Limit Thinking

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Limit Thinking<br>Words by<br>David Jordan

The ColumnIssue 08

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Limit Thinking<br>To make things better, first prove how good they can possibly be.

Example H2

Example H3

In 1712, Thomas Newcomen, a Baptist preacher and ironmonger from Dartmouth, Devon, constructed his newly designed “atmospheric engine” for Coneygree Coalworks. Each stroke of the 20-ton machine would raise about 37 liters of water from the flooded mines below. It worked continuously and without tiring, unlike the dozen horses it replaced.<br>Like those horses, however, Newcomen’s engine needed to be fed, burning copious amounts of the very coal that it was helping to unearth. Although these engines were cost-effective compared to horses or humans, they were still expensive to operate, and improvements to their efficiency were especially sought after.<br>In 1763, while working as a mathematical instrument maker at the University of Glasgow, James Watt was tasked with repairing the university’s scale model of the Newcomen engine. As he worked, Watt envisioned ways to improve the efficiency of the design. In 1776, he unveiled an engine with seemingly extraordinary modifications: it consumed 75-80 percent less fuel than Newcomen’s. Tasks that would have burned 100 kilograms of coal could now be done with a mere 20.

A Newcomen atmospheric engine. Schematic by Louis Figuer, 1868.Although a triumph of engineering, Watt’s engine was nowhere near its optimal performance. The difference between Newcomen's and Watt’s engines, in fact, is between 0.5 percent and 2.5 percent efficiency. But these inventors could not have known this because the concept of a theoretical limit — asking how efficient an engine design could be, in principle — had not yet been imagined.<br>What was missing was a notion we might call “Limit Thinking.” This abstract approach forces one to focus solely on the features of a system essential for its performance, so that one can make predictions or evaluations, regardless of the specifics of how each individual system is built. It grounds problems in mathematics — as one cannot calculate limits without being precise about what is being measured and in what units. Once such calculations have been determined, they often drive rapid progress, signaling not only when we have reached diminishing returns, but also just how far we can aspire.<br>To understand the power of this approach, let’s begin with engines, returning to the work of Nicolas Léonard Sadi Carnot.<br>{{signup}}<br>Carnot, named after the Persian poet Saadi, was born in Paris in 1796 and attended the École Polytechnique before serving as an officer in the engineering arm of the French army. In 1819, at the age of 22, he took a part-time job in the army at half pay as a travailleur scientifique (scientific worker) and attended lectures at the Sorbonne, the Collège de France, and the Conservatoire des Arts et Métiers in his spare time. It was during this period that Carnot’s interest turned to heat engines.<br>In 1824, when Carnot was just 28 years old, he published Reflections on the Motive Power of Fire and on Machines Fitted to Develop this Power, in which he aimed to determine the fundamental limits of how heat could be converted into mechanical work. This 118-page booklet, of which he printed 600 copies at his own expense, represents essentially the whole of his scientific output (regrettably, Carnot died at 36 years old from a combination of scarlet fever and cholera). Reflections was largely ignored until 1834, when the French engineer and physicist Émile Clapeyron highlighted its importance in his own work, Memoir on the Motive Power of Heat.1<br>Carnot’s work showed that what mattered for the efficiency of a heat engine was the temperature differential between the hot and cold reservoirs, rather than any particular design feature. He says of the motive power of heat: “Its quantity is fixed solely by the temperatures of the bodies between which it is effected.” In retrospect, this simple fact explained why the Watt engine was superior to the Newcomen engine; the separate condenser allowed for a larger difference between the reservoirs.

Watt’s steam engine was complex and had many moving components.<br>An axial cross section of Carnot’s heat engine. Note the extreme simplicity and abstract nature of his design.Carnot’s work would go on to inspire not only Clapeyron but also a generation of prominent physicists such as Rudolph Clausius, James Joule, and Lord Kelvin himself. It laid the groundwork for the modern theory of thermodynamics and even presented an outline of what would later become its second law. Otto Diesel directly refers to the theory of Carnot2 as the rationale for his design of the engine that now bears his name (a design which achieved an efficiency of an astounding 26 percent when first tested in 1897). While the 65 years between Newcomen and Watt had seen efficiency improve roughly two percent, in the 65 years...