The Maximum Power Principle

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Lotka (1925, 1956) formulated the maximum power principle. He suggested that systems prevail that develop designs that maximize the flow of useful (for maintenance and growth) energy, and Odum used this principle to explain much about the structure and processes of ecosystems (Odum and Pinkerton, 1955). Boltzmann (1905) said that the struggle for existence is a struggle for free energy available for work, which is a definition very close to the maximum exergy principle introduced in the next section. Similarly, Schrodinger (1946) pointed out that organization is maintained by extracting order from the environment. These two last principles may be interpreted as the systems that are able to gain the most free energy under the given conditions, i.e. to move most away from the thermodynamic equilibrium will prevail. Such systems will gain most biogeochemical energy available for doing work and therefore have most energy stored to use for maintenance and buffer against perturbations.

H.T. Odum (1983) defines the maximum power principle as a maximization of useful power. It is applied on the ecosystem level by summing up all the contributions to the total power that are useful. It means, that non-useful power is not included in the summation. Usually the maximum power is found as the sum of all flows expressed often in energy terms for instance kJ/24h.

Brown et al. (1993) and Brown (1995) has restated the maximum power principle in more biological terms. According to the restatement it is the transformation of energy into work (consistent with the term useful power) that determines success and fitness. Many ecologists have incorrectly assumed that natural selection tends to increase efficiency. If this were true, then endothermy could never have evolved. Endothermic birds and mammals are extremely inefficient compared with reptiles and amphibians. They expend energy at high rates in order to maintain a high, constant body temperature, which, however, gives high levels of activities independent of environmental temperature (Turner, 1970). Brown (1995) defines fitness as reproductive power, dW/dt, the rate at which energy can be transformed into work to produce offspring. This interpretation of the maximum power principle is even more consistent with the maximum exergy principle that is introduced in the next section, than with Lotka's and Odum's original idea.

In the book Maximum Power: The Ideas and Applications of H.T. Odum, Hall (1995) has presented a clear interpretation of the maximum power principle, as it has been applied in ecology by H.T. Odum. The principle claims that power or output of useful work is maximized, not the efficiency and not the rate, but the tradeoff between a high rate and high efficiency yielding most useful energy or useful work. It is illustrated in Figure 6.3.

Measuring For Length Balloon Garland


Figure 6.3 The maximum power principle claims that the development of an ecosystem is a tradeoff (a compromise) between the rate and the efficiency, i.e. the maximum power output per unit of time.


Figure 6.3 The maximum power principle claims that the development of an ecosystem is a tradeoff (a compromise) between the rate and the efficiency, i.e. the maximum power output per unit of time.

Hall is using an interesting semi-natural experiment to illustrate the application of the principle in ecology. Streams were stocked with different levels of predatory cutthroat trout. When predator density was low, there was considerable invertebrate food per predator, and the fish used relatively little maintenance energy searching for food per unit of food obtained. With a higher fish-stocking rate, food became less available per fish, and each fish had to use more energy searching for it. Maximum production occurred at intermediate fish-stocking rates, which means intermediate rates at which the fish utilized their food.

Hall (1995) mentions another example. Deciduous forests in moist and wet climates tend to have a leaf area index (LAI) of ~6 m2/m2. Such an index is predicted from the maximum power hypothesis applied to the net energy derived from photosynthesis. Higher LAI values produce more photosynthate, but do so less efficiently because of the metabolic demand of the additional leaf. Lower leaf area indices are more efficient per leaf, but draw less power than the observed intermediate values of roughly 6.

The same concept applies for regular fossil fuel power generation. The upper limit of efficiency for any thermal machine such as a turbine is determined by the Carnot efficiency. A steam turbine could run at 80 percent efficiency, but it would need to operate at a nearly infinitely slow rate. Obviously, we are not interested in a machine that generates electricity or revenues infinitely slowly, no matter how efficiently. Actual operating efficiencies for modern steam powered generator are, therefore, closer to 40 percent, roughly half the Carnot efficiency.

These examples show that the maximum power principle is embedded in the irre-versibility of the world. The highest process efficiency can be obtained by endo-reversible conditions, meaning that all irreversibilities are located in the coupling of the system to its surroundings, there are no internal irreversibilities. Such systems will, however, operate too slowly. Power is zero for any endo-reversible system. If we want to increase the process rate, it will imply that we also increase the irreversibility and thereby decrease the efficiency. The maximum power is the compromise between endo-reversible processes and very fast completely irreversible processes.

The concept of emergy (embodied energy) was introduced by H.T. Odum (1983) and attempts to account for the energy required in formation of organisms in different trophic levels. The idea is to correct energy flows for their quality. Energies of different types are converted into equivalents of the same type by multiplying by the energy transformation ratio. For example fish, zooplankton, and phytoplankton can be compared by multiplying their actual energy content by their solar energy transformation ratios. The more transformation steps there are between two kinds of energy, the greater the quality and the greater the solar energy required to produce a unit of energy (J) of that type. When one calculates the energy of one type, that generates a flow of another, this is sometimes referred to as the embodied energy of that type. Figure 6.4 presents the concept of embodied energy in a hierarchical chain of energy transformation. One of the properties of high quality energies is their flexibility (which requires information). Whereas low quality products tend to be special, requiring special uses, the higher quality part of a web is of a form that can be fed back as an amplifier to many different web components.

Solar Equivalents kJ / m2 h

Solar Equivalents kJ / m2 h

Energy Transformation Ratios = Embodied Energy Equivalents. kJ / m2 h

Energy Transformation Ratios = Embodied Energy Equivalents. kJ / m2 h

Figure 6.4 Energy flow, solar equivalents, and energy transformation ratios = embodied energy equivalents in a food chain (Jorgensen, 2002).

Figure 6.4 Energy flow, solar equivalents, and energy transformation ratios = embodied energy equivalents in a food chain (Jorgensen, 2002).

A good down to earth example of what emergy is, might be the following: in 1 year one human can survive on 500 fish each of the size of 500 g, that may have consumed 80,000 frogs with the size of 20g. The frogs may have eaten 18 X 106 insects of the size of 1 g. The insects have got their food from 200,000 kg dry matter of plants. As the pho-tosynthetic net production has an efficiency of 1 percent, the plants have required an input of ~3.7 X 109 J, presuming an energy content of plant dry matter of 18.7kJ/g. To keep one human alive costs, therefore 3.7 X 109 J, although the energy stock value of a human being is only in the order 3.7 X 105 J or 10,000 times less. The transformity is, therefore, 10,000.

H.T. Odum has revised the maximum power principle by replacing power with emergy-power (empower), meaning that all the contributions to power are multiplied by a solar equivalent factor that is named transformity to obtain solar equivalent joules (sej) (see Box 6.2). The difference between embodied energy flows and power, see Equation 6.1, simply seems to be a conversion to solar energy equivalents of the free energy.

Box 6.2 Emergy

"Emergy is the available energy of one kind previously used up directly and indirectly to make a service or product. Its unit is the emjoule [(ej)]" and its physical dimensions are those of energy (Odum, 1996). In general, since solar energy is the basis for all the energy flows in the biosphere, we use solar emergy (measured in sej, solar emjoules), the solar energy equivalents required (directly or indirectly) to make a product.

The total emergy flowing through a system over some unit time, referenced to its boundary source, is its empower, with units [sej/(time)] (Odum, 1988). If a system, and in particular an ecosystem, can be considered in a relatively steady state, the empower (or emergy flow) can be seen as nature's "labor" required for maintaining that state. The emergy approach starts from Lotka's maximum power principle (1922, 1956) and corrects the function, which is maximized, since not all the energy types have the same ability of doing actual work. Thus power (flow of energy) is substituted by empower (flow of emergy), that is "in the competition among self-organizing processes, network designs that maximize empower will prevail" (Odum, 1996).

Transformity is the ratio of emergy necessary for a process to occur to the exergy output of the process. It is an intensive function and it is dimensionless, even though sej/J is used as unit.

Emergy can be written as a function of transformity and exergy as follows (i identifies the inputs):

While transformity can be written as

*k = Emk /Exk even though it is often calculated as

By definition the transformity of sunlight is equal to 1 and this assumption avoids the circularity of these expressions. All the transformities (except that of solar energy) are, therefore, greater than 1.

Transformities are always measured relative to a planetary solar emergy baseline and care should be taken to ensure that the transformities used in any particular analysis are all expressed relative to the same baseline (Hall, 1995). However, all the past baselines can be easily related through multiplication by an appropriate factor and the results of an emergy analysis do not change by shifting the baseline (Odum, 1996).

Emergy and transformity are not state functions, i.e. they strongly depend on the process that is used to obtain a certain item. There are transformities that are calculated from global biosphere data (i.e. rain, wind, geothermal heat) and others that, being the result of more complex and variable processes have high variability: for example, electricity can be generated by many processes (using wood, water, coal, gas, tide, solar radiation, etc.) each with a different transformity (Odum, 1996).

In general transformity can be seen as a measure of "quality": while emergy, following "memorization" laws, can in general remain constant or grow along transformation chains, since as energy decreases, transformities increase. On the other hand, when comparing homologous products, the lower the transformity, the higher the efficiency in transforming solar emergy into a final product.

Emergy is a donor-referenced concept and a measure of convergence of energies, space and time, both from global environmental work and human services into a product. It is sometimes referred to as "energy memory" (Scienceman, 1987) and its logic (of "memorization" rather than "conservation") is different from other energy-based analyses as shown by the emergy "algebra". The rules of emergy analysis are:

• All source emergy to a process is assigned to the processes' output.

• By-products from a process have the total emergy assigned to each pathway.

• When a pathway splits, the emergy is assigned to each 'leg' of the split based on its percentage of the total energy flow on the pathway.

• Emergy cannot be counted twice within a system: (a) emergy in feedbacks cannot be double counted; (b) by-products, when reunited, cannot be added to equal a sum greater than the source emergy from which they were derived.

For in depth discussion of this issue and the differences between energy and emergy analysis see Odum (1996).

Embodied energy is, as seen from these definitions, determined by the biogeochemi-cal energy flow into an ecosystem component, measured in solar energy equivalents. The stored emergy, Em, per unit of area or volume to be distinguished from the emergy flows can be found from:

¡=i where is the quality factor which is the conversion to solar equivalents, as illustrated in Figure 6.4, and ci is the concentration expressed per unit of area or volume.

The calculations reduce the difference between stored emergy (= embodied energy) and stored exergy (see next section), to the energy quality factor. The quality factor for exergy accounts for the information embodied in the various components in the system (detailed information is given in the next section), while the quality factor for emergy accounts for the solar energy cost to form the various components. Emergy calculates thereby how much solar energy (which is our ultimate energy resource) it has taken to obtain 1 unit of biomass of various organisms. Both concepts attempt to account for the quality of the energy. Emergy by looking into the energy flows in the ecological network to express the energy costs in solar equivalents. Exergy by considering the amount of biomass and information that has accumulated in that organism. One is measure of the path that was taken to get to a certain configuration, the other a measure of the organisms in that configuration.


Second law dissipation acts to tear down structure and eliminate gradients, but ecosystems have the ability to move away from thermodynamic equilibrium in spite of the second law dissipation due to an inflow of energy from solar radiation. Even a simple physical system as a Bernard cell is using an inflow of energy to move away from ther-modynamic equilibrium. A Bernard cell consists of two plates, that are horizontally placed in water a few centimeter from each other. The lower plate has higher temperature than the upper plate. Consequently, energy is flowing from the lower to the upper plate. When the temperature difference is low the motion of the molecules is random. When the temperature exceeds a critical value the water molecules are organized in a convection pattern, series of rolls or hexagons. The energy flow increases due to the convection. The greater the flow of energy the steeper the temperature gradient (remember that work capacity = entropy times temperature gradient) and the more complex the resulting structure. Therefore, greater exergy flow moves the system further away from thermodynamic equilibrium—higher temperature gradient and more ordered structure containing information corresponding to the order. The origin of ordered structures is, therefore, openness and a flow of energy (see Chapter 2). Openness and a flow of energy are both necessary conditions (because it will always cost energy to maintain an ordered structure) and sufficient (as illustrated with the Bernard cell). Morowitz (1968, 1992) has shown that an inflow of energy always will create one cycle of matter, which is an ordered structure. Openness and a flow of energy is, however, not sufficient condition for ecosystems (see Chapter 2), as additional conditions are required to ensure that the ordered structure is an ecosystem.

Biological systems, especially, have many possibilities for moving away from thermodynamic equilibrium, and it is important to know along which pathways among the possible ones a system will develop. This leads to the following hypothesis (Jorgensen and Mejer, 1977, 1979; Jorgensen, 1982, 2001, 2002; Jorgensen et al., 2000): if a system receives an input of exergy, then it will utilize this exergy to perform work. The work performed is first applied to maintain the system (far) away from thermo-dynamic equilibrium whereby exergy is lost by transformation into heat at the temperature of the environment. If more exergy is available, then the system is moved further away from thermodynamic equilibrium, reflected in growth of gradients. If there is offered more than one pathway to depart from equilibrium, then the one yielding the highest eco-exergy storage (denoted Ex) will tend to be selected. Or expressed differently: among the many ways for ecosystems to move away from thermodynamic equilibrium, the one maximizing dEx/dt under the prevailing conditions will have a propensity to be selected.

Rutger de Wit (2005) has expressed preference for a formulation where the flow of exergy is replaced by a flow of free energy, which of course is fully acceptable and makes the formulation closer to classic thermodynamics. However, eco-exergy storage can hardly be replaced by free energy because it is a free-energy difference between the system and the same system at thermodynamic equilibrium. The reference state is therefore different from ecosystem to ecosystems, which is considered in the definition of eco-exergy. In addition, free energy is not a state function far from thermodynamic equilibrium—just consider the immediate loss of eco-exergy when an organism dies. Before the death the organism has high eco-exergy because it can utilize the enormous information that is embodied in the organism, but at death the organism loses immediately the ability to use this information that becomes, therefore, worthless. Moreover, the information part of the eco-exergy cannot be utilized directly as work; see the properties of information presented in Section 6.2.

Just as it is not possible to prove the three laws of thermodynamics by deductive methods, so can the above hypothesis only be "proved" inductively. A number of concrete cases which contribute generally to the support of the hypothesis will be presented below and in Chapters 8 and 9. Models are often used in this context to test the hypothesis. The exergy can be approximated by use of the calculation methods in Box 6.3. Strictly speaking exergy is a measure of the useful work which can be performed. Conceptually, this obviously includes the energetic content of the material, i.e. biomass, but also the state of organization of the material. One way to measure the organization is the information content of the material, which could be the complexity at the genetic or ecosystem levels. Currently, the organizational aspect of exergy is expressed as Kullbach's measure of information based on the genetic complexity of the organism:

where B is the biomass, R the gas constant, T the Kelvin temperature, and K Kullbach's measure of information (further details see Box 6.3). The exergy of the organism is found on basis of the information that the organism carries:

where Ex, is the exergy of the ith species, fii a weighting factor that considers the information the ith species is carrying in c, (Table 6.2). Jorgensen et al. (2005) show how the fi-values have been found for different organisms. A high uncertainty is, however, associated with the assessment of the fi-values, which implies that the exergy calculations have a corresponding high uncertainty. In addition, the exergy is calculated based on models that are simplifications of the real ecosystems. The calculated exergy should, therefore, only be used relatively and considered an index and not a real absolute exergy value.

Box 6.3 Calculation of eco-exergy

It is possible to distinguish between the exergy of information and of biomass (Svirezhev, 1998). pt defined as ctIB, where n

¡=i is the total amount of matter in the system, is introduced as new variable in Equation 2.8:

As the biomass is the same for the system and the reference system, B ~ B o exergy becomes a product of the total biomass B (multiplied by RT) and Kullback measure:

where pt and pi o are probability distributions, a posteriori and a priori to an observation of the molecular detail of the system. It means that K expresses the amount of information that is gained as a result of the observations. If we observe a system that consists of two connected chambers, then we expect the molecules to be equally distributed in the two chambers, i.e. p1 = p2 = 1/2. If we, on the other hand, observe that all the molecules are in one chamber, we get p1 = 1 and p2 = 0.

Specific exergy is exergy relatively to the biomass and for the ith component: Sp. ex.i= Exi/c. It implies that the total specific exergy per unit of area or per unit of volume of the ecosystem is equal to RTK.

For the components of the ecosystem, 1 (covers detritus), 2, 3, 4 N, the probability, p1 o, consists at least of the probability of producing the organic matter (detritus), i.e. p1 o, and the probability, p, a, to find the correct composition of the enzymes determining the biochemical processes in the organisms. Living organisms use 20 different amino acids and each gene determines on average a sequence of ~700 amino acids (Li and Grauer, 1991). pia, can be found from the number of permutations among which the characteristic amino-acid sequence for the considered organism has been selected.

The total exergy can be found by summing up the contributions originating from all components. The contribution by inorganic matter can be neglected as the contributions by detritus and even to a higher extent from the biological components are much higher due to an extremely low probability of these components in the reference system. Roughly, the more complex (developed) the organism is the more enzymes with the right amino-acid sequence are needed to control the life processes, and therefore the lower is the probability pt a, The probability pt a, for various organisms has been found on basis of our knowledge about the genes that determine the amino-acid sequence. As the concentrations are multiplied by RT and ln (pt/p, o), denoted fi; a table with the fi-values for different organisms have been prepared (see Table 6.3). The contribution by detritus, dead organic matter, is in average 18.7kJ/g times the concentration (in g/unit of volume). The exergy can now be calculated by the following equation:

Exergy total = ^ fit q (as detritus equivalent)

The fi-values are found from Table 6.3 and the concentration from modeling or observations. By multiplication by 18.7, we get the exergy in kilojoules. Notice that n

Exbio = ^ ci (as detritus equivalent)

while n

Consistency of the exergy-storage hypothesis, as we may call it, with other theories (goal functions, orientors; see Sections 6.2 and 6.3) describing ecosystem development will be demonstrated as a pattern in a later section of this chapter. It should, however, in this context be mentioned that exergy storage in the above-mentioned main hypothesis can be replaced by maximum power. Exergy focuses on the storage of biomass (energy) and information, while power considers the energy flows resulting from the storages.

Ascendency (Box 4.1) is a complex measure of the information and flows embodied the ecological network. The definition is given in Chapter 4. At the crux of ascendency lies the action of autocatalysis (Chapter 4). One of the chief attributes of autocatalysis is what Ulanowicz (1997) calls "centripetality" or the tendency to draw increasing amounts of matter and energy into the orbit of the participating members (Chapter 4). This tendency inflates ascendency both in the quantitative sense of increasing total system activity and qualitatively by accentuating the connections in the loop above and beyond pathways connecting non-participating members. At the same time, increasing storage of exergy is a particular manifestation of the centripetal tendency, and the dissipation of external exergy gradients to feed system autocatalysis describes centripetality in an almost tautological fashion.

In retrospect, the elucidation of the connections among ascendency, eco-exergy, and aggradation (Ulanowicz et al., 2006) has been effected by stages that are typical of theory-driven research. First, it was noted in phenomenological fashion how quantitative observations of the properties were strongly correlated; the correlation coefficient, r2, was found for a number of models to be 0.99 (Jorgensen, 1995). Thereafter, formal definitions were used to forge theoretical ties among the separate measures. Finally, the perspective offered by these new theoretical connections facilitated a verbal description of the common unitary agency that gave rise to the independent trends that had been formalized as separate principles. Eco-exergy and ascendency represent two sides of the same coin or two different angles in the description of ecosystem development. A simple physical phenomenon as light requires both a description as waves and as particle to be fully understood. It is, therefore, understandable that ecosystem developments that are much more complicated than light require multiple description. Exergy covers the storage, maximum power the flows, and ascendency the ecological network and all three concepts contribute to the overall aggradation, moving away from thermodynamic equilibrium. All three concepts have well-structured roots in the theoretical soil. Their shortcomings are, however, that calculations of exergy, maximum power, and ascendency always will be incomplete due to the enormous complexity of ecosystems (see Section 6.1).

Ecosystems can also be understood as a (high) number of interacting gradients, which are formed by self-organizing processes (Mueller and Leupelt, 1998). Gradient maintenance costs exergy that is transformed by decomposition processes to heat at the temperature of the environment, i.e. the exergy is lost. The gradients can be classified in various ways, but we could also distinguish three types of gradients corresponding to the three growth forms (see Section 6.2): gradients due to organisms in the ecosystems (trees are good illustrations), gradients due to formation of a more complex network (for instance the spatial distribution of more or fewer niches), and gradients due to information (the level of information could be used directly as illustration). The first-mentioned class of gradients requires the most exergy for maintenance, while information gradients require very little or no exergy for maintenance. Gradients summation is captured in the exergy measure since work capacity is an extensive variable times a gradient (see Chapter 2).

Exergy storage is the simplest of the three concepts to calculate; but clearly the assessment of the ^-values has some shortcomings. The latest list is more differentiated than the previous ones (Jorgensen et al., 1995; Fonseca et al., 2000) and is based on the latest results of the entire genome analyses for 11 species plus a series of complexity measures for a number of species, families, orders, or classes. The list will most probably be improved as genetics gains more information about the genomes and proteomes of more species. The total information of an ecosystem should furthermore include the information of the network. All ecological models that are used as basis for the exergy calculations are much simpler than the real network and the information contained in the network of the model become negligible compared to the exergy in the compartments. A calculation method to assess the information of the real ecological network is needed to account for the contribution to the total ecosystem exergy.

Power is very difficult to assess because the ecological observations are mostly based on concentrations and not on flows, which implies that it is hardly possible to validate the flow values resulting from ecological models. In addition, the number of flows in the real ecological network is magnitudes higher than the few flows that can be included in our primitive calculations.

Calculations of ascendency have the same shortcomings as calculations of power. The three concepts may all have a solid theoretical basis but their applications in practice still have definite weaknesses that are rooted in the complexity of real ecosystems. Based on an integration of the three concepts, we are able to expand on the earlier hypothesis based on exergy alone and let it comprise acendency and power in addition to exergy.

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Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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  • danielle smith
    What is meant by "solar radiation equilibrium" in the related ecology principle?
    2 years ago

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