The bookkeeping ofenergy analysis can be used to allocate many other kinds of indirectness. Starting with energy, we extend the method to other entities. To illustrate this, a hypothetical two-compartment system at steady state is used (Figure 1). This is complex enough to allow feedback (recycling), yet simple enough to allow using standard algebra. All that is done here can be couched in matrix notation, and shorthand is useful for systems with many compartments, but algebra is more transparent.

Figure 1 shows the flow of something, say energy, in a two-compartment system containing producers (e.g., green plants) and consumers (e.g., herbivores). (See Table 1 for definitions of terms.) The input to producers comes from outside the system (the Sun), and the export from consumers leaves the system. There is a variable amount of feedback from consumers to producers; it is possible for some plants to eat animals. In all diagrams of this type we decide which flows convey the direct and

FEEDBACK

indirect influences we deem important; our judgement is required. For example, the standard energy intensity concept is that the energy losses (e.g., low-temperature heat) are assumed to be embodied in the remaining flows of high-quality metabolizable biomass. This would give a modified diagram (Figure 2).

The remaining flows thus convey the input, which we wish to account for. The missing losses are now implicitly embodied in the flows that remain, and the energy intensities carry this formally. The parallelism and difference between Figures 1 and 2 crystallize the entire import of indirectness presented in this article. The intensities are thus the conceptual, dimensional, and numerical bridge between system input and flows. In this system, the units of intensities will be calories of gross primary production

INPUTn

INPUTc

Consumers

LOSSn

LOSSc

Figure 1 Energy flows (cal/day) in a two-compartment system at steady state. INPUTp is gross primary production of biomass by photosynthesis.

FEEDBACK *

INPUTn

FEEDBACK *

Producers |
Consumers | |

INPUTC*£P |

Figure 2 Embodied energy flows (cal GPP/day) in a two-compartment system. The energy intensities £ (cal GPP/cal) convert energy flows (cal/day) of Figure 1 to embodied energy flows (cal GPP/day).

EXPORTc *

Producers

Symbol |
Definition |
Unitsa |

A tj |
Time step |
day |

Ej |
Energy input to compartment j |
cal/day |

"j |
Energy intensity of compartment j's output |
cal/cal |

ECOL |
Energy cost of living for a household |
Btu/yr |

Household average energy intensity (=ECOL/Y) |
Btu/$ | |

"imp/ |
Energy intensity of imported material functionally identical to that produced by compartment j |
cal/cal |

EXPORT- |
Export from compartment j |
cal/day |

FEEDBACK |
Flow from consumers to producers |
cal/day |

gloof |
Generic term for system input which is allocated by method used in this paper |
?/day |

GPP |
Gross primary production, the energy fixed by plants |
cal/day |

IMPORT,- |
Import of material functionally identical to that produced by compartment j |
cal/day |

INPUT; |
Input to compartment j |
cal/day |

LOSS, |
Loss from compartment j |
cal/day |

N |
Nutrient input to compartment j |
g/day |

Vj |
Nutrient intensity of compartment j's output |
g/cal |

Vimpj |
Nutrient intensity of import of material identical to that produced by compartment j |
g/cal |

output, |
Output of compartment j |
cal/day |

PLj |
Path length of compartment j |
dimensionless |

Sj |
Stock of compartment j |
cal |

Tpj |
Trophic position of compartment j |
dimensionless |

tj |
Isolated-compartment residence time for compartment j |
day |

In-system residence time for compartment j |
day | |

Xij |
Flow from compartment i to compartment j |
cal/day |

X |
OUTPUTj |
cal/day |

Yi |
Annual household expenditure for consumption category i |
$/yr |

Y |
Sum of all annual household expenditures |
$/yr |

Z |
FEEDBACK |
cal/day |

aGrams, calories, and days are arbitrarily chosen as the units of mass, energy, and time for the examples in this article. Other units, as appropriate, are used for the applications.

(i.e., the sunlight fixed by photosynthesis, abbreviated GPP) per calorie of producer biomass or consumer biomass, but diverse units are possible depending on the system and the question asked. For example, in economic systems, energy intensity is measured in Btu/dollar (1 British thermal unit = 1055J).

The key assumption is that embodied energy is conserved in every compartment: embodied energy in = embodied energy out. The general process is summarized in Figures 3 a and 3b. Because the method can be used to allocate many things besides energy, the generic term 'gloof is used in Figure 3a. Figure 3b shows the balance equation for energy. Figures 3a and 3b allow for imports to inject embodied energy into the system. An example is Howard Odum's study of Silver Spring, Florida, where tourists fed bread to fish whose normal food was plants growing in the spring. Another example of imported embodied energy is America's importing of clothes made in China.

The energy intensities are obtained by solving the (linear) equations implied by Figure 3b:

because of their production elsewhere. This is discussed in more detail in the section on dynamic indicators. Let us apply this to a specific set of flows shown in Figure 4a. For conciseness, feedback will be denoted by Z.

Equation [1] gives one equation for each compartment:

The solution is "p = 10

2Z, "c = 20, both expressed in cal GPP/cal biomass. Substituting these energy intensities in Figure 3 b gives the embodied energy flows shown in Figure 4b. The entire system is in embodied energy balance, a consequence of the assumption that each individual compartment is in balance.

Now suppose we want to track the flow of nutrient. Nutrient intensities (r) will be expressed in grams of nutri-ent/cal of biomass. Here we will (arbitrarily) assume that nutrient is taken up only by producers and passed on without loss to consumers, but that some nutrient is leaked from consumers during metabolism. Specifically, assume that the nutrient intensity of the consumer metabolic loss is half that of the consumer export flow. This is shown in Figure 5.

The balance equations are then:

The sum is over all within-system inputs. In Figure 3b E is the energy from the Earth, or at least external to the system. The implicit energy intensity of energy itself is 1.0. Imported materials already have an energy intensity

In-system: gloof embodied in incoming in-system flows

Output:

gloof embodied in output flow

Source:

gloof generated internally plus that embodied in import flow

which are solved to yield rp = (1/10)(15 + Z)/(15 - Z), rc = 2/(15 — Z). Using these intensities yields Figure 6 for embodied nutrient flow.

100-

Producers |
Consumers | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

E SiXij i = in-system inputs S Xj Ej + Simpj IMPORT, Figure 3 (a) Generic scheme for allocating the embodied generic system input gloof for steady-state system. Gloof may be energy, nutrient, or even time (for calculating residence time). In economic applications, gloof may be money, labor, or pollution assimilation. For each compartment j, embodied gloof in = embodied gloof out, by assumption. (b) Embodied energy balance equation for compartment j, which follows from (a). Xtj are in-system inputs to j; Xj is j's total output; Ej is the (direct) energy input to j. 100-
Figure 4 (a) Explicit energy flows (cal/day) for a two-compartment system. Feedback, Z, can vary between 0 and 5 cal/day. Example is arbitrary, but the output/input ratio of 0.1 for producers is appropriate for many real systems. (b) Embodied energy flows (cal GPP/day) in system shown in (a) as a function of feedback, Z. Both compartments are in embodied energy balance, as is the entire system. Figure 5 Embodied nutrient flows (g/day) in system of Figure 4a. ^ (g/cal) are the nutrient intensities. 5n c Figure 5 Embodied nutrient flows (g/day) in system of Figure 4a. ^ (g/cal) are the nutrient intensities. Steady-State Analysis: Other Indicators Below are discussed three other indicators: TP, PL, and residence time. In Table 2, the equations for each are listed. The possibility of imports is explicitly allowed for. An example is American household electronics, most of which are made abroad. In Table 3, the specific equations for the example system in Figure 4a and their solutions are listed.
Figure 6 Embodied nutrient flows (g/day) in system shown in Figure 4a as a function of feedback, Z. Both compartments are in embodied nutrient balance, as is the entire system. 8 15 Figure 6 Embodied nutrient flows (g/day) in system shown in Figure 4a as a function of feedback, Z. Both compartments are in embodied nutrient balance, as is the entire system. Prod. nutr. intens. Prod. energy intens. Feedback, Z Prod. nutr. intens. Prod. energy intens. ## Figure 7 Energy and nutrient intensities vs. Z.The system is in embodied nutrient balance, but the flows have a surprising feature: for Z >0, the nutrient flow from producers to consumer (an internal flow) exceeds 1 g/day, the system's input flow. Critics have called this apparent contradiction a damning flaw of the method, but actually it is to be expected. Feedback speeds up a system's flows: more molecules pass a given point per unit time. Because here embodied nutrient is actual nutrient, the effect could be measured experimentally. This validates the method generally. Energy intensities and nutrient intensities as a function of feedback, Z, are shown in Figure 7. As feedback increases (and hence loss from consumers decreases), the two intensities approach equality. When Z = 5cal/day, consumers have no losses and the two compartments have functionally merged. ## Trophic PositionTrophic levels apply to a linear chain picture of feeding patterns: A eats nothing but B; B eats nothing but C, etc. If there are n compartments in the chain, then there are n integral trophic levels, and trophic level is the number of steps from the Sun + 1. Thus for producers and consumers in a chain, trophic levels = 1 and 2, respectively. With omnivory and the resulting web interactions, this view breaks down unless nonintegral TPs are allowed. Simply put, a compartment's TP is the (energy) weighted average of the TPs of each of its inputs plus 1. Caution: trophic interactions are always expressed in energy flows, so here one must use energy flows only, not nutrients or other flows. (There is also a dual approach, which results in an infinite series of integral trophic levels, which is not covered here). The standard convention of setting TPSun = 0 is used. From Table 3, the TPs are 1 + 2Z/100 and 2 + 2Z/100 for producers and consumers, respectively. Feedback increases TP of both. For Z = 0, TPp = 1, TPc = 2, as expected for a straight food chain. Path Length This can be expressed in two ways: • Looking backward in time. A molecule is just now leaving compartment j. How many intercompartment transits has it made between its entering the system and now? • Looking forward in time. A molecule is just now leaving compartment j. How many intercompartment transits will it make on average before exiting the system? For a steady-state system, these are equally easy to calculate. For a dynamic system, the backward-looking PL is preferable because it can be calculated without knowing the future. Therefore we calculate only the backward-looking PL. In words, PL is the weighted sum of the quantity (PL + 1) for each input. Imports do not figure in PL, which is based upon internal flows only. The input flows need not be energy, but they must all be in the same units so that the weighted average can be calculated.
Xj, flow of i to j; Ej, energy flow to j; Nj, nutrient flow to j. See Table 1 for additional definitions. Table 3 Explicit balance equations and solutions for five indicators Indicator Refer to figure Equations Solutions Units Energy intensity (e) cal GPP/cal Trophic position (TP) Path length (PL) Residence time (t) TPp10 2Z 100 PL is almost the same as TP. For a system with only sunlight as energy input, TP,- = PL,- + 1. If there are other system energy inputs such as imported feed, the difference between the two is more significant. As shown in Table 3, PL = 2Z/100 and 1 + 2Z/100 for producers and consumers, respectively. For Z = 0, PLp = 0 because there are no in-system inputs to producers. Residence Time (r) As with PL, this can be expressed looking either backward or forward in time, but here only the former is treated: a molecule is just now leaving compartment j. How long has it been in the system.? It is assumed that we know the isolated compartment residence times t, typically defined as the ratio of stock to throughflow. (Unlike all the indicators so far discussed, this requires that we know the stocks at steady state.) The system residence time is a function of these isolated-compartment residence times and the degree of connectedness of the compartments. In words, residence time for compartment j is the weighted average of the residence time of each input + j. From Table 3, the residence times are (1 + Z/100)tp + (Z/ 100)tc and (1 + Z/100)(tp + tc) for producers and consumers, respectively. Without feedback, the residence time for producers, rp, is just tp because the only input is from outside the system. Consumer residence time, rc, is tp + tc, because a molecule leaving consumers has passed exactly once through producers and consumers. TP and residence time are graphed versus Z in Figure 8. Indicators in Dynamic Systems ## Calculating Dynamic IndicatorsMost of energy analysis and systems ecology stressing indirectness has assumed a steady state in which flows and stocks are constant over time. Yet real systems are almost always dynamic. All the indicators addressed in this article can have a dynamic interpretation, as long as we use the back-looking form. Any dynamic analysis must be explicit about stocks, flows, and time steps. The elements of the dynamic view are shown in Figure 9. Cons. res. time Cons. trophic pos. Prod. trophic pos. Feedback, Z Figure 8 Trophic position and residence time vs. Z. The isolated compartment residence times are 1 and 5 days for producers and consumers, respectively. Figure 9 summarizes the assumption that in a time step at, the gloof embodied in the inflows and in the stock is distributed over the final stock and outflows. At the end of the time step, there has been the mathematical equivalent of perfect mixing, so that the energy intensity of stock and output are the same. Figures 10a and 10b illustrate this in detail for energy intensity and residence time. The flows are multiplied by the time step at for dimensional commensurateness with the stocks. Output can include a change in stock (inventory change in economic terminology), so that the new stock is the old stock plus this change. Figure 10a shows that energy intensity at time t + at is a function of the flows at time t + at, and the stocks and energy intensities at time t. If one knows the initial energy intensities and stocks, and one has a dynamic model to specify stocks and flows over time, one can use the equation implied by Figure 10a to calculate dynamic energy intensities: thAt At i=in-system inputs H ej Sj Similarly, from Figure 10b, one obtains for dynamic residence time: |

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