The longest thermal entrance belongs to neither metal nor gas

The heat the eddies do not carry found where the Reynolds analogy breaks. In a turbulent pipe the eddies carry heat and momentum across the flow together, and for air, water and oil the temperature profile follows the velocity profile closely enough that a friction factor predicts a heat transfer coefficient. A liquid metal breaks that. Its molecules conduct heat so well that the eddies never become the main carrier, and its temperature changes across the whole pipe as a laminar flow’s would while its velocity is as turbulent as anyone else’s. The boundary between the two behaviours was a Péclet number of about four hundred, at every Reynolds number.

That essay closed on the consequence for the length of pipe a heated flow needs before it settles. A liquid metal’s velocity develops in the few diameters turbulence takes, but its temperature is carried by conduction, and in how far before the heat arrives a laminar thermal entrance was $0.05\,\mathrm{Re}\,\mathrm{Pr}$ diameters long — a laminar flow at a Péclet number of a thousand needs fifty diameters. It asked whether a liquid metal in a turbulent pipe follows that scaling, and at what Péclet number it switches to an ordinary fluid’s few diameters.

It does follow it, with a different constant, and it switches where the earlier essay’s threshold says it should. But the consequence is not the one the question expected. A liquid metal’s thermal entrance is usually short, because its Péclet number is small, and the longest thermal entrance in a turbulent pipe belongs to the fluids between the metals and the gases.

The Graetz problem with eddies in it #

The calculation is the classical one with a single change. A fluid flows down a long pipe with its velocity fully developed; at one station the wall’s temperature steps to a new value and stays there. Downstream the fluid near the wall has adjusted and the fluid at the axis has not, and the heat flux through the wall falls from infinity at the step to a settled value as the adjustment spreads inward. Leo Graetz solved this in 1883 for a laminar flow, and the settled Nusselt number, 3.6568, is the least eigenvalue of the operator that carries heat across the pipe.

The single change is that the operator now includes the eddies. The velocity profile and the eddy viscosity are the earlier essay’s: a mixing length taken from Nikuradse’s pipe measurements and damped at the wall by van Driest’s factor. The eddy diffusivity of heat is the eddy viscosity divided by a turbulent Prandtl number of 0.85, borrowed, as before. The temperature is marched down the pipe from the step on a radial grid clustered at the wall, and the local Nusselt number is computed at each station from the wall’s temperature gradient and the mixing-cup temperature of the flow. The entrance length is the distance from the step at which the local Nusselt number has come within 5 per cent of its settled value, which is computed separately as the least eigenvalue of the same operator, so the march’s end and the eigenproblem check each other.

The first figure is the result for four friction Reynolds numbers, from 500, a bulk Reynolds number of 17,000, to 5,000, about 230,000, across Prandtl numbers from a few ten-thousandths to ten. It plots the entrance length in diameters against the Péclet number. At the left of the figure every pipe lies on a straight line of slope one, the entrance length growing in proportion to the Péclet number. Each curve leaves the line, rises more slowly to a peak of between ten and fifteen diameters, and then falls: air, at a Prandtl number of 0.7, settles in nine to eleven diameters; water, at seven, in three and a half.

A slug that conducts #

The line on the left of the figure has a constant, 0.029, and it is not the laminar one. Graetz’s flow is a Poiseuille profile, a parabola, and the same 5 per cent criterion applied to it gives 0.034 Péclet numbers of diameters. The turbulent pipe’s constant is the one belonging to a fluid moving down the pipe as a solid block — a slug — with conduction alone carrying heat across it.

That is what a liquid metal at a small Péclet number is. Its velocity profile is turbulent, and a turbulent profile is flat across most of the pipe and falls to zero only in a thin layer at the wall; at a friction Reynolds number of a thousand, the velocity a twentieth of a radius from the wall is already 65 per cent of its value at the axis, where a parabola would have reached a tenth. Its heat, meanwhile, crosses the pipe by conduction, which does not care about the velocity profile except through how much heat each radius carries downstream. So it behaves as a conducting block, and its settled profile is the slug’s own, the Bessel function $J_0(2.405\,r/R)$, whose first zero sits at the wall.

The second figure divides the entrance length by the Péclet number, which turns the slope-one line into a horizontal one. At the left all four Reynolds numbers collapse onto a single curve, tending to the slug’s 0.029 and lying just above it, because the velocity’s thin wall layer makes the real flow slightly slower near the wall than a block. This is the first half of the answer. While conduction carries the heat, the Péclet number alone decides the entrance, and the Reynolds number does not appear. That is the laminar scaling, as the earlier essay predicted, but with a profile that is neither laminar nor turbulent: the flow is turbulent and the heat transfer is laminar, and the constant is that of a flow which is neither.

Where the scaling breaks #

Each curve in the second figure leaves the slug’s constant. Taking the departure to be where the entrance length has fallen to half of what the slug’s constant predicts, it happens at a Péclet number of 465 for the lowest Reynolds number and 598 for the highest. The earlier essay’s threshold — the Péclet number below which molecular conduction beats the eddies everywhere in the pipe — is 358 and 480 at the same two Reynolds numbers. The ratio is 1.25 to 1.30 across the whole range.

The agreement is not a coincidence, and the reason it is a ratio rather than an equality is worth stating. At the threshold, the eddies at their strongest, about half-way from the wall to the axis, carry heat exactly as fast as conduction does. Everywhere else in the pipe they carry less. So at the threshold the eddies have begun to matter in one place, and the pipe as a whole is still mostly conducting; it takes a little more Péclet number before they have doubled the rate at which heat crosses the pipe on average and halved the entrance. The threshold is the onset and the departure is the effect, and the two are tied together by the shape of the eddy viscosity, which the earlier essay found to have the same peak, about 0.082 times the radius in wall units, at every Reynolds number. That fixed shape is why the ratio is fixed.

So the second half of the answer is the threshold itself. The entrance leaves the laminar scaling at about 1.3 times the threshold Péclet number of the heat the eddies do not carry, and the Reynolds number enters only through that threshold’s slow drift — four hundred, give or take a fifth, across a decade of Reynolds number.

What the developed profile looks like #

The third figure shows the settled temperature profile — the eigenfunction the march converges to — at three Prandtl numbers, at a friction Reynolds number of a thousand. At a Prandtl number of 0.002, a Péclet number of 76, it lies on the slug’s Bessel function to within one per cent of its axis value everywhere. At 0.02, a Péclet number of 760, past the departure, the eddies have begun to flatten the core: the profile is still spread across the pipe, but it is fuller in the middle than conduction alone would make it. At 0.7 the profile is a turbulent one, flat across the core with its whole change compressed into a thin layer at the wall, the shape the other layer would recognise as a thermal boundary layer.

The entrance length is the distance over which the profile, starting flat at the inlet temperature, becomes this shape. A profile that is spread across the whole pipe takes the time to conduct across the whole pipe. A profile confined to a thin wall layer only has to build that layer, which the eddies outside it keep supplied. That is the whole of the contrast between the two ends of the first figure, and it explains the third behaviour, the peak, as well.

Why the middle of the range is the slowest #

The fourth figure follows the local Nusselt number along the pipe from the step at a friction Reynolds number of a thousand, for four fluids. Water’s falls inside the 5 per cent band within 3.5 diameters. Air’s takes 9.6. A fluid of Prandtl number 0.1 — a liquid metal at a high Reynolds number, or close to the mixtures of helium and xenon whose Prandtl numbers fall to about 0.2 and which compact gas-cooled reactor designs have considered — takes 11.6. And a fluid of Prandtl number 0.005, which is sodium’s neighbourhood, takes 4.1, less than half air’s.

The reason the middle is slowest is that it is caught between the two mechanisms. A fluid at a high Prandtl number settles fast because its thermal layer is thin; a fluid at a low Péclet number settles fast because the whole pipe is only a short conduction distance across at so small a Péclet number. A fluid in between has a thermal profile thick enough to take time to form and a Péclet number large enough that conducting across the pipe is slow, and the eddies are just beginning to help. The peak of the first figure sits at Prandtl numbers from 0.2 at the lowest Reynolds number to 0.04 at the highest — at Péclet numbers from 3,400 to 9,300 — and its height grows slowly with the Reynolds number, from ten diameters to fifteen.

This is what the question about liquid metals misses. “Conduction carries the heat, so the entrance is long” is true per unit Péclet number. But the Péclet number is the Reynolds number times the Prandtl number, and sodium’s Prandtl number is a hundredth of air’s, so at the same Reynolds number its Péclet number is a hundredth of air’s too. A liquid-metal entrance is a laminar entrance at a small Péclet number, and a laminar entrance at a small Péclet number is short. It overtakes air’s only when the pipe’s Reynolds number is high enough to push sodium’s Péclet number past a few hundred: at a friction Reynolds number of 5,000, a bulk Reynolds number of 230,000, the Prandtl-0.005 fluid needs 11.2 diameters against air’s 10.7.

The settled value and the borrowed correlation #

The fifth figure shows the settled Nusselt number the entrance converges to. At low Péclet numbers every pipe tends to a value just below the slug’s $j_{0,1}^2 = 5.783$, again because the thin wall layer makes the flow there slower than a block’s: 5.2 at a Péclet number of 17 and a friction Reynolds number of 500. Above a few hundred the eddies take over, the curves rise as roughly the Péclet number to the 0.8 power and separate by Reynolds number.

Drawn beside them, as borrowed, is the correlation Seban and Shimazaki fitted in 1951 to liquid-metal pipes at a uniform wall temperature: $\mathrm{Nu} = 5.0 + 0.025\,\mathrm{Pe}^{0.8}$. Its intercept, 5.0, is the slug’s limit less the wall layer’s correction, and the closure here reaches 5.4 at a Péclet number of 38. At higher Péclet numbers the closure runs above the correlation — 26 against 23 at a Péclet number of 3,800 — which is the known fault of a turbulent Prandtl number of 0.85 for a liquid metal. The eddies are credited with carrying more heat than measurements show they do.

The number nobody has pinned down #

That fault is the one place where the entrance lengths here are not secure, and it is worth measuring how far they move. The turbulent Prandtl number of 0.85 is measured in air and water. For liquid metals the measurements are scattered, and have been read to imply values from about one to well above two at low Péclet numbers, rising as the Péclet number falls, because an eddy that is conducting heat away from itself as it moves carries less heat than its momentum suggests.

The sixth figure repeats the entrance length at a friction Reynolds number of a thousand with the turbulent Prandtl number set to 0.85, 1.5 and 2.5. Below a Péclet number of a hundred the three agree to within a tenth, because there the eddies carry almost nothing whatever their Prandtl number, and the slug’s line does not move at all. Above it they part. A larger turbulent Prandtl number weakens the eddies’ share, so the entrance follows the laminar line further before it leaves: the departure moves from a Péclet number of 507 to 894 to 1,490, in proportion to the turbulent Prandtl number, just as the threshold does, so the ratio of 1.3 between them survives. And the peak grows: 11.6 diameters, then 20.5, then 34.

So the calculation’s two firm results are the slug’s line and the ratio to the threshold. The peak’s height is the closure’s, and for the fluids in the middle of the range a designer would want an entrance length measured rather than computed. The peak’s existence is firm, since it follows from the two scalings on either side of it, but how tall it is depends on a number that the liquid-metal experiments have not settled.

What the calculation was checked against #

The march and the eigenproblem were checked on the two flows whose answers are known exactly. With the velocity set to Poiseuille’s parabola and no eddies, the developed Nusselt number is Graetz’s 3.6568 to a thousandth of a per cent, the march ends on the eigenvalue to one part in a hundred million, and at the station $x/(D\,\mathrm{Pe}) = 0.05$ the local Nusselt number stands 1.55 per cent above the developed value, against the 1.45 that how far before the heat arrives read from Graetz’s series. The tenth of a per cent between them is the march’s first-order step, which errs on the slow side and so makes every entrance length here very slightly long. With the velocity set to a uniform slug, the developed Nusselt number is $j_{0,1}^2$ to a thousandth of a per cent. Halving the radial grid and the number of steps moves the developed Nusselt number by a thousandth of a per cent and the entrance length by 2.4 per cent, which is the crossing of the 5 per cent band being located between geometric steps.

The velocity profile underneath is the earlier essay’s, with its own checks: the laminar limit’s $48/11$ and $64/\mathrm{Re}$, and Prandtl’s friction law within 1.2 per cent across two decades of Reynolds number. The checks also refuse a Prandtl number that is zero or negative and a pipe with no Reynolds number.

What the calculation leaves out #

Axial conduction. The march drops conduction along the pipe, which is exact for laminar flow only when the Péclet number is large. Below a Péclet number of about a hundred, heat conducts upstream through the step and the wall’s temperature begins to be felt before it arrives; the entrance is then shorter still, and the slug’s line here is the long side of the truth.

The velocity’s own entrance. The velocity is fully developed at the step. In a real heated pipe the velocity develops too, in the length how far before a duct forgets worked out for a laminar duct and the few diameters turbulence needs, and a duct that forgets everything but one number found what the developed flow keeps of its inlet. For a liquid metal whose thermal entrance is short, as at the left of the first figure, the two overlap and this calculation is the case where the hydrodynamic entrance is upstream of the heating.

A uniform wall temperature. The earlier essay used a uniform wall flux, and a real wall holds neither exactly, as a pipe cannot hold its gas at the wall’s temperature found for a gas line buried in the ground. The two boundary conditions give different developed Nusselt numbers — the slug’s 5.78 for a fixed temperature against 8 for a fixed flux — and somewhat different entrances; the scalings and the threshold are the same.

The closure’s wall layer. The van Driest damping is fitted to air and water. A liquid metal’s conduction layer is far thicker than the viscous sublayer, so the damping’s details matter less for it than for water, and more for the fluids in the middle.

One number, read twice #

What of order one is worth set out what a threshold in a dimensionless number is good for: it marks where one term of an equation overtakes another, and the useful question is always how far either side of it the behaviour changes. The Péclet threshold of the earlier essay was such a term ratio — eddy conduction against molecular — and here it reappears as the place where a measurable length changes its scaling, at a fixed ratio of 1.3 to it. That is the best a threshold can do: be the onset of something measurable, by a known factor. Three buffer layers, one friction found that the details of the wall layer hardly matter to the friction; here they matter more, because heat, unlike momentum, can cross the wall layer by a different route.

The practical reading is a warning about intuition. “Conduction-dominated, therefore slow” is a statement about the Péclet number, not about the fluid, and a fluid with a small Prandtl number has a small Péclet number at any Reynolds number a pipe is likely to run at. Sodium in a reactor’s fuel-pin bundle runs at Péclet numbers of a few hundred, right on the departure, which is where its entrance is longest relative to the slug’s line and where the turbulent Prandtl number’s uncertainty begins to matter.

Who worked it out #

Graetz’s laminar entrance of 1883 was solved again by Nusselt in 1910, and the eigenvalues are tabulated in every heat-transfer text. The slug-flow version, with its Bessel functions, is the conduction problem of a moving rod, and it was the first model of liquid-metal heat transfer, used by Martinelli in 1947 and by Lyon in 1951, whose correlation for a uniform flux sits beside Seban and Shimazaki’s for a uniform wall temperature. Sleicher and Tribus worked out the turbulent Graetz problem with an eddy diffusivity in 1957, and Notter and Sleicher’s correlations of the 1970s are the modern reference; Kays’s review of the turbulent Prandtl number for liquid metals is where its uncertainty is set out.

Still open: the wall that leaks heat into the metal #

Every number here has the wall’s temperature fixed. A real liquid-metal pipe has a wall of steel, whose conductivity is a tenth of sodium’s, so the wall itself is part of the conduction path and its temperature is not fixed but set by the heat it passes. That is the conjugate problem, and it matters most exactly where conduction in the fluid is strongest. The next calculation couples a wall of finite thickness and conductivity to the turbulent Graetz problem and asks how much a steel wall lengthens a sodium pipe’s thermal entrance, whether the slug’s line survives with a new constant, and at what ratio of wall to fluid conductivity the wall rather than the fluid sets how far the heat has to travel before it settles.