The gas that cools by being let go

Force a gas slowly through a porous plug from a high pressure to a low one. Insulate everything, so no heat crosses any boundary. No piston moves, so no work is done on anything outside. The gas comes out at a different temperature from the one it went in at.

That is the Joule–Thomson effect, it is the basis of every gas liquefaction plant and every domestic refrigerator, and its sign is not always the one wanted.

Why the enthalpy is what is conserved #

The bookkeeping is worth doing because it explains why an ideal gas does nothing here at all.

Consider a fixed parcel of gas being pushed through. Upstream, the gas behind it does work $P_1V_1$ pushing it in. Downstream, it does work $P_2V_2$ pushing the gas ahead of it out. No heat enters, so the change in internal energy is the net work done on the parcel:

$$U_2 - U_1 = P_1V_1 - P_2V_2$$

Rearranged, $U_1 + P_1V_1 = U_2 + P_2V_2$. The combination $U + PV$ is the enthalpy, and it is what stays constant through a throttling.

A throttling cannot be drawn as a path on a pressure–volume diagram at all. It is irreversible, the states between the two ends are not equilibrium states, and there is no curve to enclose an area under — so the work interpretation that makes such diagrams useful simply does not apply. What is conserved is the enthalpy, and it is conserved by a bookkeeping argument about what crosses the boundary rather than by anything about the path.

For an ideal gas the enthalpy is a function of temperature alone, since $U$ depends only on $T$ and $PV = nRT$. Constant enthalpy therefore means constant temperature, and an ideal gas throttled through a plug comes out at exactly the temperature it went in at.

It is worth being clear that “no work is done” refers to the outside world. Work certainly is done inside: the gas upstream pushes the parcel in and the parcel pushes the gas downstream out, and the two are not equal because the volume changed. That imbalance is exactly the $P_1V_1 - P_2V_2$ above, and it is why the internal energy is not the conserved quantity here — which is the distinction between the two kinds of expansion that took Joule and Thomson two attempts to get at.

So the whole of the Joule–Thomson effect is a departure from ideality. That makes it an unusually direct measurement of intermolecular forces: it is not a small correction to a large effect, it is an effect whose entire size is the correction.

Two effects, opposite signs #

Real gases depart from ideality in two ways, and the two push the temperature in opposite directions.

Molecules attract one another. Separating them as the gas expands costs potential energy, which has to come from the kinetic energy, so the gas cools. This dominates at low temperature, where molecules pass slowly enough for the attraction to bend their paths appreciably.

Molecules take up room. The volume available is less than the container’s, so expanding releases less energy than an ideal gas would, and there is a term that warms. This dominates at high temperature, where the attraction has stopped mattering and only the finite size remains.

Van der Waals’s equation carries exactly those two corrections and nothing else, which makes it the natural model here. Working through it to first order in the density gives the coefficient as proportional to $2a/RT - b$: the attraction term $a$ divided by the temperature, against the volume term $b$ that does not depend on temperature at all.

That expression changes sign at $T = 2a/Rb$, which is the inversion temperature. Below it the attraction wins and the gas cools on throttling; above it the excluded volume wins and it warms.

The dome, computed #

At finite pressure the criterion is a curve rather than a temperature, and it is the locus of the maxima of the constant-enthalpy curves.

Since throttling moves a gas leftward along a constant-enthalpy curve, and since the temperature change is the height change, the sign of the effect is the sign of the curve’s slope. A curve’s maximum is therefore exactly where the effect changes sign, and the inversion curve is the set of those maxima.

Computing it that way rather than from a formula is what the figure does: ninety enthalpies, each scanned for its turning point, and the resulting points compared with the published expression $P_r = 24\sqrt{3T_r} - 12T_r - 27$. They agree to three per cent of one reduced pressure unit.

The curve’s features are all pure numbers. It meets zero pressure at 0.75 and 6.75 times the critical temperature and peaks at three times the critical temperature and nine times the critical pressure — three numbers that come out of the equation of state with nothing put in by hand, and that are the same for every gas the model describes.

The upper number is the one that matters industrially, and the statement it makes is stark: a gas can be liquefied by throttling only if it starts below 6.75 times its own critical temperature.

Which gases can and cannot #

Nitrogen’s critical temperature is 126 kelvin, so 6.75 times it is 851 and the measured inversion temperature is 621. Either way, room temperature is comfortably below it, and air throttled from a compressor cools.

The critical temperature is the scale everything here is measured against, and the useful number is a ratio: the inversion temperature is between five and six times it for most gases. That is why hydrogen and helium warm on throttling at room temperature and nitrogen cools — not because they differ in kind, but because room temperature sits on opposite sides of their inversion temperatures.

Hydrogen’s critical temperature is 33 kelvin and its inversion temperature is 202. Room temperature is above it, so hydrogen throttled from a room-temperature compressor gets hotter. Helium’s inversion temperature is 43 kelvin, so the same is true and worse.

That is why the history of liquefaction runs in the order it does. Linde liquefied air by throttling in 1895. Dewar needed liquid air to pre-cool hydrogen below 202 kelvin before throttling would work on it, and got liquid hydrogen in 1898. Kamerlingh Onnes needed liquid hydrogen to pre-cool helium below 43 kelvin, and got liquid helium in 1908 — which is what made superconductivity discoverable three years later.

The sequence has a moral about how a field advances that is worth stating. Each step needed the previous one’s product as a consumable, so the interval between them is set by the time it took to make the previous liquid routinely rather than by any conceptual difficulty. The physics of throttling was fully understood by 1862; the thirteen years between liquid air and liquid helium were engineering, and what was found at the end of them was not something anybody had gone looking for.

The measured ratios of inversion temperature to critical temperature are 4.9 for nitrogen and carbon dioxide, 6.1 for hydrogen and 8.7 for helium. The model says 6.75 for all of them. That is a factor well under two across a range of substances, from a model with two parameters, which is the usual bargain with van der Waals: the mechanism exactly and the coefficient roughly.

Reading the isenthalps #

The opening figure repays more attention than a single glance, because the shape of the curves carries several statements at once.

Isenthalps connect states rather than describing how a system passes between them, and that is worth being explicit about because they are drawn as curves. Each one is a locus of states with the same enthalpy, not a trajectory — a throttling jumps from one point on such a curve to another, and what happens in between is not on the diagram and not in equilibrium.

The curves are traversed from right to left, because the pressure falls. That has to be stated rather than read off, because an isenthalp is a set of states rather than a trajectory: the throttling is irreversible and passes through nothing that could be drawn as a point on this diagram.

A curve’s slope is the coefficient, so a gas on a rising part cools and one on a falling part warms. A gas starting to the right of a curve’s maximum therefore warms at first, reaches the maximum, and cools thereafter — so a single large pressure drop can produce a net cooling even from a starting point where the initial effect is a warming.

The curves flatten at low pressure, which says the coefficient tends to a finite limit rather than to zero: throttling from one bar to half a bar still changes the temperature, just by less. The zero-pressure limit of the coefficient is the quantity most directly related to the intermolecular potential, and it is what the two zeros of the inversion curve are.

And the spacing between curves is a heat capacity. Two isenthalps differing by a known enthalpy are separated in temperature by that enthalpy over $C_p$, so the diagram carries the heat capacity as well — which is why an enthalpy–pressure chart is the working diagram of refrigeration engineering rather than a pressure–volume one.

What a refrigerator does with it #

The effect is not efficient, and understanding why explains the shape of every real liquefier.

A throttling is not reversible at all: it generates entropy and extracts no work. So a throttling-based cooler falls well below the reversible ceiling, and the gap is not an engineering imperfection but the process itself. That is the trade the Linde cycle makes — a valve is simple, robust and cheap, and it wastes most of the available work.

Throttling is irreversible. Entropy is generated and no work is recovered, so as a cooling process it is thermodynamically wasteful compared with an expansion engine, which extracts work and cools much more per unit of pressure drop.

It is used anyway, for two reasons. It has no moving parts in the cold region, which matters enormously when the cold region is at 4 kelvin and any bearing would be a heat leak. And its cooling per pass increases as the gas gets colder, because the coefficient grows as the temperature falls — so it works better as it works, which is the opposite of most cooling processes.

The trick that makes it practical is the counterflow heat exchanger: the cold gas leaving the plug is used to pre-cool the incoming gas. Each pass then starts colder than the last, and the process cascades down to liquefaction from a modest pressure ratio. Linde’s machine is that idea, and every cryocooler since is a variation on it.

A domestic refrigerator uses the same throttle in a different regime. There the working fluid is already partly condensing at the valve, so most of the cooling is latent heat rather than Joule–Thomson, and the valve is chosen for flow control rather than for the coefficient.

The pipeline that has to be heated before it is let down #

The largest industrial encounter with this effect is not in a cryogenic plant. It is at the head of every gas well, where the coefficient’s sign is simultaneously a hazard and a separation process.

Gas arrives from a reservoir at pressures of hundreds of bar and has to be let down to pipeline pressure through a choke. Methane’s inversion temperature is well above ambient, so it cools — by roughly half a degree per bar. A hundred-bar let-down therefore drops the temperature by forty or fifty kelvin, from a warm reservoir to well below freezing, in a valve with no refrigeration in it at all.

That is a problem, and the problem is not ice. Natural gas from a reservoir carries water vapour, and at high pressure water and methane together form a hydrate — a crystalline solid in which methane molecules are caged inside a lattice of water, stable well above zero degrees. At seventy bar, methane hydrate is stable up to about ten degrees Celsius, so a stream cooled by throttling can be twenty degrees above freezing and still forming solid.

A hydrate plug blocks a pipeline completely, and clearing one is a serious operation: the plug is mechanically strong, it can be metres long, and depressurising one side of it can accelerate it down the pipe as a projectile.

Three remedies are used and each is a direct response to the arithmetic. The gas can be dried before the choke, removing one of the two ingredients. It can be heated before the choke — a line heater at the wellhead, sized so that the temperature after the pressure drop is above the hydrate stability line. Or an inhibitor can be injected, usually methanol or monoethylene glycol, which shifts the hydrate curve to lower temperatures in the same way an antifreeze shifts a melting point. Subsea gas fields circulate glycol continuously through pipelines hundreds of kilometres long, and recovering and drying it again is a substantial fraction of the plant.

The same cooling is also used on purpose at the same valve. A gas stream cooled by forty kelvin drops out its heavier components as liquid, so a choke followed by a separator is a distillation column with no column: propane, butane and heavier hydrocarbons condense and are drawn off, leaving drier methane for the pipeline. Low-temperature separation units of that kind are standard, and their refrigeration is entirely the effect on this page.

One valve, one coefficient, and two consequences that have to be engineered against each other.

The cold end of a real machine #

The essay’s argument for throttling — thermodynamically wasteful, and used anyway because it has no moving parts where it is cold — is the reason a modern cryocooler has the architecture it does.

Reaching a few kelvin in a closed cycle requires something to move gas about at low temperature, and the choices differ in where the moving part is. A Stirling or Gifford–McMahon cooler puts a displacer in the cold region: it works, it reaches ten or twenty kelvin reliably, and it has a sliding seal down there that wears and a reciprocating mass that vibrates.

A pulse tube removes the cold displacer and replaces it with a column of gas whose oscillation performs the same function. Nothing solid moves below the room-temperature valve. That change costs some efficiency and buys two things that are worth more in most applications: essentially unlimited life, since there is no cold wearing part, and very low vibration.

A Joule–Thomson stage goes further still — a counterflow heat exchanger and an orifice, with no moving part anywhere in the cold end and nothing to fail. It is the least efficient of the three and it is what is used for the last step down to four kelvin and below.

The consequences are visible in what flies. An infrared telescope in space needs its detectors at a few kelvin for a decade or more, with no maintenance and with vibration low enough not to spoil the pointing — so the coolers are pulse tubes for the upper stages and a closed-cycle Joule–Thomson loop for the coldest, precisely because the JT stage has nothing in it that can wear out. Space observatories have run such systems continuously for years.

The same logic reaches the hospital. A modern magnetic resonance magnet is kept cold by a two-stage cryocooler intercepting the heat leak rather than by boiling off liquid helium, which is why such a machine now needs no helium deliveries — and the cold head is a pulse tube for the same reason a telescope’s is.

The trade is the one the essay names, made explicitly by every designer: throttling wastes work, and it wastes it in a place where nothing else has to be. Where efficiency is the constraint, an expansion engine is used; where reliability is, the wasteful process wins.

The other expansion, and why it is different #

There is a second experiment that sounds like this one and is not, and the pair together are what settled the question.

In a free expansion into vacuum there is no wall to strike on the far side, so no work is done at all — and an ideal gas expanding that way does not change temperature. That is the other expansion, and the contrast is exact: throttling conserves enthalpy and free expansion conserves internal energy, the two coincide for an ideal gas, and everything in this essay lives in the difference between them for a real one.

Joule’s free expansion lets a gas expand into an evacuated chamber with the whole system insulated. No work is done on anything and no heat enters, so the internal energy is constant rather than the enthalpy. For an ideal gas the temperature is unchanged, and Joule measured no change — which was taken as evidence for ideality until it was realised that the water bath’s heat capacity swamped the effect.

The two experiments measure different derivatives of different quantities, and the coefficients differ. The free-expansion coefficient is negative for every gas at every temperature, because it responds only to the attraction; the throttling coefficient has both terms and therefore has a sign that changes. That is why the second experiment is the useful one, and why Joule and Thomson devised it after the first proved too insensitive.

There is a third member of the family worth naming for contrast. A gas expanding against a piston — a reversible adiabatic expansion — cools substantially and for a reason that has nothing to do with intermolecular forces: it does work on the piston, and the energy comes from the molecules’ kinetic energy. That is the cooling everybody’s intuition supplies, and it is a much larger effect than either of the other two. Confusing it with the throttling is the commonest error here, and it is what makes “a gas cools when it expands” a statement that is true, false or meaningless depending on which expansion is meant.

Both are irreversible, both generate entropy, and neither is a path on an equation-of-state diagram — which is why the isenthalps in the opening figure connect states rather than describing a trajectory through them.

What the pictures cannot show #

Van der Waals is a two-parameter model. It cannot reproduce the critical exponents, the shape of the coexistence curve or the compressibility factor to better than tens of per cent, and its inversion curve is correspondingly approximate. Real inversion curves are measured, tabulated and slightly different in shape.

The isenthalps are drawn for a single-phase fluid. Real throttling in a liquefier ends inside the two-phase region, where a fraction of the stream condenses and the temperature is pinned to the boiling point; the interesting arithmetic there is what fraction liquefies, which is a different calculation.

Nothing here is a mixture. Air is not a substance, and the Joule–Thomson coefficient of a mixture is not the average of its components’; the interaction between unlike molecules contributes a term of its own, which is why mixed refrigerants are designed rather than blended.

The gas is monatomic-ish. A heat capacity of 2.5R was used for the isenthalps, which is a diatomic gas at moderate temperature; changing it moves the curves and, as the last figure shows, leaves the inversion curve exactly where it was. Only the inversion curve is a statement about the equation of state alone.

And the process is assumed slow and steady. A real valve produces a jet, turbulence and kinetic energy that has to be dissipated, so the enthalpy balance holds between well-separated points upstream and downstream rather than across the valve itself.

The ladder from here #

Later rungs on this anchor: the exact thermodynamic expression for the coefficient in terms of the thermal expansivity, and what measuring it says about the equation of state; mixed-refrigerant cycles, where the inversion behaviour of a mixture is engineered; the dilution refrigerator, which reaches millikelvin by a mechanism with no gas-phase analogue; and the Joule–Thomson coefficient’s use as a probe of intermolecular potentials, which is one of the older ways of measuring one.

The neighbouring ladders are the part of the curve no fluid follows, which is the same equation of state where it stops describing anything, the point at which the two become one, which is the temperature everything here is measured against, and pressure is a rate of arrival, where the ideal gas that does nothing under throttling is built.