Fundamentals of renewayble energy processes

They used energy only as food, probably at a rate somewhatbelow the modern average of 2000 kilocalories per day, equivalent to 100 W.Later, with the discovery of fire and an improved diet

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09 10 11 12 13 14 15 16 5 4 3 2 1

Foreword to the Second Edition

The public’s widespread desire to become informed about energy has been,

in part, satisfied by excellent media coverage and by a plethora of goodbooks on the subject Most of these books are, quite naturally, journalisti-cally slanted and treat technology superficially Granted that of the variouscomponents of the problem—technology, economics, politics—technologyrepresents only a small fraction of the total, but it is the one fraction thatmust be tackled first

Those who need to understand the limitations of technical solutionsrequire a good scientific grasp of what is being proposed This book tries

to explain how each energy process discussed actually works A reasonabledegree of mathematics is used to unify and clarify the explanations Bydiscussing fundamentals more than the state of art, it is hoped to delay theobsolescence of this writing, especially in this moment of very fast evolu-tion of ideas Those who wanting to labor in this field may find this bookuseful in preparing themselves to comprehend more specialized articles onwhatever energy process that especially interests them

In spite of its fundamentalist approach, this book will eventuallybecome dated, not because fundamentals change but because different fun-damentals will be invoked This second edition discusses several scientificareas that only recently have been recruited to resolve energy problems.After more than two centuries of intense development, even verymature technologies such as heat engines (Chapter 2) can still find newand improved forms This is the case of the free-piston Stirling enginewhose high efficiency and very long mantenance-free life has made it now

a favorite for generating electricity in remote, unmanned locations, such

as in spacecraft and in planetary exploration This second edition expandsthe seven pages of the first edition dedicated to Stirling engines, and theseultramodern free-piston devices are included

Thermoelectrics (Chapter 5) has also progressed in recent years with abetter understanding of artificially created nano materials and superlatticesthat, in a way, get around the limitations of the Wiedemann–Franz–Lorenzlaw, allowing the synthesis of materials that have large electric conductivitybut small heat conductivity

xv

Fuel cells have matured substantially Those described in the firstedition, though adequately light and efficient, were short-lived and expen-sive Catalysis problems were responsible for these shortcomings Thesecond edition has a much expanded discussion of chemical kinetics anddescribes very recent work (late 2008) that completely avoids precious met-als as catalysts, while substantially outperforming these metals.

Hydrogen production, a fairly old technique, is now beginning to lean

on photolytic processes that were of only marginal interest when the firstedition was prepared

It is perhaps in biomass that the most dramatic evolution has occurred.Public enthusiasm for ethanol and biodiesel has propelled biomass from aminor energy source into one that can contribute markedly to the fueling

of our vehicles Biomass will be firmly entrenched in such a role if the nomical hydrolysis of cellulose can be achieved The second edition delvesdeeper into the mysteries of the required biochemistry

eco-Utility-size photovoltaic plants expanded in the last few years at a tained rhythm of over 40% per year They now face a moment of decision:

sus-to continue with efficient but expensive silicon devices or sus-to adopt cheap,though much less efficient, plastic cells It may all hinge on finding a way

to improve the life span of plastic cells The second edition discusses thechemistry and technology of these polymer cells

Finally, wind energy has established itself as a major player in energyproduction Wind farms are expanding at the same 40% per year rate

as photovoltaics, but having started from a much higher base are nowbeginning to make significant contributions to the energy mix When thefirst edition was prepared, wind energy played a minor role, and it wasnot entirely clear which type of turbine (horizontal or vertical axis) wouldwin out It is now clear that the horizontal axis (propeller-type) is thedominant solution The second edition treats the fundamentals of thesemachines (Betz limit, Rankine–Froude law, wake rotation, etc.), subjectsthat were omitted in the first edition

This book is based on class notes created in the teaching of damentals of Energy Processes at Stanford since 1976 As both the cost

Fun-of energy and our dependence on foreign suppliers have risen, so has theinterest in these lectures, reflecting the mood of the American people

Aldo Vieira da Rosa

<darosa@ee.stanford.edu>

Palo Alto, CAAugust 2008

Foreword to the First Edition

This book examines the fundamentals of some nontraditional energyprocesses Little effort is made to describe the “state of the art” of the tech-nologies involved because, owing to the rapidity with which these technolo-gies change, such description would soon become obsolete Nevertheless, theunderlying principles are immutable and are essential for the comprehen-sion of future developments An attempt is made to present clear physicalexplanations of the pertinent principles

The text will not prepare the student for detailed design of any cific device or system However, it is hoped that it will provide the basicinformation to permit the understanding of more specialized writings.The topics were not selected by their practicability or by their futurepromise Some topics are discussed solely because they represent good exer-cises in the application of physical principles, notwithstanding the obviousdifficulties in their implementation

spe-Whenever necessary, rigor is sacrificed in favor of clarity Although it isassumed that the reader has an adequate background in physics, chemistry,and mathematics (typical of a senior science or an engineering student),derivations tend to start from first principles to permit the identification

of basic mechanisms

Energy problems are only partially technical problems—to a largeextent economics and politics dominate the picture In a limited fashion,these considerations are included in the discussions presented here.The organization of the book is arbitrary and certainly not all-encompassing Processes that can be considered “traditional” are gener-ally ignored On the other hand, the list of “nontraditional” processesconsidered is necessarily limited

xvii

I wrote this book Without Aili, I could not

My thanks to Dr Edward Beardsworth who, incessantly scanning theliterature, alerted me to many new developments

My gratitude also to the hundreds of students who, since 1976, haveread my notes and corrected many typos and errors

xix

1.1 Units and Constants

Although many different units are employed in energy work, wheneverpossible we shall adopt the Syst`eme International (SI) This means joules and watts If we are talking about large energies, we’ll speak of MJ, GJ,

TJ, PJ, and EJ—that is, 106, 109, 1012, 1015, and 1018joules, respectively,(See Table 1.1)

One might wish for greater consistency in the choice of names and

symbols of the different prefixes adopted by the SI The symbols for multiplier prefixes are all in lowercase letters, and it would make sense if the multipliers were all in uppercase letters, which they are not All sym-

sub-bols are single letters, except the one for “deca” which has two letters(“da”) Perhaps that explains why deciliters are popular and decaliters areextremely rare Unlike the rest of the multipliers, “deca,” “hecta,” and

“kilo” start with lowercase letters The names of the prefixes are derivedmostly from Greek or Latin with some severe corruptions, but there arealso Danish words and one “Spanish” word—“pico”—which is not listed

in most Spanish dictionaries Some prefixes allude to the power of 1000 ofthe multiplier—“exa” (meaning six), for instance, refers to 10006: othersallude to the multiplier itself—“kilo” (meaning one thousand) indicates themultiplier directly

We cannot entirely resist tradition Most of the time we will express

pressures in pascals, but we will occasionally use atmospheres because most of the existing data are based on the latter Sometimes electronvolts are more convenient than joules Also, expressing energy in barrels of oil

or kWh may better convey the idea of cost On the whole, however, we

shall avoid quads, BTUs, calories, and other non-SI units The reason forthis choice is threefold: SI units are easier to use, they have been adopted

by most countries, and they are frequently better defined

Consider, for instance, the calorie, a unit preferred by chemists Doesone mean the international steam table calorie (4.18674 J)? or the meancalorie (4.19002 J)? or the thermochemical calorie (4.18400 J)? or the caloriemeasured at 15 C (4.18580 J)? or at 20 C (4.18190 J)?

Americans like to use the BTU, but, again, there are numerous BTUs:steam table, mean, thermochemical, at 39 F, at 60 F The ratio of the BTU

to the calorie of the same species is about 251.956, with some variations

in the sixth significant figure Remember that 1 BTU is roughly equal to

1 kJ, whereas 1 quad equals roughly 1 EJ The conversion factors between

1

Table 1.1 SI Prefixes and Symbols

1024 Y yotta corrupted Italian otto = eight, 10008

1021 Z zetta corrupted Italian sette = seven, 10007

1018 E exa corrupted Greek hexa = six, 10006

1015 P peta corrupted Greek penta = five, 10005

10−3 m milli from Latin mille = thousand

10−12 p pico from Spanish pico = little bit

10−15 f femto from Danish femten = fifteen

10−18 a atto from Danish atten = eighteen

10−21 z zepto adapted Latin septem = seven, 1000 −7

10−24 y yocto adapted Latin octo = eight, 1000 −8

Table 1.2 Fundamental Constants

Avogadro’s number N0 6.0221367 × 1026 per kmoleBoltzmann’s constant k 1.380658 × 10 −23 J K−1

Charge of the electron q 1.60217733 × 10 −19 C

Gravitational constant G 6.67259 × 10 −11 m3 −2kg−1

Planck’s constant h 6.6260755 × 10 −34 J s

Permeability of free space μ0 4π × 10 −7 H/m

Permittivity of free space 0 8.854187817 × 10 −12 F/m

Speed of light c 2.99792458 × 108 m s−1

Stefan–Boltzmann constant σ 5.67051 × 10 −8 W K−4m−2

the different energy and power units are listed in Table 1.3 Some of thefundamental constants used in this book are listed in Table 1.2

1.2 Energy and Utility

In northern California, in a region where forests are abundant, one cord ofwood sold in 2008 for about $150 Although one cord is a stack of 4 by 4 by

Table 1.3 Conversion Coefficients

Energy

British thermal unit (Int Steam Table) joule 1055.04

British thermal unit (mean) joule 1055.87

British thermal unit (thermochemical) joule 1054.35

British thermal unit (39 F) joule 1059.67

British thermal unit (60 F) joule 1054.68

Calorie (International Steam Table) joule 4.18674

Horsepower (550 foot LBF/sec) watt 745.70

Other

LBF stands for pounds (force)

8 ft (128 cubic feet), the actual volume of wood is only 90 cubic feet—therest is empty space between the logs Thus, one cord contains 2.5 m3 ofwood, or about 2200 kg The heat of combustion of wood varies between

14 and 19 MJ/kg If one assumes a mean of 16 MJ per kilogram of woodburned, one cord delivers 35 GJ Therefore, the cost of energy from woodwas $4.3/GJ in northern California

Still in 2008, the price of gasoline was about $3 per gallon ($1.2 perkg) although if fluctuated wildly Since the heat of combustion of gasoline

is 49 MJ/kg, gasoline energy used to cost $24/GJ, over five times the costfrom burning wood.

In California, the domestic consumer of electricity paid $0.12 per kWh,

or $33/GJ

From these statistics, it is clear that when we buy energy, we are willing

to pay a premium for energy that is in a more convenient form—that is,

for energy that has a higher utility Utility is, of course, relative To stoke

a fireplace in a living room, wood has higher utility than gasoline and, todrive a car, gasoline has higher utility than electricity, at least for the timebeing For small vehicles, liquid fuels have higher utility than gaseous ones.For fixed installations, the opposite is true

The relative cost of energy is not determined by utility alone Onebarrel contains 159 liters, or 127-kg of oil With a heat of combustion of

47 MJ/kg, this corresponds to 6 GJ of energy In mid-1990, at a price of

$12/barrel or $2/GJ, oil cost less than wood (then at $3.2/GJ) standing oil being, in general, more useful However, oil prices are highlyunstable depending on global political circumstances The 2008 price ofoil (that peaked well above $100/barrel, or $17/GJ) is now, as one mightexpect, substantially higher than that of wood and is one of the drivingforces toward the greening of energy sources Perhaps more importantly,there is the dangerous dependence of developed nations on oil from coun-tries whose interests clashes with those of the West

notwith-Government regulations tend to depress prices below their freemarket value During the Carter era, natural gas was sold in interstatecommerce at the regulated price of $1.75 per 1000 cubic feet This amount

of gas yields 1 GJ when burned Thus, natural gas was cheaper than oil orwood

1.3 Conservation of Energy

Energy can be utilized but not consumed.†It is a law of nature that energy

is conserved We degrade or randomize energy, just as we randomize mineralresources when we process ores into metal and then discard the product as

we do, for example, with used aluminum cans All energy we use goes intoheat and is eventually radiated out into space

The consumable is not energy; it is the fact that energy has not yetbeen randomized The degree of randomization of energy is measured bythe entropy of the energy This is discussed in some detail in Chapter 2

†It is convenient to distinguishconsumption from utilization Consumption implies

destruction—when oil is consumed, it disappears, being transformed mainly into carbon dioxide and water, yielding heat On the other hand, energy is never consumed; it is utilized but entirely conserved (only the entropy is increased).

Short-wave radiation Solar

radiation

173, 000

TW

Figure 1.1 Planetary energy balance

1.4 Planetary Energy Balance

The relative stability of Earth’s temperature suggests a near balancebetween planetary input and output of energy The input is almost entirelysolar radiation, which amounts to 173,000 TW (173,000 × 1012 W).Besides solar energy, there is a contribution from tides (3 TW) andfrom heat sources inside the planet, mostly radioactivity (32 TW).Some 52,000 TW (30% of the incoming radiation) is reflected back

to the interplanetary space: it is the albedo of Earth All the remaining

energy is degraded to heat and reemitted as long-wave infrared radiation.Figure 1.1 shows the different processes that take place in the planetaryenergy balance mechanism

The recurrence of ice ages shows that the equilibrium between ing and outgoing energy is oscillatory It is feared that the observed secularincrease in atmospheric CO2might lead to a general heating of the planet,resulting in a partial melting of the Antarctic glaciers and consequent flood-ing of sea-level cities The growth in CO2concentration is the result of the

incom-combustion of vast amounts of fossil † fuels and the destruction of forests

in which carbon had been locked

1.5 The Energy Utilization Rate

The energy utilization rate throughout the ages can only be estimated in

a rough manner In early times, humans were totally nontechnological, noteven using fire They used energy only as food, probably at a rate somewhatbelow the modern average of 2000 kilocalories per day, equivalent to 100 W.Later, with the discovery of fire and an improved diet involving cookedfoods, the energy utilization rate may have risen to some 300 W/capita

†Fuels derived from recent biomass, such as ethanol from sugarcane, do not increase

the amount of carbon dioxide in the atmosphere; such fuels only recycle this gas.

In the primitive agricultural Mesopotamia, around 4000 B.C., energyderived from animals was used for several purposes, especially for trans-portation and for pumping water in irrigation projects Solar energy wasemployed for drying cereals and building materials such as bricks Percapita energy utilization may have been as high as 800 W.

Harnessing wind, water, and fire dates from early times Sailboats havebeen used since at least 3000B.C.and windmills were described by Hero ofAlexandria aroundA.D.100 ByA.D.300, windmills were used in Persia andlater spread to China and Europe Hero’s toy steam engines were appar-ently built and operated Vitruvius, the Roman architect whose book, firstpublished in Hero’s time, is still on sale today, discusses waterwheels used

to pump water and grind cereals In spite of available technology, ancientslimited themselves to human or animal power Lionel Casson (1981), a pro-fessor of ancient history at New York University, argues that this was due

to cultural rather than economic constraints Only at the beginning of theMiddle Ages did the use of other energy sources become “fashionable.” Thesecond millennium exploded with windmills and waterwheels

The widespread adoption of advanced agriculture, the use of fireplaces

to heat homes, the burning of ceramics and bricks, and the use of wind andwater led to an estimated energy utilization rate in Europe of 2000 wattsper capita in A.D. 1200 Since the popular acceptance of such activities,energy utilization has increased rapidly Figure 1.2 illustrates (a wild esti-mate) the number of kilowatts utilized per capita as a function of the date

If we believe these data, we can conclude that the annual rate of increase

of the per capita energy utilization rate behaved as indicated in Figure 1.3.Although the precision of these results is doubtful, it is probable that thegeneral trend is correct: for most of our history, the growth of the energyutilization rate was steady and quite modest With the start of the indus-trial revolution at the beginning of the nineteenth century, this growthaccelerated dramatically and has now reached a worrisome level

The increase of the worldwide per capita energy utilization rate wasdriven by the low cost of oil before 1973 when it was substantially lowerthan now.†Perez Alfonso, the Venezuelan minister of oil in 1946, was among

those who recognized that this would lead to future difficulties He wasinstrumental in creating the Organization of the Petroleum Exporting Coun-tries (OPEC) in 1954, not as a cartel to squeeze out higher profits but to

“reduce the predatory oil consumption to guarantee humanity enough time

to develop an economy based on renewable energy sources.” Alfonso alsoforesaw the ecological benefits stemming from a more rational use of oil

†In 1973, before the OPEC crisis, petroleum was sold at between $2 and $3 per barrel.

The price increased abruptly, traumatizing the economy In 2000 dollars, the pre-1973 petroleum cost about $10/bbl (owing to a 3.8-fold currency devaluation), a price that prevailed again in 1999 However, in 2006, the cost had risen to over $70/bbl In 2008, the price of an oil barrel peaked at more than $140.

1990 A.D.

12 10 8 6 4 2 0

Mesopotamia (primitive agriculture)

West Africa

Europe

(advanced agriculture)

USA

Europe (industrial)

10 6 B.C.

10 5 B.C.

8000 B.C.

1000 A.D.

1600 A.D.

Figure 1.2 A very rough plot of the historical increase in the per capita energyutilization rate

1250

Year Energy utilization rate Rate of increase (% per year)

The recent effort of less developed countries (LDCs) to catch up withdeveloped ones has been an important factor in the increase in energydemand Figure 1.4 shows the uneven distribution of energy utilization ratethroughout the world; 72% of the world population uses less than 2 kW/capita, whereas 6% of the population uses more than 7 kW/ capita

an energy crisis, while the opposite trend occurs during a financial crisis.Further industrialization of developed countries may not necessarilytranslate into an increase in the per capita energy utilization rate—thetrend toward higher efficiency in energy use may have a compensatingeffect However, in the United States, the present decline in energy uti-lization† is due mainly to a change in the nature of industrial production.

Energy-intensive primary industries (such as steel production) are phasingout owing to foreign competition, while sophisticated secondary industries(such as electronics and genetic engineering) are growing

Technological innovation has led to more efficient energy use ples include better insulation in houses and better mileage in cars Alter-nate energy sources have somewhat alleviated the demand for fossil fuels.Bioethanol is replacing some gasoline It is possible that the development

Exam-of fusion reactors will, one day, bring back the times Exam-of abundant energy.Introduction of a more efficient device does not immediately result

in energy economy because it takes considerable time for a new device to

be widely accepted The reaction time of the economy tends to be long.Consider the privately owned fleet of cars A sudden rise in gasoline price

†American industry used less energy in 1982 than in 1973.

has little effect on travel, but it increases the demand for fuel efficiency.However, car owners don’t rush to buy new vehicles while their old ones arestill usable Thus, the overall fuel consumption will only drop many yearslater, after a significant fraction of the fleet has been updated.

Large investments in obsolete technologies substantially delay theintroduction of more efficient systems A feeling for the time constantsinvolved can be obtained from study of the market penetration function,discussed in Section 1.7

1.6 The Population Explosion

In the previous section we discussed the per capita energy utilization rate.

Clearly, the total rate of energy utilization is proportional to the planetarypopulation, which has been growing at an accelerated rate:†

The most serious problem that confronts humankind is the rapidgrowth in population The planet has a little more than 6 billion inhab-itants, and the growth rate these last few decades has been around 1.4%per year Almost all projections predict a population of about 7 billion bythe year 2010 even if, right now, everyone were to agree on a limit of twochildren per family Under present-day actuarial conditions, the populationwould eventually stabilize at around 11 billion by the year 2050 Populationgrowth alone could account for a 1.4% annual increase in energy demand Infact, the recent growth rate of energy use exceeded the population growthrate The worldwide rate of energy use was 9 TW in 1980 and 15.2 TW

in 2008, a yearly growth of 1.9% The Energy Information Administration(EIA) has used this constant 1.9% per year growth rate to estimate anenergy usage rate of slightly over 22 TW in 2030 Clearly, supplying thismuch energy will not be an easy task

The constant population increase has its Malthusian side About 10%

of the world’s land area is used to raise crops—it is arable land (See

“Farming and Agricultural Technology: Agricultural Economics: Land,Output, and Yields,” Britannica Online.) Roughly 15 million km2 or

1.5 × 109 hectares are dedicated to agriculture Up to the beginning ofthe twentieth century, on average, each hectare was able to support 5

people (Smil), thus limiting the population to 7.4 billion people More

arable land can be found, but probably not enough to sustain 11 billionpeople What limits agricultural productivity is nitrogen, one kilogram ofwhich is (roughly) needed to produce one kilogram of protein Although it

is the major constituent of air, it is, in its elemental form, unavailable to

†On October 12 1999, a 3.2-kg baby was born in Bosnia Kofi Annan, general secretary

of the United Nations, was on hand and displayed the new Bosnian citizen to the TV cameras because, somewhat arbitrarily, the baby was designated as the 6 billionth inhab- itant of this planet.

plants and must either be fixed by appropriate microorganisms or be added

1.7 The Market Penetration Function

The enormous body of literature accumulated throughout the centuriesmakes it impossible for even the most assiduous readers to have studied thewritings of all scientists and philosophers of the past Hence, modern writershave built up a large roster of “often cited, rarely read” authors whose ideasare frequently mentioned even when only nebulously understood This is,for instance, the case of Thomas Robert Malthus We all have an idea that

he had something to say about population growth In 1846, Pierre Fran¸coisVerhulst put this population growth idea in the plausible mathematical

form known now as the Verhulst equation This equation is an excellent

starting point to understand the problem of technological substitution, that

is, the question of how a more advanced technology will replace a morecumbersome older one

Adapting the Verhulst equation to this problem, we have

1

f

df

wheref is the fraction of the market supplied by the new technology (hence

constrained to 0 ≤ f ≤ 1), t is time, and a is a constant In words, the Verhulst equation states that the fractional rate of change in f (represented

by f1df dt ) must be proportional to that fraction of the market, (1 −f), not yet taken over by the new technology This makes intuitive sense The Verhulst

equation is a nonlinear differential equation whose solution is

b being an integration constant.

One can solve for f:

f = exp(at + b)

and plotf as a function of t.

Figure 1.5 The Verhulst function.

Figure 1.5 shows the result for arbitrarily chosen a = 1, b = −5.

Equation 1.3, illustrated in Figure 1.5, is an example of the logistics

function One defines a takeover time interval,t s, as the length of time

for the function to go fromf in, when its value is 0.1 (at time t in) to, f out,when its value is 0.9 (at timet out)

In 1970, two General Electric scientists, J C Fisher and R H Pry,applied the Verhulst equation to the problem of new technology mar-ket penetration Each case considered involved only two competingtechnologies—an old one having a fraction,f1, of the market and a modernone having a fraction,f2, of the market They used the Verhulst equation

in the form of Equation 1.2 (ln1−f f as a function of t), not in the form of

Equation 1.3, because Equation 1.2 yields a straight-line plot and is thusmuch easier to handle analytically and to extrapolate

Figure 1.6 shows the rates of penetration of the oxygen steel cess into the market previously dominated by open-hearth and Besse-mer technologies It can be seen that the penetration rate was fastest

pro-in Japan (a takeover time pro-interval of 8.3 years) In Germany and theUnited States, the takeover time was 11.8 years, while in the former SovietUnion it was 14.0 years The results show surprisingly small dispersion—the observed points fall all very nearly on the straight-line regression

Figure 1.6 The penetration of the oxygen–steel technology in the steel duction market.

pro-As an example, in the case of Germany and the United States, theregression is

wheret is the time in years, expressed as 19xx.

The graph shows that the takeover times were different in differentcountries, and, as Fisher and Pry (1970) point out, the technique appears

to work even in a centralized command economy In addition, it turnsout, as expected, that even in the same country, substitutions of differenttypes of products have different penetration rates Some substitutions occurrapidly, whereas others take a long time Fisher and Pry list a number ofsubstitutions with takeover times ranging from just over 8 years for thecase of detergents replacing natural soaps in Japan to 58 years for the case

of synthetic rubber replacing natural rubber in the United States

The plots of Figure 1.6 depend on two parameters,a and b Parameter

a is related to the takeover time and determines the slope of the graph If

instead of using the actual time,t, as the ordinate, one uses a dimensionless

normalized variable,τ, then all plots will have the same slope and can be

integrated into a single graph See Figure 1.7 showing how 17 differentsubstitutions fit a single straight line with surprising accuracy

Figure 1.7 Fisher–Pry plot for 17 different substitutions.

The normalized time variable, τ ≡ t −t h

t s /2, is introduced as follows.Fort = t h , f = 0.5 and

ln f

1− f =at h+b = 0 ∴ b = −at h (1.5)Fort = t in , f = 0.1 and

ln f

1− f =at in+b = −2.2. (1.6)Owing to the symmetry of the logistics curve, the takeover time,

t s, is twice the interval, t h − t in, hence, subtracting Equation 1.6 from

A remarkable and useful property of the market penetration function

is its insensitivity to many factors that profoundly affect the overall ket Thus, although variations in political or geopolitical circumstances cansubstantially affect market volume, they frequently have a minimal effect

mar-on the fractimar-onal market share, probably because they influence ously all competing technologies

simultane-In the case of a simple substitution of an old technology by a newer one,

it is clear that if the newer is progressively increasing its market share, thenthe older must be progressively abandoning the market If ln( f2

then ln( f1

1−f1) = −at − b, because the total market is f = 1 Since

the behavior of the technology being replaced is, in this case, simply amirror image of the advancing technology, graphs are shown only for thelatter This is not true for the case of multivariate competition when severaldifferent technologies compete to fill market needs

Fisher and Pry’s results are entirely empirical Cesare Marchetti(1976), working at the International Institute for Applied System Analysis(IIASA) in Austria, still using an empirical approach, extended the Fisher–Pry idea in two meaningful ways One was to make it possible to considercases in which the market for a given product was supplied by more thantwo technologies To this effect, Marchetti introduced the rule of first in,first out The other was the application of the Fisher–Pry procedure to the

case of various energy sources This allows the prognostication of the

mar-ket share of individual energy-supplying technologies In his 1976 paper,Marchetti presents the available data on the market share of different fuelsused in the United States during the 1850 to 1975 period From these data,

he obtained the different coefficients of Equation 1.2 for each fuel Wood

was already abandoning the market with a characteristic abandonment

time of 60 years being replaced by coal (takeover time of 66 years) This

substitution was driven by the much greater usefulness of coal in drivingthe locomotives of the expanding railroad system Increasing use of oil andnatural gas, beginning at the turn of the twentieth century, caused theturnaround of the coal share (but not of the total coal use) Coal startedabandoning the market with a characteristic time of 99 years Initially, oil’stakeover time was 52 years, but then rose to 135 years and showed signs ofturning around in the early 1970s All this is illustrated in Figure 1.8

As an exercise in prognostication, Marchetti, using the trend lines of

Figure 1.8 derived only from data before 1935, calculated the behavior of

the oil market share, employing the formula,

f oil= 1− (f coal+f gas). (1.9)The results are displayed in Figure 1.9 They are very accurate, a factthat led Marchetti to comment: “we were able to predict the fractionalmarket share of oil in the U.S.A up to 1970 with a precision of betterthan one percent.” Alas, “It’s tough to make predictions, especially aboutthe future.”† If we extend Marchetti’s graph to 2008, we find that the nice

regularity of the behavior of the coal and gas shares, on which the nostication is based, breaks down badly in modern times (see Figure 1.10)

prog-To understand why, we must refer to the work of V´aclav Peterka (1977),which, dropping the empiricism of previous authors, tried to put the anal-ysis of market penetration on a firmer theoretical basis Peterka carefullydefines the conditions under which the empirical models hold

†Attributed to numerous sources, from Mark Twain to Niels Bohr to Yogi Berra.

to the OPEC crisis in the early 1970s.

Peterka argues that any scientific forecasting must be based oncertain a priori assumptions In the case being discussed here, a funda-

mental assumption is that there be no external infusion of capital once the

technology has established itself This is self evident—if during the tration period, a substantial increase in capital becomes available, this willalter the rate of penetration, even though it may not increase the profitabil-ity of the enterprise It would be of great value if it were possible to estimatehow much it would cost to accelerate the penetration by a given amount.Unfortunately, this is not yet possible The assumption above implies thatwhen a technology starts to penetrate the market, it must already be welldeveloped and its degree of maturity determines the eventual penetrationrate Thus, “the magnitude of the original external investment actuallydetermines the initial conditions for the model” (Peterka, 1977).

pene-The market penetration rules discussed in this subsection provide apowerful tool for planning, but must be used with caution and with closeattention to possible violations of implicit assumptions

1.8 Planetary Energy Resources

In Section 1.5, we pointed out that the rate of per capita energy utilizationrose rapidly in the last century This, combined with the fast increase inpopulation mentioned in Section 1.6, leads one to the inescapable con-clusion that we are facing a serious challenge if we hope to maintain

synthesis

Photo-Limnic

Oceanic

Direct conversion Wind

Direct combustion Pyrolysis Fermentation Digestion Gravitational Salination Evaporation Waves Currents Thermal Difference Osmotic

GEOTHERMAL

GRAVITATIONAL

FOSSIL

Coal Oil Gas Shale

MINERAL Fission

Fusion

Figure 1.10 The energy resources of Earth

these trends in the foreseeable future To investigate what can be done

to resolve this difficulty, we must first inquire what energy resources areavailable (Section 1.8) and next (Section 1.9) how we are using the resources

it is possible to take advantage of the stability of the ground temperature

a few meters below the surface The ground can thus be used as a source

of heat in the winter and of cold in the summer

Gravitational energy—that is, energy from tides (see Chapter 16)—has been used in France Tides can only be harnessed in certain specificlocalities of which there is a limited number in the world Gravitationalenergy is also important in all hydroelectric plants

Of the renewable resources, solar energy is by far the most abundant

A small part of it has been absorbed by plants and, over the eons, has beenstored as coal, oil, and gas

Estimates of reserves, fossil or nuclear, are extremely uncertain andare sure to be greatly underestimated because of incomplete prospect-ing Table 1.4 gives us only a very rough idea of our fossil fuel reserves,and Table 1.5 shows an even more uncertain estimate of reserves of fissile

Table 1.4 Known Fossil Fuel ReservesMethane clathrate >100, 000 EJ (1998)

from precise They probably represent a lower limit, because people who

estimate these numbers tend to be conservative, as testified by the secular

increase in proved reserves: proved reserves of dry natural gas, 2200 EJ

in 1976, rose to 6200 EJ in January 2007, notwithstanding the substantialconsumption of gas in the intervening years A similar situation exists withrespect of proved oil reserves: 7280 EJ in 2002 and 7900 EJ in 2007 Foroil and gas, the table lists the sum of proved reserves, reserve growth, andundiscovered reserves

Proved reserves are fuels that have been discovered but not yet

pro-duced Proved reserves for oil and gas are reported periodically in the Oil and Gas Journal.

Reserve growth represents the increase in the reserves of existing fieldsowing to further development of these fields and to the introduction ofbetter technology for their extraction

Undiscovered reserves represent the best possible guess of the tude of plausible new discoveries

magni-Reserve growth and undiscovered reserves are estimated by the U.S

Geological Survey (USGS) For example, in 2002 the Oil and Gas nal reported proved reserves of oil of 7280 EJ, and the USGS estimated

Jour-a growth of 4380 EJ Jour-and undiscovered oil reserves Jour-amounting to 5630 EJ,adding up to the total of 18, 900 EJ listed in the table For coal, the tableshows only proved reserves The total reserves for this fuel are thus sub-stantially larger than listed

One number that is particularly uncertain is that referring to rated methane William P Dillon, a geologist of the USGS, testified in the

hyd-U.S House of Representatives in 1998 that “the amount of methanecontained in the world’s gas hydrate accumulations is enormous, but esti-mates of the amounts are speculative and range over three orders-of-magnitude from about 100, 000 to 270, 000, 000 trillion cubic feet [100, 000

to 270, 000, 000 EJ] of gas.” We, being ultraconservative, listed the lowerfigure

Methane Clathrate

Clathra is the Latin word for bar or cage.

Atoms in a number of molecules group themselves in such a fashionthat a cavity (or cage) is left in the center The most famous of thesearrangements is the “buckyball,” a molecule consisting of 60 carbonatoms arranged as a hollow sphere capable of engulfing a number ofsubstances Buckyballs, discovered in the early 1980s, are not aloneamong “hollow” molecules Under appropriate circumstances, waterwill freeze, forming a cage consisting sometimes of 20 water molecules,but more commonly of 46 water molecules The configuration is unsta-ble (it decays into a common ice crystal) unless certain gases becometrapped in the central cage of the large molecule Gases commonlytrapped are methane, ethane, propane, isobutane, n-butane, nitrogen,carbon dioxide, and hydrogen sulfide

The ice crystal consisting of 46 water molecules is able to trap up

to 8 “guest” gas molecules (a water-to-gas ratio of 5.75:1) In naturaldeposits, methane is by far the most abundant and the one of greatestinterest to the energy field Usually, up to 96% of the cages are fully

occupied These solid hydrates are called clathrates.

The density of the clathrate is about 900 kg/m3 This means thatthe methane is highly compressed See Problem 1.28 Notwithstandingits low density, water ice clathrate does not float up from the bottom

of the ocean because it is trapped beneath the ocean sediment

Clathrates form at high pressure and low temperature under seaand are stable at sufficient depth The methane is the result of anaer-obic digestion of organic matter that continuously rains down on theocean floor See Chapter 13

There is no mature technology for the recovery of methane fromclathrates Proposed processes all involve destabilizing the clathrateand include:

1 Raising the temperature of the deposits

2 Depressurizing the deposits

3 Injecting methanol or other clathrate inhibitors

The third process may be environmentally undesirable

(Continues)

There are dangers associated with methane clathrate extraction.The most obvious ones are the triggering of seafloor landslides andthe accidental release of large volumes of methane into the Earth’satmosphere where it has a powerful greenhouse effect Some scientistsattribute the extinction that marked the end of the Permian era (300

to 250 megayears ago) to an enormous bubbling up of methane ThePermian-Triassic extinction (P-Tr extinction) was the worst catastro-phe to hit the biosphere of Earth—96% of all ocean species disap-peared, together with 70% of land species

of the energy used to generate electricity in the United States came fromrenewable sources Of these, 83% came from hydroelectrics Thus, only 2%

of the total came from the remaining renewables

Other

Oil 40.0%

Gas 22.5%

Nuclear 6.5%

Hydro 7.0%

Coal 23.3%

Figure 1.11 Energy sources in the world

Oil 39.4%

Gas 23.3%

Coal 22.2

Nuclear 8.2

Renewable 6.1%

USA, 2008

Figure 1.12 Energy sources in the United States

Data from EIA

Electric energy

in the United States, 2008

Renewable sources of electric energy, United States, 2008

Solar 0.2%

Geothermal 4.1%

Hydroelectric 75.3%

Wind 12.8% Biomass 7.5% Renewable

8.3%

Oil 1.4%

Coal

49.9%

Gas 20.3%

Nuclear 19.9%

Figure 1.13 Sources of electric energy in the United States

Disappointingly so far, the contribution of solar and wind energy hasbeen very small But since about 1964, this has begun to change signifi-cantly.†

For all sources of energy, the cost of the plant is proportional to theinstalled capacity, while the revenue is proportional to the energy gener-

ated The plant utilization factor is the ratio of the energy produced

†For fairly up-to-date statistics on the production of renewable energy, consult

<http://www.earth-policy.org/Indicators/index.htm>.

to that which would be produced if the plant operated uninterruptedly atfull capacity (Table 1.6) Observe the extremely high utilization factor ofnuclear plants Wind generators operate with a rather small plant factor(≈ 30%) as a result of the great variability of wind velocity Although

specific data for solar plants are not available, they also suffer from a lowutilization factor owing to the day/night cycle and the vagaries of meteo-rological conditions

It is of interest to know which are the main users of energy in theUnited States (See Figure 1.14) American residences account for nearly20% of all energy used Most of it is employed for ambient heating, an area

in which considerable economy can be realized, especially through betterhome design and use of geothermal sources

Waste heat from electric power plants can be used for residential and

commercial water and space heating and constitutes a form of district

heating The other form uses dedicated centrally located boilers, not a

cogeneration scheme Some decades ago, when steam plants had an averageefficiency of 30%, a whopping 70% of the fuel energy was either thrownaway or was piped as hot water to consumers The latter option increasedthe overall system efficiency to more than 50% Currently, steam plants

Table 1.6 Electric Energy Use, United States 2005

Used Capacity Utilization

Residential 19.2%

Commercial 13.4%

Industrial 41.2%

AVR

Figure 1.14 The different users of energy in the United States

Table 1.7 Relative Merit of Different Light Sources

† The actual efficiency of commercially available white LEDs is

sub-stantially less than the listed 22%; it is about 13% However, laboratory

prototypes have demonstrated the efficiency shown in the table.

have, by themselves, efficiencies that exceed 50%,† and district heating can

boost the overall efficiency to some 90% This comes at some considerableinitial cost, so the economic benefits are only realizable in the long term.District heating requires the location of power plants in densely populatedareas; consequently, it is nonadvisable in the case of nuclear plants andlarge fossil-fueled installations However, fuel cell plants (see Chapter 9),being noiseless and pollution free, can be placed in a downtown area.Although the largest district heating system in the world is the oneoperated by Con Edison Steam Operations, active since 1882, a subsidiary

of Consolidated Edison of New York, the technology is, in relative terms,much more popular in Europe

It is probably in the transportation sector (25% of the total energyuse) that modern technology can have the most crucial impact We arestill enamored of heavy, overpowered cars that realize less than half the fuel

efficiency possible with current technology Thus, a cultural change would

be desirable The transition to more rational personal transportation hasstarted with the introduction of hybrid cars, soon to be followed by plug-

in hybrids and perhaps by both electric and fuel cell cars, both of whichpromise to increase automobile efficiency while reducing pollution

Another area in which efficiency can be improved by nearly one order

of magnitude is illumination, an important use in both residential and

com-mercial energy use About 10% of the energy used by residences go towardillumination We are in a period of transition from the extremely inefficientincandescent bulbs to the compact fluorescent lamp to the super-efficientwhite light-emitting diodes (LEDs)

In Table 1.7, “Merit” is defined as the product of the efficiency(expressed in percent) by the lifetime in hours According to this arbitrarycriterion, LEDs have the potential of being 4000 “better” than incandescentbulbs Of course, a number of technical problems as well as the high cost

of the LEDs must be addressed before they become the standard source oflight

†The following is a quote from General Electric: “GE’s H System—one of the world’s

most advanced combined cycle system and the first capable of breaking the 60 percent efficiency barrier—integrates the gas turbine, steam turbine and heat recovery steam generator into a seamless system.”

American industry’s relative use of energy may decrease even in theface of an expansion of this sector because of the progressive shift of empha-sis from an energy-intensive industry, such as iron and steel, to more sophis-ticated activities that have a low-energy demand per dollar produced.

1.10 The Ecology Question

We have shown that there is an almost unavoidable trend toward increasingenergy utilization We have also pointed out that at present the energy used

is at least 85% of fossil origin Finally, we have observed that the fossil fuelreserves seem ample to satisfy our needs for a good fraction of the nextmillennium So, what is the problem?

Most of the easily accessible sources of oil and gas have already beentapped What is left is getting progressively more expensive to extract.Thus, one part of the problem is economical Another is political—most ofthe fuel used by developed nations is imported (using the large Americanreserves is unpopular, and politicians hesitate to approve such exploration).This creates an undesirable vulnerability The major problem, however, isecological Fossil fuels are still the most inexpensive and most convenient

of all energy resources, but their use pollutes the environment, and weare quickly approaching a situation in which we can no longer dismiss theproblem or postpone the solution

By far, the most undesirable gas emitted is carbon dioxide whose gressively increasing concentration in the atmosphere (from 270 ppm in thelate 1800s to some 365 ppm at present) constitutes a worrisome problem

pro-It is sad to hear influential people (among them, some scientists) dismissthis problem as inconsequential, especially in view of the growing signs

of a possible runaway ecological catastrophe For instance, in the last fewdecades, the thickness of the north polar ice has decreased by 40% and inthe first year of the current millennium, a summertime hole appeared in thepolar ice Since increased concentrations of CO2 can lead to global warm-ing, some people have proposed increasing the emission of SO2 to stabilizethe temperature because of the cooling effect of this gas Even ignoring thevegetation-killing acid rain that would result, this proposal is equivalent tobalancing a listing boat by piling stones on the other side

Public indifference to the CO2 problem may partially be due to thefocus on planetary temperature rise Although the growth in CO2 concen-tration is very easily demonstrated, the conclusion that the temperaturewill rise, though plausible, is not easy to prove There are mechanisms by

which an increase of greenhouse gases would actually result in a cooling of

Earth For instance, increasing greenhouse gases would result in enhancedevaporation of the tropical oceans The resulting moisture, after migratingtoward the poles, would fall as snow, thereby augmenting the albedo of theplanet and thus reducing the amount of heat absorbed from the sun

The Kyoto Treaty aims at curbing excessive carbon dioxide emissions.

It is worth noting that China, the world’s major CO2 emitter,† is exempt

from the treaty’s restrictions, as are numerous other countries, such asIndia and Brazil The United States and Australia have never ratified thetreaty and are, therefore, also exempt There are many contributors to

CO2 pollution, but by far the largest single culprit is the coal-fired powerplant, which emits 30% of the amount of carbon dioxide dumped into theatmosphere.††The problem will not go away unless the coal-to-electricicity

situation is corrected

Some scientists and engineers who are less concerned with politicalcorrectness are investigating techniques to reduce (or at least, to stabilize)the concentration of atmospheric carbon dioxide This can, in principle, beaccomplished by reducing emissions or by disposing of carbon dioxide insuch a way as to avoid its release into the air Emissions can be reduced bydiminishing overall energy consumption (an utopian solution), by employ-ing alternative energy sources, by increasing the efficiency of energy use,and by switching to fuels that yield more energy per unit amount of carbonemitted It is known that 1 kmole of methane, CH4, when burned yieldingliquid water and carbon dioxide, releases 889.6 MJ and emits 1 kilomole

of carbon: it generates heat at a rate of 889.6 MJ per kilomole of carbon.n-heptane, C7H16, which can represent gasoline, releases 4820 MJ of heatper kilomole burned and emits 7 kilomoles of CO2—a rate of 688.6 MJper kilomole of carbon Clearly, the larger the number of carbon atoms inthe hydrocarbon molecule, the lower the ratio of the heat of combustion

to the amount of carbon dioxide emitted because the ratio of hydrogen tocarbon decreases This is one reason for preferring methane to oil and oil

to coal

Renewable forms of energy are attractive but, at least for the present,they are too expensive to seriously compete with fossil fuels Hence, meth-ods for reducing carbon dioxide emission are under intense investigation.All these methods have two stages: carbon dioxide capture and carbon diox-ide disposal or sequestration The capture stage is described, superficially,

in Chapter 10 In lieu of sequestration, the captured and purified gas gan

be sold to, for instance, the carbonated drink industry But this can onlytake care of a minute fraction of the total CO2 involved

In order to select a technique, for carbon dioxide disposal, it is tant to inquire where nature stores the existing carbon Table 1.8 showsthe estimated amount of carbon stored in different places

impor-Methods to dispose of CO2could include the following

†In mid-2007, China surpassed the United States as the major carbon dioxide emitter.

††It is somewhat surprising that a 1 GW coal-fired plant can emit 100 times more

radioactive isotopes (because of the radioactive traces in common coal) than a nuclear plant of the same power.

Table 1.8 Stored Carbon on Earth

Fossil fuels 10× 1015kgOrganic matter 2.4 × 1015kgAtmosphere 0.825 × 1015kg

Ziock et al (2000) propose the use of magnesium silicates to sequestercarbon dioxide at the point where fossil fuels are burned Enormousdeposits of magnesium oxide-rich silicates exist in the form of olivines andserpentines

For serpentine, the net reaction involved is

Mg3Si2O5(OH)4+ 3CO2→ 3MgCO3+ 2SiO2+ 2H2O

Notice that the end products are materials that already exist naturally

in great abundance Substantial additional research is needed to improvethe proposed disposal system and to make it economical

1.10.3 Subterranean

CO2 can be sequestered underground as the oil industry has been doing(for secondary oil recovery) for more than 50 years The volume of theexhaust gases of a combustion engine is too large to be economically storedaway It is necessary to separate out the carbon dioxide, a task that is noteasy to accomplish One solution is proposed by Clean Energy Systems,Inc of Sacramento, California The suggested equipment extracts oxygenfrom air (a well-developed process) and mixes this gas with the fuel Com-bustion produces steam and CO2 at high temperature and pressure anddrives several turbines at progressively lower temperatures The water inthe final exhaust is condensed and recycled leaving the carbon dioxide to

be pumped, at 200 atmospheres, into an injection well At present, no bines exist capable of operating at the high temperature (over 3000 C) ofthe combustion products See Anderson et al (1998)

tur-1.10.4 Undersea

The Norwegian government imposes a stiff carbon dioxide emission tax thathas made it economical to install disposal systems that pump the gas deepinto the ocean It appears that liquid carbon dioxide can be injected intothe seas at great depth and that it will stay there for a long time Again,more work is required to determine the feasibility of the scheme

Carbon in the Atmosphere

How much carbon is there in the atmosphere?

The surface area of Earth is 510× 1012m2, while the scale height

of the atmosphere is around 8800 m (see the section on Boltzmann’slaw in Chapter 2) Consequently, the volume of air (all of it compressed

to 1 atmosphere pressure) is 510× 1012× 8800 = 4.5 × 1018 m3.Present-day atmospheric CO2concentration is 13.5 × 10 −6kmol/

m3 The atmosphere contains 13.5 × 10 −6 × 4.5 × 1018 = 61× 1012

kmol of CO2and, therefore, 61× 1012kmol of carbon Since the atomicmass of carbon is 12 daltons, the mass of carbon in the atmosphere is

0.73 × 1015kg Compare with the 0.825 × 1015 kg of the table

A simpler way to achieve about the same result is to considerthat the atmospheric pressure at sea level is 1 kg/cm2 or 104 kg/m2.Consequently, the total mass of the atmosphere is 510× 1012× 104=

510× 1016 Of this, 360× 10 −6is carbon dioxide and 12/44 is carbon.

The carbon content of the atmosphere is 510× 1016× 365 × 10 −6 ×

12/44 = 0.51 × 1015 kg, a result comparable with the previous one

1.11 Nuclear Energy

Chemical fuels, such as oil or methane, release energy when the atoms intheir molecules are rearranged into lower energy configurations The ener-gies involved are those of molecular binding and are of the order of tens of

MJ/kg When the components of an atom are arranged into lower energy

configurations, then the energy released is orders of magnitude larger dreds of TJ/kg) because of the much larger intra-atomic binding energies.The internal structure of atoms can be changed in different ways:

(hun-1 An atomic nucleus can be bombarded with a neutron, absorbing it

A different atom emerges

2 An atom can spontaneously change by emitting either electrons(beta-rays) or helium nuclei (alpha-rays) Such radioactive decayreleases energy, which can be harvested as, for instance, it is done

in Radioisotope Thermal Generators (RTGs) (See Chapter 5).

3 Atoms with large atomic numbers can be made to break up into

smaller atoms with the release of energy This is called nuclear

fission and requires that the atomic number,Z, be larger than 26.

4 Atoms with low atomic numbers can be assembled into a heavier

one, releasing energy This is called nuclear fusion and requires

that the final product have an atomic number smaller than 26.†

Currently, only two techniques are used to produce energy fromnuclear sources: the RTG mentioned above and nuclear fission reactors.††

But, nuclear energy has developed a bad reputation, especially after theChernobyl accident in 1986 Nevertheless, it is a source of substantialamounts of energy in many countries According to the Energy Informa-tion Administration, EIA, since 1998, the number of nuclear plants in theUnited States has remained unaltered at 104 Nevertheless, there has been

a 2% per year secular increase in the generation of nuclear electricity owingmostly to an improvement of the plant utilization factor from 78.2% in 1998

to over 94% in 2007 It appears that after 2008, a number of new reactorsmay be purchased.In 2007, the United States led the world in installedcapacity—104 GW—followed by France (63 GW) and Japan (47.6 GW).The utilization factor of nuclear plants that year was excellent In theUnited States, it was over 94%, in France, 77.5%, and in Japan, 68.9%

Of the total electricity generated, nuclear plants in the United States(2008) contributed a relatively modest 19.9%, while in France, heavilyreliant on this form of energy, the contribution was 76.1% In Japan, itwas 34.6% In 2000, Germany decided to phase out its 19 nuclear powerplants Each one was assigned a 32-year life after which they would bedeactivated Many plants have already operated more than half of theirallotted lifetime

The cost of nuclear electricity is high, about double that from fossilfuel In the United States (1996), it was 7 cents/kWh, whereas that of astate of the art natural gas plant was 3 cents/kWh (Sweet, William #1).Advanced reactor designs may bring these costs down considerably whileensuring greater safety (Sweet, William #2) This promised reduced costcombined with the ecological advantage of no greenhouse gas emission—agrowing concern—may lead to a renewed popularity for nuclear generators.The major objection to fission-type reactors is not so much the danger

of the operation of the power plants (the Chernobyl accident was perfectlyavoidable), but rather the problem of disposing of large amounts of long-lived radioactive by-products If the need for such disposal can be avoided,then there is good reason to reconsider fission generators as an impor-tant contributor to the energy supply system, especially if they are notrestricted to the use of the rare236U fuel the way present-day reactors are

†All are transmutations, the age-old dream of medieval alchemists.

††Cannons preceded by centuries the invention of heat engines Nuclear bombs were

used before nuclear reactors—fusion has for decades been used in thermonuclear bombs, but its use in reactors still seems far into the future.

Specifications of new-generation nuclear fission reactors might include (notnecessarily in order of priority), the following items:

1 Safety of operation (including resistance to terrorist attacks)

2 Affordability

3 Reliability

4 Absence of weaponizable subproducts

5 Absence of long-lived waste products

6 Ability to transmute long-lived radioactive waste products from oldreactors into short-lived radioactive products

The U.S Department of Energy is funding research (2004) in nologies that might realize most of these specifications One of these is the

tech-heavy-metal fast breeder reactor technology It appears that this type

of reactor may be able not only to produce waste with relatively shorthalf-lives (100 years contrasted with 100,000 years of the current waste),but in addition may be able to use current-type waste as fuel Further-more, because heavy-metal reactors operate at high temperatures (yet atlow pressures), the thermolytic production of hydrogen (see Chapter 10) foruse in fuel cell-driven automobiles looms as a good possibility For furtherreading on this topic see Loewen (2004)

The waste disposal problem is absent in fusion devices Unfortunately,

it has been impossible to demonstrate a working prototype of a fusionmachine, even after several decades of concerted research

To do even a superficial analysis of the technical aspects of nuclearreactions, we need to know the masses of the atoms involved (see Table 1.9).Most of the mass values are from Richard B Firestone Those marked with

a ♣ are from Audi and Wapstra (1993), and the one marked with a · is

from a different source It can be seen that the precision of the numbers

is very large This is necessary because, in calculating the energy released

in a nuclear reaction, one uses the small difference between large numbers,which is, of course, extremely sensitive to uncertainties in the latter.The listed values for the masses of the nucleons (the proton and thealpha in the table) are nearly the values of the masses of the correspondingatoms minus the mass the electron(s) On the other hand, there is a largedifference between the the mass of a nucleon and the sum of the masses ofthe component protons and neutrons Indeed, for the case of the alpha, thesum of the two protons and the two neutrons (4.03188278 daltons) exceedsthe mass of the alpha (4.001506175 daltons) by 0.030376606 daltons—

about 28 MeV of mass This is, of course, the large nuclear binding

energy necessary to overcome the great electrostatic repulsion between the

protons

1.11.1 Fission

There are at least four fissile elements of practical importance: 233U,

235U, 239Pu, and 241Pu Of these, only 235U is found in nature in usable

Table 1.9 Masses of Some Particles Important to Nuclear Energy

† The dalton is not yet the official name for the atomic mass unit.

Table 1.10 Uranium Isotopes

Isotope Abundance Lifetime

238U 99.283 4.5 × 109

235U 0.711 7.1 × 108

234U 0.005 2.5 × 105

quantities; 233U, 239Pu, and 241Pu must be created by transmutation of

“fertile” materials, respectively,232Th,238U and240Pu The240Pu elementmust itself be created artificially from239Pu

Uranium isotopes cover the range from 227 to 240 daltons, but naturaluranium contains only a small percentage of the fissile material:

It is estimated that the Western world has reserves of uranium oxide(U3O8) amounting to some 6× 109 kg, but only 34× 106 kg are fissile,corresponding to an available energy of 2600 EJ Compare this with the40,000 EJ of available coal energy

Nuclear fission reaction (with a corresponding release of energy) occurswhen a fissile material interacts with neutrons Consider235U:

(life-fission; that is, under the proper circumstances, 23592U absorbs a neutron,and the resulting atom splits into smaller nuclei simultaneously releasing,

on average, 2.5 neutrons and about 3× 10 −11 joules of energy:

235

92U + 10n→ 2.5 1

0n + fission products + 3× 10 −11 J. (1.11)Per kilogram of 23592U, the energy released is

3× 10 −11 J

atom× 6 × 1026 atoms

kmol

235kmolkg = 77 TJ/kg.

However, the situation is somewhat more complicated than suggested

by the equation above because more energy and additional neutrons areproduced by the radioactive decay of the fission products These additional

neutrons are called delayed neutrons Compare this with chemical

reac-tions that involve energies of the order of a few tens of MJ/kg

When Otto Hahn, demonstrated uranium fission in 1939, it becameimmediately obvious that a sustained “chain” reaction might be achiev-able—all that was needed was to cause one of the emitted neutrons to split

a new uranium atom Using natural uranium, this proves difficult because ofthe small percentage of the fissile235U The emitted neutrons have a muchgreater probability of being absorbed by the abundant238U—the reactionsimple dies out The solution is to “enrich” the uranium by increasing thepercentage of235U This is a complicated and expensive process because onecannot use chemistry to separate the two isotopes since they are chemicallyidentical Any separation method must take advantage of the minute massdifference of the two isotopes If the enrichment is carried out far enough,you can build a nuclear bomb Reactors in the United States use uraniumtypically enriched to 3.7%; this is insufficient to sustain a chain reaction

An additional trick must be used

Neutrons resulting from a235U fission are high-energy particles (some

1 MeV), and their absorption cross section is about the same for both nium isotopes However, slow thermal neutrons (say at 0.05 eV) happen to

ura-be absorura-bed much more readily by the fissile uranium than by the morestable isotope Thus, some of the emitted neutrons have to be slowed down

by making them move through a low atomic mass substance called a

mod-erator Graphite or water will do If water is used as a coolant and

heat-extraction medium, then it contributes to the moderation process.†

Fast neutrons may be absorbed by impurities in the fuel or in themoderator Of course, 23892U is a major “impurity” in the fuel; it absorbs

†No enrichment is needed if the moderator is heavy water (D2O) as used in the

CANDU reactor This is the CANadian Deuterium Uranium, pressurized heavy water

reactor that uses natural (unenriched) uranium and heavy water as both moderator and coolant.

some of the fast neutrons To reduce neutron losses, it is necessary to placethe fuel into a number of long rods embedded in a mass of moderator Thisconfiguration allows most of the fast neutrons to escape the fuel region andreach the moderator where they are slowed and may eventually reenter one

of the fuel rods They now interact with the235U perpetuating the reaction

It is essential that exactly one of the released neutrons is, on average,used to trigger a new fission If more than one, the reaction will growexponentially; if less, it will die out Control systems are used to adjustthis number to precisely one

When all is said and done, the only useful output of a fission reactor

is heat, which has to be removed by a coolant and transferred to a turbine.Most American reactors use liquid water for this purpose This limits thetemperature to about 300 C, leading to low thermal efficiency Even then,pressurization is required to keep the water in the liquid phase (hence the

label pressurized water reactor) Remember that the vapor pressure of

water at 300 C is 85 atmospheres Any rupture can cause loss of coolantand can lead to a meltdown Reactors of the class operating in the UnitedStates have a number of disadvantages that may be absent in more moderndesigns:

1 Scarcity of fuel because only the rare isotope,23592U, is burned

2 Production of dangerously radioactive “ashes”

3 Safety concerns

4 Production of weaponizable materials such as plutonium

The scarcity of fuel problems can be circumvented by using one of theother fissile materials such as plutonium,23994Pu, and uranium,23392U Theseelements are not found in nature but can be obtained by the transmutationthat occurs in any type of reactor

By using plutonium, all uranium can be made to yield energy: 320,000

EJ become available Even larger amounts of energy could be derived fromthorium One type of reactor that can greatly improve the efficiency of fuel

use (by some two orders of magnitude) is the heavy-metal fast breeder

reactor Its initial fuel load must contain enough enriched uranium or

plutonium not to need moderators; it must operate with fast neutrons so

as to transmute235U into plutonium (Equation 1.12) at a rate larger thanthat at which fissile fuel is used—it breeds more fuel than it uses (as long

as the abundant supply of fertile uranium lasts) Subsequent fuel loads

(during the 60-year life expectancy of the machine) can contain waste fuel,

natural uranium, or even depleted uranium Because fast neutrons are

needed and moderators must actually be avoided the coolant must have

a large atomic mass; otherwise, it would itself act as a moderator.† In

addition, the coolant must be liquid (have a low melting point) and musthave a high boiling point so that high temperatures can be achieved atlow pressures It is also desirable that it be relatively inert chemically This

is one of the disadvantages of using sodium as a coolant, since it reactsexplosively when it comes in contact with water One material that fulfillsthese requirements is a lead–bismuth alloy, Pb-Bi that, notwithstanding itselevated boiling point of 1679 C, melts at slightly over 100 C

Eutectics

From my online dictionary:

Relating to or denoting a mixture of substances (in fixed tions) that melts and solidifies at a single temperature that is lower than the melting points of the separate constituents or of any other mixture of them.

propor-Wood’s metal, 50% bismuth, 25% lead, 12.5% tin, and 12.5% mium, melts at 73 C It is used to, among other things, fashion tea-spoons that melt when dipped into a hot cup of brew, startling theunweary

cad-Lead melts at 327.5 C, and bismuth at 271.5 C The alloy proposedfor the heavy-metal reactor melts at≈ 100 C.

Pb-Bi reactors operate at 800 C ensuring good efficiency In addition,they may contribute to the solution of the difficult nuclear waste problem

In current nuclear plants, 235U is consumed until the amount left

in the fuel rods becomes insufficient to sustain the chain reaction cally below 1% Remember that American reactors use enriched uranium

(typi-in which the percentage of the fissile variety is, say, some 3.7%.) It is thennecessary to replace the fuel rods, the spent ones being immersed in a boricacid pool where they cool down for a number of months until the short-life radioactive materials have decayed sufficiently Then they are classi-fied as waste, in spite of consisting mostly of 238U and some plutonium

and other transuranic elements (actinides, such as neptunium, amercium,

and curium) Discarded fuel rods also contain some medium lifetime fissionproducts dominated by 90Sr and 137Cs The short-lived fission products

†A fast neutron will lose more energy when colliding with a light nucleus, which

recoils a lot, than with a heavy one.

have, by now, mostly died out So the waste consists mostly of fuel thatcan be used by the heavy-metal reactors and do not have to be stored awayfor thousands of years as dictated by the current U.S policy.

Heavy-metal fast breeders probably can be designed for passive safety,

so that their safety does not depend on control systems or human decisions.Loss of coolant must interrupt the chain reaction automatically

1.11.2 Fusion

Fusion reactors may overcome many of the objections to fission The tion that is, by far, the easiest to ignite is†

To estimate the reaction energy released, one calculates the amount

of mass lost The mass of 3T ion is 5.008271 × 10 −27 − 9.10939 ∗ 10 −31=

5.007360 × 10 −27 kg as given by the table at the beginning of this

subsec-tion Notice that we subtracted the mass of the electron from the mass ofthe tritium atom The mass of the deuterium ion is 3.344497 × 10 −27 −

9.10939 × 10 −31= 3.344497 × 10 −27 kg, so that the mass of the left side of

the equation is 8.350946×10 −27kg On the right-hand side of the equation,

the sum of the masses of the alpha particle (the helium ion) and the neutron

is 8.319590 × 10 −27 kg, a deficit of 3.135569 × 10 −29 kg When multiplied

byc2, this yields an energy of 2.818 × 10 −12 joules per deuterium/tritium

pair The reaction yields 337 TJ per kg of tritium/deuterium alloy or 562

TJ per kg of tritium

The energy released by the reaction is carried by both the alphas andthe neutrons The conversion of the neutron energy to usable forms has anefficiency of only some 40% because the particles are uncharged and heatmanagement and mechanical heat engines are involved On the other hand,the alphas can be directly converted to electricity at a much higher effi-ciency (≈ 90%) See Rostoker, Monkhorst, and Binderbauer (1997); Moir

and Barr (1973); Momota et al (1992); Yoshikawa, Noma, and Yamamoto(1992); and Bloch and Jeffries (1950) In addition, the heavy neutron fluxcreates serious radioactivity and material destruction problems Conse-quently, it is important to know how the released energy is divided betweenthe alphas and the neutrons This can be done by assuming that themomenta are equally divided between the two types of particles:

†The larger the atomic number, Z, the greater the difficulty of causing a reaction

owing to the large electrostatic repulsion between nuclei.