The outlet is the inlet, a while ago

A velocity nobody has establishes what a superficial velocity is and what it is not, and where Darcy stops finds the Reynolds number at which the linear law gives out. Both are statements about the average flow through a bed, and both are right.

This essay is about a question the average cannot answer, and it is the question a process engineer actually has: not how fast the fluid goes through, but when a particular piece of it comes out.

A bed is a filter, in the signal sense #

Put a step change of concentration into the inlet of a packed bed and watch the outlet. What comes out is not a step; it is a smeared step, and the smear is the whole subject.

The reason is that the fluid does not go through in single file. One molecule takes a wide pore between two grains and gets through quickly; another takes a narrow one, or a dead end, or spends time in the slow layer at a grain’s surface. They entered together and they leave at different times, and the distribution of those times is a property of the bed.

Write that distribution as E(t) — the fraction of a pulse leaving per unit time. Then the outlet concentration is the inlet concentration convolved with E. The bed is a linear filter whose impulse response is E, and its mean residence time is the first moment of that response and nothing more.

Everything in this essay follows from noticing that a first moment is one number about a function.

The distribution, and why this one #

The model used here is the simplest one that is not a lie: the tracer advects at the interstitial velocity and disperses with a constant dispersion coefficient, and E is the distribution of the time it first reaches the outlet plane.

That first-passage time has a closed form — the inverse Gaussian — and it is worth having its exact properties written down because every check below is against one of them. Its mean is exactly the mean residence time. Its variance is exactly twice the square of that time over the Péclet number — at a Péclet number of 40 the computed area is 1.0000000000, the mean 1.0000000000 and the variance 0.0500000000, which is 2/40 to ten places, where the Péclet number is the interstitial velocity times the bed length over the dispersion coefficient. And its Laplace transform is a closed expression too, which matters because that is what a first-order reaction converts by.

So there is one dimensionless number governing the shape: Pe. A high Péclet bed is a nearly perfect plug; a low one is nearly stirred.

Six beds, one mean #

Here are six beds with Péclet numbers from 4 to 400 and identical mean residence times. Every one of their breakthrough curves passes through a half at the same instant, which is what having the same mean means and is the only agreement between them.

The rest of the curves are nothing alike. The loosest bed’s outlet reaches a tenth of the inlet step at 0.36 of the mean time; the tightest reaches it at 0.91. At the other end, the loosest is still climbing at 1.89 mean times when the tightest finished at 1.09.

The number a filter is actually sized on is the first breakthrough, not the mean, because that is when whatever the bed was catching begins to appear downstream. Across these six it runs from 0.20 of the mean residence time to 0.85 — a factor of four in the quantity that decides when the bed has to be changed, with no change at all in the quantity that gets quoted.

How much of the past is in the outlet #

The same information reads as a memory, and this is the reading that puts the essay beside its neighbours.

At any instant the outlet stream is a mixture of fluid that entered over a spread of earlier times. The width of E between its tenth and ninetieth percentiles is that spread — the window of inlet history the outlet is currently averaging over:

All six have a mean residence time of exactly 1. The window spans a factor of 8.5 across the sweep and the first arrival a factor of 4.2, while the mean does not move at all — which is the whole of what a mean cannot tell anybody about a bed.

For the loosest bed it is 1.53 mean residence times. For the tightest it is 0.18. So the same physical question — what was the inlet doing when this fluid went in? — has an answer that is a smear across a day or a smear across two hours, depending on a property of the packing that the mean residence time does not record.

This is why a bed cannot be used as a fast sensor of its own inlet, and why a spike of contaminant arriving at the inlet of a loose bed appears at the outlet as a long low plateau rather than a spike. The bed does not delay the signal; it forgets the shape of it, and the two are different failures.

What the spread costs a reaction #

The consequence with money attached is chemical.

Run a first-order reaction in these beds with the rate constant chosen so that plug flow would convert 86.5 per cent. The six beds convert 76.9, 81.9, 84.5, 85.6, 86.1 and 86.3 per cent.

Every one of them is below plug flow, and that is not an accident of the numbers. The unreacted fraction is the average of an exponential over the residence times, and an exponential is convex, so the average of it exceeds the exponential of the average. Spreading the residence times about a fixed mean can only make conversion worse. The fluid that leaves early has not had long enough, and the fluid that stays late cannot make up for it because it is already nearly all reacted.

The shortfall closes as the bed tightens, and it closes fast: it is 71 times smaller at Pe 400 than at Pe 4. That is the useful engineering statement — above a few hundred, the mean residence time is enough; below about twenty, it is not — and it is a statement about a number nobody measures unless they have been told to.

What the solver computed, and how it was checked #

This section is short because the model has a closed form and the checks are against it rather than against another computation.

All six have a mean residence time of exactly one, and plug flow at the same mean converts 0.865.

The area is one to ten decimal places, which says the quadrature reaches far enough into both tails.

The mean is the mean residence time to ten decimal places, which is the check that the distribution is the one claimed — a first-passage time with drift has this property and several nearby distributions do not.

The variance is 2/Pe to ten decimal places, which is the sharpest check available because it is the one place a factor of two would hide.

And the Laplace transform matches its closed form to nine figures. That is the check that matters most, because the conversion figure is the same integral with a different integrand, and without it the conversion curve would be an unverified number drawn under a caption.

The refusals are the other half. The distribution is asked to build itself at a Péclet number of zero — a bed with no advection, which has no residence time at all — and it refuses rather than returning a grid of NaN.

Where the dispersion actually comes from #

The model takes the dispersion coefficient as given, and it is worth saying what it is made of, because that is where the Péclet number a real bed has comes from.

Mechanical dispersion dominates in beds at ordinary flow rates: the fluid takes many different paths between the grains, the path lengths differ, and the spread grows with distance. For a packed bed the resulting axial Péclet number based on the grain diameter settles at around two over a wide range of flow rates, which makes the bed-length Péclet number roughly twice the number of grain diameters in the bed. A bed a hundred grains deep is therefore a Pe of about two hundred, and a bed ten grains deep is about twenty — which is why shallow beds behave badly and why the rule of thumb about bed depth exists.

Molecular diffusion takes over at very low flow rates, and there the Péclet number falls with the velocity rather than staying put.

And the slow zones — dead ends, stagnant films, adsorbed layers — produce a tail that is not in this model at all. A single dispersion coefficient makes E symmetric on a logarithmic-ish scale; real beds routinely have a long right tail from fluid that got stuck, and a tracer test is how it is found.

The permeability essays in this ladder are about the same geometry from the other side. A permeability that is only the geometry computes what the pore structure does to the mean flow; the dispersion is what the same structure does to the spread, and the two are independent enough that beds matched on one can differ on the other.

The other machine in this field with a window of inlet history is a pipeline, and the same solver draws its kernel.

The same shape in three other places #

A window of history over which an output mixes its input is not a bed’s private property.

A layer that is an integral of everything upstream computes the same thing for a boundary layer: the momentum thickness at a station is a weighted integral of the pressure gradient behind it, and the weighting has a width.

How far downwind a surface is remembered computes it for the atmosphere, where the window is a fetch rather than a time and runs to kilometres.

And a scalar is a record of where its fluid was is the general statement: a concentration is not a state of the flow, it is a record of the flow’s own history, and the only question is how much of that history is still legible.

The bed is the cleanest of the four because its kernel has a closed form. That is worth having, because it means the width can be computed rather than fitted, and the fitting is what usually happens.

The number this belongs to #

The Péclet number here is a memory time over a process time like every other group this collection produced, and it is worth writing it that way because the reading changes.

The dispersion coefficient sets a time — the time for dispersion to spread a patch across the bed length, which is the length squared over the coefficient. The advection sets another — the mean residence time. The Péclet number is the first over the second, and every memory number is one time over another puts it on the same axis as the Deborah and Stokes numbers.

Read that way the two limits are the familiar pair. High Péclet is the frozen limit: the fluid gets through before dispersion has had time to do anything, so the parcels keep their identity and the bed is a delay line. Low Péclet is the quasi-steady limit: dispersion finishes before the fluid leaves, so the bed is a stirred tank with no memory of order at all. And the interesting band, as usual, is the two decades in between, where the bed is neither and the shape of E has to be carried.

The other essays in this ladder sit at the same place on a different axis. The bed that weighs itself is about the point at which the bed stops being a bed, and everything here assumes it has not reached it — a fluidised bed’s residence-time distribution is a different object with a different tail, because the solids are moving too.

What a smeared inlet does to whatever is downstream #

The practical failure this causes is not in the bed. It is in whatever is trying to control the process the bed sits in.

A control loop that acts on the outlet composition is acting on a quantity that is a weighted average of the inlet over the last window. That average has a delay — most of the response arrives after the first breakthrough and before the ninetieth percentile — and a loop tuned as though the delay were the mean residence time will be tuned for the wrong number, because the effective delay is nearer the first breakthrough than the mean.

Worse, the window is a low-pass filter with a corner frequency set by its own width. An inlet disturbance faster than that corner does not reach the outlet at all, so the loop cannot see it, cannot correct it, and will keep operating as though nothing happened — while the disturbance is doing whatever it does inside the bed.

That is the same structure as a boundary that only exists over a window: a feature that is real at one averaging window and absent at another, with nothing in the measurement to say which window was used.

What to measure, and what to do with it #

The measurement is a tracer test and it is cheap: inject a pulse, log the outlet, and E is measured directly.

Read the mean first and check it against bed volume over flow. If they disagree, part of the bed is not being used — channelling, or a dead volume — and no amount of modelling fixes that.

Read the variance and get Pe from it. Two over the variance in units of the mean squared. That one number then gives the whole curve, under this model.

Then check the tail against the model. A measured E with more area at long times than the inverse Gaussian has is a report of stagnant zones, and the excess area is roughly how much of the bed they are.

And size the bed on the first breakthrough, computed from the fitted Pe, rather than on the mean. That is the change this essay argues for and it costs one extra line of arithmetic.

What the picture cannot show #

The distributions are drawn against time in units of the mean residence time, which hides the thing an operator cares about: two beds at the same Péclet number and different lengths have the same shape and completely different absolute windows. The collapse is honest and it is a collapse.

The breakthrough curves are drawn for a step at the inlet. A real inlet history is not a step, and the outlet is its convolution with E — so a slowly varying inlet produces an outlet that looks like the inlet delayed, and the bed’s own character is invisible until something changes quickly.

And the conversion figure is drawn under complete segregation: each parcel is assumed to react on its own, without mixing with parcels of different age. That is one of two bounds. The other, maximum mixedness, gives a slightly different answer, and for a first-order reaction — only for a first-order reaction — the two coincide exactly, which is why the reaction chosen here is first order.

Where the model stops #

One dimensionless number. The whole shape here is set by Pe, and a real bed’s E can be bimodal — two populations of paths — which no single Péclet number can produce.

Constant dispersion. The coefficient is taken as uniform along the bed. Near the walls it is not, because the packing is looser there and the fluid goes faster, and in a bed less than about ten grains wide the wall channel is a substantial fraction of the flow.

No adsorption. A bed that holds the tracer as well as passing it has a retarded mean residence time and a different shape, and that is the case most filtration is actually in.

And no reaction feedback. The conversion here is computed by running the reaction along each streamline of a fixed flow field. A reaction that changes the fluid’s viscosity or blocks the pores changes the flow that carries it.

Who found it, and when #

Danckwerts wrote the residence-time distribution down in 1953, in a paper whose whole content is the observation this essay is built on — that a vessel has a distribution of residence times and that its mean does not determine its behaviour. Levenspiel’s textbook of 1962 turned it into engineering practice, and the axial-dispersion model with its Péclet number is from the same period.

The inverse Gaussian as a first-passage time is older and belongs to probability rather than to chemical engineering; Schrödinger and Smoluchowski both derived it in 1915, for a Brownian particle reaching a barrier, and it arrived in this subject by a different route.

What has not changed is which number gets quoted. A bed is still specified by its volume and its flow, and its residence-time distribution is still measured only when something has gone wrong.

Limits recorded rather than smoothed over #

A model distribution, not a measured one. Everything here is the inverse Gaussian. Its virtue is that it is exactly checkable and that it has one parameter; its vice is that real beds have tails it cannot produce.

The six Péclet numbers are chosen to span the range, not sampled from anything. What they demonstrate is the size of the effect across the range, and the range is representative of beds from ten to a thousand grains deep.

The conversion is first order for the reason given above, and the coincidence of the two mixing bounds does not extend to any other order. For a second-order reaction the RTD alone is not enough and the answer depends on how the fluid mixes as well as on when it leaves.

And the window is a ten-to-ninety width, which is a choice. A ten-to-ninety width is well determined from a measurement; a full width is not, because it depends on the tail, which is exactly the part of E a tracer test measures worst.