Energy limits on recycling

Ecological Economics 36 (2001) 373–384

METHODS

Energy limits on recycling

Paul P. CraigGraduate Group in Ecology, Uni6ersity of California, Da6is CA 95616, USA

Received 10 February 2000; received in revised form 13 September 2000; accepted 22 September 2000

Abstract

Increasing efficiency of resource use is a major ecological-economics goal. Opportunities for large efficiencyincreases have been found in every area carefully investigated. What are the limits? Energy analysis offers powerfulinsights. Ayres has argued that, even with unlimited available energy, over time it becomes necessary to haveever-more dilute waste reservoirs. If correct, this would significantly limit the potential of a hypothetical sustainablespaceship, and of a ‘spaceship economy’. However, we know the world is self-contained and long lived, and thatstable laboratory microcosms have been built. Thus, this assertion appears to violate reality. I argue that the allegedproblem does not exist. Given available energy, there is no concentration limit on sustainability. My result isconsistent with the work of Georgescu-Roegen and others. I estimate some numerical examples to show that presentsystems are extremely far from theoretical limits. For a long time to come, there will be vast opportunities to improveefficiency. The paper includes a classification schema for recycling based on chemical bonding and thermodynamicconsiderations. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Efficiency; Dematerialization; Eco-efficiency; Technological innovation; Spaceship earth; Sustainable development;Recycling

www.elsevier.com/locate/ecolecon

1. Introduction

If the society toward which we are developing isnot to be a nightmare of exhaustion, we must usethe interlude of the present era to develop a newtechnology which is based on a circular flow ofmaterials such that the only sources of man’sprovisions will be his own waste products.(Boulding, 1980)

The ‘spaceship earth’ metaphor of Boulding(1980, 1993a,b) provides a perspective encouraginghusbanding of resources and recycling. Bouldingconceptualized economies in terms of stocks andflows. To Boulding, a society that overly valuesthroughput relative to stocks errs. The primarygoal is to have a house, a shirt, or shoes, not theirproduction. Flow is important primarily in thecontext of basic life-support. Survival requires flowof air and food. Boulding recognized that manybiological/ecological resources are irreplaceable atany cost. He recognized that most physical reE-mail address: [email protected] (P.P. Craig).

0921-8009/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.

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P.P. Craig / Ecological Economics 36 (2001) 373–384374

sources are replaceable, given enough availableenergy.

The ‘spaceship earth’ metaphor appeals to ecol-ogists, ecological economists and industrial ecolo-gists. The metaphor provides wonderful imageryfor the emerging field of industrial ecology, a fieldthat takes as its goal the task of operating indus-trial systems with minimal external by-products(for example, Lifset, 1997; Lowenthal and Kas-tenberg, 1998; Moolensar and Lexmond, 1999;Rejeski, 1999). Industrial ecology is, in effect, thedesign of engineered subsystems that lead to re-duced resource use, reduced emissions, and re-duced environmental impact.

We live in a world focused on throughput.From an ecological-economic perspective, Ken-neth Boulding’s focus on stocks rather than flowsis a sound direction. However, so long as re-sources are used, it is incumbent upon us to usethem effectively. Recycling is a key to this. How-ever, while the directions are clear enough, detailsare inevitably murky. The futurist Willis Harmonexpressed the need for modesty:

It is difficult for any of us to fully comprehendwhat is undoubtedly true: If the basic assump-tions underlying modern society are indeedshifting in the way we have suggested, it followsthat society will, only a few generations fromnow, be as different from modern industrialsociety as that is from society of the MiddleAges. Furthermore, it will be different in waysthat we can only vaguely intuit, just as a Re-naissance futurist would have had a hard timetrying to describe modern society. (Harmon,1988, p. 168)

While we cannot know how future societies willevolve, fundamental principles, especially conser-vation laws, impose limits. My focus here is onfundamental limits to recycling. I conclude that, ifenergy is available, the theoretical limit is minute.My analysis is inspired by an important article byAyres (1999) in which he argues that theoreticalconsiderations may in principle require large‘waste baskets’ on idealized spaceships. While Idisagree with his conclusion, I confirm that he hasset the right agenda and asked the right questions.

2. How much recycling is theoretically possible?

From a physics perspective, recycling is bestviewed in terms of energy. The primary consider-ations are energy conservation thermodynamics.These principles place limits on the usability ofenergy, and allow estimates of the minimum en-ergy required for particular processes (Fermi,1936; Davidson, 1962).

In the context of ecological economics, it isuseful to note a recurring factor: NicholasGeorgescu-Roegen’s proposal of a ‘fourth law ofthermodynamics’ (Georgescu-Roegen, 1971,1980). This ‘law’ asserts the impossibility of com-plete recycling. This controversial proposal con-tinues to provide intellectual stimulus. The ideaunderlies the suggestion by Ayres (1999) that the-oretical limits to recycling exist. As he put it:‘‘One can only wonder whether an interstellarspaceship capable of internally recycling all re-sources would have to be so large that most of itsmass would have to be inactive by design’’. Muchof this paper explores thermodynamic argumentsthat show that Ayres’ conclusion is notpersuasive.

3. Principles

My reasoning derives from energy and thermo-dynamic considerations.1 Ecological economistshave long recognized the importance of this typeof approach (Georgescu-Roegen, 1980; Mayumi,1990; O’Connor, 1991, 1994; Biancardi et al.,1993; Ehrlich et al., 1993) The heart of recycling isthe separation of components. All objects (natu-ral, biological, or man-made) are constructedfrom atoms and bound by chemical or nuclearforces. The meaning of ‘bound’ is that energy isrequired to take things apart (to break chemicalbonds). The inverse is also true: energy is releasedin the formation of compounds. A consequence ofthe first law of thermodynamics, conservation ofenergy, is that chemical and nuclear reactions are

1 There have been many efforts to relate thermodynamics toeconomic considerations (for example, Mayumi, 1990; O’Con-nor, 1991, 1994; Biancardi et al., 1993).

P.P. Craig / Ecological Economics 36 (2001) 373–384 375

ideally fully reversible. Precisely the amount ofenergy released when a bond is created must beadded to take it apart.

Irreversibility, which inevitably occurs in practi-cal processes, arises from disorder. Thermody-namics is the science of disorder. Limits on energyrequirements for recycling can thus be reduced toanalysis of energy and thermodynamics. Thepower of energy analysis is well illustrated withtwo examples from outside the recycling area.These examples provide context for the recyclinganalysis that follows.

3.1. Example 1. First law of thermodynamics:minimum energy to dri6e a car around a closedpath

The theoretical limiting energy required tomove a vehicle over a closed path is zero.2 Thissurprising conclusion may be understood as fol-lows. Suppose the car is driven up a hill. Thework done against gravity is W=mgh, where m isthe car mass, g is the acceleration of gravity, andh is the height. The work done to raise the carcan, in principle, be recovered as the car descendsthe hill. There are many irreversible energy losses,but none of these are constrained by theoreticallimits. For example, frictional loss in bearings canin principle be made arbitrarily small. Energylosses due to air drag can, in principle, be madearbitrarily small as by aerodynamic design and bymoving the vehicle slowly. One can envisage lin-ear-motor propelled, magnetically levitated trainsoperating in a vacuum, using minute amounts ofenergy compared with any existing transportationsystem. Perhaps the best example of the principleis found in a satellite, which moves in a vacuumand requires no energy input whatsoever.

3.2. Example 2. Second law of thermodynamics:ideal efficiency of a heat pump

The efficiency of a heat engine can be no

greater than a limiting value established by theCarnot cycle. We use a more subtle illustration —a heat pump. Heat pumps are designed to moveenergy from a high temperature to a low tempera-ture. Examples are air conditioners and refrigera-tors. Heat pumps are constrained by the secondlaw of thermodynamics. The heat that can bemoved to a cool reservoir (e.g. an air-conditionedhouse) from a warm one (typically outside air) isproportional to the external work (or power) in-put. The amount of heat, Q, that can be movedby an amount W of external work (e.g. suppliedby a motor) is constrained by an inequality.

Q5� Th

Th−Tc

n×W (1)

where Th is the absolute temperature of the hotreservoir (the outside) and Tc is the temperatureof the cold reservoir (the room from which energyis to be pumped (removed)). Equality applies inthe ideal case, which is a limit and cannot beachieved in practice.

The beauty of this example is that it is bothsimple and surprising. The surprising result is thatthe ‘efficiency’ of the heat pump (the amount ofenergy pumped per unit external work input)increases as the temperature difference betweenthe outside and the inside decreases. The effi-ciency of an ideal heat pump is exactly the inverseof the Carnot efficiency for a heat engine. Practi-cal heat pumps are rated in terms of a ‘coefficientof performance’, which is a measure of their per-formance relative to this thermodynamically limit-ing value.

Fundamental principles allow one to calculatethe ideal or best conceivable performance. Practi-cal systems can then be assessed relative to theideal. In the next section, we apply this idea totwo simple but important recycling examples.

4. Recycling

The key to recycling is to make as much use aspossible of order that exists in anything beingrecycled. Materials should be thought of in termsof ‘building blocks’. One should keep the largest

2 The engine excepted. If the engine runs on a thermody-namic cycle there may be reversible energy losses. If the engineruns on electricity, there is no theoretical energy loss whatso-ever.


building-blocks that one can. If one is recycling acar, it makes sense to take out the copper wireand not allow it to mix with the iron of the engineand frame. The separated copper and iron canreadily be refabricated. In contrast, alloys of ironand copper have poor properties and are difficultto separate.

This deep principle of recycling is entirelyfamiliar to ecologists. Ecologists know howimportant it is to keep ecosystems intact. Oncedismantled, they are at best difficult and usuallyimpossible to reassemble.

The most difficult recycling problem occurswhen building blocks are randomly distributed.Work is required to establish order. The problemis the same as separating different gases. Thelimiting case is ideal gases with no binding energy.In the next section, we analyze this case. I showlater that the energy per atom (or per buildingblock) required for separation increases as thesystem becomes more and more dilute. Thismeans that recycling is easier if concentrations ofdesirable substances can be maintained at a highlevel.

Numerically, it turns out that, for atomic-scalebuilding blocks the minimum energy per atomrequired for separation, even for very dilute gases,is small in comparison with typical chemicalbinding energies. The vast majority of currenttechnologies are far from the limits. Theoreticalcalculations are, nevertheless, useful to providebenchmarks to measure how well one is doing.Design of any proposed self-sustaining spaceshipshould clearly benchmark against limitingcalculations.

5. Energy of separation

The most difficult recycling challenge is theseparation of randomly intermixed particles. Thisproblem can best be approached through an ‘en-tropy of mixing’ calculation. While this is themost general approach, it is not particularly trans-parent. An energy-based calculation is more read-ily understandable.

Consider two perfect gases. By definition, per-fect gases do not interact. Placed in a container,

each distributes itself throughout the entire vol-ume. To separate them, each must be compressedinto a portion of the total container. As they arecompressed, their pressures increase. In equi-librium, after compression, the pressures of thetwo gases must be equal. An impermeable mem-brane separating them experiences no force. Thiscalculation is presented in Appendix A. The en-ergy reduction when the two gases are mixed(which, except for sign, is identical to the energythat must be supplied to separate them) is (Ap-pendix A):

W=nRT× [x1× ln(x1)+ (1−x1)× ln(1−x1)](2)

where x1 and x2=1−x1 are the number fractionsof molecules 1 and 2, ln(x) denotes the naturallogarithm of x, R=8.32 J/mol/K is the gas con-stant, and T is the absolute temperature. At roomtemperature, 300 K, the energy is about 2500J/mol, or (in atomic scale units) �1/40 eV/atom.

This expression results directly from the workof compression of two ideal gases obeying Dal-ton’s law, which states that the pressure exertedby a mixture of gases is equal to the sum of thepartial pressures of all the components. Real gasescan deviate from Dalton’s law, but the approxi-mation is adequate for the ‘order of magnitude’estimates of interest here.

For low concentrations (x1�1) the energy permole associated with mixing is approximately:

w−Wn1

�RT [ln(x1)+1] (3)

This may be written in dimensionless form as

Z=W

n1×RT= [ln(x1)+1] (4)

Eq. (4) is plotted in Fig. 1 for low molarconcentrations. The energy associated with mixingcan, at low concentrations, be somewhat largerthan thermal energies, but is modest relative totypical chemical binding energies.

My idea that the ideal energy for recycling issmall can be illustrated with examples from twovery different domains: separating uranium fromsea-water and separating helium from the earth’satmosphere. Since currently available resourcesare far more concentrated than these sources, the


example is of theoretical interest only at this time.The value of the calculation is that it shows thetheoretical separation energies to be so low thatboth uranium and helium could, in principle, beextracted at a cost that would only marginallyimpact practical applications such as electricityfrom uranium fusion or helium for welding.

5.1. Example 1: uranium in ocean water

The concentration of uranium in ocean water isabout 3 parts per billion (Uranium InformationCenter, 1999). From Eq. (3), the minimum workrequired to separate one atom of uranium fromseawater is about 0.5 eV. This energy is minute ascompared with the energy release from nuclearfission of about 200 MeV. Thus, the separationprocess could be exceedingly inefficient and yet stillyield a large net energy gain in fission energytechnologies.

Practical experiments confirm this conclusion.Hiraoka (1994) has used a polymer absorbent toconcentrate uranium from ocean water. He esti-mates that, at commercial scale, the process couldproduce uranium at about 34 000 yen/kg uranium(�$300/kg). This would amount to about 11.8 yen(�$0.001) per kilowatt hour (kwh), a small frac-tion of the several cents per kwh total cost ofgenerating electricity in reactors.

Uranium ore from rocks contains up 2% ura-nium and uranium from ore bodies currently sellsfor far less than the cost of extraction from sea-wa-ter. Hence, such extraction is of no commercialinterest today.

Because the uranium content of sea water isextremely small, enormous amounts of waterwould have to be moved. The cost of moving somuch water would be substantial. It is also likelythat the environmental disruption associated withmoving so much water would be considerable. Atthe present time, the example is of pedagogicalrather than practical interest.

5.2. Example 2: helium in the atmosphere

The earth’s atmosphere contains about 700 000billion cubic feet (�2.3×1013 m3), an enormousquantity relative to annual use by society. Theconcentration is minute (0.0005%). Currently, com-mercial helium is extracted from certain natural gasreservoirs containing far higher concentrations(typically �0.3% helium). Despite the low atmo-spheric concentration, the theoretical energy re-quirement for extraction is modest. A typicalhelium tank sold in the US contains 244 cubic feet(�8 m3) of helium. Using Eq. (1), extraction of thisamount of helium from the atmosphere wouldrequire about 5 million joules. This amount ofenergy from natural gas costs less than $1, andamounts to less than 1% of the selling price of atank of helium gas (�$100).

The American Physical Society (Dunn et al.,1996) calculated that recovery of the 3.2 billioncubic feet (�109 m3) of helium used in commerceby the United States would require about 5% oftotal US energy. This is a factor of about 50 000larger than the theoretical limit already calculated.Since there has never been any need to extracthelium from air in commercial quantities, attentionhas never been given to achieving high efficiency.Should air extraction become necessary, it is clearthat attention to efficiency would increase enor-mously.

The helium example is particularly interesting inthat it shows how low theoretical limits can becompared with practical engineering, and suggeststhe possibility of enormous improvement should

Fig. 1. The energy required for separation of a dilute sub-stance (‘critical resource’) mixed with a matrix substance (fromEq. (4)). (Negative energy means work must be done toseparate the resource.) The required separative work requiredincreases rapidly at low concentrations. This is the primaryreason why it is good practice to recycle concentrated ratherthan dilute materials and to work hard to avoid unnecessarydilution. However, even at exceedingly low concentrations(e.g. �10−5 M), the separative work required is only about10RT, which at room temperature and above is small com-pared with typical chemical binding energies.


incentive appear. The literature of industrial engi-neering and efficiency engineering contains manyexamples of the successful search for major im-provements. The ‘factor X’ debate provides aparticularly helpful focus for this theme (Reijn-ders, 1998; Hawken et al., 1999).

6. Reservoirs

The presented analysis shows that the separa-tion energy requirement per mole of a valuedsubstance increases at low dilution. This meansthat it is best to run recycling systems at as higha concentration as possible. In this section, Iapproach the problem slightly differently, usingthe framework discussed by Ayres (1999). Theanalysis is described in words in the text. Themathematical details are presented in Appendix B.The example is for the simplest possible recyclingsystem. This system consists of two components, acritical resource (CR) and a matrix material (M).These substances may be located in either an‘in-service’ container or a ‘waste’ container. Arecycling system removes material from the wastecontainer for processing. Material concentrated inthe critical resource is returned to the in-servicecontainer and reject material goes to the wastecontainer. The separation process is imperfect.The output of the recycling system contains afraction f(CR) of the critical resource. In the idealcase of (perfect separation), f(CR)=1. The recy-cled stream is contaminated by a fraction f(M) ofthe matrix material. Material not returned to thein-service reservoir goes back to the waste reser-voir. In the ideal case, none of the desired mate-rial is returned to the waste stream, so f(M)=0.

Equation A4 (Appendix B) shows that, in equi-librium, the concentration of the critical resourcein the in-service container is related to that in thewaste container. The concentration ratio betweenthe two containers is equal to the recycle effi-ciency for the critical resource f(CR) divided bythe recycle efficiency for the matrix material f(M):

Concentration (in-service)

= [ f(CR)/f(M)]×concentration (waste)

In an ideal system, the critical resource is recy-cled with 100% efficiency. Thus, f(CR)=1 whilef(M)=0. In this case, the recycler allows nomatrix material to re-enter the in-service con-tainer. The in-service container therefore containsno impurities, and the waste container containsnone of the critical material. A specified concen-tration of in-service material can be obtained bysuitable choice of the three parameters on theright-hand side of the equation. If the concentra-tion in the waste container is to be kept small,then the recycler must operate with high efficiency(high ratio of f(CR)/f(M)). As already argued,however, even exceedingly dilute waste streamscan be concentrated with modest energy. There isno requirement that the waste container be large.

The discussion can be extended in a number ofdirections. Recycling can include multiple stagesand can be designed with special subunits forprocessing particular chemical forms. Processescan use waste from one process as input to others.Technology also provides many means to reduceprocess leakage.

Design can eliminate or minimize dangerous ordifficult-to-handle substances. Chlorine, for exam-ple, is a commonplace industrial feedstock that isdangerous to handle and if released to the envi-ronment can have serious consequences for hu-mans and biota. At high concentrations, chlorinecan damage respiratory tracts; at low concentra-tions, it damages the protective ozone layer. Theseadverse consequences have provided incentivesnot only for technical controls to minimize re-lease, but also for new product design to provideneeded services with greatly reduced or withoutchlorine.

All processes are inefficient and leak to somedegree. Thus materials must be replaced. Soundsystems design will do its best to limit loss and tocontrol the places where lost material goes. Re-sources, e.g. energy, spent in resource recovery areresources not spent elsewhere (Van Berkel et al.,1997).

Ecologists and industrial ecologists recognizethat the very idea of ‘waste container’ can bemisleading. Just as ‘beauty is in the eye of thebeholder’, so is it true that what is ‘waste’ to memay be feedstock to you. Indeed, some of the


most elegant examples of recycling have arisenfrom conscious efforts to design waste streamsfrom one process so they can serve as feedstock toanother (Socolow and Thomas, 1997).

7. Recycling classification

This section introduces a conceptual frameworkfor thinking about recycling. The goal is topresent a ‘schema’, not to offer recipes. The basicideas underlying the approach are the centrality ofsensitivity to chemical bonding energy and ther-modynamics. Sound system design should focuson keeping the largest possible reusable ‘chunks’in any stream, and should seek to maintain thehighest possible concentrations in all feedstocks.This approach minimizes the work (and the cost)of recycling.

All useful substances, both man-made and nat-ural, are ordered. Chemical compounds are care-fully constructed to have specific structures.Manufactured items such as cars, computers,shirts and shoes have a high degree of order.Biological systems are the most ordered of all. Asound recycling program must give the study oforder the highest priority.

The goal of any process designed to maximizerecycling must be to maintain existing order, andto introduce new order. Thermodynamics is thestudy of order and disorder. Accordingly, thermo-dynamics provides the best framework for a gen-eral discussion of recycling.

Two principles provide the framework for con-ceptualizing any recycling question.1. Concentrated materials require less energy and

are easier to recycle than dilute materials. It isgood strategy to separate materials and notallow them to mix and become dilute.

2. Ordered systems are easier to recycle thandisordered systems. The energy associated withchemical bonds can, in principle, be recoveredthrough reversible (non-entropy-creating) pro-cesses. Disorder places limits on reversibility.Increasing order requires inputs of energy (moreprecisely, available energy) in order to decreaseentropy in the subsystem of interest.

Examples of ordered flow are water in a channel

Table 1

Recyclability Easier Harder

SmallBig1. Chunk sizeLowHigh2. Concentration of

critical components3. Bonding energy ratio Tight bonding Loose bonding

(intra-component/intra-component)

or river, and flow of electrons in a wire. Orderedflow may be conceptualized as a thermal system atinfinite temperature. The reason electrical energy isa high-quality energy source is its effectively infinitetemperature. Water flow is also effectively at infi-nite temperature, which is the reason that hydro-power plants can convert mechanical energy of flowto electricity at first law of thermodynamics effi-ciencies approaching 100%.

Recyclability can be conceptualized in threecategories: (a) chunk size, (b) concentration, and (c)bonding. Multiple considerations and multiple ob-jectives are necessarily involved. This multiplecharacteristic means that no unique ranking ispossible. The ranking system presented in Table 1is to provide insight and guidance.

I illustrate these concepts with several examples:(a) motor vehicle wiring; (b) steel scrap mixed withother waste; (c) used motor oil; (d) helium; (e)uranium ore; (f) agricultural runoff to rivers; and(g) end-of-pipe waste to rivers.

7.1. Chunk size

The key idea here is that ordered systems aremore valuable than unordered systems. Situationsare common in which ordered systems are embed-ded within disordered systems. Motor vehicle wir-ing is generally made of copper wire. Under oldpractices, entire vehicles were melted down. Thismixed the copper with steel, thereby degrading thequality of the steel and making the copper inac-cessible. Modern practice uses removable wiringassemblies. The copper can be recycled with mini-mal contamination, and the steel is no longer


contaminated. The wire harness constitutes amacroscopic ‘chunk’. Steel scrap mixed withother waste (e.g. aluminum cans) can be mag-netically separated with low energy cost. Thealuminum can be separated with eddy current orflotation techniques. The ‘chunks’ in this caseare tightly bound pieces of steel or aluminum,loosely coupled to each other. Macroscopic sep-aration is generally preferable to microscopicseparation. The reason is that there is less mix-ing of subclasses of materials, thereby reducingthe need for separation at the atomic or molecu-lar level.

7.2. Concentration

The higher the concentration, the easier therecycling. As already discussed for helium anduranium, the energy per mole of the desiredsubstance required for concentration from amixed gas increases as concentration decreases.In practice, low concentrations require movinglarge amounts of host material, thereby costingenergy due to viscous forces, etc. Dumping ofmaterials into waterways decreases concentrationand complicates recycling. Agricultural runoff isa generally non-point-source and is very difficultto deal with. Waste appearing at pipe-ends isrelatively concentrated and should be dealt withprior to dumping into a river. A polluted riverjoining a less polluted river leads to lower pollu-tant concentration in the merged river. The gen-eral guideline is: when possible, do not dilute.

7.3. Bonding

A definition of an ordered subsystem is that itis tightly bound internally, and distinguishablefrom its environment. A goal of recyclingshould be to separate the ordered subsystems.The motor vehicle wiring harness is bonded tothe vehicle loosely, except for connections andhold-downs. These tight bonds are designed sothey can be easily broken in the recycling pro-cess. Aluminum cans and steel scrap are tightlybonded internally, but pieces are usually weaklybonded to each other.

8. Observations

This paper has focused on recycling. How-ever, viewed in the large, it is not the difficultyof concentrating dilute materials that mattersmost. It is rather that what is waste from oneperspective can be a critical resource to anotherpart of the system. For example, the atmosphereand the oceans have long been used by humansas dumps, yet they are critical parts of our life-support system and we pay heavily when weallow them to become contaminated. What iswaste to one part of the system is feedstuff toanother. We must learn how to keep pollutionlevels low enough as not to cause damage toany system that depends on it. The latter is akey issue for global warming. From an ecologi-cal perspective, there is little exaggeration to as-sert that there is no such thing as a wastestream.

Sound system design should seek to containmaterials to the maximum extent feasible, andto minimize waste streams. This criterion can bemet by tight containment associated with highlyefficient recycling systems taking their feed atthe highest-concentration points available. Whensubstances released to general pools (e.g. atmo-sphere, lakes, oceans) remain at low concentra-tions, they are difficult to recover but do noharm. As concentrations increase, harm in-evitably occurs (e.g. pesticide water contamina-tion and CO2 greenhouse contamination), butthe higher concentrations can, to a degree, sim-plify removal or reprocessing. There are no ‘inprinciple’ reasons why waste reservoirs need belarge.

Biological and ecological resources often fallinto a qualitatively different category from non-biological resources. They cannot be manufac-tured. Once lost, a species is gone forever.Humanity must be careful not to allow arro-gance about our ability to recycle to distractattention from our inability to recreate biologi-cal and ecological elements of our life-supportsystem. This limitation is crucially important onearth. It is a critical management considerationfor any sustainable spacecraft, including space-ship earth.


Appendix A

Thermodynamics provides a lower limit on theamount of energy required for recycling. Thislower limit is often discussed in terms of the‘entropy of mixing’ of two gases. Entropy analysisis of broad applicability, but the concepts are farless intuitive than energy analysis. The SecondLaw of Thermodynamics places important con-straints on what is and is not possible. We areinterested in the limits of recycling. That is, wewant to understand what is possible in order toprovide benchmarks against which practical recy-cling techniques may be measured.

Modern research on the Second Law of Ther-modynamics reaffirms the high place in the pan-theon of knowledge accorded it by AlbertEinstein, who wrote: ‘‘A theory is the more im-pressive the greater the simplicity of its premises,the more different kinds of things it relates, andthe more extended its area of applicability. There-fore the deep impression which classical thermo-dynamics made upon me. It is the only physicaltheory of universal content concerning which I amconvinced that, within the framework of the ap-plicability of its basic concepts, it will never beoverthrown’’ (quoted in Lieb and Yngvason,2000). Lieb and Yngvason (2000) describe anentirely new way of looking at the Second Law.The approach focuses on comparison of states,with entropy appearing as a key variable andtemperature emerging in a natural way as aderived concept. The approach avoids statisticalmechanical models. Thermodynamic reasoning re-garding recycling is as fundamental as is energyanalysis in work on energy efficiency.

Our goal here is to establish limits. We find ituseful to frame recycling in energy terms via anelementary energy analysis for a perfect gas.(Fermi, 1936; Davidson, 1962, p. 60).

The molecules of a perfect gas do not interact.The pressure produced by two perfect gases in acontainer is thus the sum of the pressures thateach would produce alone (Dalton’s Law). Con-sider two gases at the same pressure but in sepa-rate containers. Place the container in anisothermal bath so it can absorb or reject heat.Because the pressures are equal, an impenetrable

barrier separating the two gases will experience noforce. Now remove the barrier. Each gas expandsto fill the entire region. As the gases expand, theydo work. This work appears as heat, which flowsinto the surrounding bath. After expansion, thesystem of two gases has a lower energy.

The work of compression to separate the twomixed gases is calculated as follows. The perfectgas law is PV=nRT, where T is constant, R is thegas constant (8.2 J/mol) and n is the number ofmoles.

A.1. Initial state: both gases occupy 6olume V

V1(initial)=V2(initial)=V(initial)

The partial pressures differ, and are given by:

P1(initial)=n1×RT/V

P2(initial)=n2×RT/V

A.2. Final state: both gases are at the samepressure

P1(final)=P2(final)=P(final)

V1(final)=n1×RT/P1(final)

V2(final)=n2×RT/P2(final)

The molecular fraction of the first gas is

x1=n1

n1+n2

The molecular fraction of the second gas is

x2=n2

n1+n2

=1−x1

The final pressure is

P(final)=n1×kT/[V1(final)]

=n2×RT/[V2(final)]

=n2×RT/[V−V1(final)]

Using the molecular fractions x1 and x2, thevolumes of the two components are:


V1(final)=x1×V

and

V2(final)=x2×V= (1−x1)×V

The work done on the first and second gases isW1 and W2. The total work of separation isW=W1+W2:

W=W1+W2=&

(n1×RT/V1) dT

+&

(n2×RT/V2) dT

=RT{n1× ln[V1(final)/V ]+n2× ln[V2(final)/V ]}

=nRT [x1× ln(x1)+x2× ln(x2)]

=nRT [x1× ln(x1)+ (1−x1)× ln(1−x1)] (A1)

The work per mole of gas 1 is w=W/n1

w=W/n1=RT{ln(x1)+ [(1−x1)/x1]

× ln(1−x1)} (A2)

This calculation is for a perfect gas compressedreversibly. If the process is irreversible, additionalwork may be required.

For low concentration, x1�1, ln(1−x1)�−x1, and the work per mole (Eq. (A2)) reduces to:

Wn1

�RT [ln(x1)+1]

For an isothermal reversible process, the en-tropy change is given by S=work/T. This yieldsthe standard expression for entropy of mixing,which is valid even for irreversible processes:

S=nR [x1× ln(x1)+ (1−x1)× ln(1−x1)] (A3)

Appendix B

This example is the simple two-reservoir recy-cling model presented by Ayres (1999). The nota-tion has been slightly changed for clarity. Thesystem, shown in Fig. 2, consists of two materials,two reservoirs, and a materials recovery (recy-cling) unit.

The materials consist of a critical resource (CR)and a matrix material (M). These materials flowfrom an in-service reservoir (in-service) to awastebasket reservoir (waste).

The masses MCR and MM of the resource andmatrix materials in the in-service reservoir are:MCR(in-service) and MM(in-service). The concen-tration of the critical resource of the in-servicereservoir is:

Concentration(in-service)

=MCR(in-service)/MM(in-service)

The waste reservoir contains massesMCR(waste) and MM(waste). The waste concen-tration is:

Concentration(waste)=MCR(waste)/MM(waste)

Both the critical resource and the matrix mate-rial flows from the active to the inactive reservoir.The critical resource flows at a rate d(CR) (per s).The mass flow of the critical resource is d(CR)×MCR(in-service). The matrix material flows at arate d(M) (per s) so the total mass flow is d(M)×MM(in-service).

Material flows from the waste container to therecovery unit at rate w (per s). The rates of flow ofboth the critical resource and the matrix materialare identical. The mass flow of the critical re-source is w×MCR(waste). The mass flow of thematrix material is w×MM(waste).

The recovery unit has less than 100% efficiency.Of every unit of critical resource flowing in, afraction f(CR) is transferred to the (in-service)stock and the remainder 1− f(CR) is returned tothe waste reservoir. Ideally, f(CR)=1.

Fig. 2. Flow with imperfect recycling in a two-componentsystem consisting of a critical resource (CR) and a matrixmaterial (M). See Ayres (1999) and Appendix B.


Similarly, a fraction f(M) of the matrix flowsfrom the recovery unit to the in-service reservoir,the remaining fraction 1− f(M) returning to thewaste reservoir. Ideally, f(M)=0.

These equations may be solved dynamically,either analytically or using a computer techniquesuch as STELLA (for example, Ford, 1999) For thespecial case where the system is linear, the re-sponse to a step change in a stock is exponentialapproach to a new equilibrium.

In steady-state, mass must balance for both thecritical resource CR and the matrix M.

For the critical resource CR, the mass balanceequation is:

dMCR(in-service)= f(CR)×w×MCR(waste)

For the matrix material MM, the mass balanceequation is:

dMM(in-service)= f(M)×w×MM(waste)

The ratio of the critical resource [CR] concen-tration in the in-service container to that in thewaste container is found to equal the ratio of therecycle efficiency for the critical resource dividedby the recycle efficiency for the matrix material:

Concentration(in-service)/concentration(waste)

= f(CR)/f(M) (A4)

The choice of recycle technology drives theratio of the concentrations of the critical resourcein the in-service and the waste reservoirs. There isno term involving the concentration of the criticalresource in either reservoir.

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