# Emergy and co-emergy

## Emergy and co-emergy

G Model ARTICLE IN PRESS ECOMOD-7317; No. of Pages 19 Ecological Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect Ecologica...

G Model

ARTICLE IN PRESS

ECOMOD-7317; No. of Pages 19

Ecological Modelling xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Emergy and co-emergy Stephen E. Tennenbaum ∗ Department of Mathematics, The George Washington University, United States

a r t i c l e

i n f o

Article history: Available online xxx Keywords: Emergy Co-emergy Transformity Co-transformity Embodied energy

a b s t r a c t We introduce a method of calculating emergy that requires only ordinary algebra without any reliance on special rules to account for co-production. This is accomplished by using an intermediate computation “co-emergy”, and treating co-production as a problem of scale. In addition, we compare emergy calculations using inputs to the system with emergy calculations using what was used up in the system. It is shown that this can lead to slightly different results. We show how these methods can be used to compute emergy in systems at steady state, with imports and exports and with changes in stocks. These techniques allow direct comparison of competitive species, industries, or technologies using standard methods of linear algebra. It also enables us to include the efﬁciencies of various processes explicitly, which can help in the formulation and testing of conjectures about the relationships between emergy and local and system-wide efﬁciencies. © 2014 Elsevier B.V. All rights reserved.

1. Introduction It is often observed that traditional methods of economic analysis are inadequate to deal with valuing the contribution of environmental resources to human welfare, let alone the well being of the planet. One of the major problems is that money is not used for transfers of materials, chemicals, or energy in nature. The only thing universally used in transfers of both nature and man, even including ideas and information, is energy itself. However there is no strict analogy between money and energy since money ostensibly maintains its potency while passing through a system whereas energy’s capabilities degrade (Odum, 1973). The total energy inputs, of one type, as a measure of the work done by nature to generate resources, both natural and manmade, can be viewed as a rough analogy to the total requirements of input–output analysis. Rough because ﬁrst, we are dealing with external inputs and ﬂows through the system rather than cycling within it. And second, we need to put all those inputs on an equivalent basis of energy of one type (most economies run on only one currency). This paper is about, from one point of view, an accounting procedure to assess the total energy inputs, of one type, to a system required to support the activities or production of some associated subsystem. The numbers obtained from these assessments can be used to compare system-wide efﬁciencies of competing processes, and can be used as proxies for the importance of those subsystems to the system as a whole. In order to equitably compare processes

∗ Tel.: +1 2029946235. E-mail address: [email protected]

with vastly different inner workings, or differing in their position in the hierarchy of the system, energy inputs need to use a common metric. Typically energy inputs of multiple types are traced back to their origins as solar inputs to the earth. If that is neither possible nor practical then the inputs are put on some commensurate basis to solar inputs by comparisons to transformations to a common energy form acting under certain efﬁciency constraints (e.g. petroleum to electricity compared to solar to wood to electricity).1 “In order to put the contributions of different kinds of energy on the same basis, we express all resources in terms of the equivalent energy of one type required to replace them. A new name is deﬁned: EMERGY (spelled with an “M”) is deﬁned as the energy of one type required in transformations to generate a ﬂow or storage” (Odum, 1988). Another deﬁnition of emergy is given as “...the available energy of one kind previously used up directly and indirectly to make a service or product. Its unit is the emjoule” (Odum, 1996). Although these two deﬁnitions appear on the surface to be of triﬂing difference, they are on closer reﬂection quite different. And although the deﬁnition is usually given in the latter form, it is the former one that is the basis of almost all calculations heretofore. In this paper we show how emergy using both deﬁnitions can be calculated using the methods described herein and how both deﬁnitions have their strengths and usefulness in analyzing some system properties.

1 By our use of the terms “type” and “solar inputs” we mean at a macroscopic level, and averaged over various dimensions of availability and use respectively. These details are dealt with elsewhere, see for example, Tilley (2003), or Odum (1996)

S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

(2014),

G Model ECOMOD-7317; No. of Pages 19

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2

Fig. 1. The labeling scheme for a single sector of a system or economy with the main ﬂows discussed in this analysis. We use Odum’s energy systems language Odum and Environment (1971) Circles – “source” symbol, i.e. external resources (in terms of energy); box – an economic sector or other well deﬁned unit consisting of production and storage; chevron – “interaction” symbol, i.e. sector production function; “tank” symbol – storage; lines – energy ﬂows. Lines leading to ground symbol are heat and other second law losses from production (h) and depreciation and decay of stocks (). See text for further details.

The method outlined here is a review, elaboration and development of the “track summing method” (Odum, 1996) ﬁrst developed using linear algebra techniques in Tennenbaum (1988). We will try to cover all the major conceptual issues involved in applying our method. In Section 2.1 we establish basic notation and labeling scheme. In Section 2.2 we discuss the calculation of source emergy and transformities. In Section 2.3 we discuss the calculation of sink emergy and transformities. Using standard matrix algebra, in Section 2.4 we show how the special case of computing the emergy of co-products might be dealt as an issue of scale and without any recourse to special computational rules. In Section 2.5 we generalize the situation to include trade and changes in stocks. And ﬁnally we include an appendix where we demonstrate one possibility for how these computations might be used to compare the system wide efﬁciencies of competitive subsystems.

Table 1 Labeling conventions and some deﬁnitions of symbols used in this paper.

2. Methods

a The gross production coefﬁcients (ai,j = qi,j /pj ) are different from the IO technical coefﬁcients (ai,j = qi,j /qj ) available for many states and countries economies. The relationship between the two is, in the simplest case, ai,j = ωj ai,j where ai,j is the production coefﬁcient, ai,j is the published technical coefﬁcient, and ωj is the storage efﬁciency of compartment j. In addition we include explicit coefﬁcients for environmental inputs and labor, and may include trade, capital expenditures, or other components of ﬁnal demand, etc.

2.1. Model For our ﬁrst example we assume a simple case of a system where each sector’s output (product) is treated in the aggregate so that we avoid complications dealing with co-products. All sources are in solar units and there is a single external source, if any, per unit. For economic systems, our “sectors” will explicitly include all components of GDP (including personal and government expenditures, changes in inventories, imports and exports, etc.) and value added (wages, salaries and other forms, of income that represent human labor and effort). In addition environmental inputs and services to the economic system are included regardless of whether any human endeavor or effort was made in obtaining them. See Fig. 1 For a system of n compartments we have the following energy ﬂows related to a particular sector i (ﬂows to the n + 1st “unit” are exports2 ). Energy (or energy content of raw materials as well as non-competitive imports) are denoted by xi and diagramed coming from a circle outside the boundary of the system and ﬂowing to the interaction (production symbol). Energy content of newly produced product (gross production) is denoted by pi and is diagramed as coming from the interaction symbol to the tank symbol (storage). Energy ﬂow out of storage i being used by any components of the system, including self use and exports, is qi , and the portion of that ﬂow to another particular sector j is denoted qi,j .

2

Imports (goods and materials that are also produced within the system) and changes in storage will be dealt with in Section 2.5.

Relationship

Explanation

qi =

Total of useful outﬂow of product to all sectors and exports

n+1 j=1

ri = xi +

qi,j

n j=1

qj,i

Total inﬂow of energy, material & product from all sectors

dQ

i = pi − qi − wi dt fi,j = qi,j /qi

Rate of change of stored product Fraction of total useful output of unit i used by unit j (

n+1 j=1

fi,j = 1)

Gross production coefﬁcientsa Production losses (energy costs) Energy conversion efﬁciency Storage losses (or deaths; a function of stock level Qi ) Storage (or survival) efﬁciency Total energy losses of unit i Overall efﬁciency of the unit i

ai,j = qi,j /pj hi = ri − pi = Hi (ri ) i = pi /ri wi = pi − qi = Wi (Qi ) ωi = qi /pi ui = hi + wi i ωi = qi /ri

We will call this output distributed production, and it equals selfuse production (qi,i ) plus net production.3 Lines leading to ground symbol are losses from production hi and depreciation and decay of stocks independent of the production process wi . Storage of energy (or the energy content of capital stocks) is denoted Qi . We also have the following relationships (Table 1) At steady state we have dQ i /dt = 0 = pi − qi − wi , so that pi = qi + wi . Also at steady state pi = ri − hi and ri = qi + hi + wi . 2.1.1. Energy balance for a simple system at steady state For a system of three compartments or sectors (see Fig. 2 for example) the energy of gross production can be equated to the energy inputs less production losses, p1

= x1 + q1,1 + q2,1 + q3,1 − h1 ,

p2

= x2 + q1,2 + q2,2 + q3,2 − h2 ,

p3

= x3 + q1,3 + q2,3 + q3,3 − h3 .

3 We use net production in the economic sense of the word as opposed to the ecological sense. The ecological meaning takes into account any production exceeding respiration and includes production allocated for growth.

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Fig. 2. Energy ﬂows in a simple 3 sector economy. Units are arbitrary.

At steady state gross production is also equal to the distributed output plus losses from storage (depreciation), p1

= q1,1 + q1,2 + q1,3 + q1,4 + w1 ,

p2

= q2,1 + q2,2 + q2,3 + q2,4 + w2 ,

p3

= q3,1 + q3,2 + q3,3 + q3,4 + w3 . In general

xi +

n 

qj,i − hi = pi =

j=1

n+1 

qi,j + wi .

(1)

j=1

We can substitute output fractions times the total distributed production for each individual ﬂow between compartments on the left, that is qj,i = fj,i qj , and the total distributed production on the

n+1

right ( xi +

j=1

n 

qi,j = qi ), which gives

fj,i qj − hi = pi = qi + wi .

(2)

j=1

Or, alternatively, we can substitute production coefﬁcients times gross production, that is qj,i = aj,i pi , for the individual ﬂows xi + pi

n  j=1

aj,i − hi = pi =

n 

ai,j pj + qi,n+1 + wi .

(3)

j=1

2.2. Source emergy and transformity calculations 2.2.1. Source emergy Now to calculate the emergy, that is the total amount of energy of one type input to the system in support of some subsystem, we draw the following crude analogy. We have a system of water barrels connected by pipes with some constant inﬂow, and each barrel has ﬂow through those pipes to other barrels and also has an open tap on it. One way of determining how much of the water input “upstream” of a particular barrel is required to produce the outﬂow from that barrel is to turn off all the taps in the system and catch the outﬂow of the target barrel over a given interval of time. In calculating emergy we perform a similar feat mathematically by Please cite this article in press as: Tennenbaum, http://dx.doi.org/10.1016/j.ecolmodel.2014.09.012

eliminating heat and depreciation losses of the system and recalculating the ﬂow from the sources through to the target component (and any exports). We will assume, for the time being, that all source inputs to the system are of a single energy type (solar energy), or that we have some method of calculating the solar energy required to produce that source input (or calculating the solar energy of an equivalent process).4 These inputs of one type (e.g. solar energy equivalents with units of solar emjoules or sej) to unit i will be denoted by Ei . The total system use of energy of one type, upstream of and including that used by a particular unit j, is denoted Mj and is called the emergy of unit j . And what we might call the redirected ﬂow passing through an intermediate unit k enroute to the target unit j is called co-emergy, and is denoted Ck,[j] . (Note that we use square brackets around the target index to stress that it is an ultimate not proximate destination. Note also that we deﬁne Cj,[j] ≡ 0 since we do not want to double count ﬂow through the target). We will also assume that all the ﬂows of co-emergy from a particular compartment are distributed in proportion to the distributed production of that compartment. It should be noted that the total co-emergy output of a compartment is dependent on both which compartment it is and which compartment is the ultimate target (i.e. which compartment we are computing the emergy of). As an example for the ﬁrst unit we have, E1 + f2,1 C2,[1] + f3,1 C3,[1]

= M1

E2 + f2,2 C2,[1] + f3,2 C3,[1]

= C2,[1]

E3 + f2,3 C2,[1] + f3,3 C3,[1]

= C3,[1]

For the second unit we have, E1 + f1,1 C1,[2] + f3,1 C3,[2]

= C1,[2]

E2 + f1,2 C1,[2] + f3,2 C3,[2]

= M2

E3 + f1,3 C1,[2] + f3,3 C3,[2]

= C3,[2]

4 For example the solar equivalent of coal can be determined by analogy of coal production of electricity with wood production of electricity wherein we can calculate the solar energy inputs to produce the wood (see Odum, 1996 for details).

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And for the third unit we have, E1 + f1,1 C1,[3] + f2,1 C2,[3]

= C1,[3]

E2 + f1,2 C1,[3] + f2,2 C2,[3]

= C2,[3]

E3 + f1,3 C1,[3] + f2,3 C2,[3]

= M1

E1

−f2,1

1

E3

(1 − f1,1 )

(1 − f3,3 )

⎢ ⎥ ⎢ ⎣ E2 ⎦ = ⎣ −f1,2

−f3,1

1

−f3,2

−f1,3

0

(1 − f3,3 )

E3

E1

(1 − f1,1 )

−f2,1

⎢ ⎥ ⎢ ⎣ E2 ⎦ = ⎣ −f1,2

−f2,3

−f1,3

E3

C1,[2]

⎥⎢

C1,[3]

⎡ 2 10/25 5/25 5/25 ⎢ 5 ⎢ ⎢ 50/100 30/100 ⎥ = ⎢ 1 F = ⎣  10/100 ⎦ ⎢          1 3 1 3 3 ⎣ 10 / / 1/

⎤ ⎥

colj

F

=

[k]

0

if

j= / k

if

j=k

M3

.

(4)

Ik =

0

F

= FT Ik

[k]

(6)

f1,2

f1,3

F = ⎣ f2,1

f2,2

f2,3 ⎦

f3,1

f3,2

f3,3

and

T

T

colk C = and



0 f2,1

f3,1

0 f2,3

f3,3

f1,1

f2,1

0

f1,3

f2,3

0

−1

I − FT

M1

⎢ C = ⎣ C2,[1] C3,[1]

[k]

C1,[2] M2 C3,[2]

1 6

3 10

1 6

2 3

⎥ ⎥ ⎥. ⎥ ⎦

30, 000

⎦ = ⎣ 70, 000 ⎦ . 0

80, 000

60, 000

78, 571.4

36, 000

67, 142.9

88, 000 171, 428.6 ⎦ .

These ﬂows (the off-diagonal elements of matrix C) and the emergy (the main diagonal elements) are distributed as shown in Figs. 3–5.

T

f1,1

0

f1,3

0

f3,1

⎢ ⎥ F = ⎣ f1,2 0 f3,2 ⎦ [2] f3,3

⎢ ⎥ F = ⎣ f1,2 f2,2 0 ⎦ [3] Then

100 × 700

180, 000

⎢ ⎥ F = ⎣ 0 f2,2 f3,2 ⎦ [1] ⎡

f1,1

E=⎣

12, 000 + (100 × 180)

C = ⎣ 200, 000

For example

2

1 2

Using these and the procedure above we get the emergy/co-emergy matrix

then

T

2

1 5

(5)

i= / j or i = j = k

if

4

0

i=j= / k

1 if

2

1 5

The ﬁrst and second rows do not sum to one since 20% of agricultural and 10% of industrial output is exported respectively. For the source inputs to agriculture there is 12,000 units solar energy input directly (left hand side of Fig. 2), and 180 units of a non-renewable source with a solar equivalence factor of 100 (top of Fig. 2). There is also 700 units of this same non-renewable source input to industry. The source vector is therefore

Or using a modiﬁed identity matrix where the kth diagonal element is replaced with 0



4

[k]

colj FT



column replaced with zeros.



(8)

n

If the vector E = [Ei ] ,and the matrix F = [fi,j ] then let column j

of the matrix FT be deﬁned as F transposed and then the kth

T

[k]

T diagE

to compartment j, and Mj = M . i=1 i,j For the system shown in Fig. 2 the matrix of output fractions looks like

0 ⎦ ⎣ C2,[3] ⎦ 1

−1

I − FT



C3,[2]

⎤⎡

 rowk

The entries of M Mi,j are the emergy contributions of source i

⎥⎢ ⎥ ⎦ ⎣ M2 ⎦

0

(1 − f2,2 )

colk M =

C3,[1]

⎤⎡

0

E1

M1

⎥⎢ ⎥ ⎦ ⎣ C2,[1] ⎦

−f3,2

−f2,3

0

⎤⎡

−f3,1

⎢ ⎥ ⎢ ⎣ E2 ⎦ = ⎣ 0 (1 − f2,2 ) ⎡



In matrix notation we have

The entries of the main diagonal of C are the emergy requirements of each component, and the off diagonal entries are the co-emergies in column j for target component j . If we keep the individual contributions of each source separate we can build a matrix of only these emergy values

E

C1,[3]

(7)

⎥ C2,[3] ⎦ M3

2.2.2. Source transformity Alternatively we can use gross production coefﬁcients. These are related to the output fractions by the identities, fi,j qi = qi,j = ai,j pj = ai,j ωj−1 qj . The gross production coefﬁcients are recipes for production. Each one tells the amount of a particular type of input necessary for a unit output of product. They have an advantage over the output fractions in that they are ﬁxed for a given technology, but the have the disadvantage that they do not act like probabilities so the mathematical manipulation of them (such as sums or products) do not always have an intuitive interpretation. Using gross production coefﬁcients though, we can calculate transformities directly. To do this the easiest way is just to make direct substitutions for the output fractions in the emergy calculations given in the previous section. For the ﬁrst unit we have, E1 + f2,1 C2,[1] + f3,1 C3,[1]

= M1

E2 + f2,2 C2,[1] + f3,2 C3,[1]

= C2,[1]

E3 + f2,3 C2,[1] + f3,3 C3,[1]

= C3,[1] .

S.E.,

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and

co-emergy.

Ecol.

Model.

(2014),

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Fig. 3. Co-emergy ﬂows and emergy of agriculture in a simple 3 sector economy. All source inputs are converted to solar equivalent units (e.g. joules to solar em-joules) Output from agriculture is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of agriculture derived from that source.

Substituting for fi,j : fi,j = ai,j qj /(ωj qi ). We have

Dividing each row i by qi gives

E1 + a2,1

q1 q1 C2,[1] + a3,1 C3,[1] ω1 q2 ω1 q3

= M1 ,

C2,[1] C3,[1] ω1 E1 + a2,1 + a3,1 q1 q2 q3

=

E2 + a2,2

q2 q2 C2,[1] + a3,2 C3,[1] ω2 q2 ω2 q3

= C2,[1] ,

C2,[1] C3,[1] ω2 E2 + a2,2 + a3,2 q2 q3 q2

=

E3 + a2,3

q3 q3 C2,[1] + a3,3 C3,[1] ω3 q2 ω3 q3

= C3,[1] .

C2,[1] C3,[1] ω3 E3 + a2,3 + a3,3 q3 q2 q3

=

ω1 M1 , q1 ω2 C2,[1] q2 ω3 C3,[1] q3

, .

Fig. 4. Co-emergy ﬂows and emergy of industry in a simple 3 sector economy. All source inputs are converted to solar equivalent units (e.g. joules to solar em-joules) Output from industry is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of industry derived from that source.

S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

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Fig. 5. Co-emergy ﬂows and emergy of consumers in a simple 3 sector economy. All source inputs are converted to solar equivalent units (e.g. joules to solar em-joules) Output from consumers is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of consumers derived from that source.

Similar substitutions are made for the other units. In keeping with our “co”-lexicon we deﬁne the co-transformity as the co-emergy per unit output of an intermediate system component (k,[j] = Ck,[j] /qk ), its deﬁnition is parallel to that of transformity (the emergy per unit output of the target component; Tj = Mj /qj ). Replacing qi with its equivalent ωi pi , = ω1 T1

E2 + a2,2 2,[1] + a3,2 3,[1] p2

= ω2 2,[1]

E3 + a2,3 2,[1] + a3,3 3,[1] p3

= ω3 3,[1] ,

ω1

⎢ ⎥ ⎢ ⎣ E2 /p2 ⎦ = ⎣ 0 E3 /p3

E1 /p1

0

⎢ ⎥ ⎢ ⎣ E2 /p2 ⎦ = ⎣ E3 /p3

E1 /p1

⎢ ⎥ ⎢ ⎣ E2 /p2 ⎦ = ⎣ E3 /p3

[k]



=

colj AT

[k]

(ω2 − a2,2 )

−a3,2

−a2,3

(ω3 − a3,3 )

T1

⎤⎡

(ω1 − a1,1 )

0

−a3,1

−a1,2

ω2

−a3,2

−a1,3

0

(ω3 − a3,3 )

1,[2]

(ω1 − a1,1 )

−a2,1

0

a2,2

a2,3 ⎦ ,

a3,1

a3,2

a3,3

−a1,2

(ω2 − a2,2 )

0

−a1,3

−a2,3

ω3

T A

[1]

0 a2,1

0

0

=⎣ 0

ω2

0

0

0

ω3

a3,1

= ⎣0

a2,2

a3,2 ⎦

0

a2,3

a3,3

ω1

a1,1

a2,1

0

a1,3

a2,3

0

T A

⎥ ⎦,

a1,1

0

a3,1

= ⎣ a1,2

0

a3,2 ⎦

a1,3

0

a3,3

[2]

⎢ ⎥ A = ⎣ a1,2 a2,2 0 ⎦ [3]

⎤ Then



−1

colk  =  − AT

3,[2]

1,[3]

A = ⎣ a2,1

T

⎥⎢ ⎥ ⎦ ⎣ T2 ⎦

⎤⎡

a1,3

⎥⎢ ⎥ ⎦ ⎣ 2,[1] ⎦ 3,[1]

(9)

(10)

a1,2

and

⎤⎡

,

= AT Ik .

a1,1

−a3,1

if j = k

0

For example

−a2,1

if j = / k

or

and

A

A

and so on. In matrix notation we can write E1 /p1

colj

T

T

E1 + a2,1 2,[1] + a3,1 3,[1] p1

deﬁned as the transpose of A with the kth column replaced with zeros.

⎥⎢ ⎥ ⎦ ⎣ 2,[3] ⎦

T1

1,[2]

 = ⎣ 2,[1] 3,[1]

T3

If the vector ␣ = [Ei /pi ], the vector ω = [ωi ], the matrix  = diagω and the matrix A = [ai,j ] then let column j of the matrix AT[k] be

[k]

1,[3]

(11)

⎤ ⎥

2,[3] ⎦

T2 3,[2]

T3

The entries of the main diagonal of  are the transformities of each component’s distributed production, and the off diagonal entries are the co-transformities in column j for target component j . S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

(2014),

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Again referring to the system shown in Fig. 2 the matrix of gross production coefﬁcients is

⎡ 2 10/35 5/110 5/2 ⎢ 7 ⎢ ⎢ 10/35 ⎥ 50/110 30/2 = ⎢ 2 A = ⎣  ⎦ ⎢ 1 1 ⎣ 7 /35 /110 1/2 ⎡

4

4

1 140

1 22 5 11 1 440

5 2

⎥ ⎥ 15 ⎥ ⎥. ⎦ 1 2

The entries here are ingredients of a single unit of gross production and therefore do not, in general sum. We also need the diagonalized entries of storage efﬁciencies,

⎡ 25

⎢ ⎢ ⎢ =⎢ 0 ⎢ ⎣

35

0

0

0

⎡5

⎥ ⎢7 ⎥ ⎢ ⎥ 0 ⎥=⎢0 ⎢

⎥ ⎦ ⎣

100 110

3/2

0

0

0

10 11

0

0

3 4

0

2

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

The source input coefﬁcients are also part of the ingredient list, but are exogenous ingredients. For agriculture there is 30,000 units solar energy units for 35 units of gross production. There is also 70,000 units solar energy units input to industry for 110 units of gross production. The source vector is therefore

30, 000/35

857.1429

␣ = ⎣ 70, 000/110 ⎦ = ⎣ 636.3636 ⎦ . 0

0/2

Using these and the procedure above we get the transformity/ co-transformity matrix

3, 200

2, 400 880

1, 714.29

120, 000

24, 000

44, 761.93

 = ⎣ 2, 000

3, 142.86

⎦.

The co-transformities of outputs of each compartment are tabulated in the column owned by the particular target compartment on the diagonal cell of that column (which contains the transformity of that compartment). 2.3. Sink emergy and transformity calculations 2.3.1. Sources and system boundaries In the above sections we assumed that all source inputs to the system were of a single energy type (solar energy), or converted to equivalent (solar) units by multiplying the original physical units by its co-transformity. This may seem to pose a circular problem that may be avoided by calculating the transformities 5 of sources through other methods see for example Odum (1996, 2000), Odum et al. (2000), or Odum and Collins (2003), Collins (2003). In the following section, we discuss the energy degradation of sources by individual compartments. In this case we need to push the system boundary back to include the amount of the energy that was degraded before entering the system as a source’s co-emergy equivalent. 2.3.2. Energy degradation by individual compartments 2.3.2.1. Expanding the system boundary. In the following we will be talking about the expanded the system boundary to include the primordial solar energy, or the solar equivalent of other energy input

5

The transformity of a source will be the same or nearly the same as the cotransformity if feedbacks from the the larger system are non-existent or trivial respectively.

7

types. The examples will still use a base network of three compartment with two different types of energy input (x(),i ) for illustration  = 1 is solar energy,  = 2 is a higher quality input such as petroleum. The greek letter xi in parentheses denotes the enumeration of different source types, the i refers to a compartment that that source feeds. Using the total efﬁciency of each compartment we can write the energy balance equations for the original system as, x(1),1 + x(2),1 + f1,1 q1 + f2,1 q2 + f3,1 q3

= (1 ω1 )−1 q1

x(1),2 + x(2),2 + f1,2 q1 + f2,2 q2 + f3,2 q3

= (2 ω2 )−1 q2

x(1),3 + x(2),3 + f1,3 q1 + f2,3 q2 + f3,3 q3

= (3 ω3 )−1 q3 .

The expanded system will add a compartment which will include the aggregate biogeochemical processes and compressed time of producing petroleum from phytoplankton and ultimately sunlight. We will let this be the ﬁrst compartment and all others are promoted one index number. x(1),1

= (1 ω1 )−1 q1

x(1),2 + f1,2 q1 + f2,2 q2 + f3,2 q3 + f4,2 q4

= (2 ω2 )−1 q2

x(1),3 + f1,3 q1 + f2,3 q2 + f3,3 q3 + f4,3 q4

= (3 ω3 )−1 q3

x(1),4 + f1,4 q1 + f2,4 q2 + f3,4 q3 + f4,4 q4

= (4 ω4 )−1 q4 .

Petroleum inputs are now sector one output (x(2),2 + x(2),3 + x(2),4 = q1 ). Distribution of ﬂows from sector one are the fractions of crude oil source inputs (in the  original unexpanded system) distributed to units (f1,2 = x(2),2 / i x(2),i , etc.). The transof petroleum

formity −1 is the reciprocal of the efﬁciency of formation (1 ω1 ) = T(2) .6 Finally the amount of solar input to sector one, in the expanded system, is the tranformity of oil times the total oil source inputs to the unexpanded system (x(1),1 = T(2) i x(2),i ). For example expanding the boundary of the system shown in Fig. 2 would appear as in Fig. 6 The associated co-emergy/emergy equation is the same as before colk C =



−1

I − FT

[k]

E

(12)

with F being the expanded system matrix of output fractions, and E the increased dimension vector of solar inputs as described above. The emergy/co-emergy matrix of our “3 expanded to 4” sector economy would now be

88, 000

88, 000

88, 000

88, 000

57, 600

180, 000

36, 000

67, 142.9

⎢ 42, 000 80, 000 60, 000 78, 571.4 ⎥ ⎥ ⎣ 36, 000 200, 000 88, 000 171, 428.8 ⎦ ,

C=⎢

with column 1 – petroleum, column 2 – agriculture, column 3 – industry, and column 4 – consumers. 2.3.3. Sink emergy Suppose now we want to ﬁnd out how much energy is dissipated in the production of a particular component of our system, starting from a commensurate quality of input (e.g. solar energy). To do this, instead of using the solar inputs, we calculate exactly as we would with solar inputs on the expanded system but use the dissipation of each compartment as “input” and ignore the actual solar inputs. The total energy loss of unit i is ui = hi + wi .. For example, let Nk

6 We are, for the sake of simplicity and illustration, ignoring the extraction and reﬁning industries here. In an actual analysis the all the “output” of crude oil formation would ﬁrst go to a extraction industry compartment then a reﬁnery, and so on.

S.E.,

Emergy

and

co-emergy.

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Fig. 6. Energy ﬂows in a simple 3 sector economy expanded to internalize the production of high quality energy inputs. Units are arbitrary.

and Di,[k] be the emergy and co-emergy respectively, computed in this way. Then in an extended system with only solar inputs, for compartment 2 we have u1

= D1,[2]

u2 + f1,2 D1,[2] + f3,2 D3,[2] + f4,2 D4,[2]

= N2

u3 + f1,3 D1,[2] + f3,3 D3,[2] + f4,3 D4,[2]

= D3,[2]

u4 + f1,4 D1,[2] + f3,4 D3,[2] + f4,4 D4,[2]

= D4,[2] .

The vector with the elements [ui ] is u, and colk D = Also



−1

I − FT



colk N =

[k]

 rowk

u.

(13)

−1

I − FT

[k]

source inputs “source emergy”, “source transformities”, “source coemergy”, and “source co-transformities”, and that calculated with degraded energy (on the expanded system) “sink emergy”, “sink transformities”, etc. For example, the output fraction matrix, energy degradation vector, and sink emergy/co-emergy matrix corresponding to Fig. 6 are respectively,

⎢ ⎢ ⎢0 ⎢ F=⎢ ⎢ ⎢0 ⎢ ⎣ 0

T diag (u)

(14)

The entries of N (Ni,j ) are the energy degradation requirements of compartment i needed for output by compartment j, and Nj = N . i=1 i,j

AT[k]

are as before. The transformities and co-transformities matrix we will call Sk , i,[k] respectively (matrix ). Thus



−1 [k]

700 880

2 5

1 5

1 10

1 2

1 6

1 6

87, 120

0

⎥ ⎥ ⎥ ⎥ ⎥, 3 ⎥ ⎥ 10 ⎥ ⎦ 1 5

2 3

⎢ 12, 175.25 ⎥ ⎥ ⎣ 655.25 ⎦ , 34.5

and

2.3.4. Sink transformities To calculate transformities using energy degradation we only need to substitute the ratio of energy degradation to gross production for the vector ␣ = [Ei /pi ], again on the expanded system. But ui /pi = (−1 − ωi ) so let’s call these inputs ϕi (vector ␾). The i vectors ω = [ωi ], the matrix  = diagω, the matrix A = [ai,j ], and the

180 880

u=⎢

n

colk =  − AT

0

87, 120

⎢ 44, 355 ⎣ 39, 940

D=⎢

62, 662.5

87, 120

87, 120

79, 992.86

60, 025

199, 921.43

87, 980

180, 032.79

36, 118.5

87, 120

⎥ ⎥ 171, 329.29 ⎦ 78, 546.96

67, 142.68

These sink co-emergy (off-diagonal elements) and the sink emergy (main diagonal elements) ﬂows are distributed as shown in Figs. 7–9. 2.4. Co-products

(15)

Note that this formulation connects the transformity of a particular compartment explicitly with the efﬁciencies of all the compartments of the system! This becomes important when calculating system wide power relations. When we need to make a distinction, we will call the emergy, transformities, co-emergy and co-transformities calculated with Please cite this article in press as: Tennenbaum, http://dx.doi.org/10.1016/j.ecolmodel.2014.09.012

Co-products are products of a single compartment that have different uses in the larger system, the classic example being sheep wool and meat. Each output has a potentially different transformity and Odum’s diagraming method designates these as separate pathways emanating from a compartment rather than branches of a single pathway. The emergy required to produce the meat and the wool over the life of the sheep is equal to the emergy required to S.E.,

Emergy

and

co-emergy.

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Model.

(2014),

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9

Fig. 7. Sink co-emergy ﬂows and sink emergy of agriculture in a 3 sector economy. All source inputs are backtracked (or converted) to solar equivalent units (e.g. joules to solar em-joules) Output from agriculture is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of agriculture derived from that source.

produce the sheep as a whole. On the face of it this seems straightforward enough, however, in traditional emergy accounting, when co-products are used as inputs to some other process the accounting requires some ﬁnesse since each of the co-products, such as wool and meat, required the same inputs to the sheep over its life. Care must be taken to discount the lesser of the co-product inputs (in terms of emergy) to a given process so that double-counting of the emergy does not occur. Meticulous attention to this issue in a tabular accounting is necessary, but not particularly troublesome, in insuring valid emergy computations. However, their presence in matrix accounting procedures is more problematic. The reason is that a simple application of matrix algebra imposes a conservation

constraint on the calculus. A simple matrix has one type output for each node. Input–output methods can deal with multiple output types but, without additional manipulation or special techniques, cannot identify, down the line, what is being double counted (for example see Odum and Collins, 2003; Bastianoni et al., 2011; Baral and Bakshi, 2010, or Corre and Truffet, 2012). The other way of dealing with co-products is to treat them as an issue of scale Tennenbaum (1988), Tiruta-Barna and Benetto (2012). In other words, for any system, say a national economy, how we regard the outputs of any sector in that economy depends on the level of aggregation. If we have a three or four sector economy with agriculture as a single compartment, then agricultural

Fig. 8. Sink co-emergy ﬂows and sink emergy of industry in a 3 sector economy. All source inputs are backtracked (or converted) to solar equivalent units (e.g. joules to solar em-joules) Output from industry is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of industry derived from that source.

S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

(2014),

G Model ECOMOD-7317; No. of Pages 19

ARTICLE IN PRESS S.E. Tennenbaum / Ecological Modelling xxx (2014) xxx–xxx

10

Fig. 9. Sink co-emergy ﬂows and sink emergy of consumers in a 3 sector economy. All source inputs are backtracked (or converted) to solar equivalent units (e.g. joules to solar em-joules) Output from consumers is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under each source is the amount of the emergy of consumers derived from that source.

output can be taken as a single undifferentiated product used by all other sectors. As such we would, in the methodology formulated in this paper, calculate a single, average transformity for agricultural output itself, and a single co-transformity for agriculture in the analysis of some other sector of the economy. However, if we want to be a bit more detailed in our analysis then we might want to specify “ﬁber” (e.g. cotton, ﬂax, etc.) to industry and “food” (e.g. potatoes, corn, etc.) to consumers. We can then specify a more detailed economy with two sectors (food and ﬁber) in the place of the one (agriculture). Similarly, in the case of sheep products, we can rather than specifying “sheep” as a compartment with co-products wool and meat, we can disaggregate the sheep into its physiological and histological relevant components – muscle, skeleton, skin, hair follicles, etc. – and treat each as a compartment in of itself interacting with the rest of the sheep and the compartments of the larger system. Due to the high degree of connectivity between all of the sheep’s components, the emergy of each of these component-compartments would have the same, or nearly the same, emergy as the sheep as a whole (over corresponding time intervals). That’s because to measure the emergy we, by deﬁnition, take each component (meat, wool, or sheep) in turn as the only intrinsic absorbing state for the entire (steady-state) system.7 However, when we are looking at the emergy of some other (non-sheep) system component, we calculate the co-emergy of the sheep as a whole, or the co-emergy of the wool and the co-emergy of the meat. The co-emergy is that portion of the system energy use attributable to, or “passing through”, one compartment in support of the target compartment. The co-emergy is not the emergy of the co-products, and is, by construction, summable in the mathematical framework given above. By treating each physiological system as a separate compartment in the larger system, we avoid double counting naturally within the proposed computational method. The difﬁculty though, is that although this is conceptually a simple approach, practically it may be difﬁcult or impossible to identify all the energy pathways, structures, storages,

7

Exports are extrinsic absorbing states.

connections, inputs and outﬂows comprising an organism, or any other complex, highly integrated, and self contained entity. The real utility of this method though, is not only in the computation of the emergy of the co-products themselves, but in computing the emergy of a compartment downstream that uses energy or material ﬂows deriving from more than one of those co-products in varying proportions and degree of cycling within the network. Again a simple hypothetical example may help illustrate. We will use a system with sheep, or more speciﬁcally meat and wool, and a shepherd Fig. 10. We compose the system with four compartments. In reverse order we have the fourth compartment – the shepherd, the third – wool, the second – muscle (meat), and the ﬁrst – all other anatomical and physiological functions of the sheep (which we summarize with a interaction/work-gate symbol). The source emergy of any target compartment is how much of the solar equivalent energy entering the system is required by any that compartment.

1000

⎢ 0 ⎣ 0

C=⎢

0

1250

5000

10, 784.31

1000

4000

8627.45

250

1000

75

1090.91

⎥ ⎥ 2156.86 ⎦ 1000

These sink co-emergy and the sink emergy ﬂows are distributed as shown in Figs. 11–13. 2.5. Competitive imports, exports, and changes in stocks Imports, exports, and changes in stocks can be easily incorporated into the framework already outlined above. When storage has a net increase over time interval t then ( Qi / t) > 0, and an amount Qi is exported in time t from the system. We will denote the average rate of increase in storage Qi+ = ( Qi / t) > 0. The increase in storage comes out of gross production ﬂows, thus pi − Qi+ = qi + wi if ( Qi / t) > 0 . We also denote pi = qi + wi as the amount of gross production

necessary to balance distributed production plus depreciation pi = pi + Qi+ . When storage decreases, ( Qi / t) < 0, and an amount Qi is imported in time t and ﬂows back into the system. We will denote S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

(2014),

G Model ECOMOD-7317; No. of Pages 19

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11

Fig. 10. Hypothetical energy ﬂows in a sheep and shepherd system. Input is of a single energy type only (e.g. solar) and the number is arbitrary.

Fig. 11. Co-emergy ﬂows and emergy of meat. Source input is a single type (e.g. solar equivalent units) Output of meat is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface.

Fig. 12. Co-emergy ﬂows and emergy of wool. Source input is a single type (e.g. solar equivalent units) Output of wool is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface.

S.E.,

Emergy

and

co-emergy.

Ecol.

Model.

(2014),

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Fig. 13. Co-emergy ﬂows and emergy of a shepherd using meat and wool. Source input is a single type (e.g. solar equivalent units). Output of shepherd labor is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface.

the average rate of decrease in storage Qi− = −( Qi / t) > 0 If ( Qi / t) < 0 then we augment the distributed production so that the total output is qi = qi + Qi− , where qi is the current distributed production, and pi = qi + wi . Note that if there is no net change in storage then pi = pi , qi = qi , and pi = qi + wi as it was originally. Also note that the changes in storage are net changes so that if there is a net increase Qi+ > 0 then Qi− = 0, and if there is a net decrease Qi− > 0 then Qi+ = 0. Competitive imports are goods and materials from outside the boundaries of the system that are comparable to goods and material produced within the system. These imports are input directly into consuming compartments and the ﬂows will be designated by yij where i is the index of the comparable internal compartment and j is the destination compartment. For a 3 × 3 example the set of equations is Q1 t Q2 t Q3 t

= x1 + q1,1 + y1,1 = x2 + q1,2 + y1,2

= x3 + q1,3 + y1,3

+ q2,1 + y2,1 + q2,2 + y2,2

+ q2,3 + y2,3

+ q3,1 + y3,1 + q3,2 + y3,2

+ q3,3 + y3,3

− h1 + q1 + w1

− h3 + q3 + w3

− = x1 + q1,1 + Q1,1 + y1,1

p2

− = x2 + q1,2 + Q1,2 + y1,2

p3

− = x3 + q1,3 + Q1,3 + y1,3

− + q2,1 + Q2,1 + y2,1 − + q2,2 + Q2,2 + y2,2 − + q2,3 + Q2,3 + y2,3

− + q3,1 + Q3,1 + y3,1 − + q3,2 + Q3,2 + y3,2 − + q3,3 + Q3,3 + y3,3

= q1 + w1 = q1,1 + q1,2 + q1,3 + q1,4 + w1 ,

p2 − Q2+

= q2 + w2 = q2,1 + q2,2 + q2,3 + q2,4 + w2 ,

p3 − Q3+

= q3 + w3 = q3,1 + q3,2 + q3,3 + q3,4 + w3 .

(16)

pi

fi,j =

qi,j qi

=

− Qi,j

Qi−

=

− qi,j + Qi,j

qi + Qi−

=

qi,j qi

.

(17)

And let fQ − =

Qi−

(18)

qi

.

− h1 , − h2 , − h3 .

And on the output side with increase in storage we have p1 − Q1+

pi

for the fraction of gross production that does not lead to an increase in compartment i storage. The fraction of gross production that does lead to an increase in storage is therefore 1 − gi = Qi+ /pi . If there is a decrease in storage we will assume that the distribution of the net decrease is exactly the same as the current distributed production. That is

i

On the input side with decrease in storage p1

gi =

,

− h2 + q2 + w2 ,

in storage or inventory there is no change in output fractions but there is a split in gross production. Let

2.5.1. Source emergy calculations with trade and changing stocks In order to accommodate the additional ﬂows from trade and changing stocks we will need to expand our deﬁnition for output fractions, and deﬁne a few new variables. First if there is an increase Please cite this article in press as: Tennenbaum, http://dx.doi.org/10.1016/j.ecolmodel.2014.09.012

for the fraction of total output (distributed production plus decrease in storage) that the decrease in compartment i storage represents. What we need as far as computation is an output fraction that inﬂates the current distributed production by the amount of the additional storage used. Competitive imports can also be implicitly accounted for by including the ﬂow in the numerator of the output fraction  fi,j =

qi,j + yi,j qi

=

− qi,j + Qi,j + yi,j

qi

=

fi,j 1 − fQ −

+

i

yi,j qi

.

(19)

This inﬂates the ﬂow to a speciﬁc compartment in proportion to the amount of the imported material.8

8 This assumes that the processes in the originating, foreign system are comparable to the domestic system, so that the co-transformities (including transportation, transaction, and regulatory costs, etc.) of the local and imported goods are the same. If not, the co-emergy of the import should be calculated independently. Ideally the entire two systems should be analyzed linked together.

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co-emergy.

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The inputs to gross production can then be written in terms of the primed output fractions and ﬂows. p1

 q x2 + f1,2 1

 q + f2,2 2

 q + f3,2 3

p2

=

p3

 q + f  q + f  q − h = x3 + f1,3 3 1 2,3 2 3,3 3

 C  E2 + f2,2 2,[1] + f3,2 C3,[1]

= g2−1 C2,[1] ,

 C  E3 + f2,3 2,[1] + f3,3 C3,[1]

= g3−1 C3,[1] .

g1

0

0

G=⎣ 0

g2

0

0

0

g3

⎥ ⎦.

The columns of the source emergy/co-emergy matrix are given by colk C =



G−1 − FT

−1 [k]

E,

(20)

where F is matrix of augmented output fractions; and



colk M =



rowk

−1

G

T

− F

−1 [k]

T

diagE

.

(21)

The entries of M i.e. Mi,j are the portions of the emergy of

n

compartment j attributable to compartment i, and Mj = M . i=1 i,j These include the foreign sources of competitive imports and use of storages. When there is a use (decrease) of storage in the target compartment, the emergy calculated is for current amount of distributed production. The additional amount due to the decrease in storage is equal to (Qi− /qi )Mi = (fQ − /(1 − fQ − ))Mi . The emergy of the total i

Q˙ i ( ) d ,

(22)

t0

and

We deﬁne the matrix G as diag (g) , for example

t−t0

i,[k] ( ) Q˙ i ( ) d t0

The emergy calculations are essentially as before using current distributed production but now using the primed output fractions as well. The emergy and co-emergy is for distributed production so we need to adjust for any increase in storage – a split in the gross production. We do this by dividing the right hand side of the equations by the fraction of gross production that does not go to increasing storage (g1 ). The equations for compartment 1 are, = g1−1 M1 ,



t−t0

 i,[k] (t) =

− h2

 C  E1 + f2,1 2,[1] + f3,1 C3,[1]

from the most recent point in time when stock levels were zero or very nearly zero (t0 : Qi (t0 ) ≈ 0) to the time of use (t). That is



 q + f  q + f  q − h = x1 + f1,1 1 1 2,1 2 3,1 3

13

C i,[k] (t) =  i,[k] (t) qi (t)

Even a simple linear rate of depreciation of stocks (i.e. wi = i Qi ) can result in major changes in efﬁciency of storage if large changes in storage occur and output does not keep pace. In this case the efﬁciency is ωi = qi /(i Qi + qi ) which varies considerably with the size of Qi unless qi also grows proportionally to Qi . Since multiple compartments will experience changes in stocks and efﬁciencies over time, and since the times at which they deplete stocks may be very different, the most practical way to deal with these changes is dynamic modeling approaches. However, when the changes are small or when technologies and stocks have changed in the distant past, the average co-transformities approach the instantaneous values ( i,[k] (t) → i,[k] ). In this case we can also use the approach outlined in the last paragraph. For large changes in co-transformities over time resulting from major systemic changes in technologies and efﬁciencies, and/or when the age of products is essential to its probability of survival and use then more complex accounting methods need to be employed (which will be addressed in future work). 2.5.2. Source transformity calculations with trade and changing stocks The transformity calculations can be derived from the emergy equations using gross production coefﬁcients. These do not change since they are the “recipes” for the gross production of a particular compartment. But ﬁrst we need to derive an expression for the storage efﬁciency. Noting that the storage efﬁciency when storage decreases (Fig. 15) is ωj− =

i

approximation based on the assumption that the storage takes place close in time to the use (the technologies and efﬁciencies have not changed signiﬁcantly) or the changes that did occur happened long ago, and the differences in co-transformities of the production from time of production to time of use is negligible. Under these assumptions the amount of inﬂow to storage of Qi+ = Qi− /ωi is necessary to produce an amount of outﬂow from storage equal to Qi− assuming that ωi is now the fraction of production conserved. The gross production co-emergy is Ci,[i] , and the co-transformity is p−1 Ci,[i] so i

that the co-emergy of the storage is Qi+ p−1 Ci,[i] which upon eveni

Ci,[i] = Qi− q−1 Mi . Thus the tual use yields Qi− (ωi pi )−1 Ci,[i] = Qi− q−1 i i co-emergy of the stored gross production Ci,[i] yields numerically the same amount of emergy for an equitable use of storage (energy equal to (1 − ωi ) Qi+ is lost to depreciation). For larger changes in co-transformities over time, if we can assume that the depreciation and use of stored products is not a function of age, then we can employ an averaging procedure. Here the co-emergy (or emergy) of output at a given point in time is a product of the output times the time averaged instantaneous cotransformities (or transformities respectfully) of the entire stock Please cite this article in press as: Tennenbaum, http://dx.doi.org/10.1016/j.ecolmodel.2014.09.012

qj

qj − Qj−

=

pj

pj

qj − Qj−

=

pj − Qj+

,

(24)

the denominator of the far right hand side from the fact that Qj+ = 0 if the stock is decreasing. And when storage increases (Fig. 14)

i

amount of output (qi ) is therefore 1/(1 − fQ − )Mi . Note that this is an

(23)

ωj+ =

qj

qj

=

pj

pj − Qj+

qj − Qj−

=

pj − Qj+

,

(25)

the numerator of the far right hand side from the fact that Qj− = 0 if the stock is increasing. Since both storage efﬁciency during a decrease in storage and storage efﬁciency during an increase in storage can be formulated with the exact same expression, we can just write storage efﬁciency as the completely general expression

 ωj =

qj pj

=

qj − Qj− pj − Qj+

=

1 − fQ − j

gj

 qj pj

.

(26)

Let’s relate the primed output fractions to the gross production coefﬁcients by the following identities. −   fi,j qi = qi,j + Qi,j + yi,j = ai,j pj

(27)

So  fi,j =

S.E.,

ai,j pj qi

.

Emergy

(28)

and

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14

Fig. 14. A single sector of a system modiﬁed to include increases in storage, inventory, biomass, etc. Showing the net increase in energy ﬂow coming directly from gross production.

The transformity calculations can be derived from the emergy equations. For example, for the ﬁrst compartment in the system above,

Looking at the term ω1 M1 /q1 we can manipulate it to get it in the proper form given above, ˛1 + a2,1 2,[1] + a3,1 3,[1]

= ω1 T1

 C  E1 + f2,1 2,[1] + f3,1 C3,[1]

= g1−1 M1 ,

˛2 + a2,2 2,[1] + a3,2 3,[1]

= ω2 2,[1]

 C  E2 + f2,2 2,[1] + f3,2 C3,[1]

= g2−1 C2,[1] ,

˛3 + a2,3 2,[1] + a3,3 3,[1]

= ω3 3,[1]

 C E3 + f2,3 2,[1]

 C + f3,3 3,[1]

=

g3−1 C3,[1] .

The columns of the transformity/co-transformity matrix are given by



Substituting E1 +

a2,1 p1 a3,1 p1 C2,[1] + C3,[1] q2 q3

= g1−1 M1 ,

a2,2 p2 a3,2 p2 E2 + C2,[1] + C3,[1] q2 q3 E3 +

=

a3,3 p3 a2,3 p3 C2,[1] + C3,[1] q2 q3

The last equality since

ωj pj

=

C2,[1] C3,[1] E1 + a2,1  + a3,1  q2 q3 p1

=

C2,[1] C3,[1] E2 + a2,2  + a3,2  p2 q2 q3

=

C2,[1] C3,[1] E3 + a2,3  + a3,3  p3 q2 q3

=

and

pj

= gj pj then ωj gj pj =

[k]

␣,

(29)

where  is the diagonalized vector of storage efﬁciencies. This is identical in form to the original Eq. (11). The instantaneous rates of change of emergy and co-emergy of storages (label them Z˙ k , and S˙ i[k] respectively) can be calculated by

the relations Z˙ k = ωk Q + − Q − Tk , and S˙ i[k] = ωi Q + − Q − i[k] as

g2−1 C2,[1] ,

= g3−1 C3,[1] . qj

−1

colk  =  − AT

k

qj .

ω1 M1 q1 ω2 C2,[1] q2 ω3 C3,[1] q3

k

t becomes very small. Then, for example Z˙ 1

i

i

= E1 + q2,1 + y2,1 2,[1] + q3,1 + y3,1 3,[1] − q1 T1 ,

S˙ 2[1]

= E2 + q2,2 + y2,2 2,[1] + q3,2 + y3,2 3,[1] − q2 2,[1] ,

S˙ 3[1]

= E3 + q2,3 + y2,3 2,[1] + q3,3 + y3,3 3,[1] − q3 3,[1] .

2.5.3. Sink emergy calculations with trade and changing stocks We develop the computation for sink emergy with trade and changing stocks following the same procedures as before. The system boundary is expanded, and ﬂows and output fractions are

Fig. 15. A single sector of a system modiﬁed to include decreases in storage, inventory, biomass, etc. The energy ﬂow coming from a net decrease in storage is distributed proportionally to the current net production. Energy inﬂows due to competitive imports (denoted yji ) are distributed directly to users.

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adjusted to account for creation or use of surplus storage and competitive imports. The examples will still use a base network of three components with 2 different types of energy input for illustration  = 1 is solar energy,  = 2 is petroleum. Using the total efﬁciency of each compartment we can write the energy balance equations for the original system as, (30)

The expanded system uses primed output fractions and distributed output for the inputs to a given compartment, 1 ω1 x(1),1



=

= q2

 q + f  q + f  q + f  q g3 3 ω3 x(1),3 + f1,3 1 2,3 3 3,3 3 4,3 4

= q3

 q + f  q + f  q + f  q g4 4 ω4 x(1),4 + f1,4 1 2,4 4 3,4 3 4,4 4

= q4





u1

    w2 + g2 h2 + f1,2 D1,[2] + f2,2 D2,[2] + f3,2 D3,[2] + f4,2 D4,[2]

    w3 + g3 h3 + f1,3 D1,[2] + f2,3 D2,[2] + f3,3 D3,[2] + f4,3 D4,[2]

w4 + g4 h4 +

 f1,4 D1,[2]

+

 f2,4 D2,[2]

+

 f3,4 D3,[2]

+

 f4,4 D4,[2]

(h1 + w1 )     + f1,2 D1,[2] + f2,2 D2,[2] + f3,2 D3,[2] + f4,2 D4,[2]

= N2 = D3,[2] = D4,[2] .

G−1 − F

−1

T [k]

=

 rowk

h + G−1 w ,

T −1

G−1 − F

4 

h4 + g4−1 w4 +

 fi,3 Di,[2]

= g3−1 D3,[2]

 fi,4 Di,[2]

= g4−1 D4,[2] .

4  ai,2 p2

h2 + g2−1 w2 +

(31)

4  ai,3 p3

h3 + g3−1 w3 +

[k]

diag h + G−1 w

4  ai,4 p4

h4 + g4−1 w4 +

= D1,[2]

h2 + g2−1 w2 p2 h3 + g3−1 w3 p3 h4 + g4−1 w4

+

hi pi

ωi ). Also

=

Nk p k

= g2−1 N2,[2]

Di,[2]

= g3−1 D3,[2]

Di,[2]

= g4−1 D4,[2] .

4 

=

D1,[2] q1

ai,2

= g2−1

N2,[2]

i,[2]

ai,3

= g3−1

D3,[2]

i,[2]

ai,4

= g4−1

D4,[2]

i,[2]

i=1 4

+

 i=1 4

+

p4 But

Di,[2]

(h1 + w1 ) q1



p2

p3

p4

i=1

= (−1 − 1) and i

wi gi pi

x(2),1 = q1 , ω1 = 1, =

N ωk qk k

4 

−1 − ω2 + 2

=

wi p

i Di,[k]

gi pi

.

= 1 − ωi so =

Di,[k] p

i

= ωi

(hi +g −1 wi ) i pi Di,[k] q

i

= ωi

= (−1 − i i,[k]

and

= ωk Sk . Therefore

−1 − ω1 1

=

1,[2]

ai,2

i,[2]

= ω2 S2

ai,3

i,[2]

= ω3

3,[2]

ai,4

i,[2]

= ω4

4,[2] .

i=1 4



−1 − ω3 + 3

i=1 4



−1 − ω4 + 4

i=1

In matrix form this yields



−1

colk =  − AT 2.5.4. Sink transformity calculations with trade and changing stocks The sink transformity calculations can be derived from the sink emergy equations in the same way as we did with source transformities, substituting expressions with gross production

qi

i=1

(32)

where all matrices and vectors are for the expanded network. Ni,j is the energy degraded by compartment i to produce current dis4 tributed production by compartment j, and Nj = N . i=1 i,j

qi

i=1

T

qi

i=1

Nk gk pk

and



h3 + g3−1 w3 +

g2−1 N2,[2]

In matrix notation this is

colk N =

= g2−1 N2,[2]

i=1

= D1,[2]

= g4−1 D4,[2] .

4 

= D1,[2]

    h4 + g4−1 w4 + f1,4 D1,[2] + f2,4 D2,[2] + f3,4 D3,[2] + f4,4 D4,[2]

= g3−1 D3,[2]

colk D =

 fi,2 Di,[2]

(h1 + w1 )

    h3 + g3−1 w3 + f1,3 D1,[2] + f2,3 D2,[2] + f3,3 D3,[2] + f4,3 D4,[2]



4 

h2 + g2−1 w2 +

= D1,[2]

yeild

Rearranging to put in a form amenable to matrix calculations gives

h2 +

T(2) − 1 x(2),1

i=1

Of the co-emergy inputs plus the production losses (hi ), only a fraction gi is required to produce the co-emergy of gross production used to replace current output. The co-emergy of the current output is that plus the storage losses (wi ) . The primed output fractions already inﬂate the output from current production to account for additional use from storage and/or competitive imports. Thus we have for unit 2 emergy

q1

 q + f  q + f  q + f  q g2 2 ω2 x(1),2 + f1,2 1 2,2 2 3,2 3 4,2 4

g2−1 w2

 q = a p . So the coefﬁcients for the output fractions, i.e. fi,j i,j j i equations

i=1

gi i ωi ri = qi

15

[k]

(33)

Thus our designation of ﬂows shown in Figs. 14 and 15, and our choice of notation for for fractional ﬂows of gross and distributed production given in Eqs. (16)–(19) allows us to write the matrix equations without alteration (see Eq. (15)).

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3. Discussion In this paper we have shown how the relative output structure of network can be used to determine emergy values for that system’s components using energy inputs of a single type or the energy used up in a system with expanded boundaries. Similarly we can use a modiﬁed input–output matrix to calculate transfomities directly. In addition we have shown that the issue of co-production need not be an algebraic chimera, but by treating it as a problem of scale, ordinary linear algebra methods can be used. Of course this does not mean that the computations become simpler. To the contrary, by using the intermediary co-emergy (or its intensive partner co-transformity) we need to, not only perform a separate analysis for each target compartment of interest, but tease apart the inner workings of any imbedded subsystem where co-products are used in the larger system. Co-emergy and co-transformity have their own use. Co-emergy, in that it accounts for all the non-redundant ﬂows to a compartment downstream of co-product production when energy and material ﬂows from those co-products re-converge. In addition if some effort was made to catalogue co-transformities of direct inputs (each one would have to be indexed by both what input it was and what target is using it), these could then be used in traditional tabular methods without concern for double counting. The importance of these techniques is that they allow direct comparison of competitive species, industries, or technologies using standard methods of linear algebra. It also can provide the analytical tools to formulate and test conjectures about local and system-wide efﬁciencies. By using the methods developed here we can, for example, modify gross production coefﬁcients to investigate changes in the recipes for production that result from new or alternative technologies. Technical coefﬁcients are readily available for many states and countries economies, and the input–output tables of those provinces provide for the possibility of analyzing up to hundreds of sectors at once. These I-O tables can be modiﬁed to include natural inputs to an economy.9 By changing just one coefﬁcient we can see the cascading effects through the entire system. From a more theoretical standpoint, the direct inclusion of efﬁciencies in the formulation of sink emergy computations allows for investigations of system wide optimization of efﬁciencies (see Eq. (15)). For example this will permit the analytical formulations of hypotheses surrounding maximum power. The methods discussed here, in conjunction with dynamical modeling and game theory will allow us to determine if, and at what scales maximum power may operate under constant boundary conditions to produce evolutionary stable strategies. Once these questions are answered the door is opened to other questions, such as, faced with limited and variable resources, lag times in transfer of materials and energy, and the possibility of conﬂicts of optimization operating between different compartments or at different scales, do systems settle with sub-optimal efﬁciencies at one scale but optimal at another, or does cycling in the system result in tradeoffs in time of optimization between components? Questions such as these require both local and system wide measures of efﬁciency and energy ﬂows that can be computationally tracked. Sink emergy

9 There are numerous pitfalls with using these accounts, modiﬁcations are necessary before they can be used in tracking energy ﬂows, and care needed in interpreting results (Bullard and Herendeen, 1975). For example BEA’s I/O sector deﬁnitions are based on general industry deﬁnitions with mixed production. All transactions are in dollars, not physical units, which can be a problem if prices are not constant for all purchasers, one result being that dollar transactions do not guarantee conservation of energy. Capital goods are not included in inter-industry transactions but are listed as sales to ﬁnal demand, and so on. For an extensive list of I/O limitations in energy analysis see Bullard and Herendeen (1975).

provides just such a measure and can be computed concurrently with dynamical simulation models of systems or even analytically in the case of very simple systems.

Appendix A. Comparison of competitive services The introduction of a new technology, organization, or structure of an industry, or introduction of a new genotype and its phenotypic expression in an organisms potentially results in changes in efﬁciencies of use of resources, redirection of ﬂows of energy, and changes in inventories or biomass. To assess the effects that these changes may have on the emergy ﬂows and transformities of different components in a system, we can evaluate the system with both new and old components deﬁned explicitly as separate compartments. In addition we can evaluate the emergy and/or transformity of both compartments simultaneously. In order to do this for two competing compartments k and k + 1, we zero out the k and k + 1st columns of the transposed F or A matrices.

colj

colj

T F

[k,k+1]

T A

[k,k+1]

 =

colj FT 0

 =

colj AT 0

if

j= / k ork + 1

if j = k or k + 1

if

j= / k ork + 1

if

j = k or k + 1

.

(A.1)

,

(A.2)

Or using a modiﬁed identity matrix where the kth and k + 1st diagonal elements are replaced with 0

Ik,k+1 =

⎧ ⎨ 1 if i = j =/ k or k + 1 ⎩

i= / j

0

if

0

if i = j = k or k + 1

(A.3)

then

T F

[k,k+1]

= FT Ik,k+1

(A.4)

Similarly

T A

[k,k+1]

= AT Ik,k+1

(A.5)

As an example let us break down the agriculture sector given in Fig. 2 into two competing types of agriculture, the ﬁrst (agriculture 1) is a lower energy practice and the second (agriculture 2) has high non-renewable subsidies. Both have the same land area and renewable (solar) inputs but the second has four times the net output of the ﬁrst. We also assume much higher industrial inputs to the “industrial” agriculture – nine times higher than the low energy practice, and the low energy practice is much more labor intensive than industrial agriculture – four times as much. The speciﬁc details of revised network are given in Table 2 and Fig. 16. To analysis this system we proceed as detailed above by simultaneously zeroing out the two agricultural compartments when analyzing the emergy of agriculture and proceeding in the regular fashion for the other two compartments. We have the following matrices needed for the input emergy and transformity calculation S.E.,

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Table 2 Energy ﬂows in a three sector economy with two competing agricultural practices (Agr. 1: low-tech farming and Agr. 2: industrial farming). Energy matrix

(1) Agr. 1

(2) Agr. 2

(3) Ind.

(4) Con.

Export

q

w

p

h

Source 1 Source 2 (1) Agr. 1 (2) Agr. 2 (3) Ind. (4) Con. Total

6000 20 3 0 1 0.2 6024.2

6000 160 0 7 9 0.05 6176.05

0 700 1 4 50 0.25 755.25

0 0 1 4 30 1 36

1 4 10 0 15

6 19 100 1.5

2 8 10 0.5

8 27 110 2

6016.2 6149.05 645.25 34

Energy values are hypothetical and arbitrary. Source energies are in raw units. Source 1 is solar (transformity = 1) and source 2 is non-renewable (transformity = 100).

8000

The entire emergy co-emergy matrix is then

⎢ ⎥ E = ⎢ 22, 000 ⎥ ⎣ ⎦

⎡ ⎢ ␣=⎢ ⎣

1000

0/2

⎢ ⎢ ⎢ 0 ⎢ F=⎢ ⎢ 1 ⎢ 100 ⎣ ⎡

0

1 6

7 19

4 19

9 100

1 2

1 2

1 2 15 30 3/8

⎢ ⎢ 0 A=⎢ ⎢ 1/8 ⎣  1 5



⎡ 0.375 ⎢ ⎣

=⎢

0

⎥ ⎡ 0.5 ⎥ ⎥ ⎢ ⎥=⎢ 0 ⎥ ⎣ 3 ⎥ 0.01 ⎥ 10 ⎦ 0.1333

4 19

1 6

/8

⎤0

1 6

9/27

50/110

  1 4

/27

/110

0.009091

0.5

0.25926

0.036364

2 15 0.5

0.002273

0

0

0

19 27

0

0

0

100 110

0

0

⎤ ⎥ ⎦

0.2105 ⎥ 0.3

0.1667 0.6667

36, 000

67, 142.86

6

⎢0

 =⎣

⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

0

⎤−1 ⎡

0

0

0

100

0

0

0

0

1.5

120, 000

34, 000

⎥ ⎢ 46, 000 ⎦ ⎣ 200, 000

19

0

5, 666.67

1/2

0

⎢ 2, 421.05 =⎣ 2, 000

4/2

180, 000

4, 266.67

3, 238.10

1, 933.33

3, 119.05

880

1, 714.29

24, 000

44, 761.91

25, 600

19, 428.57

36, 733.33

59, 261.90

88, 000

171, 428.57

36, 000

67, 142.86

⎤ ⎥ ⎦

⎥, ⎦

or by using the production coefﬁcient method. The transformity of industrial agriculture, as we have deﬁned it, is less than the “low energy” agriculture since, although the emergy is 1.35 times greater, the output is also about three times greater (Fig. 17). If we had treated each agricultural practice as a separate sector, analyzing the emergy separately we get the following emergy/coemergy matrix and transformity/co-transformity matrix.

52, 455.09

99, 860.14

25, 600

19, 428.57

312, 215.57

294, 125.87

36, 000

67, 142.86

⎢ 91, 586.83 56, 223.78 36, 733.33 59, 261.90 ⎥ ⎥ ⎣ 282, 634.73 271, 328.67 88, 000 171, 428.57 ⎦

C∗ = ⎢

3/2

2

The emergy of the agriculture sector, the ﬁrst column in the emergy/co-emergy matrix, is calculated as

180, 000

This is a 4 × 3 matrix. Each column contains the emergies and co-emergy for each target sector (agriculture, industry, and consumers), and each row of that column is the emergy or co-emergy for each compartment in the network (agriculture 1, agriculture 2, industry, and consumers). The transformity/co-transformity matrix is found either by pre-multiplying the above by the reciprocal of the diagonalized output vector:

⎥ ⎥ ⎥ 30/2 ⎥ ⎦

0

0

0.0333

4/110

0.001852

0

0.5

7/27

0.333333 0.454545

⎢ ⎢ ⎢0 ⎢ =⎢ ⎢ ⎢0 ⎢ ⎣

0.09

1/2

0.025 8

0.2105

1/110



0.1667 0.1667

0.3684

2 3

0.125

⎡6

0

0

1 20

19, 428.57

C=⎢

⎢ 22, 000 ⎥ ⎥ ⎥ ⎢ ⎢ 27 ⎥ 22, 000/27 ⎥ = ⎢ ⎦ ⎢ 7000 ⎥ ⎥ 70, 000/110 ⎣ 11 ⎦ 8000/8

25, 600

⎢ 46, 000 36, 733.33 59, 261.90 ⎥ ⎥ ⎣ 200, 000 88, 000 17, 1428.57 ⎦ .

70, 000 0

34, 000

1 0 −

1 100

⎢ ⎢ ⎢0 1 − 9 ⎢ 100 (col1 C)[1,2] = ⎢ ⎢ ⎢0 0 1 ⎢ 2 ⎣ 0 0 −

3 10

2 ⎤−1 − 15 ⎥ ⎡ ⎤ ⎡ ⎤ 8000 34, 000 1 ⎥ ⎥ − ⎢ ⎥ ⎢ ⎥ 30 ⎥ ⎥ ⎢ 22, 000 ⎥ = ⎢ 46, 000 ⎥ , ⎣ 70, 000 ⎦ ⎣ 200, 000 ⎦ 1 ⎥ − ⎥ 6 ⎥ 0 180, 000 ⎦ 1 3

Note that this give the emergy of agr. 1 and agr. 2 and the coemergy of industry and consumers for the target agriculture. Please cite this article in press as: Tennenbaum, http://dx.doi.org/10.1016/j.ecolmodel.2014.09.012

8, 742.51

⎢ 4, 820.36 ⎣ 2, 826.35

 =⎢ ∗

208, 143.71

16, 643.36

4, 266.67

2, 959.15

1, 933.33

2, 713.29

880

196, 083.92

24, 000

3, 238.10

⎥ ⎥ 1, 714.26 ⎦ 3, 119.05

44, 761.91

By analyzing the emergy of the competing compartments together we implicitly assume that the outputs are used in a similar fashion and for similar purposes. This essentially treats the net output of the compartments as a “split”. By analyzing the emergy of the competing compartments separately we implicitly assume that the outputs are used in a differing fashion or for differing purposes. This essentially treats the net output of the compartments as “coproducts”. As can be seen assumptions about similarities of use and purpose have important consequences for the values computed for emergy and transformity of system components. S.E.,

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Fig. 16. Energy ﬂows in a simple 3 sector economy with the agricultural sector broken down into low-tech farming (Agr.1) and industrial farming (Agr. 2). Units are arbitrary.

Fig. 17. Co-emergy ﬂows and emergy of agriculture in a simple 3 sector economy with the agricultural sector broken down into low-tech farming (Agr.1) and industrial farming (Agr. 2). All source inputs are converted to solar equivalent units (e.g. joules to solar em-joules) Output from agriculture is labeled with emergy in bold, all other ﬂows are labeled with co-emergy in normal typeface. The numbers in parentheses under the non-renewable source is the total amount of the non-renewable emergy used by each agriculture process. Renewable (e.g. solar) emergy inputs are the same for both processes.

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