The cylinder that must not be cooled

The heat that changes no temperature tells how Joseph Black discovered latent heat in the 1760s — the heat that water absorbs as it boils and gives back as it condenses, without any change in temperature — and how James Watt, then an instrument maker at the University of Glasgow, used it. Watt had been asked to repair the university’s model of a Newcomen engine and found that it consumed absurd quantities of steam. Knowing from Black how much heat a little steam carries, he could see where it was going. That essay closes on the improvement, a factor of four in fuel. This one opens the cylinder.

The question is worth taking in detail because the answer is more than an engineering fix. Newcomen’s engine did not waste steam because it was badly made. It wasted it because of what it asked one vessel to do, and the waste could not be reduced by any adjustment of the cycle without losing work in another way. Watt’s change removed a contradiction, and the contradiction is the second law of thermodynamics in its earliest and most concrete form: that heat flowing across a temperature difference without doing work destroys the possibility of work.

Newcomen’s cycle #

A Newcomen engine was a large vertical cylinder, open at the top, with a piston connected by a chain to one end of a great rocking beam; the other end of the beam worked a pump in a mine shaft. The cylinder sat above a boiler, which supplied steam at slightly above atmospheric pressure. With the piston at the top, steam filled the cylinder. A valve then admitted a jet of cold water from a tank into the cylinder. The steam condensed, the pressure in the cylinder fell to a fraction of an atmosphere, and the atmosphere outside pushed the piston down, lifting the pump rods at the other end of the beam. Steam was admitted again; the weight of the pump rods pulled the piston back up; the cycle repeated about twelve times a minute.

It was an atmospheric engine. The steam’s pressure did no work at all; it only displaced the air from the cylinder and then, condensing, made room for the atmosphere to push. The work per stroke was the atmosphere’s pressure, less whatever pressure remained in the cylinder, times the piston’s area and stroke. And the pressure remaining in the cylinder was set by how cold the injection made it.

Newcomen’s dilemma #

The vapour pressure of water falls steeply as it cools. Water at 100 °C boils under one atmosphere, as a boiling point is a pressure explains; at 70 °C its vapour pressure is less than a third of an atmosphere, at 40 °C about a fourteenth.

So a cylinder cooled to 40 °C leaves little vapour behind and the atmosphere pushes with nearly its full weight, 0.93 of an atmosphere. But the cylinder’s iron has been cooled with it, and the next charge of steam must reheat the iron to 100 °C before any steam can stay in the cylinder as steam. The heat comes from condensing steam on the cold walls: each kilogram condensed gives up 2.26 megajoules. A cylinder left at 70 °C wastes half as much steam on its walls, but leaves enough vapour in the cylinder that the atmosphere pushes with only 0.69 of an atmosphere. Every setting of the injection trades steam against work. The engineers of the eighteenth century ran their engines in between, and the trade was the best the cycle allowed.

The source of the dilemma is that one vessel was being asked to be two things in succession: hot enough to receive steam without condensing it, and cold enough to condense it completely. No cycle that alternates a single vessel between those two states can avoid paying, on every stroke, for the heat that moves back and forth through its walls.

Where the steam went #

Putting numbers on the waste shows how large it was.

A cylinder twenty-one inches across and eight feet long, the size of the larger Newcomen engines of the 1760s, holds a third of a kilogram of steam at atmospheric pressure. That is what each stroke needs to fill it. But its walls, cooled by the injection to around 45 °C, must be brought back to 100 °C before the cylinder holds steam, and in the model that costs two kilograms of steam condensed on the iron — six times the steam that does the useful job of filling the cylinder. Watt’s own measurements on the Glasgow model gave a smaller multiple, a factor of about four, which in a small model with relatively more wall per unit volume is if anything conservative; either way, most of the coal burned in a Newcomen engine heated iron that was about to be cooled again.

Watt’s remedy, conceived on a Sunday walk on Glasgow Green in 1765 by his own later account, was to keep the cylinder permanently hot, wrapped in a jacket of steam, and to condense the steam in a separate vessel kept permanently cold, connected to the cylinder by a valve and emptied by an air pump. When the valve opened, the steam rushed into the cold condenser and condensed there, and the cylinder, never cooled, was ready for the next charge with no reheating at all. A hot vessel loses a little heat to its surroundings, and the steam jacket costs a little steam, but the waste on the walls was gone.

The skin of iron each stroke reheats #

How much iron the injection cools is not the whole cylinder but a skin, because heat diffuses only a short way into the wall in the few seconds a stroke lasts.

A surface whose temperature swings periodically drives a temperature wave into the material behind it that dies away exponentially over a penetration depth $\sqrt{2\alpha/\omega}$, where $\alpha$ is the thermal diffusivity and $\omega$ the angular frequency of the swing. It is the same physics as the summer that reaches the cellar in December, where the annual swing of the surface temperature reaches a few metres into the ground, lagging by months. For iron and a stroke of four seconds the depth is about five millimetres. The inner surface swings through the full range between injection and steam; a centimetre in, the iron barely notices the strokes and sits at a steady temperature in between.

That skin is still a substantial mass. Over the whole inner surface of the cylinder it is nearly two hundred kilograms of iron, and swinging it through fifty-five degrees on every stroke takes the two kilograms of steam computed above. It also sets the engine’s speed: a slower stroke lets the cold reach deeper into the wall and wastes more steam, while a faster one reduces the waste but the injection then has less time to condense the steam fully. Newcomen engines ran slowly, and their efficiency, which depended on the ratio of the reheated skin to the cylinder’s volume, rose with their size. Larger cylinders were more economical because their volume grew as the cube of their size and their surface only as the square.

The condenser and its pump #

The separate condenser freed the vacuum from the cylinder’s temperature, but it did not free it from the condenser’s. The pressure left in Watt’s condenser, and so in the cylinder connected to it during the down-stroke, was the vapour pressure of water at the condenser’s temperature — the same curve as Newcomen’s dilemma, now read without the penalty on the walls. A condenser kept at 40 °C by a steady flow of cold water held about a fourteenth of an atmosphere, and the colder it was kept, the better the vacuum. Watt’s engines ran their condensers as cold as the water supply allowed.

A condenser also collects everything that is not steam: the condensed water itself, the injection water, and air, which leaks into any vessel below atmospheric pressure and dissolves in boiler water and comes out with the steam. Air does not condense, and left to accumulate it would raise the condenser’s pressure until the vacuum was gone. Watt added an air pump, worked from the beam, to draw off air and water from the condenser every stroke. The pump cost a little of the engine’s work and was indispensable; it is the ancestor of the vacuum pumps on every steam turbine’s condenser today.

Modern power stations still end their cycles in a condenser, with the same physics in it. A large steam turbine exhausts into a surface condenser kept at about 30 to 40 °C by cooling water from a river, the sea or a cooling tower, at a pressure of about a twentieth of an atmosphere, and the temperature of that condenser is the cold end of the station’s Carnot ceiling. The temperature an engine really takes its heat at shows how the averaged temperatures at which a real cycle takes in and rejects heat replace Carnot’s two, and a condenser held a few degrees colder raises a station’s efficiency by a few tenths of a per cent — which is why stations are built beside cold water, and why their output falls in a heatwave.

Why Newcomen engines ran for seventy years anyway #

Newcomen engines were not displaced quickly. Hundreds were built through the eighteenth century, and many ran into the nineteenth, long after Watt’s engines were available. The reason is economic, and it shows what efficiency is worth. A Newcomen engine pumping a coal mine burned coal from the mine itself, the small coal and slack that could not be sold, at almost no cost; its inefficiency was irrelevant there. Watt’s engines prospered in Cornwall, where the tin and copper mines had to buy coal shipped from Wales, and every bushel saved was money. Boulton and Watt’s royalty, a third of the value of the coal saved, made the bargain explicit: the engine’s value was its efficiency multiplied by the price of fuel.

That calculus has not changed. Efficiency matters in proportion to what the fuel costs and what its waste costs, and the history of heat engines is a history of irreversibilities removed one after another, each when its removal became worth paying for.

The work lost to a cold wall #

The waste is not merely a matter of heat, and seeing why requires the second law of thermodynamics, which the ceiling on every engine introduces through Carnot’s limit: an engine working between a hot and a cold temperature can convert at most a fraction $1 - T_c/T_h$ of the heat it takes in into work.

When steam at 100 °C condenses on a wall at 45 °C, its latent heat flows into the wall across a temperature difference of fifty-five degrees and does no work on the way. In the language developed a century after Watt, entropy is created: the heat leaves at a high temperature and arrives at a low one, and the increase of entropy multiplied by the temperature of the surroundings is work that could have been done and now never can. For a kilogram of steam condensing on a wall at 45 °C, that lost work is 307 kilojoules — most of the 484 kilojoules that a kilogram of steam condensing at 100 °C could at best deliver to an engine rejecting heat at 20 °C. The heat is not lost; it is still in the wall, and next stroke it heats the injection water. But it is heat at a temperature too low to do much, and the drop in temperature was the loss.

This is the same accounting that the work left in two buckets of water applies to mixing a bucket of hot water with a bucket of cold: no energy disappears, and yet the capacity to do work does. Watt’s condenser does its saving by ensuring that the steam meets only metal at its own temperature in the cylinder and only cold water in the condenser — never a cold surface in the place where it is meant to be hot. The second law had no name in 1765, but the principle it formalises, that heat should not be allowed to fall across a temperature difference without doing something, was the whole of the invention.

A century of steam, by its coal #

The improvement can be read from the engines’ records, which the eighteenth century kept in a unit of its own: duty, the number of pounds of water raised one foot by a bushel of coal.

A typical Newcomen engine of the mid-century raised about five or six million foot-pounds of water per bushel, an efficiency of less than one per cent. John Smeaton, who measured and improved Newcomen engines systematically, raised the best of them to about nine and a half million, or one per cent. Watt’s engines of the late 1770s, with the separate condenser, reached about twenty million, and his later double-acting rotative engines more; Boulton and Watt charged mine owners a royalty of a third of the coal they saved, which made the duty a matter of commercial record. In the 1830s the high-pressure engines of Cornwall, which used steam at several atmospheres and let it expand in the cylinder — the principle that the engine that expands further than it compresses traces to its descendants — reached sixty million on average and over a hundred at their best.

Each step attacked a different irreversibility. The separate condenser removed the cold walls. Expansive working let the steam do work as its pressure fell, instead of being exhausted at full pressure. Higher pressures raised the temperature at which the heat came in, and so raised Carnot’s ceiling itself. None of them reached the ceiling. The engine that has to finish explains why no real engine can: an engine that does its cycle in finite time must let heat flow across finite temperature differences, and the losses Watt removed at the cylinder walls reappear, in smaller form, everywhere heat enters or leaves.

One cylinder and one sinusoid of temperature #

The cylinder dimensions are those of a large mid-eighteenth-century engine, the stroke period is four seconds, and the cylinder is treated as iron of uniform properties with its inner surface swung between the injection temperature and 100 °C; real cylinders were cast iron or brass, their temperature swings were not sinusoidal, and the injection water itself, collecting at the bottom, heated and was drained between strokes. The waste on the walls is estimated as the heat to swing a skin one penetration depth deep through the full range, a stated simplification of the periodic heating of a thick wall. The vapour pressure is the Antoine equation for pure water, ignoring the air that leaked into real cylinders and spoiled their vacuum further. The duty figures are approximate published values for representative engines; duties varied widely between engines and between trials, and their conversion to efficiency assumes thirty megajoules a kilogram for coal of uncertain quality.

Still open: how much of an engine’s loss is still at its walls #

Wall losses did not disappear with Watt; they shrank and moved. In a modern internal-combustion engine, heat lost from the hot gas to the cooled cylinder walls during combustion and expansion is still one of the largest single losses, a fifth to a third of the fuel’s energy in many engines, and it is the reason for the persistent interest in insulated, low-heat-rejection engine designs with ceramic coatings. Those designs have repeatedly disappointed, partly because hotter walls heat the incoming charge and reduce the mass of air the cylinder takes in, a trade that echoes Newcomen’s. Whether a coating that swings its surface temperature in step with the gas — hot during combustion, cool during intake — can escape that trade is an active question in engine research.

The habit worth carrying away is to ask whether a vessel is being asked to be two temperatures. Newcomen’s cylinder had to be cold to make a vacuum and hot to receive steam, so each stroke condensed several cylinders’ worth of steam on a skin of iron five millimetres deep — and condensing a kilogram of steam on a 45 °C wall throws away 307 kJ of possible work, which is why Watt’s separate condenser, by keeping the hot hot and the cold cold, cut the coal by a factor of three or four. The second law was a design rule before it was a law.