The Concept of Exergy

Energy is considered a sort of a priori concept in thermodynamics; it is 'an extensive property of a system such that its change in value between two states is equal to the adiabatic work between these states'. But energy is known to manifest itself in several forms, each endowed with a different 'quality'. The concept of quality can be fully explained only recurring to the second law, but in general we can say that the quality of an energy flux is a measure of its capacity to cause change. Referring interested readers to the books by Tadeusz Kotas and by Adrian Bejan, George Tsatsaronis and Michael Moran for a more detailed discussion, we can identify 'ordered' or 'high-quality' energy forms (potential, kinetic, mechanical, electrical) that can be ideally converted into each other with 100% efficiency, so that even after a number of transformations, as shown in Figure 1a, the output flux of useful energy Enout is quantitatively equal to the flux of useful input Enin. There are other forms though (internal energy, chemical energy, thermal radiation, turbulent kinetic energy) that cannot be converted into high-quality energy without an intrinsic efficiency loss, so that (Figure 1b) the useful output Enout is always lower than the useful input Enin. Notice that since energy is conserved (this is a basic postulate in all physical sciences), the difference Enin - Enout must be accounted for: it is well known that this difference is 'dissipated' into a low-temperature flux absorbed or provided by the environment. The 'environment' is a system so large that its properties (pressure, temperature, chemical composition, kinetic or potential energy, etc.) are not affected by the interaction with the man-made system.

The concept of exergy has been developed to provide a congruent and coherent quantification to the quality of an energy form. Imagine that at a certain time to we place a system S identified by a certain set of thermodynamic properties (V, zB, p, T, c, m, p,.. .) in contact with the environment O (we adopt the symbol 'O' for the environment, from the Greek 'OtKot; (home) from where all the

Hydro turbine

Elect. motor

Elect. S/ generator


Elect. motor

Elect. S/ generator



Hydro turbine


Qh rMA/Vn

Thermal plant

Compressed air

Environment (A)

Compressed air


Kotas Exergy

Environment (C)

Figure 1 (a) A chain of ideal energy conversions of high-quality energy forms into each other. (b) Examples of conversion of low-quality into high-quality energy forms. Adapted from Kotas T (1985) The Exergy Method of Thermal Plant Analysis. London: Butterworths, Academic Press.

Environment (B)

Environment (C)

Figure 1 (a) A chain of ideal energy conversions of high-quality energy forms into each other. (b) Examples of conversion of low-quality into high-quality energy forms. Adapted from Kotas T (1985) The Exergy Method of Thermal Plant Analysis. London: Butterworths, Academic Press.

'eco-' prefixes have originated), and arrange things in such a way that S may exchange mass and energy only with O (Figure 2): our intuition tells us, and experiments confirm, that after some time trelax which depends on the extension of S and on the difference between its initial properties and those of O, S will come to (a possibly dynamic) equilibrium with O, that is, even dynamic interactions will occur at a macroscopically stationary state, and will result in no net change in the mass and energy contents of either system. 'Exergy is defined as the maximum work developed in this ideal process'. Notice that,

Figure 2 A system S and its environment O.

since the interaction is between O and S, and we assume that all processes can be described by a succession of quasi-equilibrium states (so that the time interval trelax drops out from the problem formulation), exergy will be a function only ofthe initial and the final state ofS (the final being equal to that of the environment), so that it is an attribute of the pair (S, O), and not of S alone. S and O exchange energy fluxes under different forms: kinetic (all parts of S in relative motion with respect to E come to rest), potential (the barycentric position of S reaches a fixed elevation in the gravity field of O), thermal (heat flows from S to O or vice versa depending on the initial temperature difference TS - TO), work interaction (O performs or receives work from its boundary interactions with S), mass exchange (mass fluxes from S to O and vice versa carry different energies). Let us consider here only stationary interactions between S and O in which flows ofenergy and matter from S to O are continuous and (on the average) steady in time: by applying energy and entropy balances to the complex system composed of S and O we obtain yYl - -Trj Qj - W + m'e' "output = Ex [1]

j \ j / in out where the first term represents the thermal exergy flow, the second is external work, and the third and fourth the exergy in- and outflows due to material exchanges and Ex is the exergetic destruction. In [1] we make use of the definition of specific exergy:

e = h - ho - To (s - so) + Ek [Ag. + RTq * ln ck/qj0)]

From eqns [1] and [2] we can draw the following conclusions:

1. The initial kinetic and potential exergy of S are entirely 'recovered' into useful work.

2. Work exchanges between S and o are entirely 'recovered' in the exergy balance. In fact, work 'is' exergy and, as such, its transmission is only affected by viscous-type losses, lumped here in Ex. If we define a 'quality' or 'exergetic factor' as the ratio between the energy and the exergy content of an energy flux, all of the 'high-quality' forms (mechanical and electrical work, kinetic and potential energy) have an exergetic factor equal to 1.

3. Thermal energy has an exergetic factor equal to its associated Carnot efficiency: this reflects the fact that, under the given assumptions, the maximum work that can be extracted from a quantity of heat Qavailable at a certain temperature T is equal to Wq = (1 — TO/T) Q.

4. If S is initially at a lower temperature than O, thermal energy will flow from O to S, with a corresponding exergy flow equal to Eq= (1 — TS/TO)Q This amount is always positive.

5. For a simple substance whose states are defined by temperature and pressure (like all gases for most practical engineering purposes), exergy is always positive, except for particular combinations of pS/pO and TS/ TO (for details, see Kotas again).

6. The chemical potential of the elements in S cannot be entirely converted to work: since S must come to equilibrium with O, the most we can 'recover' for the generic kth component is the difference between the values of its Gibbs function in S and in O, each weighted by the respective concentration. In other words, we can ideally transform only a portion of the initial AGS into useful work, because the products of reaction must be at their respective environmental concentrations and chemical potentials at TO and pO.

7. Exergy is not conserved: in every real process there is an exergy destruction Ex caused by irreversibility. In the simplest case (work)work transformation) this exergy destruction is equal to the irreversible entropy generation due to 'dissipative' effects: this can be seen by considering that S exchanges only adiabatic work with O, and the loss is equivalent to frictional heat. If there is only one medium participating in the exchange, from [1] and [2] we obtain

Ex = - W + m(es - eO)= - W + m\hs - ho - To (Ss - So)] = - To(SS - So )(W)

Since SS — SO must be negative (S may spontaneously evolve only toward higher entropies), the destroyed exergy flux Ex is positive and equal to the 'lost work' defined by Gouy and Stodola.

8. Equation [1] can be rewritten in such a way that the different 'components' of exergy are explicitly represented:

where the suffixes indicate the usual denomination of the various terms: physical, chemical, kinetic, potential exergy. If other forms of energy fluxes are involved (magnetic, nuclear, etc.), one needs only to explicit the suffix j and apply the corresponding definition.

9. Exergy is an extensive property, so that

10. If two systems Sa and Sb interact, the maximum work obtainable from their exclusive interaction is |Ea — Ej.

11. If a stream a undergoes a series of transformations i,j, k. . ,,z in which it receives or delivers the exergies Ej... Ez, its final exergy content is the sum of Ea + Ej + Ej + Ek +••• + Ez. Exergy is thus an additive property.

For engineering (and exergoeconomic) calculations it is convenient to define a 'standard environment' (a state at T0, p0, cj0, z0, V0, ...) to which all exergy values can be referred.

The exergy concept has important practical applications: it can be shown that a 'second-law efficiency' defined on the basis of exergy fluxes provides more information than the 'first-law efficiency' based on energy fluxes. For example, consider the cogeneration plant shown in Figure 3. From the design data given in Table 1, the efficiencies shown in Table 2 can be calculated, and the following considerations apply:

1. The first-law efficiency attains a value of 0.655: this means that 65.5% of the fuel chemical energy is converted into useful energy output. This is misleading though, because it does not account for the temperature-dependent quality of the heat flux Qcog. An exergetic analysis, in which all inputs and outputs are expressed in exergetic terms, provides a = 0.347, correctly indicating that a substantial portion of the chemical exergy of the fuel has been 'degraded' into heat, that is, into a low-quality energy form.

2. Component exergetic efficiencies (^n) also provide interesting information, especially when compared with their energetic counterparts (^j). For example, while the quasi-adiabatic compressor and turbine display similar

Figure 3 Schematic representation of a cogeneration plant.

values for •qj and •qjj, both the heat recovery boiler and the condenser have •qjj ^ This indicates that it is useless to invest excessive resources to improve boiler performance, because this component is quite efficient in transmitting the heat flux it is required to (here, ^JB = 0.81). The boiler's low exergetic efficiency is intrinsic and not due to design errors: it degrades a high-T, high exergy heat flux into a low-T, low exergy one. Similar considerations apply to the combustor: it is the degradation of the relatively high-quality chemical energy into heat, and not its energy-transfer capacity, that lowers its exergetic efficiency. Energywise, the performance of a combustor is more than satisfactory («0.94). A system that exploits the AG of the fuel by means of chemical reactions, without recurring to combustion, like the fuel cells, for example, has a much higher •qjj than that of a combustor.

section titled 'Scope and function of an ecological indicator', and thus the question that remains to be answered is: does E\ measure the environmental impact of a process or of a series of processes that represent the 'signature' of a specific community on the environment? We can anticipate that the answer is negative, but to justify this conclusion we must analyze the problem in more detail.

Thermal Dissipation

Consider first a situation in which the system S exchanges only thermal energy with the environment O, which in this case can be conveniently split without loss of generality into a 'hot' and a 'cold' portion (Figure 4). The heat flux Qh enters the system (e.g., solar radiation, geothermal heat), and the quantity Qc leaves the system (e.g., waste heat). The respective exergies are a function of both the temperature of the heat flow and of the environment. Let us assume for simplicity's sake that the entire Qh is provided at a temperature Th and the entire Qc is provided at a temperature Tc, both higher than TO. For the exergy destruction we obtain e\ = eQh - eQc = qT^(TT-TT^

With Te = 293 K, Th = 500 K, and Tc = 308 K, eqn [4] tells us that, for every kW of thermal input, 0.365 kW (or 36.5%) are destroyed in this thermal cascade.

We can conclude that E\ is a correct indicator of the adverse ecological impact of transferring energy from high to low temperatures.

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