Elusive Entropy

partial mixWe’ve all heard it. We think we understand it: entropy is a measure of disorder. Combined with the Second Law of Thermodynamics—that the total entropy of a closed system may never decrease—it seems we have a profound statement that the Universe is destined to become less ordered.

The consequences are unsettling. Sure, the application of energy can reverse entropy locally, but if our society enters an energy-scarce regime, how can we maintain order? It makes intuitive sense: an energy-neglected infrastructure will rust and crumble. And the Second Law stands as a sentinel, unsympathetic to deniers of this fact.

A narrative has developed around this theme that we take in low entropy energy and emit a high entropy wake of waste. That life displays marvelous order—permitted by continuous feeding of this low entropy energy—while death and decay represent higher entropy end states. That we extract low entropy concentrations of materials (ores) from the ground, then disperse the contents around the world in a higher entropy arrangement. The Second Law warns that there is no going back: at least not without substantial infusion of energy.

But wait just a minute! The preceding paragraph is mostly wrong! An unfortunate conflation of the concepts of entropy and disorder has resulted in widespread misunderstanding of what thermodynamic entropy actually means. And if you want to invoke the gravitas of the Second Law of Thermodynamics, you’d better make darned sure you’re talking about thermodynamic entropy—whose connection to order is not as strong as you might be led to believe. Entropy can be quantified, in Joules per Kelvin. Let’s build from there.

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Heat Pumps Work Miracles

Part of the argument that we cannot expect growth to continue indefinitely is that efficiency gains are capped. Many of our energy applications are within a factor of two of theoretical efficiency limits, so we can’t squeeze too much more out of this orange. After all, nothing can be more than 100% efficient, can it? Well, it turns out there is one domain in which we can gleefully break these bonds and achieve far better than 100% efficiency: heat pumps (includes refrigerators). Even though it sounds like magic, we still must operate within physical limits, naturally. In this post, I explain how this is possible, and develop the thermodynamic limit to heat engines and heat pumps. It’s a story of entropy.

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Exponential Economist Meets Finite Physicist

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Some while back, I found myself sitting next to an accomplished economics professor at a dinner event. Shortly after pleasantries, I said to him, “economic growth cannot continue indefinitely,” just to see where things would go. It was a lively and informative conversation. I was somewhat alarmed by the disconnect between economic theory and physical constraints—not for the first time, but here it was up-close and personal. Though my memory is not keen enough to recount our conversation verbatim, I thought I would at least try to capture the key points and convey the essence of the tennis match—with some entertainment value thrown in.

Cast of characters: Physicist, played by me; Economist, played by an established economics professor from a prestigious institution. Scene: banquet dinner, played in four acts (courses).

Note: because I have a better retention of my own thoughts than those of my conversational companion, this recreation is lopsided to represent my own points/words. So while it may look like a physicist-dominated conversation, this is more an artifact of my own recall capabilities. I also should say that the other people at our table were not paying attention to our conversation, so I don’t know what makes me think this will be interesting to readers if it wasn’t even interesting enough to others at the table! But here goes…

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Basking in the Sun

Who hasn’t enjoyed heat from the sun? Doing so represents a direct energetic transfer—via radiation—from the sun’s hot surface to your skin. One square meter can catch about 1000 W, which is comparable to the output of a portable space heater. A dark surface can capture the energy at nearly 100% efficiency, beating (heating?) the pants off of solar photovoltaic (PV) capture efficiency, for instance. We have already seen that solar PV qualifies as a super-abundant resource, requiring panels covering only about 0.5% of land to meet our entire energy demand (still huge, granted). So direct thermal energy from the sun, gathered more efficiently than what PV can do, is automatically in the abundant club. Let’s evaluate some of the practical issues surrounding solar thermal: either for home heating or for the production of electricity.

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Growth Has an Expiration Date

Just a quickie.  A few weeks back, I tried to cram four Do the Math posts into a 20 minute talk, delivered at the Compass Summit.  For those of you who would rather watch 23 minutes of video than sit down to read four posts, here is a link to the video of the talk.  Perhaps you’ll see why I should stick to writing.

Growth Has an Expiration Date from Compass Summit on FORA.tv

Can Economic Growth Last?

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As we saw in the previous post, the U.S. has expanded its use of energy at a typical rate of 2.9% per year since 1650. We learned that continuation of this energy growth rate in any form of technology leads to a thermal reckoning in just a few hundred years (not the tepid global warming, but boiling skin!). What does this say about the long-term prospects for economic growth, if anything?

Gross World Product

World economic growth for the previous century, expressed in constant 1990 dollars. For the first half of the century, the economy tracked the 2.9% energy growth rate very well, but has since increased to a 5% growth rate, outstripping the energy growth rate.

The figure at left shows the rate of global economic growth over the last century, as reconstructed by J. Bradford DeLong. Initially, the economy grew at a rate consistent with that of energy growth. Since 1950, the economy has outpaced energy, growing at a 5% annual rate. This might be taken as great news: we do not necessarily require physical growth to maintain growth in the economy. But we need to understand the sources of the additional growth before we can be confident that this condition will survive the long haul. After all, fifty years does not imply everlasting permanence.

The difference between economic and energy growth can be split into efficiency gains—we extract more activity per unit of energy—and “everything else.” The latter category includes sectors of economic activity not directly tied to energy use. Loosely, this could be thought of as non-manufacturing activity: finance, real estate, innovation, and other aspects of the “service” economy. My focus, as a physicist, is to understand whether the impossibility of indefinite physical growth (i.e., in energy, food, manufacturing) means that economic growth in general is also fated to end or reverse. We’ll start with a close look at efficiency, then move on to talk about more spritely economic factors. Continue reading

Galactic-Scale Energy

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Since the beginning of the Industrial Revolution, we have seen an impressive and sustained growth in the scale of energy consumption by human civilization. Plotting data from the Energy Information Agency on U.S. energy use since 1650 (1635-1945, 1949-2009, including wood, biomass, fossil fuels, hydro, nuclear, etc.) shows a remarkably steady growth trajectory, characterized by an annual growth rate of 2.9% (see figure). It is important to understand the future trajectory of energy growth because governments and organizations everywhere make assumptions based on the expectation that the growth trend will continue as it has for centuries—and a look at the figure suggests that this is a perfectly reasonable assumption.  (See this update for nuances.)

U.S. total energy 1650-present (logarithmic)

Total U.S. Energy consumption in all forms since 1650. The vertical scale is logarithmic, so that an exponential curve resulting from a constant growth rate appears as a straight line. The red line corresponds to an annual growth rate of 2.9%. Data source: EIA.

Growth has become such a mainstay of our existence that we take its continuation as a given. Growth brings many positive benefits, such as cars, television, air travel, and iGadgets. Quality of life improves, health care improves, and, aside from a proliferation of passwords to remember, life tends to become more convenient over time. Growth also brings with it a promise of the future, giving reason to invest in future development in anticipation of a return on the investment. Growth is then the basis for interest rates, loans, and the finance industry.

Because growth has been with us for “countless” generations—meaning that everyone we ever met or our grandparents ever met has experienced it—growth is central to our narrative of who we are and what we do. We therefore have a difficult time imagining a different trajectory.

This post provides a striking example of the impossibility of continued growth at current rates—even within familiar timescales. For a matter of convenience, we lower the energy growth rate from 2.9% to 2.3% per year so that we see a factor of ten increase every 100 years. We start the clock today, with a global rate of energy use of 12 terawatts (meaning that the average world citizen has a 2,000 W share of the total pie). We will begin with semi-practical assessments, and then in stages let our imaginations run wild—even then finding that we hit limits sooner than we might think. I will admit from the start that the assumptions underlying this analysis are deeply flawed. But that becomes the whole point, in the end.

A Race to the Galaxy

I have always been impressed by the fact that as much solar energy reaches Earth in one hour as we consume in a year. What hope such a statement brings! But let’s not get carried away—yet.

Only 70% of the incident sunlight enters the Earth’s energy budget—the rest immediately bounces off of clouds and atmosphere and land without being absorbed. Also, being land creatures, we might consider confining our solar panels to land, occupying 28% of the total globe. Finally, we note that solar photovoltaics and solar thermal plants tend to operate around 15% efficiency. Let’s assume 20% for this calculation. The net effect is about 7,000 TW, about 600 times our current use. Lots of headroom, yes? Continue reading

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