How far before the heat arrives

A duct’s velocity profile takes about $0.05\,D\,\mathrm{Re}$ to settle, and how far before a duct forgets what was fed into it computes that. Heat takes $0.05\,D\,\mathrm{Re}\,\mathrm{Pr}$, and the extra factor is the Prandtl number — which is why oil in a heat exchanger is still developing thermally after a hundred diameters while its velocity profile settled in the first ten.

Both numbers are quoted with the same 0.05 in front, in every textbook, in the same way that Mach 0.3 is, with no tolerance attached to either. This essay works out what the thermal one delivers, and finds two things: the constant is a tolerance in disguise, and the quantity it is a tolerance on is not a term ratio at all.

The problem, and the two ways to solve it #

With the velocity profile already parabolic, the wall at a fixed temperature and axial conduction neglected, the energy equation is

$$2(1-\bar r^2)\frac{\partial\theta}{\partial\zeta} = \frac{1}{\bar r}\frac{\partial}{\partial \bar r}\left(\bar r\frac{\partial\theta}{\partial\bar r}\right), \qquad \zeta = \frac{\alpha x}{Ua^2}.$$

It is solved here twice, by routes that share no arithmetic.

As an eigenvalue problem. Separating variables gives $(\bar rR')'/\bar r + \lambda^2(1-\bar r^2)R = 0$ with $R'(0) = 0$ and $R(1) = 0$, whose eigenvalues are found by shooting: integrate from the axis with the series that removes the singularity there, and bisect on $\lambda$ until $R(1)$ vanishes. The first six come out as 2.704364, 6.679031, 10.673380, 14.671078, 18.669872 and 22.669143.

As a parabolic march. Step the equation forward in $\zeta$ from a uniform inlet, on a radial grid, with Crank–Nicolson. Nothing in it knows about eigenvalues.

The number a duct settles at is an eigenvalue #

Far downstream only the slowest-decaying mode is left, so the temperature profile becomes $R_0(\bar r)$ whatever it started as, and the Nusselt number becomes a property of that one function:

$$\mathrm{Nu}_\infty = \frac{\lambda_0^2}{2} = 3.656794.$$

Not a ratio of two terms. Not something that is one when anything balances. The smallest eigenvalue of a Sturm–Liouville problem, halved, and it is the same for every duct, every fluid and every flow rate that satisfies the assumptions.

That places it in the same category as the Rayleigh number at which a layer starts to convect and as the critical Stokes number — a threshold produced by an operator rather than by a comparison, and therefore at a value with no relation to one.

The two routes meet where nothing said they should #

Here is the result worth the whole file. The march settles at 3.6568, agreeing with the eigenvalue to five figures — which is a check that the two calculations are the same problem, and is expected.

What is not expected is the rate. The march’s departure from 3.6568 decays exponentially in $\zeta$, and the rate of that decay comes out at 18.681 per unit $\zeta$. The eigenvalues require it to be

$$\frac{\lambda_1^2 - \lambda_0^2}{2} = \frac{44.610 - 7.3136}{2} = 18.648,$$

because what is being measured downstream is the second mode’s survival alongside the first. The two agree to 0.18 per cent.

So the eigenvalue problem sets both the value a duct settles at and the speed at which it gets there, and the march — which has neither in it — reproduces both.

The entry length is a tolerance #

Now the quoted number. Because the approach to the developed state is exponential rather than algebraic, the entry length depends on the tolerance only logarithmically — which makes it well-behaved, and does not make it unique.

A factor of three and a half across that range. And the quoted 0.05 sits between the five per cent and one per cent rows, delivering 1.45 per cent — which is a perfectly reasonable engineering tolerance, and which is nowhere written down beside the number.

The consequence is not that anybody’s heat exchanger is wrong. It is that a designer who needs half a per cent and a designer who is happy with ten both use 0.05, and neither can tell from the number whether it suits them.

Why the Péclet number is not the threshold either #

The group in the entry length is $x^+ = x/(D\,\mathrm{Re}\,\mathrm{Pr})$, and $\mathrm{Re}\,\mathrm{Pr} = \mathrm{Pe}$ is the Péclet number — the ratio of advection to conduction. So the whole quantity is $x/(D\,\mathrm{Pe})$, and the folklore reading would be that something happens when it is one.

Nothing does. At $x^+ = 1$ the duct has been thermally developed for a factor of twenty, the fluid’s temperature has equilibrated with the wall to within $e^{-14}$, and there is nothing left to measure. The entry length is $x^+ = 0.05$ and not $x^+ = 1$, by a factor of twenty, and the factor comes from the eigenvalue.

There is a genuine Péclet threshold in this problem, and it is the assumption that was made to write the equation down: axial conduction was neglected. That is justified when $\mathrm{Pe}$ is large, and the usual criterion is $\mathrm{Pe} > 100$ — a threshold on the Péclet number that has nothing to do with the entry length and is quoted in the same chapter.

For liquid metals, whose Prandtl number is around 0.01 — the far end of the ratio that separates the two layers — $\mathrm{Pe}$ is small even at large Reynolds number, axial conduction matters, and the Graetz solution does not apply at all. That is the case the number genuinely decides, and it is not the one it is usually asked about.

Two conventions, and a factor of four #

Before any of the numbers can be compared with a published table, two conventions have to be pinned down, and both of them move the answer.

Which axial variable. The march runs in $\zeta = \alpha x/(Ua^2)$, built on the radius; the tables run in $x^+ = x/(D\,\mathrm{Re}\,\mathrm{Pr})$, built on the diameter. The conversion is $\zeta = 4x^+$, and getting it wrong moves every entry length by four.

Which Nusselt number. The local value $\mathrm{Nu}_x$ and the mean value averaged from the inlet are both called the Nusselt number, and near the inlet they differ by tens of per cent because the mean is dragged up by the singularity. A mean Nusselt number reaches within a per cent of 3.657 much later than a local one does, because it is still carrying the entrance region in its average.

Neither convention is wrong and both are ubiquitous. A published entry length that differs from another by four, or by a factor that changes with position, is usually these two and nothing else — which is the same failure mode as the length in the Reynolds number, arriving in a subject where the number itself is not in dispute.

What it means for something built #

Take a heat exchanger tube of 10 mm bore carrying water at 0.5 m/s. $\mathrm{Re} = 5{,}000$ — turbulent, so this laminar theory does not strictly apply, but the arithmetic makes the point. $\mathrm{Pr} = 7$, so $\mathrm{Pe} = 35{,}000$ and the one-per-cent entry length is $0.055 \times 10 \,\mathrm{mm} \times 35{,}000 = 19$ metres.

Take the same tube carrying oil at $\mathrm{Pr} = 200$ and $\mathrm{Re} = 100$. $\mathrm{Pe} = 20{,}000$, so the entry length is 11 metres — in a laminar flow, which is where the theory does hold, and in a tube that is very unlikely to be 11 metres long.

Laminar heat exchangers are essentially all entry region. The developed Nusselt number of 3.66 is a lower bound they never reach, the local value everywhere along them is higher, and a design that uses 3.66 is conservative by a factor that depends on the length in exactly the way this essay computes.

The law that governs the region a real exchanger lives in #

The section above ends with the observation that a laminar heat exchanger is essentially all entry region, and that leaves the essay quoting a developed number for a duct that never develops. The region does have its own law, it is the older half of this problem, and it is the half that transfers to flows the Graetz solution has nothing to say about.

Near the inlet the heated layer is thin compared with the tube’s radius. A thin layer does not know the tube is round and does not know the profile is parabolic; all it can feel is the shear rate at the wall, because over its own thickness the velocity is a straight line through zero. So the problem loses its eigenvalues, loses its geometry, and becomes a similarity problem of the kind this collection solves in its external-flow field: one length built from the wall shear, the diffusivity and the distance travelled, and a profile that is a function of one variable.

The result is Lévêque’s, and it is a power law rather than a constant:

$$\mathrm{Nu}_x = 1.077\,(x^+)^{-1/3}.$$

At $x^+ = 10^{-3}$ that is 10.8 against the developed 3.66 — a duct three times better at moving heat than the number a designer would look up, over the whole of its first metre.

Put the two laws side by side and something falls out that ties the essay together. They cross where $1.077\,(x^+)^{-1/3} = 3.657$, which is $x^+ = 0.0255$ — within a few per cent of the ten-per-cent entry length in the table above. That is not a coincidence and it is the cleanest available statement of what an entry length is: the station where the entrance asymptote and the developed asymptote meet. Everything upstream of it belongs to Lévêque and everything downstream to Graetz, and the “entry length” is the corner between two straight lines on a logarithmic plot rather than a property of either.

And the entrance law is the robust one. Lévêque’s derivation used only the wall shear rate, so it survives every change that leaves a thin layer over a sheared wall — a different duct cross-section, a developing velocity profile, a non-Newtonian fluid whose profile is not parabolic at all, even a turbulent flow if the shear rate at the wall is the turbulent one. Substitute the appropriate wall shear and the $-1/3$ power stands. The Graetz eigenvalues do not travel that way: change the section and every one of them changes, as the list of developed Nusselt numbers at the end of this essay shows.

Which reverses the usual order of presentation. The developed value is the number in the textbooks because it is a single constant and the eigenvalue problem is elegant; the entrance law is relegated to a footnote because it is an approximation with a fractional power in it. In a laminar exchanger the approximation applies over most of the length, generalises to fluids and geometries the exact solution cannot reach, and is the one whose error a designer can bound. The exact result describes the state the apparatus is heading towards and the approximate one describes the state it is in.

The shape of the approach, and what it rules out #

One consequence of the exponential approach deserves its own statement, because it separates this problem from most of the others in the collection.

An algebraic approach — an error falling as $1/x$ or $1/\sqrt{x}$ — never really ends. Doubling the length halves the error, and there is no distance at which a reader can say the process has finished. A boundary layer’s growth is like that, and so is the approach of a jet to self-similarity.

An exponential approach ends decisively. Each fixed distance costs the same factor, so the error falls through decades in a length that barely changes. That is why the thermal entry length is a usable engineering quantity at all, and why the same textbook that quotes 0.05 without a tolerance gets away with it: over the range of tolerances anybody uses, the answer really does not move much.

The folklore is nearly right here for a structural reason rather than by luck. And the structural reason is the eigenvalue: a discrete spectrum with a gap between its first two members produces an exponential approach, and a continuous spectrum would not.

One number from that comparison is worth carrying into a design office. Because the entrance law falls as the cube root, the mean Nusselt number over a tube of length $L$ is 1.5 times the local value at its exit — an integral of a $-1/3$ power — so a short exchanger’s average performance is higher than its worst point by exactly half again. That factor is why a bank of short tubes outperforms a single long one of the same total area, and why the comparison has to be made on mean Nusselt numbers rather than on the developed constant, which describes neither.

What the march had to be told #

Two numerical points, both of which are results rather than housekeeping.

Crank–Nicolson does not damp a discontinuity. The inlet condition is uniform fluid meeting a wall at a different temperature, which is a step, and Crank–Nicolson’s amplification factor tends to $-1$ for the modes that resolve one. The first hundred steps returned Nusselt numbers of $-739$, $+653$, $-530$, $+383$ — a saw with a decaying envelope, settling eventually to the right answer. The mean temperature was smooth throughout, because the ringing lives in the two nodes next to the wall, and the wall gradient is what a Nusselt number is made of. Forty fully implicit steps at the start damp it completely and cost nothing, since they span $\zeta = 8\times10^{-4}$ of a march that runs to 0.5.

The entry length has to be scanned backwards. Looking forward for the first station inside the tolerance band finds any station where the value happens to fall in it — and near the inlet, before the march is resolved, that includes stations where nothing has developed at all. Scanning from the far end for the last station outside the band cannot be fooled that way.

What the picture cannot show #

The inlet region is not resolved. The exact solution has $\mathrm{Nu} \propto (x^+)^{-1/3}$ as $x^+ \to 0$ — Lévêque’s similarity solution — and a uniform radial grid cannot follow a layer that thin. The march gives 23 where the asymptote gives 30 at the first recorded station, and the figures do not draw that region as though it were right.

The velocity profile is already parabolic. A real duct develops both profiles at once, and for a gas at $\mathrm{Pr} \approx 0.7$ they develop together, so the combined-entry problem — which has no closed form — is the honest one there.

Axial conduction is neglected and the wall temperature is fixed. A constant heat flux instead gives $\mathrm{Nu}_\infty = 48/11 = 4.364$, a different eigenvalue problem and a different entry length, and the two are routinely quoted in the same table without saying that they are different problems.

And it is a circular tube. Every duct section has its own eigenvalue: 7.541 for parallel plates with both walls heated, 4.861 with one, 2.98 for a square. The number 3.66 is a property of a circle.

Who found it, and when #

Graetz published the solution in 1883 and Nusselt independently in 1910, and the eigenvalues were computed by hand for decades afterwards — Sellars, Tribus and Klein produced the definitive set in 1956, and the higher ones are still quoted from asymptotic formulae because computing forty of them by shooting is tedious even now.

The constant 0.05 has no such provenance. It is a rounding of a computed entry length at some tolerance nobody recorded, and its survival is the same story as Mach 0.3: a number that is roughly right, easy to remember, and detached from the calculation that produced it.

The surprising connection is between the value and the rate. That $\mathrm{Nu}_\infty$ and the forgetting rate are both eigenvalues of the same operator is not remarkable once stated; that the first of them is $\lambda_0^2/2$ and the second is $(\lambda_1^2 - \lambda_0^2)/2$ means the number a duct settles at and the speed at which it settles are two readings of one spectrum. A duct that carried more heat when developed would also forget its inlet faster, and the two cannot be designed independently.

Where the ladder goes next #

Above this rung is the combined entry problem, where the velocity and temperature profiles develop together and there is no similarity variable to separate — the case that actually occurs in a gas, and the case with no closed form.

Beside it sits the momentum version, which is the same question with the Prandtl number removed, and the external layer, where the two thicknesses differ by $\mathrm{Pr}^{1/3}$ and the whole business of two boundary conditions on one wall began. Below it is the general account of thresholds, in which this essay supplies both an eigenvalue row and a term-ratio row from the same problem.