What it costs to take the salt out

Osmotic pressure is a counting argument: the dissolved particles behave as a gas of their own, and the pressure they exert is theirs alone. That essay computed a pressure. This one asks what it costs to work against it, and the answer turns out to be fixed before any apparatus exists.

The floor, and where it comes from #

Seawater holds about 35 grams of salt in a kilogram, which is roughly 1,150 moles of dissolved particles in a cubic metre once the sodium and chloride are counted separately. Van 't Hoff’s expression gives an osmotic pressure of

$$\Pi = cRT = 1150 \times 8.314 \times 298 = 2.85\ \text{MPa}$$

which is 28.5 bar, or about the pressure at the bottom of a 290-metre column of water.

Pushing water through a membrane against that pressure costs pressure times volume, so the first cubic metre of fresh water costs 2.85 megajoules, which is 0.79 kilowatt-hours. Nothing about that number refers to a membrane, a pump or a material: it is the free energy of mixing salt into water, read backwards.

Why mixing happens at all is a count. The mixed arrangements outnumber the separated ones so heavily that the separated state is simply never visited — not forbidden, not resisted, just overwhelmingly outnumbered. Mixing what is already mixed works out the entropy of that count, and the work computed in this essay is the same quantity with its sign reversed: unmixing is paying back what the count gave away.

Why taking more costs more #

The 0.79 figure is the cost of the first drop, and it is the number most often quoted. It is the wrong number for a plant, because a plant does not take one drop.

Taking a fraction $r$ of the feed as fresh water leaves the salt behind in the remainder, whose concentration rises by a factor $1/(1-r)$. The last drop is therefore pushed against a higher pressure than the first, and the total work depends on how the pressure is applied along the way.

Two limits bracket everything. If the pressure is raised continuously as the feed concentrates — a reversible separation — the work per unit of product is $(\Pi_f/r)\ln[1/(1-r)]$. If instead a single pump holds the whole feed at one pressure, that pressure must be the one the final concentrate needs, $\Pi_f/(1-r)$, and everything before the end is over-pressurised.

At half recovery those come to 1.10 and 1.58 kilowatt-hours per cubic metre, against 0.79 for the first drop. The gap between them is the shaded band on the first figure, and it is not friction, leakage or an imperfect membrane. It is the cost of pushing the first drop at the last drop’s pressure.

What staging buys #

If a single stage is expensive because it holds everything at the final pressure, the obvious repair is to do the job in pieces, each at its own pressure.

The excess above the reversible limit falls as one over the number of stages. That is the general law of finite-step irreversibility, and it is the same arithmetic as an engine with many small temperature drops instead of one large one: the loss comes from doing in one step what the system would have done in many.

It also explains why real plants stop at two or three passes. The first split removes half the excess, the second a third of what is left, and by the fourth the saving is smaller than the cost of another pressure vessel. Diminishing returns of a computable shape are what set the engineering optimum, and the reversible limit is the target rather than the design.

The structural analogy with a heat engine is exact and worth stating in full, because it explains why the number behaves as it does. A heat engine’s ceiling depends on the two temperatures and not on the machine; it is approached by making each step small; and it is never reached by anything running at a finite rate. Every one of those three is true here with concentrations in place of temperatures — which is why the minimum work is a property of the seawater and the product, and why a better membrane cannot lower it.

The device that made the difference #

One piece of hardware deserves a paragraph, because it is the clearest example of the shaded band on the first figure being money.

A reverse-osmosis stage produces two streams: fresh water at atmospheric pressure and concentrate still at sixty bar. Discharging that concentrate to the sea throws away most of the energy the pump supplied — at half recovery, half of the volume leaves at nearly full pressure, which is about half the input work. Early plants did exactly that, which is most of the reason they used ten kilowatt-hours a cubic metre.

An energy recovery device puts that pressure back into the incoming feed. The modern version is a rotating ceramic cylinder in which the concentrate and the feed touch directly for a fraction of a second, so the pressure transfers with almost no machinery in between; it reaches around 97 per cent transfer efficiency, which is higher than any turbine-and-pump pair would manage.

Pressure transmitted from one piston to another is the whole of an energy recovery device. The spent brine leaves a reverse-osmosis stage still at nearly the pressure it went in at, and a pressure exchanger hands that pressure directly to incoming feed instead of throttling it away. Two-thirds of the improvement in desalination energy since 1980 came from that piece of plumbing rather than from the membrane, which is the least intuitive fact in the subject.

The lesson generalises past desalination. When a process is bounded below by a reversible limit, the improvements that matter are the ones that stop discarding available work, and they are usually plumbing rather than chemistry.

How close the industry is #

A modern seawater reverse-osmosis plant uses about three kilowatt-hours per cubic metre, all-in, of which roughly two are the membrane stage and the rest is intake, pretreatment and delivery.

Against a reversible 1.10 at the recovery such plants run at, the second-law efficiency of the membrane stage is around fifty per cent, and of the whole plant around thirty-five. That is a remarkable figure for a process operating at ambient temperature, and it has a consequence that is worth stating plainly: desalination cannot get much cheaper by improving the separation. The remaining margin is a factor of two or three, not a factor of a thousand.

The history makes the point better than the number. Seawater desalination cost about 10 kWh/m³ in 1980 and about 3 today, and nearly all of that improvement came from two things: energy recovery devices, which take the pressure left in the concentrate and put it back into the feed, and larger membrane modules that allow a lower flux and therefore less irreversibility per unit area. Both are moves toward the reversible limit rather than improvements in the membrane’s selectivity.

At a working membrane the water leaves and the salt does not, so a layer of concentrated brine builds up against the surface and has to diffuse away against the flow that keeps making it. That is concentration polarisation, and it is the main reason a real stage runs above its own thermodynamic cost: the local osmotic pressure at the membrane face is higher than the bulk value, so the applied pressure has to beat a number the feed tank never reports.

The other three ways of doing it #

Reverse osmosis is not the only separation, and comparing the alternatives against the same floor is the clearest way to see what the floor means.

Distillation boils the water and condenses it, which costs the latent heat — about 630 kWh per cubic metre if the heat is used once. Multi-stage flash and multi-effect distillation reuse the heat across stages and bring that down to 10 or 15 kWh of thermal energy, which converted to work-equivalent is a few kilowatt-hours. The reason distillation was ever competitive is that low-grade heat is cheap, not that the process is efficient.

Freezing exploits the fact that ice excludes salt, and costs the latent heat of fusion — a seventh of vaporisation’s, so about 90 kWh/m³ before any heat recovery. It has been tried repeatedly and defeated each time by the difficulty of washing brine off ice crystals — and the exclusion itself is the same phase equilibrium a boiling point is, read at the other end of the temperature scale.

Electrodialysis moves the ions instead of the water, which is a good idea when there are few of them to move: its cost scales with the salt removed rather than with the water produced, so it beats reverse osmosis for brackish water and loses badly for seawater.

The comparison is what makes the thermodynamic floor useful. It is the one number that all four processes can be measured against, and it makes clear that three of them are not competing on efficiency at all — they are competing on what kind of energy they consume.

One way to hold the size of the number: seawater’s osmotic pressure of about 28 bar is the pressure at 290 metres of water. So separating a cubic metre of fresh water from the sea costs the same as lifting that cubic metre out of a very deep well — which is a large number for a household and a small one for a city, and is why the argument about desalination is always an argument about scale.

Where the same accounting appears #

In every separation in chemistry. The minimum work to separate any mixture is the free energy of mixing, and for a dilute component it grows as the logarithm of how dilute it is. Removing a contaminant present at one part per million costs about fourteen times as much per mole as removing one at one part in ten — which is why dilute waste streams are expensive to clean and why concentrating before separating is nearly always the cheaper route.

In carbon capture, where the same arithmetic sets the debate. Extracting CO₂ from the air at 400 parts per million has a thermodynamic floor several times that of extracting it from a flue gas at ten per cent, for exactly the reason above. No amount of engineering closes that gap, because it is a ratio of logarithms of concentrations.

And in the reverse direction, as a power source. Where a river meets the sea, the mixing that a desalination plant undoes is happening for free, and the free energy released is the same quantity: about 0.8 kWh per cubic metre of river water. Pressure-retarded osmosis is the attempt to collect it, and the reason it has not succeeded commercially is that the membrane area needed scales with the power, and the power density is low.

Nothing in this essay is about rates. Every number is a minimum work, achieved only in the limit of infinite slowness, and the exponential dependence that decides how fast anything actually happens appears nowhere in it. That is exactly where the remaining gap lives: the distance between a thermodynamic bound and a practical process is always a rate, which is why the last factor of two is the hardest one to close and why it is closed with area rather than with cleverness.

The number that decides the argument about water #

It is worth converting the floor into the units in which desalination is actually argued about, because the conversion changes the shape of the debate.

A cubic metre of fresh water is a thousand litres, and a person in a developed country uses about 150 litres a day for everything domestic. At the reversible 1.10 kilowatt-hours per cubic metre, a person’s domestic water costs 0.17 kilowatt-hours a day to separate from the sea — about the energy of running one incandescent bulb for two hours. At a real plant’s three kilowatt-hours it is 0.45, which is still under one per cent of a developed country’s per-person energy use.

That is the honest scale, and it is why the practical constraints on desalination are the capital cost of the plant, the disposal of the brine and the cost of moving water to where it is wanted, rather than the energy of separation. Agriculture is where the arithmetic changes: growing a kilogram of wheat takes a cubic metre of water, so irrigating from the sea costs several kilowatt-hours per kilogram of grain, which is comparable to the energy content of the grain itself.

The general habit is worth keeping. A thermodynamic floor is most useful when it is converted into the units of the decision being made, and the conversion often shows that the bounded quantity is not the one that matters.

What the membrane actually does #

The floor is indifferent to the mechanism, which is what makes it a floor. The mechanism is nevertheless worth knowing, because it is not what the word membrane suggests and the difference has consequences that are measurable.

A reverse-osmosis membrane is not a sieve. There are no pores small enough to pass a water molecule and stop a hydrated sodium ion, and nothing is being strained. The selective layer is a dense amorphous polyamide, about a hundred nanometres thick, with no continuous void space at all. Water gets through by dissolving into the polymer, diffusing across it down a gradient of chemical potential, and coming out the other side. Salt does the same, and gets through far less, because its solubility in the polymer is much lower.

That mechanism makes a prediction a sieve does not, and the prediction is what confirms it. In a sieve, raising the pressure pushes more of everything through and the composition of what emerges is unchanged. In solution-diffusion, the water flux is proportional to the excess of applied pressure over osmotic pressure, while the salt flux is proportional to the concentration difference and does not depend on pressure at all. So running harder produces more water and the same amount of salt — and the rejection improves. Every membrane’s datasheet shows exactly that, and it is not what a filter does.

The numbers are good: a seawater membrane rejects 99.5 to 99.8 per cent of the salt, turning 35,000 parts per million into a few hundred in one pass.

The other half of the design is where the invention was, and it is a nice illustration of separating two requirements that look like one. Selectivity is a thermodynamic property — how much less soluble salt is than water in the polymer — and it does not depend on the film’s thickness. Flux is a kinetic property and is inversely proportional to thickness. Early membranes were thick, so they were selective and useless.

Loeb and Sourirajan’s contribution around 1960 was to make an asymmetric membrane: a very thin dense skin supported by a thick porous layer of the same material, so the selective part is a fraction of a micrometre and the mechanical part carries the sixty bar. Cadotte’s thin-film composite of the 1970s went further, forming the polyamide skin by a reaction at the interface between two immiscible solutions, so its thickness is set by the reaction and not by any casting process.

The whole of modern desalination rests on that decoupling. Selectivity and permeability are independent properties, and the way to have both is to make the selective layer as thin as it can be made and put something else underneath to hold the pressure.

The half that is thrown away #

The recovery figures used above have a consequence that no energy calculation contains: at fifty per cent recovery, half of everything drawn from the sea leaves the plant again, at twice the salinity it came in at.

The volumes are large. A plant producing a hundred thousand cubic metres of fresh water a day takes in two hundred thousand and discharges a hundred thousand of concentrate at about seventy grams of salt per kilogram, along with whatever antiscalant and coagulant the pretreatment used, and a couple of degrees of waste heat from the pumps.

The physical problem is that the concentrate is denser than seawater and therefore sinks. Discharged without care it forms a hypersaline layer along the seabed, and the seabed is where the benthic communities are. The engineering answer is a diffuser: a manifold of nozzles that fires the brine out as a fast jet, so that it entrains many times its own volume of ambient seawater before it slows, and arrives at the bottom already diluted. Where a power station is next door, co-discharging with its cooling water does the same job for free.

There is a thermodynamic reading of the discharge that is worth stating, because it closes the essay’s own accounting. The concentrate is not waste in the energetic sense: relative to the sea it is being dumped into, it holds free energy — precisely the free energy the plant spent to concentrate it. Letting it mix back irreversibly destroys that work, and it is the last and largest irreversibility in the whole chain, larger than anything happening inside the membrane.

That is why the same industry keeps returning to the idea of recovering it, and why the recovery keeps not being worth doing. The available work is real and it is spread very thinly: about a kilowatt-hour per cubic metre, delivered as a slow flow across an enormous membrane area. The energy is there; the power density is not.

Why the membrane does not appear in the answer #

A last point of principle, because it is the one that makes the whole calculation worth doing.

Nothing in the floor mentions a membrane. The quantity computed is a difference of free energies between two states of matter — salt water on one side of the ledger, fresh water and brine on the other — and a difference between states does not care what route joined them. A membrane, a still, a freezer and an electric field are four routes, and every one of them is bounded by the same number.

That is what a thermodynamic argument is for, and it is also its limitation. It says nothing about whether any route exists, how fast it runs, or what it costs to build; a bound with no mechanism attached can be discouraging or encouraging and is never a design. The exclusion of the impossible is what thermodynamics does well, and the reason its bounds are trusted is precisely that they are indifferent to the apparatus.

The ledger is the same one every irreversible process is charged against. Separating is the removal of an entropy of mixing, removed in steps, each of which must be paid for at the temperature it is removed at — and the total is fixed by the endpoints however many steps are used. What staging buys is not a lower total; it is a closer approach to it.

What the pictures cannot show #

The van 't Hoff expression is the dilute limit. Seawater is concentrated enough that the exact chemical-potential expression gives a few per cent more, and a real calculation uses measured activity coefficients rather than a count. The shape of every conclusion is unchanged and the numbers move by less than the spread between plants.

Nothing here is a rate. The reversible curve is approached only as the process is slowed to a stop, and every real plant trades energy against throughput. The right way to read the second-law efficiency of thirty-five per cent is as a statement about how much the industry has already spent to slow itself down.

The feed is one salt at one temperature. Real seawater varies from 32 to 40 grams per kilogram, and the Gulf is at the top of that range and warm as well — which is why plants there use more energy per cubic metre than the Mediterranean’s, before any difference in equipment.

And the boundary drawn is around the separation only. Intake screens, pretreatment, chemical dosing, brine disposal and pumping to the customer are outside every curve on these figures, and together they are about a third of a real plant’s consumption and most of its environmental argument.

The ladder from here #

Later rungs on this anchor: the exact chemical-potential derivation, in which the osmotic pressure is what makes the solvent’s potential equal on both sides; concentration polarisation, and the boundary-layer calculation that decides how far a real stage sits above its own thermodynamic cost; pressure-retarded osmosis, which runs the same device backwards as a power source; and the general minimum work of separation as a function of dilution, which is where carbon capture and trace purification get their numbers.

The neighbouring ladders are the pressure that comes from counting, which is the quantity this essay integrates, mixing what is already mixed, which is where the free energy came from, and the ceiling on every engine, which is the same structure of argument applied to heat.