A steel wall lengthens a sodium entrance by half

The longest thermal entrance belongs to neither metal nor gas followed a heated liquid metal down a turbulent pipe and found that, below a Péclet number of a few hundred, its thermal entrance grows as 0.029 times the Péclet number in diameters — the constant of a fluid moving as a solid block, because conduction rather than the eddies carries its heat across. Every number there held the wall at a fixed temperature, and the essay ended on the obvious objection: a sodium pipe is made of steel, steel conducts heat about a tenth as well as sodium, and so the wall is part of the conduction path and its temperature is set by the heat it passes rather than fixed — the Péclet number’s layer with a second solid layer outside it.

The natural expectation is that the wall makes things much worse — that a slow path in series with the fluid lengthens the entrance in proportion to how slow it is. It does lengthen it, and the amount it can is bounded, by a factor of about one and a half, for a reason that says what an entrance length actually measures.

A wall in series #

The wall here is thin and conducts only across its own thickness, so it adds a conductance per unit area $h_w = k_w/(R\ln(1+t/R))$ between the fluid’s wall temperature and a fixed outer temperature. In the units of the fluid it is a wall Biot number,

$$\mathrm{Bi}_w = \frac{h_w D}{k_f} = \frac{2k_w}{k_f\ln(1 + t/R)},$$

infinite for a wall at a fixed temperature and zero for one so resistive that it fixes the heat flux instead. The temperature problem is the same Graetz march as before — a step in the outer temperature, the profile marched downstream on a radial grid clustered at the wall — with one more conductance in series at the wall. The entrance length is still where a Nusselt number comes within five per cent of its developed value, but it is read on the fluid-side Nusselt number, on the fluid’s own wall temperature. The overall one, on the outer temperature, is held near its developed value from the start by a resistive wall, which dominates it, and it says nothing about whether the fluid’s profile has settled.

The factor, against the wall #

For a slug — the low-Péclet limit of every turbulent pipe in the earlier essay — the answer is a single curve. A wall that conducts well, a Biot number above a hundred, changes the entrance by less than six per cent. As the wall’s resistance grows, the entrance lengthens, to a peak of 1.501 times at a Biot number of 2.29, and then shortens slightly, to 1.457 times, for a wall that hardly conducts at all. The slug’s constant goes from 0.0290 to 0.0435 at the peak. The entrance is never as much as half as long again, however poorly the wall conducts.

The low-Péclet line of the earlier essay therefore survives with a new constant. A liquid metal in a resistive tube still has an entrance proportional to its Péclet number below a few hundred, with the constant multiplied by a factor between one and one and a half that depends only on the wall Biot number.

What a resistive wall does to the developed state #

The reason the factor is bounded is clearest in the developed state. With a resistive wall the slug’s developed profile is $J_0(\mu r/R)$ with $\mu J_1(\mu) = (\mathrm{Bi}_w/2)J_0(\mu)$, so its fluid-side Nusselt number is $2\mu J_1(\mu)/(2J_1(\mu)/\mu - J_0(\mu))$. At an infinite Biot number $\mu$ is the first zero of $J_0$, 2.405, and the Nusselt number is $\mu^2 = 5.783$, the classical value for a slug at a fixed wall temperature. As the Biot number falls, $\mu$ falls to zero and the Nusselt number rises to exactly 8, the classical value for a slug at a fixed flux. The march’s developed values agree with the closed form to $10^{-5}$ at four walls.

That is the whole effect of the wall on the fluid: it moves the boundary condition from a fixed temperature to a fixed flux. The overall Nusselt number, on the outer temperature, is the fluid side and the wall in series, $1/\mathrm{Nu}_o = 1/\mathrm{Nu}_f + 1/\mathrm{Bi}_w$, and falls towards the Biot number itself as the wall dominates. A pipe with a resistive wall heats its fluid more slowly overall — that is a longer pipe for a given temperature rise — but that is not the same thing as a longer entrance.

The entrance is a gap between two decay rates #

An entrance length measures how long the profile takes to forget its initial shape, and the profile is a sum of modes each decaying at its own rate. The developed state is the slowest mode; the entrance is over when the next one has died away relative to it. For a slug at a fixed wall temperature the decay rates are the squares of the zeros of $J_0$, and the gap between the first two is $5.520^2 - 2.405^2 = 24.7$. At a fixed flux the first mode no longer decays at all relative to the uniform heating, and the next decays at the first zero of $J_1$ squared, $3.832^2 = 14.7$. The resistive wall slows both modes, and the gap between them shrinks only from 24.7 to 14.7, a factor of 1.68.

That is why the factor is bounded, and why it is less than 1.68: the five per cent criterion is applied to the Nusselt number, in which the second mode appears with an amplitude that itself depends on the wall, and the result is 1.46 for a fixed flux and a maximum of 1.50 near a Biot number of two, where the second mode shows most strongly in the fluid’s wall temperature. The figure shows the three Nusselt numbers falling from infinity at the step towards their developed values — 5.783, 7.392 and 7.983 — and reaching them within five per cent at 0.0290, 0.0435 and 0.0423 diameters per unit Péclet number.

The profile flattens #

The developed profiles show the change of boundary condition directly. At a fixed wall temperature the profile is $J_0(2.405\,r/R)$ and falls to zero at the wall. With a resistive wall the fluid’s wall temperature is no longer the outer one; part of the drop has moved into the wall, and the fluid’s profile falls less across the pipe. At a Biot number of 0.05 it is nearly flat: almost all the temperature drop is in the wall, and the fluid moves as a nearly isothermal block heated through its edge, which is the fixed-flux slug. A fluid with a flatter profile is, if anything, a fluid closer to settled, which is the other way of seeing why a resistive wall cannot lengthen the entrance without limit.

In a turbulent pipe #

In the turbulent pipe of the earlier essay, with its mixing-length eddy diffusivity, the same wall does less as the Péclet number rises. At the lowest Péclet number here, 76, a wall of Biot number 2.3 lengthens the entrance 1.39 times, near the slug’s 1.50; at 3042, 1.19 times. The eddies carry an increasing share of the heat across the core as the Péclet number rises, the conduction-dominated region shrinks towards the wall, and the wall’s boundary condition reaches less of the profile. At a Prandtl number of 0.01, Péclet number 380, the developed fluid-side Nusselt number rises from 8.15 to 9.55 between a fixed-temperature wall and a wall of Biot number 3, the turbulent version of the slug’s rise from 5.78 towards 8.

The fading is the same mechanism that made the liquid metal’s entrance long in the first place, run in reverse. The heat the eddies do not carry found that below a Prandtl number of a few hundredths conduction beats the eddies everywhere in the pipe, so the temperature profile is a slug’s even though the velocity profile is turbulent; the wall condition then reaches the whole profile, and the slug’s factor applies. As the Péclet number rises the eddies take the core, conduction is confined to a layer at the wall, and the profile’s slow modes are increasingly set by the turbulent core rather than by the condition at its edge. A boundary condition matters in proportion to how much of the profile is conduction-dominated, which is the same statement a layer that is an integral of everything upstream makes about a boundary layer’s memory of its wall.

The same bound in a duct the earlier essays used #

The thermal entrance has been followed down several pipes before, and the wall’s effect reads differently against each. How far before the heat arrives found the laminar entrance’s quoted 0.05 to be a tolerance rather than a property, and the factor here inherits that: on a 1.45 per cent criterion the slug’s peak is a little different, because the criterion weights the second mode differently. How far before a duct forgets what was fed into it is the velocity’s version, where there is no wall condition to change — a no-slip wall is a no-slip wall — and so no factor at all. And the momentum layer’s exact solutions with a wall that moves are the closest analogue to a wall whose temperature moves: an oscillating plate’s layer has a depth set by the fluid alone, and a wall that responds to the fluid, as this one does, shares the layer with it.

What all of them share is the reading of an entrance as the decay of everything but the slowest mode. A boundary condition changes which modes there are, and by how much the second outlasts the first; it cannot make the profile forget more slowly than its slowest pair of modes allow. For a slug those pairs are fixed by Bessel functions whose zeros differ by a bounded amount whatever the boundary condition between the two classical ones, and that is the whole of why the factor stops at one and a half.

Why a sodium loop’s designer still cares #

Half as long again is not nothing. A sodium heat exchanger’s tubes are typically tens of diameters long, and the earlier essay found sodium entrances of order ten diameters at the Péclet numbers such exchangers run at. A factor of 1.4 on that is four more diameters in which the local heat transfer is not the developed value, and an exchanger sized on developed correlations over a short tube is sized wrongly by that much. The developed fluid-side Nusselt number changes too, rising by about a sixth as the wall’s resistance pushes the condition towards a fixed flux — which, read the other way, means a correlation measured in a copper tube and applied to a steel one over-predicts the wall’s share of the resistance and under-predicts the fluid’s coefficient. Neither error is large; both are systematic, and both come from the same series conductance.

Real tubes #

With stated conductivities — sodium near 70 watts per metre-kelvin at 450 °C, austenitic steel near 20, lead–bismuth 13, mercury 8.5, water 0.6 — the wall Biot numbers of real pairs fall in a revealing order. Sodium in a steel tube with a wall a tenth of its radius is at 6.0, and with a wall a fifth of its radius at 3.1: either side of the peak at 2.3. It is the one common combination that sits where the wall lengthens the entrance most, and there it does so by 1.4 to 1.5 times in the low-Péclet limit. Lead–bismuth in steel is at 32 and mercury at 49, where the effect is a few per cent to fifteen. Sodium in copper is at 114. Water in steel, at 699, has a wall at a fixed temperature for every practical purpose, which is why the conjugate question never arises for ordinary fluids: their own conduction is so poor that any metal wall is a perfect conductor beside it. That ordering is the Péclet number’s other layer seen from the outside — the thinner the fluid’s thermal resistance, the more the wall’s matters.

How the wall was checked #

The wall’s conductance is added in series with the half-cell between the last grid node and the wall, so the fixed-temperature problem is the infinite-Biot limit of the same operator; at a Biot number of $10^{12}$ the slug’s entrance constant is 0.02896, the isothermal march’s to five figures. The developed fluid-side Nusselt number agrees with the Bessel closed form to $10^{-5}$ at Biot numbers of 100, 6, 1 and 0.1. The overall and fluid-side numbers satisfy the series rule to rounding. And the closed form reaches the two classical limits, $j_{0,1}^2$ and 8. The march and its grid are those of the earlier essay, whose own checks — Graetz’s 3.6568 and his entrance constant, the slug’s — carry over unchanged.

The convention: which Nusselt number #

The entrance length is read on the fluid-side Nusselt number, defined on the difference between the fluid’s mixing-cup temperature and its own wall temperature, and the five per cent tolerance is applied to it. The Biot number is on the diameter and the fluid’s conductivity: one more number that decides which physics applies, here which of two boundary conditions the fluid sees. The wall is thin and conducts only radially; the outer temperature steps at the start of heating and stays fixed. A different definition of the entrance — on the overall Nusselt number, or on the bulk temperature’s approach to an exponential — gives different numbers, and the overall one in particular gives an entrance that shortens as the wall resists, which is the quantity’s definition talking rather than the flow.

What the wall in this model cannot do #

Conduct along its length. A real wall carries heat axially as well as across, and for a liquid metal at low Péclet number the fluid’s own axial conduction is already significant; a steel wall’s axial conduction pre-heats the fluid upstream of the step and smears the step itself, which a thin radial wall cannot represent. Have a thermal capacity. Everything here is steady; a wall heated in time adds its own time constant. Be heated unevenly. A real exchanger has a heat flux that varies along and around the tube, and a resistive wall then spreads it. And the turbulent pipe keeps the earlier essay’s mixing-length closure and its turbulent Prandtl number, whose value for liquid metals is uncertain by a factor of two, which moves every turbulent number here and not the slug’s.

Who solved the pieces #

Graetz solved the entrance with a fixed wall temperature in 1883, and Nusselt in 1910; the slug-flow versions with fixed temperature and fixed flux, with their Nusselt numbers of 5.78 and 8, are in every heat-transfer text, as is the third-kind condition, through a wall Biot number, that joins them. The conjugate problem — the wall’s conduction coupled to the fluid’s — was formulated for laminar ducts in the 1960s and studied for liquid metals in the fast-reactor programmes of the same decade, where sodium in stainless steel made it unavoidable. The bound on the entrance, read as a gap between two decay rates that both move, follows from those classical pieces; the peak near a Biot number of two is a property of the five per cent criterion on the Nusselt number and would move with a different criterion.

Still open: a wall that conducts along its length #

The radial wall here cannot carry heat upstream, and at the Péclet numbers where a liquid metal’s entrance is longest, a steel wall’s axial conduction is comparable to the fluid’s own. The next calculation gives the wall a thickness and an axial conductivity, couples it to the fluid along the whole pipe with the step applied on its outside, and asks how far upstream of the step the fluid starts to warm, whether the entrance measured from the step then shortens rather than lengthens — the heat having started its journey before the step — and at what ratio of wall to fluid conductance the wall’s axial path, rather than the fluid’s, sets where the heating begins.