Peclet number — where it appears
Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.
Two slow things make a fast one
Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.
The other layer, and the one number that separates them
A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.
How far a parcel gets
Turbulent transport is one integral. A parcel carried by a fluctuating velocity goes in a straight line while it still remembers its own motion and performs a random walk once it has forgotten, and the crossover is the correlation time — so an eddy diffusivity is not a property a flow has, it is the limit of a measurement that has run for long enough.
How far before the heat arrives
A duct's thermal entry length is quoted everywhere as x/(D·Re·Pr) = 0.05, with no tolerance attached. Working out what it delivers gives a Nusselt number 1.45 per cent above the developed value — and the developed value itself is not a term ratio at all but an eigenvalue, 3.6568, which is also the rate at which the duct forgets its inlet.
The outlet is the inlet, a while ago
A bed has a mean residence time and everybody quotes it. Six beds with the same mean let their first hundredth through at 0.20 and at 0.85 of it, mix a window of inlet history between 1.53 and 0.18 wide, and convert a first-order reaction by amounts the mean cannot distinguish.
The scalar has its own cascade
Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.
The heat the eddies do not carry
In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.
The best a tube can do is its own radius
Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.
Transport with nothing transported
Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.
A washed filter works as long as it rests
A batch filter that has to be emptied between runs has a best cycle, and the rule for it is exact: the time the cake is responsible for equals the time the filter stands idle. Washing the cake does not break the rule, because a wash through a finished cake costs time in proportion to the square of its thickness, just as filtering it did. What a wash changes is the price, and the price is set by the slurry's concentration and by how far the wash front smears, not by the cake's softness.
The longest thermal entrance belongs to neither metal nor gas
A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.
A steel wall lengthens a sodium entrance by half
A liquid metal's thermal entrance is long because its heat crosses the pipe by conduction, and a steel tube conducts a tenth as well as sodium, so the wall is part of the path. Put it in series with the fluid and the entrance does grow — but by half at the very most, at a wall Biot number near two, which is where sodium in a steel tube happens to sit. A wall that resists more changes the boundary condition from a fixed temperature to a fixed flux, not the length.
A bubble is throttled by its age, and a sheet by its length
A cavity cools the liquid it boils, and the cooling lowers the vapour pressure it needed. Whether that matters was a question about one number, the B-factor — vapour made over liquid cooled — and no real cavity's was known. Computed, the two kinds a pump has are nothing alike. A travelling bubble cools a thin shell that thins further as it stretches, so its B grows as the square root of its age and reaches the critical value at Brennen's thermal time: a millisecond in water at 60 °C, four microseconds at the boiling point. An attached sheet keeps renewing the liquid it cools, so its B is a few, fixed by its length and speed. In warm water the bubbles are suppressed forty degrees before the sheet.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitDiffusionDispersionDimensionlessMeasurementTransportHeat transferPrandtl numberResidence timeEigenvalueMixingNusselt number