The margin friction lends a siphon

The siphon that does not break finds a cancellation with no approximation in it. Lower the source of a frictionless siphon by a metre and the rise to its crown grows by a metre, which costs $\rho g$; the drop to its outlet shrinks by a metre, which hands back exactly $\rho g$ of velocity head. The crown pressure is a constant containing the crown’s height above the outlet and nothing else, and a siphon that starts cannot break by draining.

That essay also says what it left out, and says it carefully. Friction, it notes, reduces the velocity at the crown, which raises the margin, and costs pressure along the rising leg, which lowers it; the two have opposite signs and neither was computed. Anybody who has run a siphon through a long garden hose has an intuition about which of those matters more, and the intuition is that a long, thin, rough hose brings the crown nearer to breaking.

That intuition is wrong in every hose there is, and the reason is one line of algebra.

Two energy balances and one substitution #

The flow through a siphon with losses in it is set by an energy balance from the free surface of the source, where the water is at rest at atmospheric pressure, to the jet leaving the outlet, which is also at atmospheric pressure. The whole drop $\Delta$ goes into the jet’s kinetic energy and into the losses on the way:

$$g\,\Delta = \tfrac{1}{2}V^2 K, \qquad K = 1 + K_{\text{in}} + K_{\text{bend}} + f\,\frac{L}{D}.$$

The one is the jet’s own kinetic energy. $K_{\text{in}}$ is the loss where the water enters a plain hose end, taken as a half; $K_{\text{bend}}$ is the crown bend, taken as 0.3; and $fL/D$ is the friction of the whole length of hose, with the friction factor from the exact laminar law or from Colebrook’s equation according to the Reynolds number.

The crown pressure is the same balance stopped at the crown. By then only the losses upstream of the crown have been paid, and the water is carrying its kinetic energy:

$$p_c = p_{\text{atm}} - \rho g\,(z_c - z_s) - \tfrac{1}{2}\rho V^2 K_{\text{up}}, \qquad K_{\text{up}} = 1 + K_{\text{in}} + f\,\frac{L_{\text{up}}}{D}.$$

Substituting the first into the second removes the velocity, and with it the friction factor, the bore and the viscosity:

$$p_c = p_{\text{atm}} - \rho g\,(z_c - z_e) + \rho g\,(1 - \varphi)\,\Delta, \qquad \varphi = \frac{K_{\text{up}}}{K}.$$

The first two terms are the frictionless constant: the atmosphere less the crown’s height above the outlet. The third is what friction does, and φ is the share of the hose’s whole resistance — the kinetic energy counted as a resistance — that lies before the crown.

Why the sign cannot come out the other way #

φ is a part divided by a whole. Every term in $K_{\text{up}}$ is also a term in $K$, and $K$ has more: the friction of the falling leg and the bend. So φ is at most one, $1 - \varphi$ is never negative, and friction can only raise the crown pressure — not usually, not in the cases drawn, but in every siphon whatever its layout, since the argument uses nothing except that the crown lies somewhere between the inlet and the outlet.

The two effects the frictionless essay named are both inside that one term, and it is worth seeing why one of them always wins. Friction on the rising leg does cost pressure before the crown — the $\tfrac12\rho V^2 fL_{\text{up}}/D$ in the balance. But friction anywhere in the hose also slows the whole flow, and a slower flow arrives at the crown carrying less kinetic energy, which is pressure the crown no longer has to give up. The slowing is set by the resistance of the whole hose; the cost is set by the part before the crown. The saving exceeds the cost, always, by exactly the share of resistance that lies after the crown.

The energy reading of the frictionless case carries straight over and makes the same point without the algebra. There, a parcel’s kinetic energy at the crown was borrowed against a fall it had not yet taken, and the borrowing was measured from the outlet. With friction, part of that fall will be spent on the far side of the crown, fighting the falling leg’s walls, so less of it could be borrowed in advance — and kinetic energy that was never borrowed does not have to be repaid in pressure at the top.

Where the pressure goes along the hose #

The reference installation throughout is the geometry the frictionless essay drew — a crown two metres above the starting surface and an outlet six metres below it, so the crown stands eight metres above the outlet — joined by eighteen metres of 12.5 millimetre hose, nine metres up to the crown and nine down from it.

Without friction the water would leave the outlet at 10.8 metres a second and the hose would pass 79.9 litres a minute, which is the frictionless siphon’s peculiarity shown plainly: a speed at which the hose’s walls would in fact be taking almost everything. With Colebrook’s friction factor at a Reynolds number of 21,843, the hose’s resistance $K$ comes to 38 velocity heads — the jet gets one of them and the walls and fittings the other thirty-seven — and the flow is 12.9 litres a minute.

The crown is at 51.5 kilopascals with friction and 23.0 without. φ is 0.516, so a little over half of the resistance lies before the crown and a little under half after it, and the crown has been lifted by $\rho g \times 0.484 \times 6$ metres, which is 28.4 kilopascals.

The profile shows where that comes from. Near the inlet the water with friction is moving slowly and is at a higher pressure than the frictionless water at the same place, because it has less velocity head to carry. On the rising leg it loses pressure faster, to gravity and to the walls both. At the crown it has still lost less in total. And on the falling leg the pressure rises towards atmospheric more gently, because the walls are holding the falling water back — water held back presses back up on the crown, and that is the physical picture of the margin.

The margin is a ratio of lengths #

The friction factor, the bore and the viscosity cancelled out of the crown pressure, and φ is where they went. When the friction of the hose is large against the fittings and the jet — when $fL/D$ is most of $K$ — both $K_{\text{up}}$ and $K$ are dominated by the same friction factor over two lengths of the same hose, and

$$\varphi \;\to\; \frac{L_{\text{up}}}{L}, \qquad p_c - p_{c,0} \;\to\; \rho g\,\Delta\,\frac{L_{\text{down}}}{L}.$$

The lift friction gives the crown is then set by where the crown sits along the hose, and by nothing else. Not the bore, not the roughness, not the fluid, not the flow.

With the crown three tenths of the way along a thirty-metre hose, friction adds 40.3 kilopascals; half way, 28.8; seven tenths, 17.4 — against a limit of 41.1, 29.4 and 17.6. The four bores lie almost on top of the limit line and on top of each other.

This is the surprising connection, and it is one an electrical engineer would recognise on sight. The crown is the tap on a potential divider. The available head $\rho g\Delta$ is dropped across two resistances in series, the hose before the crown and the hose after it, and the pressure at the tap is set by the ratio of the two and not by their size. What makes the analogy better than an analogy is that it survives the nonlinearity. Turbulent friction is not Ohm’s law — the loss goes as nearly the square of the flow — but both halves of the divider are the same hose at the same velocity with the same friction factor, so whatever the law is, it multiplies both halves alike and cancels from their ratio.

The practical rule follows at once and is not the one intuition gives. Put the crown early. A crown near the source, with most of the hose after it, has most of the resistance downstream of it, and the downstream resistance is what holds the crown up.

Bore changes the flow, and barely the margin #

The limit is only a limit, and the bores in the last figure did separate slightly. The reason is the one resistance in the divider that is not the hose.

Across bores from six to fifty millimetres the flow changes by a factor of 263 and the margin by fifteen per cent. The margin falls as the bore widens because the fittings and the jet’s kinetic energy — the one and the 0.5 and the 0.3 in $K$ — are a fixed number of velocity heads, while the hose’s own friction in a wide bore is a smaller number of them. They take a larger share of a smaller whole, and since the inlet’s share lies before the crown, φ creeps up and the margin creeps down.

So the bore is a decision about flow, and the layout is a decision about margin, and the two can be made separately. Roughness is in the same position as bore: it changes the friction factor, the friction factor multiplies both halves of the divider, and a hose whose roughness the flow cannot feel and one thoroughly fouled lend the crown nearly the same pressure while passing very different amounts of water.

That separation is unusual in pipe flow, where almost every quantity of interest moves with the friction factor. It is closer to the result that a sudden enlargement’s loss contains no viscosity: a quantity that looks as though it should depend on the details of the wall turns out to be set by the geometry of the arrangement, because the wall’s details multiply everything and divide out.

What the draining tank takes back #

The margin is $\rho g(1 - \varphi)\Delta$, and $\Delta$ is the drop from the current surface to the outlet. As the tank drains the drop shrinks, and the margin goes with it.

Friction stretches the drain from under five hours to a day and a half, and the shape of the stretching is worth reading. For most of the drain the flow is turbulent and the level falls roughly as it would without friction, only slower. With 31 centimetres left, twenty-six hours in, the Reynolds number falls out of the turbulent range; with 7.3 centimetres left it is laminar, the flow becomes proportional to the level, and the time to take the last of it grows as a logarithm — a laminar siphon never quite finishes, and the integration stops at a centimetre rather than pretending otherwise.

Through all of it the crown pressure comes down, as the first figure shows. The margin is 28.4 kilopascals at the start, 9.8 after thirteen hours with two metres left, 3.4 after twenty-two hours, 1.2 after twenty-seven, and with a centimetre of level left the crown is at 23.06 kilopascals against the frictionless 23.01. Friction lent the crown a margin and the draining tank has taken back every pascal of it.

That settles the question the frictionless essay could not. The worst crown pressure a real siphon ever sees is exactly the frictionless constant, and it sees it at the very end of the drain, when the flow is at its slowest. The frictionless design rule — measure the crown’s height above the outlet, not above the source, and keep it under the barometric height — is therefore the right rule for a siphon with friction as well, and friction’s margin is something a siphon enjoys at the start and cannot keep.

A siphon that starts, and breaks later #

The frictionless essay’s firmest design statement was that the margin does not drift: a siphon breaks at the instant it starts or not at all. With friction that stops being true, and the exception is exactly the siphon an operator would most want to trust.

Raise the crown to five metres above the starting surface, so that it stands eleven metres above the outlet. That is more than the barometric height, and the frictionless constant is −6.4 kilopascals — below zero absolute, nearly nine kilopascals below the vapour pressure, a column that could not be started by any means short of stripping the water of everything it could boil on. Run the same siphon through twenty-four metres of 12.5 millimetre hose and friction lends its crown 28.7 kilopascals at a six-metre drop, so the crown starts at 22.3, comfortably above the vapour pressure. It starts, and it runs, visibly and steadily, for seventeen hours.

Then the tank takes the margin back. With 1.81 metres of level still above the outlet the lent margin has shrunk to less than the gap between the frictionless constant and the vapour pressure, the crown reaches the vapour pressure, and the column parts — with nothing about the installation changed since it was set running, and with the tank still far from empty.

A frictionless siphon breaks when it starts or never; this one starts and breaks later, which is the ending the cancellation ruled out, supplied by the effect that looked like a detail. It is also an unusually misleading failure. A siphon that has run for most of a day has passed every test an operator is likely to apply, and it passed them only because it was running fast; the moment it slowed, it became again an installation that should never have worked.

So the frictionless design rule becomes stronger here rather than weaker. The crown’s height above the outlet must be kept under the barometric height whether or not the siphon starts, because friction will start siphons the rule forbids and then abandon them part-way through the drain.

Near the end of the drain #

With thirty centimetres of level left the two profiles have nearly converged at the crown. The flow is 2.3 litres a minute at a Reynolds number of 3,944, in the band where the friction factor is not known and the model interpolates rather than claims, and the crown sits 1.5 kilopascals above the frictionless constant. The rising and falling legs still differ from their frictionless counterparts along their lengths, because the velocity heads still differ; the crown, which is where the two effects meet, is where they have almost finished cancelling.

The same picture explains what happens to a siphon whose inlet is higher than its outlet, which is the arrangement of every siphon draining a tank over its rim into a drain below. That siphon stops by swallowing air when the surface reaches the inlet, with the drop still equal to the inlet’s height above the outlet. It stops with part of the margin still lent, and never reaches the frictionless floor at all — which is the one case in which friction’s margin is kept.

What the energy balance leaves out #

It is quasi-steady. Each instant is solved as though the flow were steady at the current level. Over a drain of a day and a half that is excellent, and at the start — where the column accelerates from rest in a time set by how long a message takes to travel the hose and by its inertia — it says nothing.

The transitional band is an interpolation. Between Reynolds numbers of 2,300 and 4,000 the friction factor is a straight line joining the laminar law to Colebrook’s value, which is a statement that the value there is not known rather than a claim about it. A real hose in that band can flicker between the two states, and a critical Reynolds number is a number for how quiet the flow was rather than a property of the hose.

The fittings are two numbers. The inlet’s half and the bend’s 0.3 are handbook values for a plain end and a smooth bend; a real crown may be a sharp kink, and a sharp kink is a larger resistance after the crown, which helps.

The hose is rigid. A soft hose with a crown at 23 kilopascals absolute has 78 kilopascals of atmosphere pressing on its outside, and a thin-walled hose flattens under less. That is a structural limit on the crown, and it arrives from outside the fluid mechanics entirely.

And there is no air in the water. Every figure here is for water that holds all its dissolved gas, and the crown is exactly where it would come out.

The identity the essay rests on is checked at every step of every drain rather than once, because φ changes with the Reynolds number all the way down and a single point would not see the laminar tail. The crown pressure from the two-balance calculation matches the frictionless constant plus the friction term to a part in a million at each of six hundred levels; the frictionless drain reproduces the constant to a part in a billion; and the margin with a centimetre left is under three per cent of the margin at the start.

A result with no name #

Pipe friction has been computable since Weisbach’s form of the loss in the 1840s and Darcy’s measurements a decade later, and Colebrook’s equation for turbulent flow is from 1939. The siphon’s energy balance is older than any of it. Nothing in this essay needed a method invented after the Second World War.

The result nonetheless does not appear to have a name or a standard statement, and the reason is probably the one that kept the frictionless cancellation unnamed: once the two balances are written, the substitution is a line, and nobody who writes it down needs to state what it means. What it means is a design rule that reverses the obvious one — a long hose after the crown is a safety feature — and a potential divider hiding in a garden hose. Neither is in the handbooks, and both follow from arithmetic that is.

Still open: what a slowing flow does with the air in the water #

The drain’s most exposed moment has turned out to be its end, when the crown has been handed back to its lowest pressure and the flow has slowed to a trickle. That is also the moment the water at the crown is most over-full of dissolved air, and the moment the flow is least able to carry a bubble away.

That is the air that breaks a siphon nothing else can: Henry’s law at the crown gives the gas budget, a bubble’s own rise speed in a tube decides whether that gas gathers or is swept out, and the comparison of the two puts the break in a definite place in the drain.

Beside it is the case the crown’s lowest pressure points at directly. A crown sitting at 23 kilopascals is far above zero, and a liquid kept free of anything to boil on can go much further — below zero entirely, which is how a siphon runs with no atmosphere at all and how a column taller than the barometric height can be held together when nothing in it gives way.