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Commerce · II.06 · MMXXVI · daylight

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TEN CONCEPT BRIEFS · Chapter II.06 — Thermodynamics and Economic Flow

One page each. A reader who reads only these ten pages has the chapter.

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BRIEF 1 — Exergy: the quantity that is actually consumed

The idea. Energy is conserved. Exergy is not. Exergy is the part of an energy flow that could, in principle, be turned into work — and it is what a household means when it says the fuel was used up.

The first law says a joule into a building equals a joule out of it. That is true and it is why an energy audit of a boiler tells you almost nothing. The second law says that in every real process some of the capacity to do work is destroyed, permanently, and never comes back. That destroyed capacity is the cost. Everything else is bookkeeping.

For a flow of heat Q at absolute temperature T, measured against a reference environment at T₀:

        B  =  Q · (1 − T₀/T)

Worked example. One kilojoule of heat, twice. As a flame at eighteen hundred degrees against a ten-degree environment, the Carnot factor is 0.8634 — it is almost entirely work waiting to happen. As room warmth at twenty-one degrees against a freezing outside, the factor is 0.0714. Same joule, same first law, and about fourteen times less capacity to do anything at all.

Why it matters. Every commercial decision about energy is really a decision about exergy, and the standard reporting does not contain the word. A firm that adds one column — delivered temperature — can compute it from data it already has.

You already know this because you have never confused a full bath with a kettle, even though the bath holds far more heat. You have always known that grade matters. Exergy is that knowledge with a unit attached.

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BRIEF 2 — The dead state, and why it must be declared

The idea. Exergy is not a property of a thing. It is a property of a thing relative to an environment, and you have to say which environment.

The reference is called the dead state: the temperature, pressure and chemical composition at which the system can do no further work. Change the dead state and every exergy figure in the report moves. Heat at ninety degrees carries a factor of 0.2203 against a ten-degree environment; against a warmer reference it carries less.

This is the single most common way an exergy account goes wrong, and it does not go wrong through error. It goes wrong through drift — somebody updates the ambient assumption between two reporting periods and a series becomes incomparable without anybody lying.

Worked example. A site declares 283.15 K — ten degrees, its annual mean — as its dead state and fixes it in writing. A stack at three hundred degrees then carries a Carnot factor of 0.5060 and a ninety-degree delivery carries 0.2203, every quarter, for as long as the account runs. If someone proposes changing the reference, the change is a restatement and is disclosed as one.

Why it matters. An exergy account used in a contract or a bonus becomes negotiable at exactly this point. Fixing the dead state in writing, and auditing it annually, is the cheapest control in this chapter.

You already know this because you already insist that a like-for-like comparison states its baseline. The dead state is the baseline of a physical measurement, and it deserves the same signature.

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BRIEF 3 — Second-law efficiency

The idea. First-law efficiency asks what fraction of the energy reached the task. Second-law efficiency asks what fraction of the work capacity reached it. The two numbers can differ by more than an order of magnitude, and only one of them tells you what to buy.

Worked example — the boiler and the heat pump.

The ratio is 4.03 times. The two appliances do the identical job — a room at twenty-one degrees — and the first law cannot distinguish them in any way that matters, because it is answering a different question.

Why it matters. This single comparison is the origin of the entire modern heating debate, and it was settled arithmetically in 1975 by a group of physicists who wrote the definition down. Everything since has been deployment.

You already know this because you would not run a diesel generator to charge a phone, even though the energy arithmetic works. You already price the mismatch between a very high-grade source and a very small task. This is that instinct, made into a ratio.

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BRIEF 4 — Gouy–Stodola: lost work has a formula

The idea. The work destroyed in any real process is exactly the dead-state temperature multiplied by the entropy that the process generated.

        W_lost  =  T₀ · S_gen

This is the bridge between the abstract statement entropy increases and a number on an invoice. Entropy generation is not a metaphor in this chapter; it is a quantity with units of kilojoules per kelvin, and multiplying it by the ambient temperature converts it directly into money-bearing work.

Worked example. The condensing boiler above. Chemical exergy in, 831.65 kJ/mol. Exergy delivered to the room, 51.55 kJ/mol. Therefore:

  exergy destroyed        780.10 kJ/mol       = 93.80 % of the input
  entropy generated       780.10 / 273.15     = 2.856 kJ/(mol·K)

Nothing left the system. Energy was conserved to the last joule. And 93.80 percent of what that fuel could have done was annihilated in the flame, before the heat ever reached the wall.

Why it matters. An energy audit always balances, which is why it never surprises anybody. An exergy audit never balances, and the gap is the finding. The gap is where the improvement lives.

You already know this because you have watched a hot meal go cold on a counter and understood that nothing was lost and everything was lost. The heat went into the room. The dinner is still ruined.

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BRIEF 5 — Georgescu-Roegen's entropy argument

The idea. The economic process is not a circular flow. It is a one-way passage of low-entropy inputs into high-entropy waste, and that direction cannot be reversed by any amount of clever arrangement.

Nicholas Georgescu-Roegen set this out in The Entropy Law and the Economic Process in 1971. The target was the textbook diagram in which firms and households exchange goods and factors forever in a closed loop with no arrow of time. Against that, he insisted on three claims, all of which stand:

1. Production transforms; it does not create. What it consumes is availability, not matter or energy.
2. The transformation is irreversible. Some of the availability is gone, and no price signal recovers it.
3. Therefore the economy has a physical direction, and a model with no such direction is missing its most important variable.

Worked example. The world put about 478.5 EJ of primary energy through itself in 2005. Every joule of it still exists. Essentially none of its work capacity does. That is the argument, stated as an accounting fact rather than as a warning.

Why it matters. Before 1971 there was no economics that treated the physical direction of production as a first-class variable. After it, there was a discipline. Everything in this chapter's arithmetic is a descendant.

You already know this because you have never expected the smoke to go back into the log.

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BRIEF 6 — The fourth law, and why physicists rejected it

The idea. Georgescu-Roegen went one step further than his own argument supported, and the step was false. Knowing exactly where it failed is what makes the rest of the programme usable.

The claim. In the late 1970s he proposed a fourth law of thermodynamics: in a closed system, available matter degrades irrevocably into unavailable matter, so complete recycling is impossible even with unlimited energy.

Why it does not hold. Bianciardi, Tiezzi and Ulgiati showed in 1993 that with a sufficient energy flux through an open system, complete material recycling is thermodynamically possible. The Earth is emphatically open: the solar flux is 173,000 TW against human primary power of 19.0 TW, a ratio of 9,105 times, computed in Chapter I.01. Robert Ayres — the fourth law's most sympathetic reader — reached the same conclusion in 1999: it is false as a law and valuable as an economics.

What survives, and it is a great deal. The practical claim is untouched. Recovering dispersed matter is brutally expensive; it simply is not a law of physics that makes it so. The difference matters commercially, because a cost curve moves when technology moves and a law does not.

Why it matters. A programme that leans on a false law gets dismissed wholesale by the people best equipped to use it. Marking the failure precisely is what allows the rest to be taken seriously.

You already know this because you have seen a good argument lose a room on one overstated sentence, and you have wished somebody had cut it.

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BRIEF 7 — Useful work as a factor of production

The idea. Put the right physical quantity into the production function and it behaves very differently from the wrong one.

The wrong one is energy. Energy is about six percent of the cost of a developed economy, and at its cost share it can account for almost none of growth. The right one is useful work — exergy actually converted into shaft power, motive power, process heat at its working grade, and light.

        U  =  E · f        →        g_U  =  g_E  +  g_f

  E  primary exergy      f  aggregate conversion efficiency

Worked example — the United States, 1900 to 1998. Ayres, Ayres and Warr's series puts aggregate conversion efficiency at 2.5 percent in 1900 and 13.0 percent in 1998 — a multiple of 5.20, or 1.682 percent a year. Primary energy went from 9.6 to 94.8 quads, 2.337 percent a year. So useful work grew at 4.019 percent a year, a multiple of 51.4 times over the century, against real output at 3.2 percent a year, or 23.0 times.

Why it matters. Something inside the economy was compounding at four percent a year for a hundred years and was not on anybody's books. It was the second law, gradually being obeyed less wastefully.

You already know this because you do not describe a worker's contribution by the calories they eat. You describe it by what they got done. Useful work is that distinction applied to fuel.

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BRIEF 8 — The Solow residual, re-read

The idea. The largest number in twentieth-century growth economics is a name for ignorance, and a good part of it has a physical explanation.

Robert Solow's 1957 accounting attributed about 87.5 percent of the growth of American output per man-hour to technical change — the residual, the part that capital and labour did not explain.

Worked example — what a factor can carry. The share of output growth a factor accounts for is its elasticity times its growth rate, divided by output growth:

The elasticity of useful work that would absorb Solow's residual exactly is 0.697. Ayres and Warr's estimates land near there.

What to be careful about — and this is not optional. An elasticity of 0.50 against a cost share of 6.0 percent is a wedge of 8.3 times. Competitive factor pricing says elasticity equals cost share. So either markets misprice energy by nearly an order of magnitude or the elasticity is an artefact of the functional form that produced it. The literature argues the first. It has not settled it.

Why it matters. This is the strongest commercial argument in the chapter and it must be carried with its weakness attached, or the first economist in the room takes it apart and takes the rest with it.

You already know this because you have seen a variance analysis where the "other" line was the biggest one, and you knew immediately that the real work was to break it up.

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BRIEF 9 — Dilution: the logarithm and the hyperbola

The idea. The cost of recovering a dispersed material is not an entropy problem. It is a mass-handling problem, and the two grow at wildly different rates.

The minimum reversible work to separate a species at mole fraction x is R·T₀·ln(1/x). That is a logarithm, and a logarithm barely moves.

Worked example — copper.

  from 0.5 % ore      13.13 kJ/mol  =  206.7 kJ/kg  =  0.0574 kWh/kg
  from seawater       57.93 kJ/mol  =  911.6 kJ/kg  =  0.2532 kWh/kg
  dilution between them                             7.042e+07 ×
  rise in the thermodynamic minimum                      4.41 ×

Dilute by seventy million times and the entropy cost rises by four. Now count what must be moved: 200 kg of ore per kilogram of copper against 4,000,000,000 kg of seawater — a throughput ratio of 20,000,000 times. Pumping alone, at a generous 0.10 kWh per tonne, is 400,000 kWh per kilogram against a thermodynamic minimum of 0.2532: a factor of 1,579,724.

Why it matters. It tells you where to intervene. You cannot repeal the handling cost, but you can decide, at design time, whether the atom ends its life in a lump or in a dispersion. Aluminium recovered from clean scrap costs 0.70 kWh/kg against 14.00 in a smelting cell — a 95.0 percent saving, earned entirely by the fact that the metal never dispersed.

You already know this because you have hunted for a dropped contact lens on a tiled floor and found it, and you would not have attempted the same search in a swimming pool. The difficulty was never the lens.

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BRIEF 10 — The cascade, and the exergy-indexed tariff

The idea. Match the grade of the source to the grade of the task, in descending order, and sell every step.

A cascade orders a site's thermal demands by required temperature and feeds them from one flow: the hottest duty first, then the next, then district heat, then a glasshouse, then a pond. Iceland's Svartsengi field runs four uses off one geothermal flow and ends in a lagoon that is a business.

Worked example — the offtake, costed. A stack rejecting 12.0 MW at three hundred degrees, of which 9.0 MW is recovered at ninety degrees:

  exergy in the stack   12.0 × 0.5060  =  6.072 MW
  exergy delivered       9.0 × 0.2203  =  1.983 MW
  cascade exergy efficiency             =  32.7 %
  first-law recovery rate               =  75.0 %

75.0 percent is what the vendor quotes. 32.7 percent is what you captured.

The clause that gets it signed. Index the heat price to its grade: price(T) = base × f(T)/f(90 °C). At a flat 38.04 EUR/MWh-thermal, a buyer of ninety-degree heat pays 172.69 EUR per megawatt-hour of exergy and a buyer of sixty-degree heat pays 253.48 — an overcharge of 46.8 percent. Indexed, the sixty-degree price becomes 25.92 EUR/MWh-thermal and the quarrel disappears into a thermometer reading.

Where it stops earning its keep. Where energy is 2.0 percent of cost, a twenty-percent exergy saving moves total cost by 0.40 percent — inside the noise of every other line. Below that threshold the exergy account is bookkeeping rather than management. Know which side of it you are on.

You already know this because you do not wash the floor with drinking water heated to boiling. You already cascade in your own kitchen. This is that, with a contract.

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All figures in these briefs are computed in lib/verify/II_06.py and printed with their inputs by python3 lib/verify.py II.06.