The exergy of a system is more probably a measure of the ability to perform some work than of the work itself. In order to realise the work, i.e. to bring about a reversible transfer of energy, and thus obtain the maximum work, it is necessary to introduce some auxiliary body (working body), which performs a certain reversible, cyclic process. For instance, this process may take place in such a way that the body and the system, between which a direct exchange of energy takes place, are at the same temperature. That is to say, the working body at temperature T is brought into contact with the system at the same temperature T and isothermally gains from it a certain energy dU. Then it is adiabatically cooled to temperature T0 and gives up this energy at this temperature to the environment at temperature T0 and finally returns adiabatically to its initial state. During this process the working body does work on external objects (environment). The cyclic process we have described is called the Carnot Cycle. Turning now to the calculation of the resulting maximum work we note that the working body need not be considered since it returns to its initial state.
In the Carnot Cycle only the transfer of energy from a warmer system to a colder environment is considered. Therefore, in Eq. (1.4) the terms connected with compression and expansion and the chemical processes can be discarded: IdAmaxI = (T - T0)dS. Since dS = dU/T, IdAmaxI = [(T - T0)/T]dU. The ratio of the work done to the amount of energy used is called the efficiency coefficient (or simply efficiency) y. The maximum efficiency is equal to
A more convenient value is the so-called coefficient of utility ut, defined as the ratio of the work done to the maximum work which could be obtained in the given conditions. It is obvious that ut = h/Hmax. Note that efficiency may be equal to 100% only if T0 = 0, i.e. at absolute zero of temperature. Even the temperature in Space is about 2.7 K!
It would be very useful to set up an exergy balance for any system. It is hardly possible to give general equations to be applied for all systems. However, it may be possible to make some general considerations on the development of exergy balances for a system. Usually this will not involve that the exergy of information is included in the analysis (about the exergy of information, see Sections 5.5 and 5.6), but that the input of first class energy in the form of, for instance, fossil fuel, electricity or other energy forms is calculated and compared with the exergy of the outputs and (or) with the loss of exergy in the form of non-useful heat from the system. Useful heat is always associated with a higher temperature than the environment or the reference state—compare this condition with the Carnot Cycle, where the temperature difference between the warm and cold reservoirs determines the efficiency. It should always be considered whether non-useful heat could be made useful, but this would of course require that the heat can utilise the temperature difference.
A machine based on the Carnot Cycle is naturally named the thermal machine. Since solar radiation can be considered as a form of heat, formally we can define a solar machine as some kind of thermal one. Following this logic, the working body of the solar machine is "hot" photons with T = Ts = 5770 K. These photons have to "heat" the machine up to the same temperature. Under such a condition the transfer of energy from the Sun to the solar machine is possible. In the process of adiabatic cooling down to the temperature of Space (T0 = 2.7 K) the solar machine performs work. It is obvious that its efficiency is almost 100%. An ideal solar machine made from a super refractory alloy, located on board some star cruiser, could attain this efficiency. The second possibility is that the machine consists of a photon gas or plasma, such as may exist on the planet Solaris created by the Polish writer Stanislav Lem.
Unfortunately (or perhaps fortunately), photons have been able to (i.e. they really have done it) heat our planet only up to ~ 290 K, and the effective temperature of photons penetrating through the atmosphere is approximately equal to this value. The climatic machine, which is undoubtedly a solar machine, operates with these photons as a source of energy, the corresponding entropy of which is equal to qin/TE (qin < 240 W/m2 is the energy flux of incoming solar radiation, TE < 290 K is Earth's mean temperature) but not qin/Ts. Note that the working body of the atmospheric climatic machine is the air, not photons, and Gibbs has pointed to this!
Let us estimate the efficiency of the climatic machine by taking into account all these considerations. Here we follow conceptually a very interesting work by Petoukhov (1985). The machine includes the troposphere, stratosphere, ocean and the effectively interacting layer of land, i.e. it covers the whole planetary surface V. The surface is divided into two (not necessary compact) subsets V+ and V_. In the first domain the system gets energy, and in the second spontaneously loses it. It is natural that the temperature in the first domain, T, is higher than in the second, and also higher than Earth's mean temperature (see also Section 11.6). Note that if the temperature is constant over the whole planet then the climatic machine does not work. Since only V+ -domains assimilate solar radiation energy, and in addition the assimilation occurs at the temperature up to which the system is heated by incoming short-wave radiation and at which long-wave radiation is outgoing, dS in the expression for infinitesimal exergy, ldAmaxl = d(Ex) = (T — T°)dS, is represented as dS =f (qi° , qout) dv = (qin — qout)dv (3.2)
Here T is the mean temperature of the "hot" domain, T < 300 K, and qin and qout are incoming and outgoing (thermal) radiation fluxes at the system's upper boundary. Since a temperature of the environment T° is the temperature of tropopause, where a jump in the temperature gradient takes place, then it is natural to set T° < 220 K. Therefore, the efficiency of the climatic machine in relation to the absorbed energy Qabs = Jv+ (qin — qout)dv will be equal to the Carnot efficiency Hmax = (T — T°)/T = (300 — 220)/300 < 26.7%. However, the real efficiency, in relation to the total solar energy Qs = J n qin dv, will be significantly lower; h = utHmax where the utility coefficient ut = Qabs/Qs. Using the mean estimation of these values (Lorenz, 1967): Jv+(<?in — qout )dv/ ¡n dv < 15 W/m2 and J"n qmdv/\ n dv < 240 W/m2 we get ut = 6.25% and h < 1.67%. It is interesting that this value of efficiency of the Earth's climatic machine is very close not only to Petoukhov's estimation, h < 1.25-1.7%, but also to the maximal empirical estimations cited by Lorenz.
The other type of solar machine is a green leaf, which in the process of photosynthesis creates new biomass. In this case the working body comprises the chemical molecules (chlorophyll, ATP, etc.) transferring the energy of photons into leaves. Since the reaction of photosynthesis is exogenous, the heat warms the system—the "leaf'—which then can be considered as a thermal machine with maximum efficiency Hmax = (Tieaf — Tair)/Tleaf. The mean daily difference between the temperatures of leaves and the surrounding air under summer conditions in the temperate latitudes is of the order of several degrees (about 3-7°C; Budyko, 1977). To determine it we set that the difference is equal to 5°C, and the temperature of the surrounding air T° = 20°C, then Hmax = 5/298 < 1.68% (see also Chapter 10).
Let us consider the so-called chemical machine, assuming that the work is performed by the system only by means of a change in the number of particles in the field of chemical potentials. Then from Eq. (1.4) we get n ldAmax l = d(Ex) = £ (Mi — |°)dNi. (3.3)
A significant difference between this machine and the thermal one is that it cannot be characterised by an efficiency coefficient like Hthem = (T — T°)/T, since the chemical potentials are not at zero. Nevertheless, we keep in mind that the change of internal energy in the case of chemical reactions is dU = Xn=1 m, dN^ then the maximum efficiency of the chemical machine can be defined as h = pn=1 (Mi— M°)dNi
As an example, we try to estimate the efficiency of such a global chemical machine as the GCC. We assume that the GCC is described only by one variable: the amount of carbon in the living biomass of global vegetation, C. This is close to reality since the global vegetation is the most important part of the GCC. In the initial stage, the environment does work on the system (the GCC) by increasing the equilibrium amount of carbon C0 up to C, so that C0 ! C = C0 + NPP where NPP is the annual net production of global vegetation. In the final stage, as a result of spontaneous destructive processes, an equivalent amount of carbon is returned into the environment. Note that the latter takes place only for stationary processes.
We already know that the chemical potential of ith substance with concentration (activity) [ci] is m = Mi(0) + RT ln[ci] where Mi(0) is a constant depending on the origin of the substance and its temperature. The concentration may be expressed in any units that are able to describe the total amount of substance in the system, but it is important that the system's volume does not change. We also assume that the GCC is an isothermal process. Then substituting the expression for chemical potential into Eq. (3.4) we have
= RT0 ln(C/Cp)dC = ln(1 + NPP/Cp) , ln(1 + NPP/Cp) (3 5) Hmax [m(0) + RT0lnC]dC a + ln(C0 + NPP) ln(C0 + NPP) ' ( ' )
where a = m(0)/RT0 > 0. The standard estimations for C0 < 610 GtC and NPP < 60 GtC (den Elzen et al., 1995; 1 Gt = 109 t), and from Eq. (3.5) we get Hmax < ln 1. 1 /ln 660 < 1.44%.
We would like to call your attention to an interesting coincidence: the efficiencies of natural machines, such as climatic and photosynthetic machines, and the GCC are all very low and close to each other: approximately 1.5%. This may just be a coincidence, but perhaps it shows some common property of natural systems on our planet?
Now we go from the global to the regional scale. Szargut (1998) calculates the exergy efficiency of technological processes by adding the exergy loss due to the emission of waste to the environment. Application of this approach is indeed recommended, as a highly exergy-effective process generally should not be used if it produces waste products which can harm the environment. As the harm can be expressed as a loss in exergy, the overall exergy efficiency also considering this loss of exergy should, of course, be used in the selection of technologies.
When contaminants, e.g. heavy metals, are widely dispersed, exergy is lost. When lead was used all over the world in petrol to obtain a higher octane number, in the order of 2 X 108 kg of lead was widely dispersed every year. It was even found in the ice pack of Greenland! This "dispersed" state can be considered as a thermodynamic equilibrium. Since the exergy loss is equal to the exergy accumulated within lead ore, A(Ex) = RT0 X ln(core/cdis)AC, where core is the lead concentration in ore (0.05 kg/kg ore), cdis a typical concentration of lead in the environment after dispersion (1 mg/kg soil), and AC is the total amount of dispersed lead (in moles). If we presume T0 = 300 K then the annual exergy lost can be found as:
AEx = 8.3 X 300 ln(0.05/10—9)(2 X 1011/207) < 42.5 GJ. (3.6)
The exergy efficiencies of many of our production processes are surprisingly low. There has been an increasing interest in reducing the exergy consumption with increasing exergy (energy) cost and decreasing access to low-cost energy. The exergy consumption of a good modern refrigerator is, for instance, only 30-40% of a 20-year-old refrigerator with the same capacity. By more comprehensive use of exergy balances, it will be possible to indicate how to reduce the overall exergy consumption in developed countries very significantly. If, on average, we could increase exergy efficiencies from 40 to 60%, it would imply a reduction of our energy consumption and the associated emissions of pollutants by 33.3%.
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