The waterfall at every river mouth

The Amazon discharges about two hundred thousand cubic metres of fresh water into the Atlantic every second. As it mixes with the ocean, the salt spreads into it and the free energy of the two separate waters — the reason mixing is the overwhelmingly probable arrangement — is given up. Where does it go? Almost nowhere that can be measured. Mixing salt and fresh water releases very little heat; what changes is the entropy, and the free energy is simply lost, as the work that could have been had from two buckets of water at different temperatures is lost when they are poured together. Nothing in a river mouth looks like a power source, and yet the energy released per cubic metre of fresh water is the same as if that cubic metre fell nearly three hundred metres. Every river mouth on Earth is a waterfall of that height that nobody can see.

The calculation is not new. R. E. Pattle pointed it out in 1954, and in the early 1970s Sidney Loeb, one of the inventors of the reverse-osmosis membrane, proposed catching the energy with the same membrane run backwards. What it costs to take the salt out priced desalination at its thermodynamic floor, about 0.8 kilowatt-hours to take a cubic metre of fresh water out of the sea. Mixing is the same process with the arrow reversed, and the same number is the most it can return.

The height of the waterfall #

The energy available from mixing a small volume of fresh water into a large body of salt solution is the solution’s osmotic pressure times the volume. The pressure that comes from counting found where that pressure comes from: the solvent molecules arriving at a membrane from the fresh side outnumber those arriving from the salt side, and the difference has to be stopped by a pressure on the salt side. For dilute solutions it is van 't Hoff’s law, proportional to the number of dissolved particles; for concentrated ones it is corrected by the osmotic coefficient, which the pressure that stops counting molecules followed as ions begin to interfere with one another.

Dividing the osmotic pressure by the weight of a cubic metre of water turns it into a height. Sea-strength sodium chloride, at 28.4 bar, is a fall of 290 metres — the height of the Eiffel Tower — for every cubic metre of river water. The brine left behind by a desalination plant, about twice as salty, is worth 617 metres. A saturated salt lake is worth nearly four kilometres. The curve bends upward because a concentrated brine is not an ideal solution: its ions are close enough together to bind water around themselves, and its osmotic coefficient rises from about 0.92 at sea strength to 1.28 at saturation.

Spread over the world’s rivers — some 37,000 cubic kilometres a year reach the sea — the 0.79 kWh per cubic metre comes to about three terawatts on average, comparable to the whole world’s consumption of electricity. Nearly all of it is inaccessible: rivers cannot be diverted wholesale through membranes, estuaries are ecosystems, and most of the flow arrives in floods. But the number explains why the idea keeps returning. A waterfall nearly three hundred metres high at the mouth of every river is not a resource anyone can ignore on principle.

Where an estuary’s energy goes #

The free energy of mixing is $\Delta G = \Delta H - T\,\Delta S$, and for a solution as dilute as seawater the enthalpy term is small: dissolving more water into brine neither releases nor absorbs much heat. Almost the whole of the 0.79 kWh is the entropy term, $T\Delta S$ — the gain in the number of ways the salt ions can be arranged once they have the river’s volume to spread into as well as the sea’s. When the waters mix in an estuary, that entropy is produced and the work is gone. No thermometer registers it. The second law has been obeyed in its quietest form, as a loss of opportunity rather than a release of heat.

That is what makes the energy hard to catch and easy to overlook. A waterfall is visible because its energy arrives as motion. The energy of a river mouth arrives as nothing at all unless something is placed in the way of the mixing — a barrier that lets one species through and stops the other, so that the urge to mix has to push against something. A semipermeable membrane is exactly that barrier, and the pressure it develops is the urge made into a force. Plants use it all the time: roots load salts into their sap and draw water in after them, building a root pressure that can push sap a few metres up a stem at night — small beside the tension that lifts water up a tall tree, which is pulled, not pushed, but the same conversion of a concentration difference into a pressure. A membrane in a river mouth is asked to do it on purpose, at the scale of a power station.

A membrane run backwards #

The device is a semipermeable membrane with river water on one side and seawater on the other — the arrangement of the membrane that almost holds, with the reflection coefficient close enough to one that water passes and salt nearly does not. Left alone, water crosses from the river side into the sea side, driven by the osmotic difference $\Delta\pi$. Now pressurise the seawater side to $\Delta P$, less than $\Delta\pi$. Water still crosses, but more slowly, against the pressure, and the extra volume it adds to the pressurised side can be let out through a turbine. The water climbs a pressure hill of height $\Delta P$ as it crosses, and the turbine collects that.

For an ideal membrane the flux of water per unit area is proportional to the net driving pressure,

$$J_w = A\,(\Delta\pi - \Delta P),$$

where $A$ is the membrane’s water permeability, about one or two litres per square metre per hour for each bar of driving pressure in modern membranes. The power per unit area is the flux times the pressure it was pushed against:

$$W = J_w\,\Delta P = A\,(\Delta\pi - \Delta P)\,\Delta P.$$

That is a parabola in $\Delta P$, zero at both ends and highest exactly halfway, at $\Delta P = \Delta\pi/2$, where

$$W_{\max} = \frac{A\,\Delta\pi^2}{4}.$$

At no pressure the water flows fastest and does no work, since it crosses no pressure hill. At the full osmotic pressure each litre would do the most work it could, 28.4 bar times a litre — but no litres cross. The best compromise does half the possible work on each litre and lets half the possible flow through. For a membrane of 1.5 litres per square metre per hour per bar, that is 8.38 watts per square metre.

The same half that every source pays #

The factor of a half is the signature of a source with an internal resistance, and it appears wherever one is asked for its maximum power rather than its maximum efficiency. A battery with internal resistance delivers the most power into a load equal to that resistance, with half its voltage across the load and half lost inside. The engine that has to finish found the thermal version: a heat engine that must deliver its work in a finite time, with finite conductances to its reservoirs, gives the most power at an efficiency well below Carnot’s, because driving heat quickly needs a temperature difference and that difference is lost. Here the membrane’s permeability is the conductance, the osmotic pressure is the open-circuit voltage, and the held pressure is the load.

In each case the reversible limit is reached only by doing nothing: an engine with no temperature difference across its conductances, a battery with no current, a membrane with no flow. And in each case the price of getting the energy at any useful rate is the same: at maximum power, half of what is available is dissipated in the conductance itself, as friction in the water forcing its way through the membrane. The membrane converts at most half the free energy that crosses it. The other half is lost as it is lost in the estuary, except that here it is lost as friction, and does warm the water a little.

A cubic metre is worth less the more are used #

The 0.79 kWh is the value of a trickle of river water into an ocean that stays at full strength. A real plant takes a finite flow of seawater and a finite flow of river water and passes them along opposite sides of a membrane, and as the river water crosses, it dilutes the seawater it enters.

Mixing $x$ cubic metres of river water reversibly into one cubic metre of seawater releases $\Delta\pi\,\ln(1+x)$ of work, for an ideal solution, and so $\Delta\pi\,\ln(1+x)/x$ per cubic metre of river water. That starts at the full 0.79 kWh for a trickle and falls steadily: with equal volumes it is 0.55 kWh per cubic metre of river water, since by the time the last of the river water arrives the seawater has been diluted to half strength. And a membrane held at one pressure throughout cannot even collect that. Water stops crossing where the diluted seawater’s osmotic pressure has fallen to the held pressure, so a single pressure $\Delta P$ collects $\Delta\pi - \Delta P$ per cubic metre of seawater; the best a single pressure can do with equal volumes is 72 per cent of the reversible limit.

This is the same staging argument that what it costs to take the salt out found in reverse: desalination pays more than the floor because it must push against the saltiest brine at the end of its run, and staging the pressure along the train recovers some of the difference. A mixing plant that wanted the last 28 per cent would step its pressure down along the membrane as the seawater dilutes, which costs pumps and complexity. Every real design settles somewhere between, and the number on the figure that matters is not 0.79 but something between 0.3 and 0.5.

Two thin layers against the membrane #

The parabola is for an ideal membrane, and no membrane is ideal in the way that matters most here. The osmotic difference that drives the water is the difference between the solutions in contact with the membrane’s two faces, not between the river and the sea, and those are not the same.

On the sea side, the fresh water arriving through the membrane dilutes the seawater in a thin layer against the membrane face, and the flow of seawater along the channel can only sweep that layer away at a finite rate, set by a mass-transfer coefficient — here 38.5 micrometres per second. The faster water arrives, the more dilute the layer. That alone costs a fifth of the power.

On the river side the problem is worse, because the membrane is not a single film. The salt-rejecting layer is a fraction of a micrometre thick and has to be supported on a porous backing hundreds of times thicker, and the river water reaches the membrane only by diffusing through that support. Salt leaking back through the membrane — no membrane rejects it perfectly — collects in the support’s pores, where no flow can sweep it out, and raises the osmotic pressure on the fresh side right where it counts. The model drawn here is Yip and Elimelech’s, from 2011, which puts the dilution, the leak and the trapping into one relation between flux and pressure; with all three the best power is 5.80 watts per square metre, and it comes at a little less than half the osmotic pressure, because the losses grow with the flux.

The more permeable membrane that gives less #

The obvious remedy for low power is a more permeable membrane, since the ideal power is proportional to $A$. The losses say otherwise.

A membrane that passes water faster dilutes the seawater at its face faster, and since the salt leak rises with the water permeability in real membranes — the more open the polymer, the less selective — it also traps more salt in its support. Both losses grow exponentially with the flux, because the concentration profile in a layer through which the water is moving is an exponential in the ratio of the flow to the diffusion. At low permeability the losses are a tenth of the power; at three litres per square metre per hour per bar, half; at eight, more than four-fifths. In this model the best power peaks at about 8.5 watts per square metre and a still more permeable membrane gives less — not because it is a worse membrane, but because it outruns the diffusion that has to feed it.

The industry’s own threshold sits on this curve. When the Norwegian utility Statkraft built the world’s first pressure-retarded osmosis prototype at Tofte in 2009, the figure it gave for a commercial plant to pay was about five watts per square metre of membrane. The model passes it at a modest permeability. The prototype reached of the order of one watt per square metre, and Statkraft stopped the programme in 2013. The difference between the curve and the plant is everything the curve leaves out: membranes fouled by the silt and organic matter in real river water, which the reverse-osmosis industry avoids by pretreating seawater at a cost a power plant cannot afford; the pressure lost in pumping two streams along hundreds of metres of narrow channel; the river’s own salt; and the energy recovered imperfectly by pressure exchangers that are themselves a few per cent lossy.

Where the gradient is steeper #

The waterfall figure points to where the economics might close. Its height grows faster than the salt content, so the same membrane between fresh water and a strong brine works under a far larger driving pressure, and the power density goes as its square. Desalination plants reject brine at about twice sea strength, worth 617 metres of head against fresh water, and coastal cities discharge treated wastewater a few hundred metres away. Pilot plants in Japan and Korea have paired the two, recovering some of the energy the desalination spent and diluting the brine before it is returned to the sea. Salt lakes and the brines of salt works are steeper still — and harder on membranes, which have to hold off pressures of a hundred bar and more. A salt-gradient lake that stores the sun’s heat at its bottom, like the pond that is hottest at the bottom, is also a store of mixing energy several kilometres high, held apart by the same stillness.

There is also a second way to catch the same free energy. Instead of letting water cross a membrane, let ions cross one: a stack of membranes alternately permeable to positive and negative ions, between alternating channels of river water and seawater, builds up a voltage across each pair of about $(2RT/F)\ln(c_{\text{sea}}/c_{\text{river}})$, which for a thirtyfold ratio is about 0.17 volts before losses — the voltage that is a chemical potential, made into a battery that the rivers recharge. This reverse electrodialysis has run as a pilot on the Afsluitdijk in the Netherlands since 2014. It faces the same arithmetic — internal resistance, maximum power at half voltage, polarisation at the membrane faces — in electrical rather than hydraulic units.

A membrane at one point, fed with clean water #

Every figure treats sea water as sodium chloride at 3.5 per cent and 25 °C, which is close but not exact: real seawater has an osmotic pressure of about 27 bar, a little below the 28.4 here, because of its magnesium and sulphate. The osmotic coefficient is interpolated from tabulated values, and the ideal-solution logarithm in the mixing-energy figure ignores the coefficient’s variation with concentration, which shifts the numbers by a few per cent.

The membrane model is one-dimensional and steady. It describes a single point on a membrane, where in a real module the seawater is diluted and the river water salted progressively along the channel, so every point sees a different driving pressure; the energy figure handles that only in its idealised form. Fouling is absent entirely, and in practice it is the dominant loss for membranes fed with untreated river water, growing over days and partly reversible by cleaning. The salt leak is scaled in simple proportion to the water permeability, where measured membranes follow a steeper trade-off between permeability and selectivity, which would bring the peak in the membrane figure lower and earlier. And the plant’s own pumping, filtration and pressure exchange — which a working design must pay from the same few watts per square metre — are not drawn at all.

Still open: whether the gradient can be harvested at a profit #

The physics is not in doubt. The free energy is there, at 0.8 kWh per cubic metre of river water and far more where brines are available, and the membranes to tap it exist. What remains open is whether a membrane can be made that resists fouling by real river water, passes water fast enough without leaking salt, and lasts long enough, at a cost that a few watts per square metre can repay. Proposals now concentrate on the steep gradients — desalination brine, salt-works brine, closed loops using thermally regenerated draw solutions — where the higher pressure buys power density, and on membranes with thinner, more open supports that shorten the diffusion path the river water must cross.

The habit worth carrying is the one the parabola teaches. A source with an internal resistance gives its most power at half its driving pressure, and pays half the available work as the price of speed; and when the source’s own losses grow with the flow, making the resistance smaller can make the power fall. The waterfall at the river mouth is real and three hundred metres high. The difficulty is that it falls through a membrane a tenth of a micrometre thick, and the water has to diffuse to it from both sides.