The ceiling on a heat engine

At its defaults it draws the ceiling on a heat engine. Maximum possible efficiency against the ratio of cold to hot reservoir temperature. Reaching 100% would need a cold reservoir at absolute zero. The line is marked at ratios of 0.9, 0.7, 0.5, 0.25, where the ceiling stands at 10%, 30%, 50%, 75%.

engine-efficiency is one function in lib/figures/thermal.js — cycles, distributions and the statistics underneath them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Maximum possible efficiency against the ratio of cold to hot reservoir temperature. Reaching 100% would need a cold reservoir at absolute zero. The line is marked at ratios of 0.9, 0.7, 0.5, 0.25, where the ceiling stands at 10%, 30%, 50%, 75%.

How much heat a three-reservoir machine can lift out of a cold space per unit of heat supplied to drive it, against the temperature of the driving heat, for 3 cold temperatures and an ambient of 300 kelvin. No work enters or leaves: the machine takes heat in at the top, takes heat in at the bottom, and rejects the sum at ambient. That it can do anything at all is the surprise — the second law allows heat to be moved up a gradient provided a larger flow is moved down one, and the accounting is a single inequality in the three entropy flows. The ceiling is the product of two familiar expressions, and at 450 kelvin driving a 253-kelvin space it is 1.79. The marks are what real machines achieve, which is a fifth to a third of it — absorption refrigeration is not efficient and is chosen when the heat is free and the silence and the absence of moving parts are worth something.

The heat crossing each boundary of a reversible three-reservoir refrigerator, per joule supplied to drive it, with the entropy each flow carries printed beside it. Heat enters at 450 kelvin and at 253, and the sum of the two leaves at 300. No work crosses any boundary in either direction, so the first law is simply that the heats add. The second law is the interesting one: the entropy arriving is 0.00931 joules per kelvin and the entropy leaving is 0.00931, and they are equal — which is exactly the condition that fixes how much heat could be lifted. A joule arriving at a high temperature carries little entropy and a joule arriving at a low one carries a lot, so the driving heat's entropy budget is what pays for the cold heat's.

The heat crossing each boundary of a reversible three-reservoir refrigerator, per joule supplied to drive it, with the entropy each flow carries printed beside it. Heat enters at 380 kelvin and at 283, and the sum of the two leaves at 305. No work crosses any boundary in either direction, so the first law is simply that the heats add. The second law is the interesting one: the entropy arriving is 0.01160 joules per kelvin and the entropy leaving is 0.01160, and they are equal — which is exactly the condition that fixes how much heat could be lifted. A joule arriving at a high temperature carries little entropy and a joule arriving at a low one carries a lot, so the driving heat's entropy budget is what pays for the cold heat's.

How much heat a three-reservoir machine can lift out of a cold space per unit of heat supplied to drive it, against the temperature of the driving heat, for 3 cold temperatures and an ambient of 300 kelvin. No work enters or leaves: the machine takes heat in at the top, takes heat in at the bottom, and rejects the sum at ambient. That it can do anything at all is the surprise — the second law allows heat to be moved up a gradient provided a larger flow is moved down one, and the accounting is a single inequality in the three entropy flows. The ceiling is the product of two familiar expressions, and at 450 kelvin driving a 240-kelvin space it is 1.33. The marks are what real machines achieve, which is a fifth to a third of it — absorption refrigeration is not efficient and is chosen when the heat is free and the silence and the absence of moving parts are worth something.

Contours of how much heat a reversible three-reservoir machine lifts per joule supplied, in the plane of the driving temperature and the temperature of the cold space, at an ambient of 300 kelvin. Two edges bound the region and neither is negotiable: the driving heat must be above ambient, or there is no engine, and the cold space must be below it, or there is nothing to refrigerate. Approaching either edge sends the performance to zero. The contours run steeply, which is the practical content: deep refrigeration from modest heat is nearly impossible, and mild cooling from hot heat is easy. That is why absorption machines are used for air conditioning, where the cold space is fifteen degrees below ambient, and almost never for freezing.

physicscheck asserts something about engine-efficiency that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.