A filter is slowed by what it has caught

A velocity nobody has sets out Darcy’s law: flow through a porous bed in proportion to the pressure across it, with a permeability that is a property of the bed. A permeability that is only the geometry derives that permeability from the size and packing of the grains. Both treat the bed as given — a column of sand, a layer of rock, a filter already made.

A filter is the one porous bed that is not given. It is made by the flow through it. Every litre of slurry pushed through a cloth leaves its solids behind on the growing cake, so the bed’s thickness at any moment is the integral of everything that has been filtered, and its resistance grows as it works. That changes the arithmetic of filtration completely, and in the case of a soft cake it changes what pressure is for.

Two resistances in series, one of them growing #

A filter at a constant pressure difference $\Delta p$ passes liquid through two resistances in series — the same liquid through each, since it has nowhere else to go: the cloth, with a fixed resistance $R_m$, and the cake, whose resistance is proportional to how much of it there is. With $\alpha$ the cake’s specific resistance — its resistance per unit mass deposited on a unit area — and $c$ the mass of solids left behind per cubic metre of filtrate, a filtrate volume $V$ on an area $A$ has built a cake of $cV/A$ kilograms per square metre. Darcy’s law through both then says

$$\frac{dV}{dt} = \frac{A\,\Delta p}{\mu\left(R_m + \alpha c V/A\right)}.$$

The rate falls as the filtrate accumulates, and the equation integrates to a statement that has a name. The elapsed time divided by the filtrate collected is a straight line in the filtrate,

$$\frac{t}{V} = \frac{\mu\alpha c}{2A^2\Delta p}\,V + \frac{\mu R_m}{A\,\Delta p},$$

Ruth’s line, from Bernard Ruth’s work of the 1930s. Its slope is the cake’s and its intercept the cloth’s.

The cake’s specific resistance comes from the grains. A cake of spheres of diameter $d$ at porosity $\varepsilon$ is a packed bed, and the Kozeny–Carman formula of the essay before gives $\alpha = 180(1 - \varepsilon)/(\rho_s d^2\varepsilon^3)$ per unit mass. For ten-micron silica at a porosity of 0.4, $\alpha$ is $6.4 \times 10^9$ metres per kilogram.

Integrated step by step with a fourth-order scheme for an hour of filtration — water at one bar through a square metre of cloth of resistance $10^{10}$ per metre, fifty kilograms of silica per cubic metre of filtrate — the filtration lies on Ruth’s line to seven parts in a thousand million. The cloth contributes a constant hundred seconds per cubic metre. The cake contributes 1,592 seconds per cubic metre for every cubic metre already collected. As resistances, the cake overtakes the cloth after the first three hundredths of a cubic metre, and after a cubic metre it is thirty-two times the cloth’s.

The square root of time, and of pressure #

Once the cake dominates, Ruth’s line says the filtrate is proportional to $\sqrt{t}$: to collect twice as much liquid the filter must run four times as long. The same square root applies to the pressure, because pressure and time enter as a product.

Integrated over long times, the exponents come out at 0.502 in time and 0.501 in pressure, the small excesses being the cloth’s share, which is still visible after an hour. Four times the pressure buys twice the filtrate. It is the same diminishing return that governs anything whose resistance grows with what it has done: the more a filter has achieved, the harder it works for the next litre.

The practical consequence is the filtration cycle. Since the rate falls continuously, a batch filter is run for a time, stopped, emptied of its cake and restarted, and the best cycle time balances the falling rate against the downtime for cleaning. The square-root law makes the balance simple: the output per cycle including downtime is greatest when the filtering time equals the downtime, a result any plant engineer can apply and which follows from nothing more than Ruth’s line.

A cycle that stops at the right moment #

The best cycle can be put in numbers for the silica slurry. Suppose emptying the cake and closing the filter again takes twenty minutes. Run the filter for twenty minutes too and it collects 0.84 cubic metres per square metre of cloth, so each forty-minute cycle averages 1.26 cubic metres an hour. Run it for forty minutes and it collects 1.20, a larger batch but only 1.20 cubic metres an hour over the hour-long cycle. The rate at which a filter produces is highest when it is stopped while still producing well, because the last litres of a long run are the most expensive ones in the whole cycle.

That rule depends on nothing but the square root. A filter whose cloth dominates — a very clean liquid, or very coarse particles — has an output nearly proportional to time and should be run as long as possible; a filter whose cake dominates should be cycled. Where a particular filter sits between the two is exactly what the intercept and slope of its Ruth line say, which is why that line is drawn from the first test a plant engineer runs.

Measuring a cake from its line #

Ruth’s line is as useful as a measurement as it is as a prediction. A small leaf of cloth dipped in a sample of slurry and run at a fixed pressure gives a time and a volume every few seconds; plotted as time over volume against volume, the points fall on a line whose slope gives $\alpha c$ and whose intercept gives $R_m$. Repeat the test at a second pressure and the slopes differ if the cake is compressible: the average specific resistance of a Tiller cake rises with pressure as $\Delta p^n$, so two pressures give the compressibility index. Three numbers from two short experiments are enough to size a plant filter, and the same arithmetic run backwards is what the figures above compute.

The flow through such a cake is slow in the sense that matters for Darcy’s law. At the start of the silica filtration the liquid passes the cloth at a centimetre a second and, once the cake has formed, at about a millimetre a second; through ten-micron pores that is a pore Reynolds number of about a hundredth, a thousand times below the value at which Darcy’s law begins to fail. Filtration is squarely in the linear regime, which is why its laws are so clean.

The coffee filter #

The everyday filter shows all of this in a few minutes. Pour water over ground coffee in a paper filter and the first cupful passes quickly; the rest slows, because the finest particles of the grounds are carried down onto the paper and form a cake over it, and because the bed of grounds is itself a cake that compacts as it is wetted. A finer grind makes a slower brew by the inverse square of the particle size, as the figure above says for silt and clay, and pressing on the grounds — as a hurried brewer sometimes does — compacts the bed against the paper and slows it further, which is the compressible cake’s lesson in a kitchen.

The espresso machine resolves the conflict the other way. It uses a very fine grind for extraction and forces water through at nine bar, a pressure at which a coffee puck behaves as a compressible cake; the shot’s flow rate responds far less to the pump pressure than a rigid bed’s would, which is why baristas control a shot by the grind and the dose rather than by the pressure.

The particle size is almost everything #

The Kozeny–Carman resistance goes as the inverse square of the grain size, and the time to filter a given volume goes, once the cake dominates, as the resistance. So the particle size is almost the whole of a filtration’s difficulty. A cubic metre per square metre of ten-micron silica takes 28 minutes at one bar. The same of one-micron clay takes 2,655 minutes, nearly two days. Hundred-micron sand takes two minutes, most of which is the cloth.

That is why fine slurries are the hard case of solid–liquid separation, and why industry spends so much effort making fine particles coarser before filtering them: adding a flocculant that makes clay particles clump into aggregates ten times larger cuts the filtration time by a hundred. It is also why a filter aid — a coarse, rigid powder such as diatomaceous earth mixed into the slurry — works: it forms a cake with a large-particle resistance through which the fine particles are caught without being allowed to form their own.

A soft cake packs itself tighter #

Everything so far has assumed the cake is rigid, with the same porosity throughout. Many are not. A cake of flocculated clay, of biological cells, of fine precipitates, is soft, and the liquid flowing through it drags on its particles. That drag is transmitted through the cake from particle to particle, so the solid stress in the cake rises from nothing at its face to the whole pressure drop at the cloth, and a soft cake compresses under it — loosest at the face, densest against the cloth.

The standard description, due to Tiller and his co-workers, makes the local specific resistance a power of the local solid stress, $\alpha = \alpha_0(1 + p_s/p_a)^n$, with $n$ the compressibility index — zero for a rigid cake, of order one for a very soft one. In the cake’s own mass coordinate Darcy’s law becomes $dp_s/dw = \mu q\,\alpha(p_s)$, and integrating across a cake of $W$ kilograms per square metre gives the flux in closed form:

$$\mu q W = \int_0^{\Delta p}\frac{dp_s}{\alpha(p_s)}.$$

The profile shows where the resistance lives. Integrating the stress through a cake of five kilograms per square metre at two bar, with a compressibility index of 0.7, the stress rises slowly through the loose outer layers and steeply near the cloth, and the porosity falls from 0.85 at the face to 0.54 against the cloth. The layer against the cloth is the densest and least permeable, and it carries most of the pressure drop. Integrated from the face with the flux from the closed form, the stress arrives at the cloth equal to the imposed two bar to a part in $10^{13}$, which is the check that the closed-form flux is right.

More pressure, and almost no more flow #

The integral has a consequence that the rigid case hides. For an incompressible cake it is proportional to $\Delta p$, and a hundredfold increase in pressure gives a hundredfold increase in flow through the same cake. With a compressibility index of a half, it gives 13. With an index of one, 2.9. With an index above one the integral is bounded however large the pressure becomes, and at 1.3 a hundredfold increase gives 1.7. The extra pressure goes into compacting the layer against the cloth, whose rising resistance absorbs almost all of it.

A compressible cake stops paying for pressure. That is the answer to the refinery engineer who doubles the pump’s pressure and finds the output barely changed: the pump is compressing the cake, not driving the liquid. The remedies are the ones that change the cake rather than the pressure — a filter aid that props the cake open, a thinner cake removed more often, or a squeeze applied mechanically at the end of a cycle, when compacting the cake is the aim rather than the side effect.

The same effect sets a limit on dewatering. A sludge pressed to remove its water compacts first against the drainage surface, sealing it, and the pressure that was meant to drive the water out holds it in instead. Belt presses and filter presses are designed around this, applying pressure in stages so the cake is never compacted faster than the water can leave it, and the last of the oil squeezed from between two plates is a relative of the same problem in a different geometry.

Where the cake’s resistance came from #

The cake is a packed bed, and the same force balance that lifts a bed when the flow through it exceeds its weight appears here in a different guise: the drag of the flowing liquid on a bed’s particles is transmitted to whatever holds the bed up. In a fluidised bed the particles are held by the flow and rise when the drag exceeds their weight. In a filter cake they are held by the cloth, and the drag presses them against it — which is exactly the stress that compacts a soft cake. The bed that weighs itself and the cake that squeezes itself are two ends of one balance.

And the cake’s structure records its history, which is where the outlet that is the inlet a while ago meets filtration. The first particles deposited sit against the cloth under the full stress for the whole filtration, the last ones at the face under almost none, so a compressible cake’s layers were each compacted by a different pressure for a different time. A cake that creeps under sustained stress keeps compacting after the flow is steady, and its resistance keeps rising at constant pressure — an effect Ruth’s line, which assumes the resistance per kilogram fixed, cannot show.

What the picture cannot show #

A uniform slurry and a flat cake. Real slurries settle as they filter, so the cake’s composition changes with depth; and a real cake cracks, especially as it dries, opening channels that short-circuit its resistance.

Particles that stay on the cake. Fine particles can migrate into the cake or through it and blind the cloth, which raises the cloth’s resistance during the filtration rather than leaving it fixed.

A compressibility law. The power-law form of the specific resistance is a fit to measurements on each material, with its index and reference stress measured in a compression cell. The conclusions about saturation hold for any law whose resistance grows faster than the stress; the numbers are for the form used.

The convention the numbers depend on #

$\alpha$ is the specific cake resistance in metres per kilogram, $R_m$ the cloth’s resistance in reciprocal metres, $c$ the mass of dry solids deposited per cubic metre of filtrate, and $V$ the filtrate per square metre of cloth. The liquid is water at a viscosity of a millipascal-second. The compressible cake follows $\alpha = \alpha_0(1 + p_s/p_a)^n$ with $\alpha_0 = 10^{10}$ m/kg and $p_a$ = 0.1 bar, and its porosity falls with the solid stress as the 0.15 power.

Who found it, and when #

Constant-pressure cake filtration and its straight-line plot are Bernard Ruth’s, from a series of papers in the 1930s that turned filtration from a craft into a measured process; the plot is still how a filter’s constants are obtained from a laboratory test. The compressible cake’s theory was developed by Frank Tiller and his students from the 1950s to the 1990s, including the local-stress formulation used here. The Kozeny–Carman resistance goes back to Kozeny in 1927 and Carman in 1937, and Darcy’s law to Henry Darcy’s measurements on the sand filters of Dijon in 1856 — which is to say that the first experiment on flow through porous media was itself an experiment on a filter.

Still open: when to stop and wash #

The square-root law makes the best cycle time for a simple batch filter a one-line result. A real filter does more between cycles: it washes the cake to recover the liquid still in its pores or to remove impurities from the solids, and the wash liquid flows through the finished cake at a constant rate set by the cake’s final resistance. The calculation that follows adds the washing stage — displacement of the pore liquid by the wash, with the dispersion a packed bed’s own flow produces — and asks for the cake thickness that maximises clean product per hour. For a compressible cake the answer should shift towards thin cakes removed often, because a thick one spends most of its life resisting its own wash.