaining to very Comlex Systems, concerning both their time evolution behavior an sace structure organization The latter consieration esecially refers to the fact that, if we want to stuy a living organism (eg a lant or an animal) or even an organ (eg, the liver or the brain) as a whole, the resence of billions an billions of cells suggests we consier a continuous system rather than a iscrete one mae u of n arts This imlies that Eq (23) becomes more general if re-written in terms of integrals instea of summations However, even if formulate in such a way, it can always be easily reuce into iscrete terms, if necessary An even if there might be some iscontinuity conitions, they can always be ealt with in the frame of Lebesques theory of integrals (Kolmogorov an Fomin, 1980), which unoubtely covers all the cases usually consiere in Literature on the subject ner such assumtions, Eq (23) can be re-written in the following synthetic an comact form Γϕ v V 3 = emv V t 3 (41) where ϕ v = the equivalent Source Term er unit volume (see Eq(14) in iscrete terms); Γ = the local structural amlification an re-normalization factor (corresoning to the rouct of the coefficients γ k an γ k in the case of iscrete form (see Eq (23)), which also accounts for the structural variations with time; em = v emv, m em v em (42), q v, w in which em v, m = C? ρ? ex is the Emergy er unit volume associate to the mass (thus transortable by mass flows)3 em, v q = the Emergy er unit volume associate to heat source terms em v, = the Emergy er unit volume associate to work source terms4 w

7 ner such conitions the Maximum Em-Power Princile may be mathematically formulate as follows Γϕ v V 3 = emv V Max t 3, D ( t) S ( t) (43) that is vali for any Domain ( D ) belonging to niversal Sace ( S (t ) ) Such a rincile, in the light of the revious consierations, may be verbally formulate as follows: Every System tens to organize its internal structure to generate rogressively increasing sring-emergy levels in orer to maximize the flow of rocesse (or useful ) Emergy 5 GENERAL CONSIDERATIONS ON THE MATHEMATICAL FORMLATION OF THE MAXIMM EM-POWER PRINCIPLE Formulation (43) may be seen as constitute by three arts: 1 st art: emv3v Max t, S (51) which corresons to the usual efinition of the Princile in terms of henomenological effects: Every System tens to maximize the flow of rocesse (or useful ) Emergy ; 2 n art: Γϕ v V 3 Max, S (52) which oints out the internal causes of such effects: Every System tens to organize its internal structure for rogressively increasing sring-emergy levels ; 3 r art: Γϕ v V 3 = emv V Max t 3, S t () (53) which emhasizes that its mathematical structure is a logical consequence of the first two mentione arts an oints out the existing irect relationshi between internal causes (52) an henomenological effects (51) The symbol = has been secifically introuce to emhasize the versus of the equivalence which goes from causes to effects5 Thus an alternative verbal formulation which is able to inclue both the first an the secon sie of Eq (53) coul be the following one: Every System tens to maximize its internal structure level in orer to maximize its Sring-Emergy Flow which, in turn, maximizes the flow of rocesse Emergy In aition it is also worth ointing out that Eq (43) (or Eq (53)) exresses a tenency Princile This imlies that: i) the symbol Max has to be consiere, in general, as an Extremum; ii) thus in the long run, Eq (43) can also be written as follows Γϕ v V 3 = emv V t 3 0, S t () (54) where the sign equal hols only when the maximum is actually achieve; iii) for very comlex systems the ynamic behaviour is usually controlle by very high time constants, so that the sloe of the increasing tren coul be so slow that, in a given time-sace winow of analysis, it woul be ossible to assume that

8 Γϕ v V= emv V t (55) It is also worth mentioning that the revious exression (43) constitutes the formulation of the M Em-P P in its general version that coul also be sai in a weak sense In fact it is also ossible to consier another an more cogent efinition (in a strong sense) as follows t 2 Γϕ vv em 3 = v V t 2 3 0, S S (56) which is vali however only for articular subsets ( S ) of niversal Sace S () t On the basis of the revious results we can now analyze the relationshi between the M Em-P Princile an the other Thermoynamic Princiles 6 THE SECOND PRINCIPLE IN THE LIGHT OF THE M EM-P PRINCIPLE As an introuctory asect let us consier a simle examle: a Comlex System, mae u of n iscrete sub-systems, in steay state conitions In this case Eq (23) may be re-written as follows m ( )= ( ) j = 1 l = 1 ( 1 ϕ ) α α β β j j Em uj l l Em y l (61) In fact, on the basis of Eqs (13) an (14), the comrehensive equivalent Source Term ( Φ ) can be exresse in terms of the equivalent inut Emergy flow, through an equivalent secific amlification (or generative) efficiency ϕ : Φ n = Φ ( m γ γ )= ϕ α α ( ) k k k u1, u2,, um j j Em uj k = 1 Such a roceure, which is always ossible on the basis of the n Emergy Balance equations escribing the n consiere sub-systems (see Giannantoni, 2001), becomes articularly simle in steay state conitions (Giannantoni, 2000a) If we now assume, for the sake of simlicity, that the System has just one inut, we get ( 1 ϕ ) Em u β β j = 1 ( )= l l Em( yl) l = 1 (62) (63) If (for ulterior simlicity) we assume equal co-rouction coefficients (β l ), equal renormalization factors ( β ) an their rouct ( β? β l Ex u Tr( yl ) = Tr( u) ( 1 ϕ ) Ex y l = 1 ( ) ( ) l l l ) equal to 1, we have (64) Eq (64) has a funamental imortance because it shows that the Transformity of any outut quantity is an amlification of the inut Transformity as a consequence oftwo istinct reasons: i)the issiation of Exergy ue to the Secon Thermoynamic Princile ( Ex yl Ex u ); l = 1 ( )< ( )

9 ii) the amlification factor (1 ϕ ), which inicates the global gain secifically ue to the generation of Emergy on behalf of internal Source Terms But what is really now worth ointing out is that, for very Comlex Systems characterize by a lot of internal co-generation an/or interaction rocesses, the contribution given by the amlification factor (1 ϕ ) is generally much higher than the one ue to Exergy losses In aition, what is even more imortant is that the amlification ue to Emergy Source Terms is in-eenent from the other one: the former can also be resent both in ieal Systems (that is even if Exergy losses ha been absent) an in real Systems (even if characterize by an increase of Exergy) This result can be consiere as a first inication of the fact that the M Em-P P is ineenent from the Secon Princile an, what is even more imortant, it inicates the effective reason for the increasing orer in self-organizing Systems, whereas the (habitually) associate loss of Exergy constitutes only a concomitant circumstance Such a conclusion can be easily (an more rigorously) rawn by starting from a very general oint of view, that is by starting from the general formulation (43) of the M Em-P Princile (as we will see in ar 9) Let us now consier another imortant asect: how it is ossible to obtain the traitional Exergy Balance Equation from the one that exresses the M Em-P Princile If, in fact, we consier Systems escribe only in terms of a traitional Thermoynamic aroach (that is by neglecting any Emergy Source term, although ever resent), our escrition reuces either to the classical Exergy Balance Equation or the Energy Balance Equation, accoring to the secific istinct assumtions consiere in the two ifferent mentione cases The Exergy Balance Equation may be obtaine by remembering that the irreversibility flow terms (mainly ue to internal, but also to external losses) which can be exresse as t emv, irr V = 3 C exirr V C exirr vrn S t ρ 3 ρ 2 (65) are generally neglecte in Emergy Analysis on the basis of the assumtion that their secific co-rouction coefficients are equal to zero (see Giannantoni, 2000a) If, vice versa, we continue to consier such contributions, but we istinguish between their concetual existence an the fact that their ertinent local co-rouction coefficients are everywhere numerically equal to zero ( c ( x, irr y, z, τ ) = 0) (ib), we may recognize that the Exergy Balance Equation can be thus easily obtaine from the Maximum Em-Power Princile by assuming that all the co-injection an co-rouction factors are equal to 1 Such an assumtion corresons to consiering ineenent inuts an all the oututs as slits ner these conitions, an without any Emergy Source Term ( Φ = 0), Eq (43) becomes, in exlicit terms wv, V ws S t ρ ex V ρex v S 0 = ρ ( kz ) ρ ( kz ) rn v θ s θ t h Ts V h Ts v S q V q S 3 2 irr 3 irr rn 2 D D (66) which exactly exresses the Exergy Balance Equation, usually written as follows

10 v 3 s 2 q θ V q θ S t ρ( u Ts) V ρ( h Ts) v S = kz 3 kz rn irr 3 irr rn 2 w v V ws S t D D ρex V ρ ex v S (67) 6 where θ is the generalize Carnot coefficient 7 THE FIRST PRINCIPLE IN THE LIGHT OF THE M EM-P PRINCIPLE As far as the relationshi between the First Princile an the M Em-P P is concerne, we may easily obtain the mathematical formulation of the former from that of the latter by following a methoological roceure analogous to the one followe in the case of the Secon Princile The only ifference now is that the First Princile oes not take into consieration what haens insie the System (which is in fact consiere as a black box ), but only accounts for quantities measure on the System frontier as simle aitive contributions ner such conitions, the local co-injection an co-rouction factors c irr ( x, y, z, τ ) ue to the various irreversibilities, are assume to be equal to zero in the whole volume not because it is assume that they o not contribute to the Balance Equation (like in the case of Emergy Balance), but because they are not seen in the assume balance ersective If then, in aition, we o not consier any form of Emergy (although ever resent), that is Φ = 0, we can easily get, in exlicit terms 0 = ρ kz ρ kz rn v t s h V h v S q V q S v, 3 s 2 w V w S (71) which is exactly the Energy Balance Equation, usually written as follows t 3 rn 2 v 3 s 2 vnc, snc, ρen V ρen v S = q V q S w V w S 3 2 (72) 7 It is now really imortant to oint out that the methoological roceure reviously followe in orer to obtain both the Exergy Balance Equation an (analogously) the Energy Balance one from the mathematical formulation of the Maximum Em-Power Princile cannot be thought of as a euction of the former equations from the latter, but rather as a reuction of the latter to the former The consiere Princiles in fact always remain in-eenent from each other The roceure followe only shows in what reuctive ersective (an associate limiting assumtions) the M Em-P P can be thought of as being equivalent to the two basic well-known traitional Thermoynamic Princiles

11 Now, before analyzing the Thir Thermoynamic Princile in the light of the M Em-P Princile, it is worth ealing with the Minimum Action Princile, because it is closely relate to the First Princile 8 THE MINIMM ACTION PRINCIPLE IN THE LIGHT OF THE MAXIMM EM-POWER PRINCIPLE The Minimum Action Princile8 is a well-known in Classical Mechanics It eals with Systems characterize by kinetic an otential Energy, in the absence of any issiative rocess ner such conitions it states that, for each System, there exists a function Lqqt which satisfies the following conition where efine as (,, ) Lqqt (,, ) = T (81) δ t2 t1 t2 Lqqt (,, ) t= δ T t ( ) = t1 q = set of geometrical coorinates 0 (82) q = set of their time erivatives T = kinetic Energy = otential Energy an the symbol δ reresents the variation of the first orer (often simly name as variation) of the integral when the function q (t) is relace by q( t) δq( t), where δq(t) sufficiently small in the whole time interval [ t ] 1,t 2 is a function which is The Minimum Action Princile is very useful to escribe the time-sace evolution of a Comlex System mae u of n arts In this case, an in the (usual) hyothesis of time uniformity, it is erfectly equivalent to the Energy Conservation Princile In fact, in these conitions, the Lagrangian function Lqqt (,, ) oes not een exlicitly on the time an Eq (82) can be re-written as follows (Lanau an Lifchitz, 1969) δ t2 t1 The equations of motion Lqqt (,, ) t= t2 t1 L L = 0 q t q L L q t q = 0 (83) t q erive from Eq (83) imly, as a main consequence, that one articular integral of the first orer (calle Energy) is constant (84) En = T = const (85)

12 In other wors the Minimum Action Princile escribes the time-sace evolution of a System (see Eqs (84)) which involves a continuous transformation between kinetic an otential Energy contributions (see Eq (82)), but in such a way as to satisfy the conition that their sum is at any time ket constant (Eq (85)) Consequently the Maximum Em-Power Princile is by no means in contrast with such a rincile: in fact it not only confirms the Minimum Action Princile, but also as something more As far as the former asect is concerne, what we sai in ar 7 may be consiere as being alreay sufficient As far as the latter is concerne, we shoul remember that the Minimum Action Princile escribes the System only in terms of aitive contributions of mechanical Energy, whereas the Maximum Em-Power Princile is able to escribe henomena in which the whole is more than its arts As an examle we can consier two elements that generate a new entity escribe by a binary function: in this case the Emergy outut, obtaine as a solution of a ifferential equation of fractional orer (see Giannantoni, 2001), is higher than the inut Emergy of its corresoning comonents The case of a hoton (consiere as mae u of a ositron an an electron (Giannantoni 2000b, 2001a,b) an the examles analyze in Giannantoni (2001) can be significantly exlanatory From a general oint of view however, if we consier an isolate System Σ, we can write Γϕ v V= Σ t Em Max (86) 3 Σ Consequently, by remembering (see Ref [10]) that we can always write the System Emergy as in the long run we have EmΣ = TrΣ Ex (87), Σ t Em t Tr Ex Tr = t Ex 0 (88) Σ Σ Σ Σ Σ Consequently, there are two ifferent ossibilities: a) if we analyze the System in terms of only Energy aitive contributions (like in the case of the Minimum Action Princile), there are no Emergy Source Terms At the same time this imlies that the System Transformity Tr is constant an Eq (88) Σ is vali with the sign equal, that is t Ex Σ = 0 (89) which eviently corresons to the Energy Conservation Princile for non-issiative systems; b) but we will immeiately see (in the next aragrah) that Emergy may generally be increasing even if Eq (89) hols This means that the Maximum Em-Power Princile, as a Quality Princile, aherently contemlates the co-existence of the Minimum Action Princile (see Eq (88)), which can consequently still be consiere as being a quantitatively ineenent Princile 9 ORDER AND DISORDER IN THE LIGHT OF THE M EM-P PRINCIPLE The M Em-P Princile also throws new light on the relationshi between orer an isorer in the niverse To this urose it is useful to start from the alication of Eq (43) to the Whole niverse, thought of as an isolate System (such an alication will also clearly illustrate the result synthetically mentione in ar 6) ner such conitions we have (see also Eq (86)) Γϕ v V= t Em Max 3 (91) S

13 Consequently, by remembering (see Giannantoni 2000a) that we can always write the niverse Emergy as Em = Tr Ex (92) an by taking the total erivative, we can then obtain (in analogy to E (88)) t Em t Tr Ex Tr t Ex = (93) If we now consier that in the long run we may use Eq (54), we can write 1 1 Tr t Tr Ex t Ex (94) which, on the basis of Eq (64), is always vali in the following form 1 1 Tr t Tr Ex t Ex >> (95) If then, on the basis of the Secon Princile, we assert that this imlies that lim S = lim Ex ( τ) τ Max (96) t ( ) = t Ex t t t 0 0 lim t (97) Consequently Eq (95) enables us to assert that Tr is always increasing (accoring to a logarithmic tren) even in these extreme conitions, because it is always 1 Tr 0 t Tr >> lim (98) t This result continues to be true in the light of the Thir Thermoynamic Princile too In fact, if we formulate the Thir Princile in terms of Exergy variations, that is Ex lim S = lim = 0 T0 0 T0 0 T 0 (wheret 0 is the absolute Temerature of the niverse), we are able to assert that, even if the Exergy variations are infinitesimal of higher orer with resect to T 0, this fact oes non imly any suerior limit to the increasing niverse Transformity (99) Tr So that the M Em-P Princile allows us to conclue that: i) not only oes the meta-mechanical orer (reresente by Tr ) increase much more raily than the mechanical isorer (exresse by the increase of Entroy); ii) but even when the latter is near its maximum, there is always a ossible increase of Tr (even in the resence of an extremely limite availability of resiual Exergy), because most of the increase in Tr (as alreay shown) is ineenent from Exergy variations, which always reresent only a concomitant circumstance

14 10 THE SO-CALLED FIFTH PRINCIPLE IN THE LIGHT OF THE MAXIMM EM-POWER PRINCIPLE The results achieve in the revious aragrah allow us to show that the hierarchical orer of the niverse (the so-calle Fifth Princile) can also be obtaine on the basis of the M Em-P Princile, but such rocess of euction is substantially ifferent from a traitional euctive rocess Such a Princile in fact cannot be thought of as being a mathematical corollary to the Maximum Em-Power Princile, but as a sort of crowning of the former, because it resents a higher Quality content in its conclusive assertions The Fifth Princile in fact asserts that Energy flows of the niverse are organize in Energy transformation hierarchy Position in the Energy hierarchy is measure with Transformities (Oum, 1994b) Alternatively, it can also be enunciate as follows: The niverse is hierarchically organize an a manifestation of Energy Transformity is a measure of the hierarchy of Energy (ib) The latter of the two quote equivalent versions is esecially inicate for our consierations In fact, if we consier the niverse as being a Whole, which is (only ieally) reeately sub-ivie into two ifferent arts, an in such a ersective we first aly Eq (91) to the whole System ΓϕvV 3 ΓϕvV 3 = ( Em1 Em2) Max, D t D t S t t 1() 1() () 101) D1 D2 an contemorarily we aly the same equation to the two istinct arts (eg in a iscrete form, such as Eq (23)), it is easy to show that: i) while the global Transformity of the niverse ( Tr ) is generally increasing (see also ar 9), ii) the rates of the Accumulate Emergy ertaining to the two sub-systems are not ientical: in fact they een on ifferent Emergy Source istributions an generally ifferent Emergy interchange flows If we then consier the corresoning equations that exress such sub-system variations in terms of Transformity an Exergy (in analogy to Eq (93) which is vali for the whole System), we can easily conclue that, even if (by hyothesis) the sub-system Transformities are originally equal, in general they rogressively ten to iffer with time This eviently shows that there is a hierarchy between the Transformities ertaining to the two sub-systems each time consiere Consequently there is a istribution of Transformities in the niverse, which escribes the content of information (or the egree of organization) of all the various ossible subsystems consiere Such a istribution is funamentally ue to two istinct reasons: the istribution of Emergy Sources (main cause), which is also resonsible for Emergy interchange flows, an the istribution of Exergy variations (as concomitant effects) associate to Energy transformations So that, on the basis of consierations similar to those that have le us to Eq (64), we can also assert that the effects ue to Emergy Sources are those which are generally ominant an, above all, ineenent from those which are ue only to the above-mentione associate circumstances Consequently, the hierarchical organization logically erive from Exergy variations (generally roortional to a timesace scale) is thus only a basic (or a first orer) istribution (as a consequence of the Secon Princile) In fact this basic effect is contemorarily moulate by a suerimose hierarchical istribution ue to Sring-Emergy Sources (Fourth Princile), which is generally much more influent than the former This allows us to reach the following general conclusions: i) the hierarchical organization tenency of the niverse is generally increasing; ii) it is essentially base on the Forth Princile, but its statement shows an increase Quality content so that such a e-uctive rocess coul be more aroriately name (by means of a neologism) as an over-euction ; iii) the Fifth Princile is thus a substantial ste ahea with resect to the traitional hierarchy of the various well-known Thermoynamic Princiles It Reresents a new jum of Quality in such a hierarchy an, consequently, an exlicit harmoniously aherent crowning of the M Em-P Princile

15 In the light of the revious consierations we coul thus reformulate such a Princile as follows: The niverse is hierarchically organize Its organization orer level has a generally increasing tenency an is a manifestation of both Emergy Source istribution an Energy transformations Transformity is a comrehensive measure of such a ynamic (or, rather, thermoynamic?) hierarchy At this stage, as a consequence of all the revious aragrahs, we may ask the following (an, in a certain sense, alreay anticiate) basic question: 11 IS THE MAXIMM EM-POWER PRINCIPLE A THERMODYNAMIC PRINCIPLE? In orer to answer this question in a comlete an correct way, we have to eal with another really reliminary question: as to whether Emergy is a state variable or not To this urose we may re-structure Eq (41) in an aroriate way or, to simlify further, we may re-structure Eq (23) in the following form; ' n m Cρex 3V γ kγ k k u u um α jα j Em uj q w βlβl Em yl q w t D Φ 1, 2,,,,,, (111), k = 1 j = 1 l = 1 where the last two terms refer to inut an outut Emergy contributions associate to heat an work flows (in fact mass flow contributions are inclue in the term on the left sie of the equation) Equation (111), if thought of as belonging to the comlete set of balance equations (mass, momentum, Energy, Exergy, etc)escribing the behavior of the System, contributes (irectly or in its integrate form) to efine State xt) ( of the System, which satisfies the require funamental conitions of uniqueness, causality, consistency an searability (see Aenix 1) This imlies that Emergy, efine by Eqs (11) an (12), satisfies (through Eq (111)) all the roerties require in orer to be a state variable Furthermore we have also shown that, uner secific assumtions, the well-known Thermoynamic Princiles can be obtaine from the M Em-P Princile Such a rocess, however, as reviously state, shoul not be consiere as a euction of the former from the latter, but rather as a sort of reuction of the latter to the former In other wors such a roceure gives us results which are strictly equivalent to the classical formulations only in quantitative terms, but not in Quality terms: in fact there is always the resence of imensional factors characteristic of Emergetic Algebra that retain their secific meaning even if they are assume to be equal to 1 This means that we are able to obtain the above-mentione Classical Princiles (in their quantitative version) by accounting for only that fraction of Emergy which corresons merely to the mechanical relationshi between the ifferent arts of the System: this can be either the art which can be integrally transforme into mechanical work (in the case of the Secon Princile) or the art which ertains to every Energy relationshi, in its various forms, ineenently on the recirocal an comlete transformability of one form into another (in the case of the First Princile) In other wors the M Em-P Princile never loses its caability of accounting for that meta-mechanical relationshi (that is: beyon the mere mechanical asect) which always characterizes hysical henomena, the resence of which is always recalle by the associate imensional factors, though (or even if) we forget or neglect its resence when accounting for only mechanical entities This fact shoul alreay be sufficient to assert that the M Em-P P is not a Thermoynamic Princile (in the usual sense of the term), because in actual fact it is much more In fact it cannot be, strictly seaking, referre to as Thermoynamic because it oes not eal only with the mere transformability of heat into mechanical work or transformability of mechanically equivalent Energy from one form into another On the other han it can still be terme as Thermoynamic (in a more general sense) if we recognize that there are forms of work that go beyon the mechanical asects (this is why they have been reviously name meta-mechanical), foun esecially in living systems, = ( ) ( ) ( ) '

16 human systems, information, artificial intelligence an so on There are in fact many forms of Energy (unerstoo in its wiest meaning) caable of sustaining an riving hysical henomena All of them can be object of analysis by means of the Maximum Em-Power Princile: the only necessary requisite is their observability, escribability an measurability 12 CONCLSIONS By starting from a rigorous mathematical efinition of Emergy (Eqs (11), (12)) an a General Emergy Balance Equation aroriately restructure (Eq (23)), we have shown that: i) A mathematical formulation of the Maximum Em-Power Princile can be state in the form given by Eq (43); ii) As a first result, such a formulation contributes to a clear an rigorous efinition of the meaning of useful (or rocesse ) Emergy; iii) The real novelty, however, of such a formulation, aart from the mathematical asect, is its constitution in three funamental arts: causes, effects, an their relationshi unerstoo as being a common tenency; iv) As a tenency Princile, the M Em-P P can also be formulate both in a weak sense (as a general rincile) an in a strong sense (in articular cases); v) If unerstoo as a general rincile, the given mathematical formulation allows us to enlighten the revious an well-known Thermoynamic Princiles, without ever losing its Quality roerties, even if it is alie accoring to a reuction roceure In fact we have shown, in articular, that: vi) The First an the Secon Princiles can be obtaine (ineenently from each other) from the given mathematical formulation of the Maximum Em-Power Princile; vii) Such a erivation however cannot be thought of as being a euction of the former Princiles from the latter, but rather as a reuction of the latter to the former; viii) The three consiere Princiles in fact always remain in-eenent from each other The roceure followe only shows in what reuctive ersective (an associate limiting assumtions) the M Em-P P can be thought of as being equivalent to the two basic well-known traitional Thermoynamic Princiles; ix) The Maximum Em-Power Princile, as a Quality Princile, aherently contemlates the co-existence of the Minimum Action Princile, so that this can still be consiere as being an ineenent (though only quantitative) Princile; x) The mathematical formulation of the M Em-P Princile also allows us to see the relationshi between orer an isorer in a new light: in fact, the meta-mechanical orer (reresente by Tr ) increases much more raily than the mechanical isorer (exresse by the increase in Entroy); in aition, most of the increase in Tr is ineenent from Exergy variations, which always reresent only a concomitant circumstance; xi) This clearly shows how the M Em-P Princile, even if it accounts for Exergy issiations (which are ever resent in any rocess), is concetually ineenent from the Secon Princile (which only eals with such concomitant circumstances); xii) The Fifth Princile, vice versa, although obtainable from the mathematical formulation of the Maximum Em-P Princile, cannot be consiere as a mathematical corollary to the latter, but as an exceeing Quality crowning of the Maximum Em-P Princile; xiii) The orer in the niverse (asserte by such a Princile), hierarchically associate to ifferent time-sace scales an basic Exergy variations, is in fact mainly characterize by a suerimose orering istribution funamentally ue to the Sring-Emergy Sources, when these are seen as art of a Whole; xiv) We also emonstrate that Emergy has all the roerties of a state variable; xiv) This ave the way to a clear assertion in favor of the M Em-P P as a Thermoynamic Princile: both when it is unerstoo in the reuctive traitional sense an in the new much more general Thermoynamic ersective oene by the concet of Emergy; xvi) As far as the last asect is concerne, we also emhasize the wiest ersective of such a Princile in ealing with observable, escribable an measurable henomena Consequently, in this context, it has an extremely wie generality an an almost unlimite ractical alication; xvii) Finally, the mathematical formulation of M Em-P Princile enable us to show that such a Princile, when seen in the light of the hierarchical rogressive increase in Quality content, in the successive assages from the traitional Thermoynamic Princile to the Fifth Princile, reresents a new fecun starting oint rather than an imortant oint of arrival, esecially

17 because of the above-mentione roerties which have been clearly ointe out as consequence of its mathematical formulation To conclue this aer it is worth ointing out that such a mathematical formulation of the Maximum Em-Power Princile is not only the result of a ersistent effort constantly gazing forth to give an elegant form to this Quality Princile: in other wors, a quality language for a quality rincile It is also, an at the same time, a sort of hymn to Quality Quality, in fact, which always lays such a funamental role in Emergy Analysis, cannot be, strictly seaking, erive It can only be recognize, escribe, accete It cannot be erive in any case whatsoever, because it is funamentally, by itself an in itself, in-erivable Quality in fact is always emerging, generative, rimary It simly aears: shows itself, resents itself, reveals itself, an it is always source of astonishment, fascination, charm It certainly has a founation: this is given by the quality of the resuositions from which it originates; but the rocess of emerging sring genesis is the one which always remains, secifically, in-erivable9 REFERENCES Brown M T, 1993 Worksho on Emergy Analysis Siena, Setember Gaggioli R A, 1988 Available Energy an Exergy Int Journal of Alie Thermoynamics, Vol 1 (No 1-4), 1-8 Giannantoni C, 1998 Environment, Energy, Economy, Politics an Rights Proceeings of Avances in Energy Stuies, Porto Venere, Italy, May E MSIS, Rome, Giannantoni C, 1999 Integrate Aroach to the Analysis of Investments by Means of Three Synthetic Economic Inicators: Energetic, Exergetic an Emergetic DCF (Discounte Cash Flow) International Conference on Inices an Inicators of Sustainable Develoment St Petersburg, Russia, July Giannantoni C, 2000a Towar a Mathematical Formulation of the Maximum Em-Power Princile Proceeengs of t h e First Biennial Emergy Analysis Research Conference niv of Floria, Gainesville (SA), Giannantoni C, 2000b Multile Bifurcation as a Solution of a Linear Differential Equation of Fractional Orer International Congress of Qualitative Theory of Differential Equations Siena (Italy), Setember Giannantoni C, 2001a Avance Mathematical Tools for Energy Analysis of Comlex Systems Proceeings of the International Worksho on Avances in Energy Stuies Porto Venere (Italy), May 23-27, 2000 E SGE, Paua Giannantoni C, 2001b The Problem of the Initial Conitions an Their Physical Meaning in Linear Differential Equations of Fractional Orer Thir Worksho on Avance Secial Functions an Relate Toics in Differential Equations - June Melfi (Italy) To be ublishe by Elsevier Science Giannantoni C, 2001 Mathematics for Quality: in Living an Non-Living Systems Secon Emergy Evaluation an Research Conference Gainesville (Floria, SA), Setember 20-22, 2001 Kolmogorov A N an Fomin S V, 1980 Elements of the Theory of Functions an Functional Analysis E MIR, Moscow Krasnov M L, Makarenko G I, an Kisele A I, 1984 Variational Calculus E MIR, Moscow Lanau L an Lifchitz E, 1969 MÈcanique E MIR, Moscow Oum H T, 1994a

18 Ecological an General Systems An Introuction to Systems Ecology Re Eition niversity Press Colorao Oum H T, 1994b Environmental Accounting Environ Engineering Sciences niversity of Floria Oum H T, 1994c Ecological Engineering an Self-Organization Ecological Engineering An Introuction to Ecotechnology Eite by Mitsch W an Jorgensen S J Wiley & Sons, Inc Oum H T, 1994 Self Organization an Maximum Power Environ Engineering Sciences niversity of Floria Oum H T,1995a Public Policy an Maximum Emower Princile Net EMERGY Evaluation of Alternative Energy Sources Lectures at ENEA Heaquarters, May 24 Oum H T, 1995b Energy Systems an the nification of Science From Maximum Power The Ieas an Alications of H T Oum C A S Hall, Eitor niversity Press Colorao Ruberti A an Isiori A, 1969 Theory of Systems E Sierea, Rome Szargut J, Morris D R an Stewar F R, 1988 Exergy Analysis of Thermal, Chemical an Metallurgical Processes Hemishere Publ Cor, SA APPENDIX 1 EMERGY AS A STATE VARIABLE The roblem can be seen in the most general context of Dynamic Systems Theory In such a context, in fact, the aroach is simlifie because Emergy Analysis is secifically oriente at escribing what haens insie the System (it is not a black box analysis), so that we can immeiately start from a System escrition which corresons to the inut-state-outut aroach Thus we first consier the set of ifferential equations that escribe the Comlex System uner consieration, that is: continuity, momentum, Energy, Exergy, etc, an Emergy Balance Equations written for each sub-system ner such conitions it is very easy to ientify those variables that are otentially able to lay the role of state variables One of those coul be Total Emergy (obviously relate, as we alreay know, to the secific Emergy of each sub-system) If we then consier the classical inut-state-outut mathematical reresentation of the System in the ertinent form given through the state transition function where ψ :( TxT) x X x X, (A1) an the outut transformation function yt () = η(, txt (), ut ()) where η : Tx X x Y (A2) where ut (), x(t), y(t) are the inut, state an outut vectors resectively such as u ( t), x ( t) X, y ( t) Y (while, X, Y are the inut, state an outut saces resectively), we can easily verify that Eq (111) satisfies the above-mentione conitions such as (see Ruberti an Isiori, 1969): a) uniqueness an causality (A3) ψ(, tt0, x0, u1 ) = ψ(, tt0, x0, u2 ) [ t, t) [ t, t) 0 0

19 ( 0 TxT ) t, t ) (, x 0 X, u, u : 1 2 u1 = u2 [ t, t) [ t, t) 0 0 that is: ientical inuts lea the System to the same an unique final state value; b) consistency ψ ( t x 0, t 0, x 0, u) = 0 t0 T, x X 0, u that is: the transition into the initial state coincies with the initial state value; c) searability (A4) ψ(, tt0, x0, u ) = ψ(, tt1, ψ( t1, t0, x0, u ), u ) (A5) [ t0, t ) [ t0, t1) [ t1, t) (, tt0 ) ( TxT), t1 ( t0, t), x0 X, u [ t t) that is: the state evolution can be analyze by means of a finite sequence of successive stes It is also worth aing that the roerty of searability has alreay been consiere as a funamental assumtion in orer to efine a reference level of Emergy (eg, Solar Emergy) In fact such a roerty immeiately allows us to write t Em Em Ex ( τ) τ (A6) = 0 t0 (see also Giannantoni, 2000a, Eq (23)) eq 0, (Footnotes) 1 Such a concet, alreay introuce in Giannantoni (2000a), will be analyze in etail in the comanion aer (Giannantoni, 2001) resente in this Conference, esecially as far as its rofoun meaning an wie consequences are concerne 2 In this form each inut an internal contribution to the System has its corresoning effectirectly an exclusively exresse in terms of outut quantities ner these conitions we always have ( β l βl) 1, for l = 1,2,Ö 3 For the sake of generality, in oen systems the secific mass Exergy is efine as 1 2 ex = ( h T0s) v gz, that is it inclues the kinetic an otential terms 2 4 The aex inicates a ositive quantity when furnishe by the System, while its algebraic sign secifies the exact versus Obviously we may use, if neee, all the consistent transformations such as A = A an so on Such a convention is articularly useful for a successive comarison with the Secon Princile formulate in terms of Exergy Balance Equation (see ar 6) Generally seaking we coul say that the convention suggests the ersective accoring to which one shoul evaluate the actual versus of each consiere flow

20 Moreover, the term ertaining to work is unerstoo as incluing a secific contribution ue to ressure work such as t V 3 in orer to have the same exression for secific mass Exergy (ex ) both in the volume an surface integrals an in the case of both a Eulerian an Lagrangian escrition At the same time the System is assume to be subjecte to conservative force fiels which are assume to be constant in time (as usually haens) 5 Even if such a symbol is not aote in writing the following equations, it will always be unerstoo as being substitute by the convention accoring to which the equations will be (generally) written: the left sie will reresent the causes an the right sie will reresent the corresoning effects Such a hysical-mathematical convention (from left to right) is in some way analogous to the toological convention aote in Emergy Analysis System Diagrams The ifference between the resence of h kz = h v gz in Eq (66 ) an of u kz = u v gz in Eq (67) eens on the fact that the term 2 mentione in note 4, whereas the term? v w oes not? w v, inclues the ressure work The ifference between Eq (71) (in which there is h kz = h v gz ) an Eq (72) 2 1 2?? (in which there is en = ukz = u v gz ) eens on the fact that the terms 2 w v, an nc w s, nc account for all the contributions ue to non-conservative forces, incluing those ue to ressure work (aart from the term mentione in note 4, which is now unnecessary) 8 The Hamilton Princile, generally known as The Least-Action Princile, has been here rename by making use of the ajective Minimum (even if less correct) only to stress the (aarent) contrast with Maximum On the other han the istinction between Minimum an Extremum is unessential to the establishment of the equations of motion (84) 9 A significantly exlicative examle is aroriately given by the above-mentione assages (or, rather, over-euctions) from the traitional thermoynamic Princiles to the Fourth Thermoynamic Princile an then, from the latter, to the Fifth Princile