Exponential growth of large self-reproducing machine systems

M&d.

Pergamon

Cornput. A4odelling Vol. 21, No. 10, pp. 55-81, 1995 Elsevier Science Ltd. Printed in Great Britain

08957177(95)00071-2

Exponential Growth of Large Self-Reproducing Machine Systems K. S. Theoretical

Division,

LACKNER

Los Alamos

Los Alamos,

C. H. Physics

Department, (Received

National

Laboratory

NM 87545, U.S.A. WENDT

University

of Wisconsin,

November

1994;

Madison,

accepted

December

WI

53706, U.S.A.

1994)

Abstract-we

address quantitatively the major issues involved in the design of self-reproducing machine systems that are capable of both rapid growth to a very large scale and the accomplishment of correspondingly large tasks. A minimal system that satisfies the growth requirement would consist of a large solar cell array and a colony of diverse and specialized machines. With solar energy, raw dirt, and air as its input, the collective purpose of the colony is to expand the solar cell array and build more machines largely without the aid of man. Once the desired size is attained, the entire production capacity of the system may be diverted to useful applications such as large scale energy collection, control of greenhouse gases in the atmosphere, and fresh water production. We consider the issues of resource availability, the suitability of current automation technology, and the required investment in land area. In the discussion of resources, we propose a high-temperature, metallurgical process for separating useful elements from raw dirt without the use of rare elements. Automation technology is judged by a formal productivity requirement in the production chain of each machine type, which must be satisfied to achieve a given overall growth rate. We estimate the time scale for exponential growth to be on the order of months, so that such a system could reach continental size in less than a decade. An area of lo6 km2 is enough to provide the key elements of a sustainable world economy. At ten percent efficiency, a solar cell array of this size can collect energy at three times the rate of today’s global energy consumption. Keywords-Automation, machine systems.

Auxons, Macroengineering,

von Neumann machines, Self-reproducing

1. INTRODUCTION The idea of self-reproducing machines was introduced almost fifty years ago by Ulam [l] and by von Neumann [2]. The character of the ensuing work was the development of mathematical formalisms which would model the process of self-reproduction. Ulam’s cellular automata, which have been much discussed in recent years [3], are mathematical objects obeying simple rules and were explicitly designed to reproduce themselves [I]. Closer to a physical implementation is von Neumann’s kinetic model of self-reproducing automata [2]. These are essentially idealized robots which reproduce themselves from an infinite supply of parts, using simple rules. Von Neumann used this model as a basis for a formal analysis of whether self-reproduction was possible. While much of the work had application to mathematics and theoretical biology, e.g., [4,5], the possibility to use self-reproducing machines for practical purposes was also discussed [6]. Seagoing We gratefully acknowledge conversations and discussions of the manuscript with B. Beyer, D. Butt, S. Colgate, F. Dyson, .I. Hopson, R. Menikoff, N. Metropolis, R. Rauenzahn, E. Ramberg and J. Solem. Typeset by d&-w 55

K. S. LACKNER AND C. H. WENDT

56

self-reproducing machines, called artificial plants, were proposed as a means of collecting natural resources [7]. In his 1970 Vanuxem Lecture, Dyson speculated about self-reproducing machines and their implications for society [8]. M ore recently, such machines have been discussed in the context of space exploration reproducing machine systems In principle,

self-reproducing

[g-13]. Our interest on Earth. machines

in the present

would offer a new and different

accomplish very large scale tasks, which include those currently design and/or building effort, as well as those that are currently scale. Examples are collection atmosphere, and desalination

paper

is in land-based, means

for society

selfto

achieved by great repetition of unimaginable because of their

of solar energy, direct removal of greenhouse gases from the Earth’s of water for irrigation. The special feature of the self-reproducing

machine approach is that effort is focused on design and construction of a small seed system, and the large scale follows from exponential growth of this system through a few tens of generations. Some human is achieved

effort could also be expended

by the machines

to be applied

as desired

themselves.

in the growth phase, but for the most part the growth

Once the initial

in new places, both quickly

design is achieved,

this allows the system

and at very low incremental

cost.

Using self-reproducing machines for large scale projects presents several advantages over the traditional approach, where the required investment in labor and equipment scales directly with the scope of the goal. Like the advantage of low incremental cost, they all follow from the premise that large amounts of human effort are not required beyond the design and construction of the seed system. For example, widespread duplication of technical expertise is unnecessary, avoiding a potential bottleneck which could slow the implementation of very large scale projects in the conventional approach. Typically the conventional approach also requires great duplication in the policing of environmental regulations, whereas a self-reproducing system admits greater focus in dealing with risks and side effects by incorporating counteracting measures in the design. Finally, the low-cost production capacity of a self-reproducing machine system permits the additional effort required for use of low grade but abundant natural resources as well as for additional processing steps designed to eliminate environmental side effects. Our focus in this paper is the viability of self-reproducing machine systems and in particular whether an exponential growth curve is possible in the size range of interest. For the purpose of discussion we propose a model system that consists of a large solar cell array which feeds energy to a large variety of specialized but primitive machines. These machines, which we call UWCO~S,~ have as their collective purpose to gather material and produce additional solar cells and more auxons. The resources used for growth could be as simple as sunlight, air, and common dirt or rock. For robustness, we imagine the operations of an auxon colony to be highly decentralized: in manifold duplication of each auxon type, in division of tasks among the auxon types, and in self-control of each auxon according to predefined rules. The self-reproductive aspect of the system arises from the interactions of the various auxon units rather than being embodied in a single complicated machine or a centralized controlling unit. There are two classes of problems that could prevent an exponential growth curve. The first class concerns the supplies for growth, both material and energy-if the accessible supplies do not grow proportionately with the system, then the growth curve will not be exponential. The energy supply scales naturally with the size in the case of solar power, and therefore does not present an obstacle to exponential growth. To address the same issue in the case of material supplies, we consider whether it is possible to extract what the system needs directly from common rock, the availability of which scales with the system’s size, without using any significant external inputs. We show that the extraction of useful materials from raw dirt can be accomplished using only plentiful materials that are in turn obtained in the same extraction process. The proposed processing scheme has direct parallels in industrial metallurgical practice and appears feasible

lFrom

the Greek C&ELY, to increase in size.

Exponential

from thermodynamical

considerations.

Growth

57

We therefore argue that the material supplies also scale

with the size of the system. The second class of problems which could prevent exponential growth concerns the automation. One may start by phrasing this as the question of whether the necessary automation is feasible. We argue that “feasibility” in this context means: (a) each task can in principle be automated; (b) it is not inordinately expensive for the system to build and run the corresponding machines; and (c) the effects of failures can be tolerated. By observing the trends in high-tech automation, e.g., in the manufacture of computer chips, one arrives at the conclusion that virtually all manufacturing processes can be automated given a sufficiently large effort, which addresses point (a). This leaves the question of the amount of effort required, that is, points (b) and (c). We formally define this effort in a way that its bearing on the exponential growth is made clear. Our analysis effectively supplies a definition for the word “Cost)” so that if each machine is shown to have a cost below a certain threshold, then we can show that exponential growth is not limited by the expense of building and running machines. If one subsystem’s cost exceeds this threshold, it can result in a production bottleneck which limits the overall growth rate. For example, one might expect that as the system grows, the cost of the transportation system will eventually limit the growth rate. The additional problem of failures (c) is absorbed into the same formalism if failures are accommodated by building replacement machines. This simply adds to the rate of building implied by the growth rate and scales in the same way. To illustrate, we indicate how well a few typical machines perform when evaluated with our cost criterion, and note that the answers are encouraging. The final issue concerns the amount of time and land that would have to be devoted to the construction of an auxon system with useful applications, and how it would compare to the expected benefits. To illustrate this point, we consider several potential applications of auxon technology. In each case, we estimate the size requirements and other constraints. We find that auxon systems can rapidly grow to any desired size and could collect energy, produce fresh water or eliminate greenhouse gases on a global scale while employing areas smaller than devoted to cropland in the United States.

2. OUTLINE

OF AN

AUXON

SYSTEM

As the basis for the quantitative discussion that follows, we imagine a specific type of system which incorporates the most obviously necessary features. The system begins with the installation of a small startup version on an area of land. One component of the system is a fixed infrastructure, which includes almost complete coverage of the area by solar energy collectors. The infrastructure also includes a simple grid of tracks for transportation and some form of energy distribution. It provides a controlled environment for the other major component, which consists of a large variety of machines (auxons). The infrastructure is the interface between these machines and the natural terrain. The collective purpose of the auxons is to extend the infrastructure onto the surrounding empty land and to make more machines like themselves. The object is to achieve exponential growth, and eventually to attain a very large size-perhaps a thousand kilometers from one side to the other. We assume that this is to be accomplished without the external supply of equally large provisions. Therefore, all significant requirements must be procured by the system itself from its natural surroundings. An example is the solar energy obtained by the collectors mentioned above. Other requirements must be satisfied from locally available water, air, and minerals. The minerals must be obtained in the form of raw dirt or rock, rather than in the traditional enriched form that results from prospecting and mining operations. Such operations would imply a highly sophisticated autonomous collection process

K. S. LACKNER AND C. H. WENDT

58

unattainable with today’s technology. In addition, the scarcity of mineable deposits could lead to supply bottlenecks which would slow the growth of a very large system. Instead, the collection of raw dirt or rock is distributed throughout the occupied area, as are the processes these raw materials into new infrastructure and new auxons.

converting

Thus, we start with a picture of machines sitting or moving on a network of tracks, some of which dig up a thin layer of dirt and pass it on to the beginning of a separation, refining and manufacturing chain. All the machines in this chain are powered by the solar energy which is collected in the surrounding infrastructure. The final products of the manufacturing chain are new tracks

and solar energy

and installed

to form additional

collectors,

which are transported

infrastructure,

and additional

to the boundary auxons

of the system

which will occupy the new

infrastructure. The auxons

themselves

will be specialized

machines,

with the minimum

level of sophistication

necessary to accomplish their assigned tasks. Only together can they form a growing system, in the same way that a factory requires many different machines for its operation. As in an automated manufacturing plant, we imagine that the auxons use physical constraints and simple rules as a basis for exchanging materials, moving around, or other maneuvers. This “reflex” approach, which is also gaining ground in robot designs [14], has the advantage of being robust without using massive computational power or sophisticated pattern recognition. The exact form of all the different auxons required is impossible to specify in advance. The many tasks to be performed include gathering raw material, separation and refining, manufacture of parts, and assembly of more auxons and structures. As a starting point, one may say that in some way all auxons must embody automated versions of known manufacturing processes, or reasonable modifications thereof. However, the raw materials available are unusual from the perspective of standard practice, as is the optimization for maximum growth rate. Additionally, such a large system must not generate any significant waste products. These peculiarities of auxon systems form the core of a more detailed picture. Using dirt as the major material input for growth means that most of the system must be made of common elements. Otherwise a much greater quantity of material would have to be processed [ll], which would slow the growth rate and introduce a large waste stream as an output. Rare elements can be used, but then only in very small amounts. Estimates for the major elemental fractions in typical rock are listed in Table 1, and the rare elements are listed in Table 2. The chemical composition of most surface materials is reasonably close to that of the Earth’s crust as a whole. Of course, there are variations among different soils and rocks, and the growth process must tolerate this variability in its inputs. Among the elements to be found in large quantity are aluminum and iron, which could be separated out and processed to manufacture metal parts. In oxide form, abundant silicon together with the other metals may be used to make ceramics, glass or cement. Silicon is also needed to make solar cells and electronic parts. Copper is an example of a traditionally important metal which is somewhat rare at less than one part in ten thousand. Exotic elements like gold, platinum or palladium would be exceedingly difficult to obtain. Some elements that are frequently needed in chemical processing but that are not common in rock, such as hydrogen, carbon and nitrogen, may be supplemented from air and rainwater. Chlorine and sulfur, often used in conventional chemical process design, are available in rock but only at the level of one part per thousand. Although such small fractions are expensive to extract, they may suffice for chemical processes that are highly specialized and that do not involve However, the bulk separation of dirt into its elements processing large amounts of material. required for the growth of the total system is itself one of the major operations, and therefore This suggests that the chemistry of separation and may not rely heavily on rare elements. refinement will show some departures from common practice. The separation is considered in more detail in Section 3.

Exponential

Growth

59

Table 1. Nominal composition of 1 kg of average crustal rock, based on [15]. The composition is close to that of sediments and igneous rocks. It contains, however, significantly more CO2 and somewhat more water than the latter. The last column gives the binding energies of the oxides as an approximate estimate of the binding energy in the mineral.

mass

metal

mass

binding

component

kl

component

SiOz

600

Si

5 130

oxide

TiOz A1203

Fez03

35

Fe0

30

MgO

kl

energy [MJ]

280

9.09

Ti

3

0.06

Al

69

2.14 0.18

Fe

48

30

Mg

18

0.45

CaO

60

Ca

43

0.68

NazO

0.11

20

Na

15

0.13

K2O

30

K

25

0.12

Hz0

30

JJ2

3

0.48

co2

30

C

8

0.27

total:

1ooog

512g

14 MJ

Table 2. Crustal abundances of the rare elements based on a compilation

mass fraction

elements

10-n to 10-s

Ti, H, P, Mn

10-s

S, Cl, Rb, F, Sr, Ba, Zr, Cr, V, Zn

to 10-d

10-4 to 10-s

in [15].

Co, Ni, Cu, W, Li, N, Ce, Sn, Y, Nd, Nb, La, Pb, Ga, MO, Th

10-S to 10-s

Cs, Ge, Sm, Gd, Be, Pr, SC, As, Hf, Dy,

10-e to 10-T

Tb, Lu, Tl, Hg, I, Bi, Tm, Ce, Ag, In

U, B, Yb, Er, Ta, Br, Ho, Eu, Sb


Se, Pd, Pt, Au, He, Te, Rh, Re, Ir, OS, Ru

As a by-product of the separation process, there will inevitably be some fraction of material not specifically needed for growth. An optimal system would make use of such material for noncritical structural items, perhaps in the form of a concrete. In any event, it could be bound in stable form as rock-like waste products in additional processing steps, mixing it with silica, alumina or other carriers. The incentive to incorporate it into the system is a strong one, however, since the growth rate is directly affected. Thus, the system’s composition should be well adapted to the availabilities of the various elements. To summarize the effects of material input constraints, it appears that many materials commonly used in manufacturing are sufficiently abundant in dirt that major changes in machine design are unnecessary. In some cases, the conventional choices of processing steps will have to be altered due to a limited supply of certain chemicals. Any procedure employing such rare elements as silver, gold, or platinum would have to be redesigned or eliminated. The largest departure from usual practice is expected to be the separation of elements from raw dirt. The maximum possible growth rate may be crudely estimated by comparing the time averaged solar power reaching a unit area (~300 W/m2 [15] or 25 MJ/ m2 per day) with the energy needed to reduce the oxides found in dirt to their metallic states (14 MJ/kg or 4.7eV per oxygen atom). Supposing that it takes about 10 kg of processed material to build a new square meter of the Ma421/10-E

60

K. S. LACKNER AND C. H. WENDT

system, one calculates an exponential growth time constant of less than three months. This estimate assumes a ten percent efficiency in the energy collection, which is typical for solar cells 116,171. A time constant as long as six months would still permit growth from 0.01 to lo6 km2 in less than a decade. The allowance of 10 kg/ m2 is large enough to include the solar energy collection system [18,19] and other infrastructure components.

One must add to this

the average density of auxons distributed throughout the system. If the total material needs are significantly larger than 10 kg/m2, then the growth rate will be proportionately smaller. Therefore, it is important to minimize the investment in the auxon component, so that at first glance the system looks like a large array of solar cells. In practice, the optimization implied is that of the auxons’ productivity, so that fewer are needed to accomplish a given work load. A crude comparison with industrial assembly times (< 20 hours for a car [20]) suggests that the auxon assembly subsystem may be able to produce many times its own weight in the time course of a few months and is therefore not likely to dominate the mass of the system. In Section 4, we will formulate a more explicit test of whether current machine technology can achieve the necessary productivity. The desire to make the infrastructure as low in mass as possible is one of several factors which influence the choice of a size scale for the individual auxons and infrastructure components. Although the optimum can only be found by experimentation, we surmise that it will be lo-30 cm for a typical unit. This is clearly quite different from conventional industrial equipment, which is optimized in the context of human builders, operators and maintenance personnel. Our relatively low estimate represents a compromise between the major factors. Again the first consideration is the average mass per unit area of the infrastructure, which limits the supports for solar cells to a fairly small height. Second, although the ratio of an auxon’s production capacity to size appears at first sight to be roughly constant, there are advantages to keeping the size small. One is simply the cost of developing and building the startup system. Savings in the size of structural supports is another advantage. In addition, lower grade materials could be used in the supports of smaller units, which would not only imply a lower energy investment but also could absorb some of the variation in the raw material composition. On the other hand, the machines must be large enough to stand up to the elements. They should also not be so small that they are difficult to create with readily available technologies. Although very large sizes could yield savings in fixed-scale items like control circuitry as well as in heat insulation, these factors are balanced by a desire to minimize the times required to complete chemical and physical separation processes. Such times tend to grow rapidly with increasing length scales, in particular for diffusion dominated chemical separation processes. Shorter processing times lead directly to higher productivity for any subsystem, allowing the amount of material invested in that subsystem to be proportionately smaller. We note that a departure from the scales used in industry does not imply a radical shift in practices, since most large-scale processes can also be implemented in small laboratory-scale models. One feature of auxon systems which must differ qualitatively from the industrial example is the approach to fault tolerance. Although many complex industrial practices have been automated, the current state of the art does not provide an appreciable level of automatic diagnosis and repair when failures occur. A machine can replace certain parts which wear out on a predictable basis, but more serious problems require human intervention. Although this may be possible for a small auxon system, human intervention for the repair of individual units is not an option once the system reaches a large size. Self-repair is not a viable option either, since we assume that the auxons are not very intelligent, exhibiting only the sophistication of machines found on an automated assembly line. In order to deal with serious malfunctions in the absence of repair personnel or their mechanical equivalents, we may stipulate that each auxon should at least have some means of diagnosing its own failure. If the task it performs is not very complex, this may be some simple verification step which is added on. Some common malfunctions may admit simple recovery procedures which

Exponential

are presumably unrecoverable,

Growth

61

built into an auxon’s design. However, more unusual corresponding to a breakdown of the auxon in question.

problems are effectively In this case, a unit could

simply retire permanently from further activity. Its mass might be recycled as raw material, but we assume that repairs are not possible. Unlike on the factory floor, the chain of processing must be sufficiently open so that the workload can be passed on to other copies of the failed unit. The proposed growing system, time

constant.

Section

strategy of junking failed auxons is not unduly expensive in an exponentially as long as the mean lifetime of an auxon is significantly longer than the growth Even this rule does not always have to be obeyed,

4. In any case, it is an easily definable

constraint

as will become

on the design of individual

apparent auxons,

in and

has several general implications. First, it drives the need for auxons to be very specialized. That is, in order to maximize the auxon lifetime, tasks should be broken down into many simple steps, one per auxon. If one were instead to combine n functions into a single unit, the mean lifetime would be reduced by a factor n. Second, the lifetime constraint becomes much more difficult to satisfy

if the growth

rate is very slow, since the minimum

allowable

auxon lifetime

larger. Such a situation would occur if much more than 10 kg/ m2 of raw material for growth. This reinforces the incentive to keep the material requirements small.

would then be were required

To summarize our picture of an auxon colony, it is a large-area system dominated by a lightm2 including the solar cells. The infrastructure is populated weight infrastructure, about 10 kg/ by auxon units. The auxons are about lo-30cm in size, are on average distributed perhaps one per 10 m2, and each performs a specialized task using energy from the solar cells (30 W/m2). An auxon has the minimum sophistication necessary to carry out its duties and check its own operation, and should have an average lifetime of at least several growth periods. The auxons dig up a thin surface layer of dirt or rock, averaging a depth of less than a centimeter, and separate it into the major elemental fractions needed for growth. Using procedures fashioned after industrial practice, they go on to manufacture more auxons and infrastructure with a growth time constant of 3-6 months.

3. EXTRACTING

THE

COMMON

ELEMENTS

FROM

DIRT

In the foregoing, we indicated that common dirt can be viewed as a low-grade ore that provides, in substantial quantit,y, the materials of which auxons and infrastructure are made. In addition, we showed that sufficient solar energy flux is available to reduce this ore to the metals at a rate sufficient to maintain rapid growth. However, an analysis of the steps involved in producing the auxons and infrastructure will reveal the need for additional resources. An example of this is any material that is used in the ore reduction and separation. This example is a key one, since all material eventually incorporated into auxons or infrastructure must pass through this process. Therefore, we now show that the main steps in the material separation can also be accomplished with the resources available, and specifically without the use of rare elements. In order to operate within this closure constraint, all elements used in the chemical separation process must derive ultimately from dirt. The difference to conventional practice is not only in the lower concentration of values in the raw material but also in the scarcity of certain materials that are often used in separation processes. Among these materials are acids for washing ores, and cryolite (NasAl Fs) used as a fluxing agent in aluminum metallurgy. Although we are limited by these constraints, we are able to construct a scheme whose basic ingredients correspond to known metallurgical processes. In this scheme, operation at the high temperatures typical of pyrometallurgy allows the major separation steps to proceed without rare materials. To a good approximation, the average mineral can be viewed as a mixture of oxides of the most abundant elements (Table 1). The separation proceeds by a number of reduction steps in which the oxygen is transferred to increasingly stronger reducing agents. At each step, those elements that have been reduced are separated by physical means. The reducing power of a material is given by the free energy of its oxidation reaction. A is reduced by B if the negative free energy of

62

K. S. LACKNER AND C. H. WENDT

the oxidation reaction of B exceeds that of A. The free energies involving the relevant elements are shown in Figure 1 as a function of temperature.

-250

z

-500

0

\

E

2

w

-750

2 -1000

O0

5oo"

looo"

1500°

temperature

2ooo"

2500'

‘C

Figure 1. The free energies of oxidation reactions relevant to the separation of raw dirt into its elemental constituents. Downward pointing triangles mark phase transitions in the metal, upward pointing triangles phase transitions in the oxide. The data are taken from the JANAF tables [21].

The approach outlined here uses silicon, aluminum, carbon and hydrogen as the reducing agents. Each of these is in turn reduced in another process step, so there is no net input of the elements in reduced form. Carbon plays a special role because it becomes gaseous when oxidized and, as a result, its reducing power increases steadily with temperature. It is moderate at low temperatures but at 2200°C carbon reduces all the other common elements. An implication of this behavior is that depending on the temperature, carbon can act as an oxygen donor as well as an oxygen acceptor. The scheme described here has as its goal the complete separation of raw dirt into its elemental constituents. In reality, much of the material would ultimately be used in its oxidized form, and in this sense complete separation and reduction is unnecessary. Therefore, the scheme proposed here is illustrative only, and we allow that it could be improved upon for the sake of efficiency. A summary of the processing steps is given in Figure 2. The diagram illustrates the fact that except for raw dirt, all substances needed are produced elsewhere in the cycle. In more detail

Exponential

Growth

63

r-l

RawDirt (1oow

Figure 2. The element separation cycle. Processing begins with raw dirt as an input at the top of the figure, and proceeds counterclockwise. The products are shown at the outside. With each step is listed the amount of heat which must be added to bring the reactant up to the process temperature and to accomplish the reaction. The heats are normalized to one kilogram of raw dirt. The electrolysis step requires electric energy which is listed separately. Ignoring recovery of heat from the final products and steps with a net energy release, the total energy consumption of the cycle is 25 MJ/kg or 7 kWh/kg.

the individual steps are as follows: (1) Culcination. The process starts with raw material which is broken up and ground to a powder. The powder is calcined by gradually increasing the temperature to about 13OO’C. Moisture is removed at -2OO”C, and water bound in minerals will be released between

64

K. S. LACKNER AND C. H. WENDT

500” and 1000°C. Carbon dioxide is completely removed before 1300°C is reached. There are many industrial prototypes for this process; the calcination of limestone for cement making is one example. Calcination should occur in an oxidizing atmosphere to remove rarer elements like sulfur and phosphorus which form volatile oxides, e.g., SO;, and PzOs. A significant fraction of the sulfur can be extracted in this way. Some sulfates, however, are stable to much higher temperatures and will only be extracted in later steps. Because their concentration is quite low, we will not discuss them any further. The apparently redundant step of first oxidizing the raw materials, which consumes about 4g of oxygen, enhances the robustness and fault tolerance of the extraction process. If for some reason unusual materials are processed, for example a piece of wood instead of mineral rock, the system will follow through with its normal operation even though it produces materials in unusual ratios. (2) Reduction with silicon. Metallic silicon can reduce iron, sodium and potassium (Figure 1). At temperatures of -1300” to 1400°C and in the presence of elemental Si, Na and K will leave the melt as metallic vapor, whereas Fe forms a liquid phase and precipitates out of the melt. Silicon must be added gradually so that it is not lost by mixing with the Fe precipitate. At 15OO”C, even Mg can be removed from the melt. Although the chemical equilibrium favors MgO over SiOs, the elemental Mg escapes as vapor from the reaction zone [22]. Above 15OO”C, this process may also affect CaO, but at a much lower rate because the free energy change is much larger. We may compare these processes with industrial use of silicon to remove residual oxygen from iron in steelmaking, and with the Magnetherm Process in which magnesium at low pressure is reduced with silicon [23]. A related process in which the vapor pressures are higher is the carbothermic reduction of magnesium which was implemented at a Kaiser pilot plant [23]. This first reduction step will also remove a number of trace elements. Ni, Co, Mn and Cr will be found in the iron phase; Cu may enter the vapor phase [24]. The iron will also dissolve a fraction of the remaining phosphorus and sulfur, part of which will also evaporate. More difficult to predict is the behavior of the halogen ions, Cl- and F-, which at this stage probably stay in the melt. A container lined with A1203 could hold the melt initially at -1300°C.

Silicon is added

gradually. In the early stages the vapor is dominated by potassium and later by sodium. The alkali vapor is transported to a different container and may be separated by fractionation making use of the 120°C difference in condensation temperature for K and Na. However, because of the chemical aggressiveness of these two elements, different means of separation may be used in a later stage and the alkali vapor could immediately be reoxidized to NazO and KsO. As the temperature is gradually increased, liquid iron separates from the melt and is removed from the bottom of the container. In a secondary step, the temperature is raised to just below the boiling point of Ca (1484°C) and the vapor rich in Mg is collected. (3) The volatilization of silicon. Silicon reacts with silicon dioxide melts to form volatile silicon monoxide: SiOz + Si + 2Si0. The process has been used at 1200” to 1300°C to remove SiOz from aluminates, but at those temperatures it requires a good vacuum because of the low SiO vapor pressure [25,26]. We suggest the use of this reaction at the higher temperature of 1800°C where the equilibrium pressure of SiO rises to about atmospheric pressure based on free energies from the JANAF tables [21]. Two possible contaminants in the SiO vapor are Ca and AlsO. The Ca vapor, however, is oxidized by SiO and may return to the melt as CaO. Al20 contamination is also expected to be minor, based on free energy calculation.

Exponential

65

Growth

Again using A1203 for the container walls one can raise the temperature until SiO begins to evaporate in large quantities.

of the melt

It may be possible to extract Ca before

this point is reached. Here we assume, however, that Ca remains behind until all silicon is removed. The process is highly endothermic. Approximately 8.2 MJ per kilogram of raw material are required in this processing step. The reduction of the resulting SiO will be (4)

discussed below. The reduction of titanium, calcium. are those of titanium,

After the removal of silicon, the only oxides left

calcium and aluminum.

Titanium

cannot be reduced with car-

bon because it forms titanium carbide [27] which is extremely stable and melts only at 314O’C [15]. Th is may offer one way of removing Ti, and another is based on the observation that TiOz can be reduced with Al. Hence, by adding aluminum to the melt we can extract

titanium,

which at a density of 4.4g/cm3 should sink to the bottom of the

container, and calcium which will evaporate. (5)

The carbothermic reduction of aluminum. The remaining material, essentially AlsOs, may be reduced by adding carbon to the mixture. The container could be made of graphite, some of which would be consumed in the process. The reduction is complicated by the fact that volatile A120 tends to escape together with the carbon monoxide [28]. Based on the free energies [21], however, it is possible to reduce most of the Al20 at temperatures between 2300” and 2400°C as it comes in contact with the graphite container walls. The remaining aluminum oxide vapors are separated from the CO at lower temperatures through reoxidation. In the process, a small fraction of the carbon monoxide is reduced to carbon. The resulting Al203 and C must then be recycled through the reduction step. In effect, the total reaction is given by Al203 + 3C + 2Al+ 3C0.

(1)

Any aluminum carbide which might be formed is decomposed above 22OO’C into the elements. The industrial feasibility of carbothermic aluminum reduction has been demonstrated [29,30]. (6) The reduction of silicon monoxide. At high temperatures elemental form SiO + C + Si + CO.

carbon reduces silicon to its (2)

Hence, by streaming SiO vapor over graphite surfaces at 2300°C, we can collect pure Si and produce carbon monoxide. The reaction is essentially energy neutral because of the similar binding energies of SiO and CO. The reduction of Si with C is indeed used in industrial processes [16,31]. In industrial applications, the electric potential between the heating electrodes allows operation at somewhat lower temperatures [32]. An alternative route to the reduction of SiO makes use of hydrogen. If cooled, the silicon monoxide becomes unstable and converts into a mixture of silicon and silicon dioxide. It should be possible to separate the Si-SiO2 mixture through physical means, e.g., crystallization of SiO:! in a Si melt. In this case, one can treat the extracted SiO2 with Hz and one obtains according to SiO2 + H2 -+ SiO + Hz0

(3)

more silicon monoxide. This is a shortcut to the basic transfer of oxygen to hydrogen which omits the intermediate oxidation of carbon. Silicon obtained in this way may be preferable as a starting point for producing highly purified silicon since its carbon contamination will be greatly reduced. However, this process has a number of disadvantages over the

66

K. S. LACKNER AND C. H. WENDT

reduction with carbon. First, the carbon reduction is a single step operation and is therefore simpler. Second, although the reduction with hydrogen is exothermic overall, it requires a very large amount of energy at 18OO’C (12.9 MJ per kilogram of raw material) and releases even more at 1500°C (16.2MJ per kilogram of raw material). As a result, this process generates large quantities of waste heat which cannot be reused within the element extraction process. Finally, hydrogen is a scarce commodity and due to its high diffusion rate the potential for leakage at 1800°C is very high. (7) The redzlction of carbon oxides with hydrogen. The carbon oxides (CO and COz) can be reduced to elemental carbon using hydrogen as the reducing agent [33] CO2 + H2 + CO + H20, CO+Hz+C+HzO.

(4)

The latter reaction is always accompanied by the disassociation of carbon 2CO-,C+COz

(5)

which typically occurs at about the same rate [33], leading to a net reaction 3C0 + Hz + 2C + CO2 + H20.

(6)

The two reduction reactions in equation (4) proceed at different temperatures. Therefore, we assume that they require two different reactor vessels. The reduction of CO occurs in the first vessel at 5OO’C in the presence of Fez03 which acts as a catalyst. The reduction of COz occurs at 1100°C in the second vessel. The gas mixture leaving the first reactor vessel is cooled to remove the steam. Then, the remaining CO2 is heated and enters the second reaction vessel. The resulting CO, Hz0 mixture is cooled to remove the water vapor, and the remaining CO is led back into the first reaction chamber. Presumably, excess hydrogen accompanies the carbon oxides through all steps of the process. The carbon and hydrogen reactions discussed here are akin to the water gas and shift conversion reactions used in industry, although these are typically operated in the opposite direction. (8) The electrolysis of water. The final step in the reduction of mineral oxides involves the release of the oxygen which started in the dirt and was then transferred from one reducing agent to the next. In our scheme, this is accomplished through the electrolysis of water at room temperature. The process is one of the major energy consumers and requires approximately 16 MJ per kilogram of water. Assuming that all the oxygen stored in the mineral is released through this process, 549 g of water must be electrolyzed per kilogram of mineral, leading to an energy consumption of 8.8 MJ in the electrolysis alone. In summary, the processing of one kilogram of raw material goes through a number of steps which are indicated in Figure 2. From the figure it can be seen that except for dirt and sunlight all the inputs needed are accounted for as a subset of the outputs. In this sense, the process is closed. All of the outputs are fully reduced forms of the common elements. The output of trace elements like sulfur and phosphorus, if it is not needed by the auxon colony for other purposes, may be bound with other outputs to make stable solids such as pyrite (FeS) and calcium phosphate (CasPzOs). Such minor elements may easily be incorporated into the mass of a growing system. Hydrogen and carbon play a major role in the separation scheme. Because their output is small compared to the amounts used internally, the available reserves of hydrogen and carbon are reused several times in a process cycle. In this sense, their availability in dirt is marginal and may also be too variable for a rapidly growing system. Fortunately these elements can be readily supplemented from air and/or rainwater.

Exponential

Growth

67

The net amount of energy that needs to be supplied in the form of heat or electricity is 25 MJ per kilogram of raw dirt input. Since the binding energy is only 14 MJ/kg it is clear that much of the energy is lost as waste heat. This is explicitly seen in steps which show a net energy release. Additional heat is released in the cooling of the final products. Some fraction of this heat could be recovered and used in other steps, but conversely there will be additional losses which are not explicitly included. For a first approximation we ignore both of these corrections which are in opposite directions. Nevertheless we consider the estimate conservative, because not all stages of this process will have to occur for all of the material.

For example, that fraction

of the aluminum which is needed for refractory bricks or high-temperature

containers is used in

the form of alumina and therefore does not require the reduction to metallic aluminum.

In this

case, the processing chain can be stopped after the removal of silicon and calcium. Maintaining materials in their oxidized form greatly reduces the energy investment in the material separation. Using the nominal figure of 25 MJ/kg, the production of 10 kg/m2 of completely separated material will take about 100 days, based on an average power supply of 30 W/m2 from the solar cell array. The process described is stil1 in a conceptual form. Nevertheless, it already points to some of the important issues. For example, the use of high temperatures implies that heat losses must be well controlled. It is possible that the scheme could be improved by eliminating some of the highest temperature steps, but in any case it is clear that a large amount of mass must be invested in insulation materials. More importantly, this discussion has shown that element separation can be performed under the closure constraint without reliance on rare elements like fluorine. In particular, it is not necessary to process much more material than is actually to be used in building the system, which would drastically inflate the energy requirements and slow the growth rate of the auxon system.

4. THE

GROWTH

RATE

Given sufficient material and energy resources for building and operating an auxon system, the remaining issue is whether machines can be designed which accomplish all the necessary tasks quickly enough to keep up with the growth rate. To answer this, we begin with the premise that, in principle, a machine can be designed that performs any task, but the question is really whether each machine is simple and efficient or complicated and inefficient. This is a measure of the level of technology available. If the available technology is inadequate, then a given output rate can only be satisfied by a large number of machines which may also be difficult to produce. Since these machines must in turn be built as part of a growing system, a fixed energy input implies that poor technology causes the available resources to be tied up in the construction of production machinery, resulting in slower growth. Thus, the adequacy of available technology boils down to an analysis of the growth rate, and the key ingredient in this analysis is the productivity of individual machines. In the following, we will make quantitative statements concerning the productivity required of individual machines which must be satisfied for use in an auxon colony. This requirement can then be compared with the performance of typical machines used in today’s automated production processes. We may observe that fault tolerance, another feature of the available technology, is manifested in a similar way. This follows from the fact that failed units must be replaced, lessening the net growth rate. As long as the average auxon lifetime is greater than the growth time constant, losses amount to small corrections in the growth rate. In any case, the formalism we present includes the auxon lifetime as a separate ingredient. To begin the analysis, we exponential growth. The area r is the growth time constant are constant in time. In order

recall that the first objective of an auxon system is to achieve A covered by the system increases with time, A(t) = AgetiT where of the system. Area1 densities such as the mass per unit area (M) to achieve exponential growth with a growth time constant r, the

68

K. S. LACKNERAND

C. H. WENDT

auxons and infrastructure in any given area must be able to produce a new version of themselves in a time I-. Thus, the total rate of production per unit area is given by M/r. The most immediately apparent limit to the growth rate is the energy required to process the materials for growth.

We have estimated this energy requirement as 25MJ/kg

materials, and the amount of mass per unit area as 10 kg/m2.

for typical

When combined with a solar

power input of 30 W/m2 which is available after photovoltaic conversion, this implies a growth time constant of about three months. In principle, however, the growth rate could be less than this for several reasons. First, some materials may require a substantially larger energy investment. Among these are any which require many energy-intensive purification steps. However, these cases may be confined to special-purpose materials which make up only a small fraction of the system, so that the growth rate would not be adversely affected. If only a small fraction E of the total raw dirt input must pass through n purification steps, while the remainder passes through only one step, then the exponential time constant is lengthened by roughly a factor (1 + en). As an example, we may consider the pure silicon required for solar ceils. Based on a thickness of a few microns [16,17], thin-film solar cells require less than log/m2 so that E = 0.001. Thus, the growth rate is affected by less than one percent even if the purification of this silicon requires ten times as much energy as other materials. Second, the figure of 10 kg/ m2 for the average mass per unit area assumes that the system is dominated by the solar panels and other fixed infrastructure. The mass of the processing and manufacturing sector should be much less than this, and is dictated by the productivity of this sector. If the productivity of the equipment building the infrastructure is measured by the amount of material processed per unit time and per unit weight of equipment, then a tenfold increase in productivity implies a tenfold decrease in equipment required. For example, suppose the growth time constant is one month. Then a subsystem that can process its own weight in 45 minutes would need to weigh only l/1000 of the mass which it processes over the course of a month. If instead it requires three days to process its own weight, then the required mass invested in the subsystem would grow to l/10 the mass of product needed in a month, and the mass of the equipment required to generate the subsystem would have to be increased by a similar factor. Finally, we mention three more factors which can reduce the growth rate of the system. These are the inefficiencies in the use of energy, material losses in the individual production steps, and losses of auxons and infrastructure due to their limited lifetime. In the following, we develop estimates of the time constant r which take into account the performance of the components that make up a given auxon system. First we indicate how in a very simple model the average productivity, losses of energy and material, and the average lifetime of the auxons would affect the growth time constant. Then we construct a more realistic framework for the description of an auxon system, allowing for the difference in productivity between different auxon types. This framework leads to bounds on the growth time constant which are expressed in terms of individual production chains, one for each auxon type. Each chain begins with raw material and ends with assembly of a particular auxon type, and must obey a design criterion which does not require knowledge of the remainder of the system. Once these criteria are met for all production chains, the desired growth rate is guaranteed for the whole system. 4.1. The Growth

Rate

for a Constant

Productivity

Model

As mentioned above, all parts of a growing auxon colony are to be produced at a rate so that their numbers increase in proportion to et/7. On top of the equipment dedicated to manufacturing the infrastructure, more equipment is needed to manufacture that equipment, and so on. The total equipment required is therefore a sum of terms, which may or may not converge to a finite result. If temporarily we make a simplifying assumption that the effective manufacturing

Exponential Growth productivity growth

is some constant

69

P, then the total mass of equipment

per unit area needed to sustain

is

The productivity

P is defined as the mass rate of output

ment, MI = 10 kg/m2 is the infrastructure Note that teristic

the series converges

of available

independent

technology,

of the available

If the available

only if P > l/r.

This makes it clear that

equip-

sets a lower limit on the growth

P, which is a charac-

time constant

r.

This

limit is

energy.

solar power is S = 30 W/m2 and the energy required

is H = 25 MJ/kg,

per unit mass of material

then growth implies

S

(MI+ME)

-z

H

Combining

per unit mass of manufacturing

mass per unit area, and r is the growth time constant.

this with the previous

r

expression,

(8)

.

we find that the growth rate is given by 1

MIH

(9)

r=S+P. The first term

is the time constant

and the second term accounts the system

for a system

that

is dominated

for growth of the other subsystems.

by the fixed infrastructure, The latter

will not dominate

and the growth rate as long as N 0.3/month.

That

is, the processing

its own mass.

and manufacturing

Not included

use, and the finite lifetime Inefficiencies amount

1.25.

Because

would increase

In contrast,

into energy

exceeds

potential either

the mass of finished

problem

is sidestepped

by recycling

structural necessity The because

to convert product,

so that the energy

finite

lifetime

by a factor less

be more dramatic. small

mass of raw material

used is also much larger.

This

that is able to make use of its own waste streams,

processing

of the equipment

the replacement

steps

or by incorporating

This characteristic

will increase

them

into low-grade

is apparently

the growth

time

constant

of failed units will use up some of the available

this should not be a problem

a practical

as long as the mean lifetime

of the system

resources.

As already

X is much longer than the

time constant.

by including

the combined

the corrections

effect of an energy efficiency

due to X and 77in equations

time constant

This estimate

assumes

to achieve

inefficiency

77, lifetime

analogous

X and productivity

that material

of the solar cells themselves

(11)

+ l/P)’

losses are cycled back into the system.

7 = 0.8, P = l/month,

If the overall system

and X = 2 years, then the original

is already

accounted

for in S.

P

to (7) and (8), giving a growth

MIH/S~/ + l/P 7 = 1 - (l/X)(MIH/Sv

2The

losses

per unit

raw dirt into new auxons or infrastructure,

which are needed in any case.

One may estimate

manages

energy

required

would increase

losses could in principle

The result could be that the required

by a system

to earlier

If average

for a viable auxon system.

mentioned, growth

them

materials

losses.

by 25%, so that the time constant

of the many steps required

greatly

units.

losses and material

of material

of at least l/3 of

in energy and material

of the solar cells2, then the power (H)

the consequences

losses at each step would multiply.

output

(9) are effects of inefficiencies

of the auxon and infrastructure

to 20% of the power output

mass of material than

sector must have a monthly

in the estimate

may be divided

(10)

naive estimate

of

K. S. LACKNER AND C. H. WENDT

70

r = 3 months which is based on the infrastructure only would grow to 6 months. Substantially worse values for these constants could result in much slower growth or no growth at all. The discussion so far has used constants

(P, H, etc.) which describe the system’s operation

in an average sense. In reality, there are a variety of machines and processes, each with different characteristics, which is the subject of the folIowing. 4.2. Superposition

Principle

An auxon colony is made up of many different machines and hence is a complicated system with many interdependencies. Although one can imagine enumerating a complete set of auxon types which are sufficient to accomplish all of the necessary tasks, the question arises whether it is possible to distribute their numbers so that the system grows uniformly. For example, can the network of interdependence

lead by itself to some sort of bottleneck, so that the system performs

much more poorly than expected from the performance of individual auxons?

Or can a small

number of slow and difficult processes create another type of bottleneck so that the growth rate is catastrophically limited? In order to answer these questions, we define a formalism which organizes the processing and manufacturing steps within an auxon colony. The formalism provides conditions on each auxon type, which taken together are sufficient to guarantee success of the whole colony and which set an upper limit on the overall time constant for growth. Also we can show that even if some auxon types violate these conditions in a benign way, the system’s operation is not too badly affected. These conditions can then be used confidently in the design phase in order to define success for each subsystem, and in order to define a measure for the merits of available technologies. The final products of the processing and manufacturing chain are infrastructure components and individual auxons. The rate at which each auxon type must be produced in order to maintain exponential growth is A&/T, where Mi represents the quantity of auxons of type i present per unit area, and r is the time constant. Similarly, new infrastructure is produced at a rate MI/T where MI is the amount of infrastructure per unit area. The amount of equipment necessary to make a given final product from scratch is proportional to the desired production rate. One can then state a superposition principle which says that the total equipment required to achieve this rate can be computed by summing the requirements for each individual product. This is represented by the equation

Each matrix element C’ij depends only on auxons of types i and j and specifies the quantity of auxon type i which is required to maintain a unit rate of production of auxon type j. The number Ci, has the dimensions of time, and is also equal to the amount of time it would take for a unit amount of auxons i to do their part in producing a unit amount of auxon j. Specifically, Cij is the inverse of the output rate which those auxons can sustain. (It is not the time delay between the input and output of the manufacturing chain.) The quantities C,’ are related, but apply to production of infrastructure, which here is considered as a single entity measured by MI. One may construct a picture by tracing each output stream backwards through all steps, starting with final assembly and proceeding to the raw material. Each step is represented as a node in the “stream diagram” (Figure 3), and is performed by a particular auxon type. The amount of material passing through a node, together with the productivity of the auxon type at the node, determines the quantity of auxons required at that node. This quantity will be strictly proportional to the final product amount as long as the time delay between each step and the final product is much less than the growth time constant, which for now we will assume to be the case. (Otherwise the coefficients C’ij must be multiplied by factors exp(&/r), where At, is the time delay for node i, and the following treatment would need to be extended.) The whole system’s stream diagram is the superposition of the individual diagrams for each output.

Exponential

Growth

71

auxon j I

7

raw dirt Figure 3. Schematic of a stream diagram indicating the flow of materials through the auxon system for the production of an auxon type j. The lines represent mate rial streams, the triangles processing auxons. Each triangle represents a particular process performed by a specific auxon type. In this example, only one auxon type (downward pointing triangle) has multiple outputs.

In the case of an auxon that produces several different output streams simultaneously, it is not automatically true that the superposition principle applies. A typical example of such an auxon is a material separation auxon. Suppose that a material separation step splits its input into equal amounts of A and B, and consider the manufacture of products 1 and 2. If product 1 requires A but not B, and product 2 requires B but not A, then the amount of material passing through the separation is given by the larger of the requirements for products 1 and 2, rather than the sum. Thus, a naive application of the superposition principle would lead to an overestimate of the number of material separation auxons, amounting to a conservative estimate of the growth rate. In this case, the coproduction of B with A is considered waste, similarly the coproduction of A with B. To the extent that A and B are used in their natural ratios, the waste is completely avoidable. This would be the characteristic of a well-designed system which has some flexibility in the choice of construction materials for many auxon types. For such a balanced system, we can obtain a more accurate estimate of the required auxon amount by resealing the Czj with the fraction of an auxon’s output that is used in the stream j. In doing so, the superposition principle is maintained while apportioning the requirements for the multiple output auxon between the consumptions of the separate products. 4.3. Limits

on the Growth

Time

Constant

For a given time constant r, equation (12) can be solved to give the necessary amounts of each auxon per unit area: -1 M,=jC(1-~)

C,‘IL!f~.

ij

3

(13)

As long as the resulting values for Mi are nonnegative, a system with the specified growth rate can exist, given a sufficiently large power input. Therefore, we consider the conditions under which Mi are guaranteed to be nonnegative. Such conditions follow from the Taylor series expansion

\

T/

7

7-L

(14) \ I

I<. S. LACKNER AND C. H. WENDT

72

As long as this series converges, the statement that C (1 - C/7)-l has no negative components. This in turn It happens that the series above converges if and only modulus less than 1. This is equivalent to the statement

has no negative components implies t,hat implies that none of the i& are negative. if all the characteristic roots of C/r have that the series converges if 7 > ror where A of

70 is the largest characteristic root of C. One can show that for an arbitrary square matrix dimension n, the modulus of the largest characteristic root X is bounded by the relations

I. Since in our case Aij = CiJ/7 2 0, these bounds in C, or if it is larger than

physical solution for the set of Mi. The column sum in particular has an intuitive total

auxon

mass necessary

imply that

if r is larger than

all row sums, then the series above converges

to maintain

interpretation.

a unit rate of output

all column

and therefore

This quantity,

of auxons

(15)

xi

sums

there is a Cij,

is the

of type j. Alternatively,

it can be interpreted as the amount of time it takes a unit amount of auxons, distributed among all nodes in the proportions required to produce auxon j, to make a unit amount of auxon j from scratch. If this time is less than some number 7~ for all j, then a physical solution for 111,exists whenever IT > rc. Note that if one changes the units in which auxon amounts are measured, the numerical values for Cij with i # j will change, giving a different value for 7~. In practice the units may be judiciously chosen to minimize rc, revealing the full allowed range of 7 for a given system. The minimum rc may be regarded as an estimate of 70 since rc 2 70. The case of unphysical Mi, when the series (equation (14)) diverges, occurs when the productivity of the system is insufficient to maintain the growth rate l/-r. This could occur as a result of a single, poorly designed process, appearing in the manufacturing chain for auxon type j so that xi Czj > r. By designing each manufacturing chain to avoid this situation, one guarantees a system which will grow at the desired rate if its energy requirements are met. The amount of solar power per unit area, S, used by the system can be found by adding the average power consumptions of each auxon type: S = C

WiMx.

up

(16)

Here Wi is the power consumption of a unit amount of auxons of type i. If S is fixed, then this equation together with equation (12) determines both r and Mt for a given system design. As already discussed, it is not obvious that a physical solution exists for equations (12) and (16). To answer this question, we may imagine that at first we consider only equation (12), and start with a very large value of ITso that all Mi > 0 (cf. equations (13),(14)). In this limit, the Mi values can also be made arbitrarily small, implying Ci W,M% < S. As r is decreased towards 70, but keeping 7 > 70, the Mi increase monotonically and therefore xi WiMi becomes larger and larger. For all systems with the exception of a set of measure zero, the energy requirement C, WiMt grows to infinity as r 4 70, and therefore it eventually meets and then exceeds S. This meeting point corresponds to a physical solution of equations (12) and (16). Note that if S is made very large, the system will exhibit a production bottleneck and further increases in S will only lead to marginal improvements in T, which will become closer and closer to the minimum value re. In the process, some Mi become very large and dominate both the system and its power needs. Note that a certain amount of power is necessary for the construction of the infrastructure, (17) Since this must be less than the total power S,

ExponentialGrowth

73

By comparing rr with 7-0which is estimated from rc, we may recognize several system types. For TO<< 71, the system is energy limited, has a time constant r M 1-1and is dominated by its infrastructure.

This is the most desirable situation, with nearly the maximum possible growth

rate. If ~0 >> 7-1, then the requirement 7 > 70 implies that the system grows very slowly, using most of its resources to construct auxons. This is the case of a “productivity bottleneck.” Between these extremes are systems limited by a combination of energy and productivity. It follows from equations (12) and (16) that the overall growth time constant is given by

This expression cannot be directly evaluated without knowing all Mj, which are not known until the full system is specified. Nevertheless, one may construct a limit on 7 if 7~ is defined so that

(20) for any possible Mi. The limit T 5 ~1 + rw then follows directly from equation (19). Note that Ci WiCik is the energy required to make a unit amount of auxon Ic, and this may be defined as the cost of this auxon. Then ctj WiCi,Cik is the cost of the equipment needed to maintain a unit rate of production for auxon k. Thus, if I-W is defined so that the cost of equipment needed to produce each auxon j with rate l/rw is less than the direct cost of auxon j, then cwicijcjk

< Tw c

(21)

Wjcjk.

j

i3

If the same holds for the infrastructure, then C

WiCi,Ci

< Tw C

fJ

WjC,‘.

(22)

j

Equation (12) guarantees that the vector Mj can be written as a linear combination with positive coefficients of the vectors Cjk, k = 1.. n and C;. Therefore, these relations imply C

WiCz,MJ I 7~ C

ij

(23)

WjMj,

3

or r 5 71 + 7sy. Thus, we have a limit on r which follows from consideration, one at a time, of the streams which produce each auxon type or infrastructure. One may also show that there is a physical system (Mi > 0) that satisfies this relation. It is reasonable to expect that some auxons will require in their manufacture some unusually expensive equipment, suggesting a much higher value of TW than would be computed for the remainder of the system. It turns out that if one leaves these special auxons out of the determination of 7~~ one can still derive an upper limit on T which is usually not adversely affected by the expensive equipment. Specifically, suppose that the criteria (21) and (22) are satisfied for all k except those in a small set X of expensive auxons. We note that while these criteria are independent of the units chosen for M, and Ml, they take a very simple form if auxon amounts are measured by the energy cost in their manufacture. This system of units is defined formally by the requirement c, WiCi, = 1 for all j. In these units, it is simply the sum cj which gives the cost of equipment needed to maintain a unit rate of production for auxon type k, and equations (21) and (22) reduce to cjk

c c c; < c,k

5 TW, rw.

k 4 X,

(24) (25)

K. S. LACKNER AND C. H. WENDT

74

One can calculate for each auxon lc and for the infrastructure the fraction of the production cost which is accounted for by the set X:

With Gx the largest of all Gf

and G,x , the growth time constant is bounded by the relations 1 n<7IrI+rW-.

l-Gx

(27)

The relations of (27) are our final bounds on the growth time constant T. The constant TW as defined in equations (24),(25) reflects the most expensive production chain outside the set X. The constant Gx reflects the size of the excluded set X. The most useful bound will result if the set X is judiciously chosen so that neither is 7-w kept large by a few uncharacteristic processes, nor is Gx close to unity. Note that we still assume that the time delay from input to output of the production process is small compared to the growth period and that auxon lifetimes are large compared to the growth period, and departures from these assumptions must be taken into account. In particular, given a desired growth period I-, the finite lifetime rj of auxon j may be allowed for without changing the analysis: replace the matrix elements and Cj with Cik and C’f which are scaled up by a factor 1 +r/rj, replace Wj by Wi which is decreased by this factor, and finally replace iVj with Mi which is increased by this factor. This change leaves invariant the condition xi WiCij = 1, so equations (24) and (25) still apply. The values of 71 and 7~ will then change because of the changes in and ci. Similarly, a time delay Atjk from auxon j to final production of auxon k is absorbed into the matrix element by multiplying with a factor exp(Atjk/r). To summarize, the growth time constant has two major ingredients. The first piece is 71, which accounts for the energy required to convert raw dirt into infrastructure. This energy includes heat used in the material separation process as well as operation of all other necessary equipment. To a first approximation, the energy requirements are dominated by the material separation, so that 7-1 is proportional to the mass of the infrastructure. The second piece is m, which accounts for production of auxons. Because TW expresses the amount of equipment required per amount of auxon produced, it reflects directly the productivity of that equipment. If productivity is higher, then fewer auxons are required and rw is lowered. For the general auxon systems being considered, TW is the analogue of l/P from the constant productivity model. Note that although TW is a ratio of energy investments, energy investments are usually dominated by material separation, so it may also be thought of approximately as a ratio of masses. The conversion factor between energy and mass is roughly 25 MJ/kg, as already discussed. Especially important is that because the definition of TW refers separately to the production process for each auxon type, the limit (equation (27)) can be applied already during the design phase, before the entire system is specified. For example, one may guarantee an overall growth rate of T < 6 months by designing components which satisfy r~ N 3 months, rw < 2 months and cjk

cjk

cjk

Gx < l/3. The w constraint is then translated into a limit on the amount of equipment needed to maintain production of each auxon type, according to equations (24) and (25). The number Gx specifies that if some auxons violate the TW constraint, they cannot be required in any large fraction to produce themselves, other auxons, or infrastructure. 4.4.

Construction

of the Track Grid

Although the design of an auxon system includes many different machines, it was already pointed out that the system’s mass is typically dominated by a fixed infrastructure. This includes the solar cell array and power grid as well as a network of tracks. The tracks provide both the equivalent of a factory floor and the backbone of a simple transportation scheme. In this section,

Exponential

Growth

75

we will consider, in outline form, the various steps by which more track is added to the growing colony. The object is to illustrate how to check that the growth of each subsystem obeys the rules for ~1 and 7~ which will guarantee that the overall system achieves a given growth time constant. In particular, for the track growth subprocess, we find that achieving the desired time constant of six months depends mainly on the energy and equipment needs of the basic material separation scheme, rather than higher-level operations. We assume that the tracks form an approximately square grid, with spacing of about one meter. With this arrangement there is no need for any auxons to move on the natural terrain, since all points in the system are accessible to machines which sit on the tracks.

Vehicles and loads are

expected to be light, up to 10 kg and 30 cm in size. This can be accommodated by a ceramic track which is a ribbed structure about 10 cm across, slightly elevated above the ground and supported at the grid intersection points. Based on the strength of typical ceramics [34] the track segments could weigh as little as 0.5 kg/ m, so that the track network would contribute only one kilogram per square meter to the infrastructure mass. Objects of large dimension, for example the track segments themselves, could be transported on such a network by a combination of two vehicles. The major steps in growth of the track grid are material separation, track fabrication, transport, and installation. In order to determine whether the sequence satisfies the constraints imposed by the desired growth rate, these steps are to be quantified in terms of the energy and equipment required. As suggested in the previous section, a goal of r < 6 months would be satisfied by 1-1 < 3 months, rw < 2 months and G x < l/3. From the constraint on TI, we find that the total energy investment in the 10 kg/m2 of infrastructure should be less than T~S, which works out to 23MJ/kg. The tracks are part of the infrastructure, so we will ask that they respect this allocation. The constraint TW < 2 months stipulates that the total amount of equipment required to maintain a unit output rate (1 unit per month) of each product should be less than 1 unit/month x TW = 2units. We will therefore add up the quantities of equipment at each step and see if the total is less than this amount. Strictly speaking these amounts should be measured as energy investments, but for now they will be approximated by using mass as the unit of measure. Then a unit output rate corresponds to 1 kg per month, and the equipment required for maintaining this output rate must be less than 2 kg. We will begin with the final step, installation, and work backwards. New tracks are installed at the boundaries of the existing track network. The process is straightforward as long as the neighboring terrain is free of large obstacles and exhibits only a moderate grade. In this case, a track laying auxon, operating from the existing track, first sinks a support block in a hole one meter away and then mounts a track segment between the new support and an old support. Once a new square is framed, other auxons will attach additional infrastructure components such as the solar panels. If, however, the neighboring square does contain obstacles or steep slopes, then it can be left unoccupied and growth proceeds in a different direction. In order to check for this possibility, a new square is first tested by extending a mechanical arm into the space, checking for clearance and measuring the elevation at the desired support block site. Note that in this way the procedure does not have to involve any grading or other preparation steps. Also, no sophisticated pattern recognition is necessary in order to deal with the terrain variations. Suppose that it takes ten minutes for a track laying crew to install the two segments which add a new square meter to the network. If five machines are involved, each consuming 20 W of power, then the energy investment is 0.06 MJ/m2, or equivalently 0.06 MJ per kilogram of track. This is clearly an insignificant fraction of the 23MJ/kg allocated in order to satisfy 71 < 3 months. In order to address TW, note that if these same five machines weigh 5 kg each, then the investment in equipment needed to maintain a unit output rate of one kilogram per month is 0.006 kg. This will be added to the requirements of other steps in growth of the track network, giving a total which must be less than the budget of 2 kg which follows from w = 2 months. The next step to consider is transportation of new track parts from the fabrication point to MCM 21/10-F

76

K. S. LACKNER AND C. H. WENDT

the installation point. The effort involved in this step depends directly on the mean transport distance, which for example we may take to be 100 meters. If vehicles move with a speed 0.5 m/set, this leads to a mean transit time of a few minutes. Suppose that a 2 kg transport vehicle carries five times it own weight and consumes ten watts of power. Then the energy investment in transportation is only O.O002MJ/kg, and the equipment investment for a unit output rate is 1.5 x 10m5kg. These numbers are insignificant in relation to the budgets for rw and 71, even if the mean transport distance is substantially larger than 100m. We may also apply these same assumptions to the transport of the entire mass of infrastructure and auxons being produced (10 kg/m2), and we then discover that the typical distance between vehicles is over 100 meters even if this mass needs to be transported several times. Thus, traffic jams would not be a problem. The transportation requirements could, however, become much more stringent in the case of a very large system. With growth proceeding only at the perimeter of such a system, transportation distances could easily amount to hundreds of kilometers. There would also be another problem, namely that the speed with which the perimeter progresses outward becomes unreasonably large. Both problems are avoided if the ratio of perimeter length to area covered is kept large. For example, such large systems might adopt a dendritic growth pattern, or they could simply be divided up into smaller subcolonies, with new subcolonies being started from complete sets of auxons made by older subcolonies (“spores”). To reduce transport distances even further, new equipment processing large amounts of material should be installed close to where it is needed for growth, namely at the periphery. As this equipment deteriorates with age, it is not replaced at the same location but at the new periphery. The interior region is then responsible mostly for energy collection, which in some form must be delivered to the perimeter. A potentially large subsystem is required for the fabrication of track segments. Ceramic tracks would be fired in kilns, and this process requires significant amounts of heat, time and insulation material. A linear kiln about eight meters long and some fifty centimeters wide could deliver 50 tracks every 2.5 hours while heating track segments for 20 hours. We estimate the energy consumption to be about 4 MJ per kilogram of track, which is a significant fraction of the budgeted 23MJ/kg. The equipment investment is dominated by the insulation required, which is about one ton for the same kiln. This translates into a need for about 0.14 kg of kiln for a unit output rate, using up almost ten percent of the TW budget. There is also a smaller amount of equipment required for forming the track segments before firing. We will not attempt to be explicit about the very first step in growth of the track system, namely preparation of raw material. This process begins with raw dirt and may include several energy-intensive chemical separation steps in order to produce a well-defined base material for the fabrication step. The remaining ~1 budget at this point is 19 MJ/kg, which is likely sufficient for preparation of ceramic, since complete reduction of dirt into the elements is not necessary. However, it is clearly possible that the bulk of the energy budget (71) is used here, and judging from the insulation mass required for the kilns mentioned above, so may be the bulk of the equipment budget (rw ). One may conclude several things from this example. Mechanical operations, such as transportation and installation, are very cheap in terms of both energy and equipment requirements. The equipment requirements are even low enough that lifetimes for this equipment could be shorter than P-, since a mean lifetime ‘TICfor auxon k is allowed for by increasing equipment requirements by the factor (1 + ~/rk). Instead, the large investments are in heat for separation and firing processes, and also in large amounts of mass invested in insulation material for those same processes. It is clearly very important to design carefully the use of furnaces and kilns, and to find efficient processes for the manufacture of insulation. These same features will also characterize the growth of other major subsystems-for example the manufacture of kilns. However, some systems will require making precise mechanical parts, a subject we did not have to discuss for the track system. The typical contributions of such processes to rr and 7~ are therefore left un-

ExponentialGrowth

77

known for the time being. It is expected that they are intermediate between efficient mechanical operations and expensive heat consuming processes. 5.

APPLICATIONS

The first useful applications of a large-scale production system should be chosen such as to make the transition from the minimal system, which does not perform any task besides growing to a specified size, as simple as possible. Consequently, we have restricted ourselves to applications that can be implemented through relatively minor modifications of auxons which are already present in the minimal system or through introduction of new units which can be built by the minimal system and which use commodities already provided. The first two examples, energy generation and desalination of water, deal with resources whose limited availability is of worldwide concern.

The third application, removal of excess carbon dioxide from the atmosphere, is an

example of a project which can only be accomplished by operating on a global scale. In each case, the required investment in land compares favorably to the expected benefit. 5.1. Solar Energy

Collection

Generation of power for human use is the most basic application of an auxon colony, since the tasks of energy collection, storage and transportation will already be performed by the minimal system. The modifications of the minimal system necessary to transform it into a very large power plant are a significant increase of the internal energy transport system, a means of delivering the energy to the outside, and an enlarged energy storage system. The optimal energy carrier is likely to differ from that most suitable for internal energy transport. One possibiiity is the delivery of electric power at high voltage. However, for a very large system a large fraction of the energy could alternatively be delivered as a chemical fuel which is shipped more easily over long distances. This fuel could be a hydrocarbon in order to accommodate existing consumption patterns. The production of hydrocarbons could be incorporated into the production system with relative ease, since the reduction of carbon monoxide with hydrogen, which is already a part of the processing scheme, can be modified to generate hydrocarbons, as for example in the Fischer-Tropsch reaction [33]. This energy output represents a time Solar cell arrays deliver approximately 30 MW/km2. average over daily and seasonal fluctuations and assumes a conversion efficiency of 10%. The number is somewhat lower for nondesert regions which are subject to significant cloud cover. Solar flux varies daily, seasonally and with the weather. If the energy is to be delivered as electricity, it is highly desirable to include an internal storage capability which can smooth out these fluctuations. Since this is already an important issue for the design of the minimal system, we expect that conversion from growth mode to an energy delivery mode would require only the enlargement of the existing energy storage subsystem. On the other hand, the requirements on internal storage would be greatly reduced if a large fraction of the capacity were used for production of fuels or for other applications which could proceed at a variable rate. An area as small as 10 km x 10 km already exceeds the output of a typical conventional or nuclear power plant. Such a small system could avoid the requirement of complete closure, using some man-made goods for its operation, and perhaps even be economically viable as is. A single system of size 1Oekm2, which is smaller than many deserts and amounts to only 0.2% of the Earth’s surface area, would generate 3 x 1Ol3 W. This exceeds the current world electric power generation [35] by a factor of 23. The total mass of such a system is approximately 1013 kg. Comparing this to the world’s annual production of iron ore (9.5 x lOi kg [36]), we may observe that true internal closure is quite advantageous for energy production on these unprecedented scales. Devoting 1% of the available land surface to energy production would provide energy at 5 times the total world energy consumption from all sources [35,37]. As a natural extension of

I<. S. LACKNER AND C. H. WEND-I

78

these ideas, one might some clay design an auxon colony for the ocean where the available surface area is much larger. 5.2. Fresh Water

Generation

and Distribution

In many regions of the world, human activities are severely limited by the available water. The oceans provide an unlimited supply but with the current price of energy, the desalination of seawater on a large scale is not economically feasible. To be useful in agriculture requires much cheaper energy. Based on current designs for desalination plants using reverse osmosis, the energy investment per cubic meter of fresh water is between 18 and 25MJ [38]. Thus, a solar cell array operating at 10% efficiency could generate enough water to cover an area of its own size with 10 to 15cm of fresh water per day. Assuming a relatively high irrigation rate of 100 cm per year [39], a solar cell array of approximately 2% of the field size could collect the energy necessary to desalinate seawater for its irrigation. An additional, although typically smaller energy cost is related to the transportation of the water. Compared to the minimal system, an auxon system generating fresh water would have in addition desalination units and an extensive system for transporting water. A major change over a minimal system is the need for a layout interspersed with human activities. Because of the advantages in local processing and energy collection, agricultural fields may actually be surrounded by narrow strips containing the water transportation system, solar cell arrays and access pathways. A twenty meter wide strip between fields one kilometer on a side should be sufficient for providing the energy and water necessary for irrigation. For a large irrigated desert area of lo6 km2, which is half the size of the U.S. cropland [36], the amount of water pumped from the sea is 30, 000m3/sec and is comparable to that of a major river [40]. For a system extending all the way to the ocean, the water take-up may, however, be spread out over a long coast line. The ready availability of fresh water and desalination at the point of use would also make it possible to prevent salt build-up in the fields, which is a common side-effect of irrigation, by reprocessing the run-off water through the same desalination process. The brine left over from the desalination is either transported back to the ocean, or alternatively it could be collected in parts of the auxon system not involved in water desalination as a ready source of raw materials like sodium and chlorine which for a purely land based system are much more difficult to obtain. 5.3. Removal

of Excess

Carbon

Dioxide

from the Atmosphere

If the increase of carbon dioxide and other greenhouse gases in the atmosphere were to cause a catastrophic warming of the global climate [41-431, only an extremely large processing system would be capable of removing greenhouse gases from the atmosphere at a rate sufficient to stop the warming trend. A dramatic worldwide increase in the rate of photosynthesis through enlarging forested areas, intensifying the growth of forests with the help of fertilizers, and growth of special crops has been suggested [44] as a possible means of extracting excess carbon dioxide from the air. These estimates indicate that halting the increase in atmospheric carbon dioxide would be possible through a large world wide concerted effort. First, however, one would have to reverse the trend towards deforestation [43,45]. Large processing systems provide a more direct approach-we estimate that an auxon colony covering an area of lo6 km2 is large enough to significantly change the carbon dioxide content of the atmosphere over the course of a few years. First we note that the energy required for the chemical reduction of 20% of the atmospheric carbon dioxide, which is approximately today’s excess amount [42], corresponds to the energy collected in five years by the solar cell arrays. An energetically more favorable approach to the extraction of carbon dioxide from the air binds the carbon dioxide in form of carbonates, thereby completely avoiding the energy investment in

Exponential

Growth

79

the reduction to carbon. In particular, iron, magnesium and calcium oxides can be transformed into the corresponding carbonates in strongly exothermic reactions. This alternative requires that processes be devised that can fix the carbonates in mineral form, preferably without going through energetically expensive purification steps. The resulting energy requirements could then be satisfied in significantly less than five years of collection time. The rate limiting step would likely be that of pumping carbon dioxide rich air through carbon dioxide extracting units. In order to pump 20% of the total atmosphere through the system, the system must process the atmosphere above it about 100 times. If the desired time for adjusting the atmospheric level of CO2 is ten years, then each day the system must process 2.7% of the atmosphere above it or a layer which is about 230m thick. These volumes have to be compared with the mixing depth of the lower atmosphere and the horizontal motion of the air, in order to verify that the same air is not immediately cycled back through the system. The effective diffusion constant for vertical mixing is about lo5 cm’/sec in the lower atmosphere [46] which implies that a vertical mixing of a layer 230 m thick requires about 1.5 hours, which is still small compared to the processing time of one day. At sixteen days, the characteristic mixing time equals the processing time at a layer thickness of about 3700m. The time for the horizontal motion to exchange the air over the processing area must not exceed this time scale. For a system 1000 km in diameter, this would imply a minimum wind velocity of 0.7m/sec which is easily exceeded in desert areas [47]. The rate limitation caused by atmospheric mixing times could also be avoided completely by dividing the system up into smaller pieces which are separated by large distances. There are 2.5 x 1015kg of carbon dioxide in the earth’s atmosphere. Thus, in extracting 20% of this amount and storing it in the form of carbonates, an auxon colony would generate a layer of “marble” approximately 50cm thick and lo6 km2 in area. It should also be noted that large scale energy production by auxons would greatly reduce the need for fossil fuels and thus would eliminate a major source of excess atmospheric carbon. Indeed, hydrocarbons synthesized by auxons would be generated from carbon dioxide extracted from air, and their use would therefore not contribute to the greenhouse effect.

6. CONCLUSION The limitations in nonrenewable resources and the need to control and limit deleterious side effects of human activities on the environment will present major challenges in the coming century. These problems are exacerbated by a growing world population and by an obligation to allow all peoples to enjoy a standard of living similar to that which is taken for granted in the industrialized countries. Energy and fresh water are two major resources which are severely limited and whose availability will greatly affect the well-being of mankind. At the same time, myriad human activities will continue to pollute the air and build up greenhouse gases in the atmosphere. The reduction of air pollution at its sources already taxes technology and politics to the limit, and stopping a potentially catastrophic runaway greenhouse effect could require operations on a scale dwarfing anything achieved in the past. Auxon technology would provide the means to operate on the global scale necessary to mitigate environmental disasters. At the same time, because of its complete reliance on renewable or virtually limitless resources, coupled with an enormous increase in productivity, auxon technology would allow the development of a sustainable economy large enough that all could benefit. The proof that auxon systems can be built will follow from practical demonstration of the separation of raw dirt, and from the step-by-step evaluation of productivity in manufacturing each auxon type and each infrastructure component. We have shown how to use this information, obtained incrementally in the design process, to measure the likely success of the overall endeavor. The study should begin with those parts which are expected to dominate the mass of the system. The energy efficiency of the material separation, for which we have outlined a promising approach, will be a key ingredient in the equation Additional issues which should be addressed concern the

K. S. LACKNER AND

80

C. H. WENDT

decision-making algorithms embodied in each auxon, for example what level of communication between auxons is generally necessary to ensure stable behavior patterns. Such issues may become more clear in the context of specific designs. Along the way to a full technology of auxon systems, one may imagine a stepping-stone of “recursive automation” which could already be competitive for some large scale projects. Recursive automation

would be characterized

by a layered structure,

with each layer consisting of many

small machines which produce yet more machines for the next layer. This arrangement can give rise to a large multiplication

of human and capital investment.

The goal of an autonomous self-

reproducing system is reached by adding more and more layers to the production scheme until finally closure is reached and productivity has reached its maximum. At this point, the size of potential projects would be limited only by the area available. Ultimately it is the large increase in productivity which makes it possible for an auxon system to rely on low-grade inputs like dirt and sunlight and therefore lead to an economy freed from the constraints of limited nonrenewable resources.

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