A broken line empties at the pace of its friction

Two ways to choke found that a duct can be choked by friction as well as by area: push gas along a long enough pipe and the flow reaches the speed of sound at the exit whatever the pressure beyond it, because friction drives subsonic flow towards Mach one. A pipe cannot hold its gas at the wall’s temperature added the heat the wall exchanges with the gas and found that in steady flow it hardly matters: the line chokes at Mach one with its gas a sixth colder than the wall, and the capacity of a long line is what Fanno’s adiabatic theory says. Both essays were about steady lines, and the second ended on the case that is not.

A line that breaks is the case engineers most need, because it is the one that happens once and badly. A pipeline ruptures, or a vessel is vented through a long line, and the question is how fast the pressure falls: how long a leak lasts, how much gas comes out in the first minute, when it is safe to approach. The usual first answer is the choke. The broken end is open to the air at a pressure far below the line’s, so the flow there reaches the speed of sound, and a choked outlet passes a mass flow set only by the conditions just inside it. That answer is correct at the instant of the break and increasingly wrong after it, and this essay computes by how much.

Marching a line that breaks #

The line is closed at one end and full of gas at rest. At time zero its far end breaks, into still air at a twentieth of its pressure. The gas obeys the unsteady one-dimensional equations — mass, momentum and energy — with the Darcy friction force of a turbulent pipe — the friction factor that the three buffer layers set — and the friction enters through one number: the line’s length measured in friction lengths, $F = fL/D$. A line of 10 km of 0.5 m pipe with a friction factor of a hundredth is two hundred friction lengths long. Time is measured in transit times, the line’s length over the gas’s initial speed of sound: about twenty-three seconds for that line full of natural gas.

The equations are solved by a second-order finite-volume scheme, with the friction and the wall’s heat applied implicitly in a separate step. The break is handled the way the theory of characteristics says it must be: information leaving the line through the break is carried by the outgoing invariant, and the break passes either the sonic state that invariant allows — choked — or, once that state’s pressure would fall below the air’s, the state at the air’s pressure. Nothing is imposed about which; the march decides.

One detail of the friction step matters enough to state. A pipe’s wall does not move, so its shear stress does no work on the gas: friction takes momentum away and turns kinetic energy into heat, and the total energy is unchanged. That is why a Fanno line keeps its stagnation enthalpy, and why a duct with friction cannot be run backwards: the heat stays in the gas as entropy. A first version of the step held the pressure instead, which quietly threw the kinetic energy away, and the check against Fanno’s steady solution failed by seventeen per cent until it was put right.

The line empties from the break backwards #

The first figure is the pressure along a line three hundred friction lengths long, at five moments. Half a transit time after the break, a rarefaction has crossed half the line: behind it the gas has started to move towards the break and its pressure has fallen; ahead of it, towards the closed end, nothing has happened, because nothing could have been told. That much is the same with or without friction, since the front of the rarefaction travels at the undisturbed gas’s speed of sound.

After that, friction takes over. With no friction the rarefaction would reflect from the closed end and the line would empty in a few transit times, the pressure nearly uniform behind each passing wave. With friction the gas can move only as fast as the pressure gradient can push it against the wall, and the pressure settles into a slope that steepens towards the break. At two transit times the closed end has lost an eighth of its pressure; at six, it is at 0.55; at fifteen, 0.21. By then the break itself sits barely above the air’s pressure — no longer choked, and no longer the thing deciding anything.

The closed end hears the break at the speed of sound #

The second figure watches the closed end. Every line holds its pressure for exactly one transit time, because the rarefaction’s head travels at the gas’s own speed of sound whatever the friction; the closed end cannot know about the break sooner. What friction changes is what arrives. With no friction the head of the rarefaction brings the full drop behind it, and the line reaches half its pressure 1.37 transit times after the break. With 30 friction lengths it takes 2.6; with 300, 6.8; with 3,000, 20. On the longest line the rarefaction’s head arrives on time and is nearly empty: friction has spent almost all of the drop within a short distance of the break, and what reaches the closed end is a slow leak along a gradient.

Half-time goes as the square root of the friction #

The third figure puts the half-time against the friction length on logarithmic axes, and on a long line it follows a square root. At 1,000 friction lengths the half-time is 12.0 transit times; at 3,000, 20.2; the ratio of each to $\sqrt{F}$ is 0.38 and 0.37.

The square root is the signature of turbulent friction, and the argument for it takes one line. On a long line the gas’s inertia is negligible against its friction: the momentum balance is the pressure gradient against the wall’s drag, $\partial p/\partial x \approx -(f/2D)\rho u|u|$. The speed the gas moves at is therefore the square root of the gradient, $u \sim \sqrt{|\partial p/\partial x| D/\rho f}$ — not proportional to it, as it would be in laminar flow — and putting that into the mass balance leaves an equation in which time appears only in the combination $t/\sqrt{F}$. Double the length of a turbulent line and its blowdown takes 1.41 times as many transit times, each twice as long — 2.8 times as long in seconds; a laminar line’s would take four times as long.

The same argument explains why the break stops mattering. The choked break can pass gas at a rate set by the pressure just inside it; the line can deliver gas to the break at a rate set by the square root of the gradient along it. On a short line the second is larger and the break is the bottleneck. On a long line the gradient is spread over a longer distance, the delivery is smaller, and the gas cannot reach the break fast enough to keep it choked.

The break is choked only at the start #

The fourth figure measures that directly. Up to about a hundred friction lengths the break stays choked through the whole time the line takes to fall to half its pressure, and the choked-outlet answer is the right one: the gas leaves at its own speed of sound, and the air outside is never heard from. At 300 friction lengths the break unchokes after three-fifths of the half-time. At 1,000 it is choked for the first 13 per cent of it, and at 3,000 for 3 per cent. For nearly all of a long line’s blowdown the pressure at its break is the air’s, and the flow there is subsonic.

A calculation that sizes a vent, estimates a leak or plans an isolation valve on the choked-outlet rate is therefore right about the first instants and wrong thereafter by a factor that grows with the line: it overstates how fast a long line empties, and it gets the shape of the decay wrong, since a choked vessel empties exponentially and a friction-limited line does not.

The wall’s heat, which did not matter before #

The fifth figure returns to the question the steady essay left open. Gas that expands cools. In the adiabatic blowdown the closed end’s gas has fallen to 0.82 of its starting temperature by the time its pressure has halved, which is what an isentropic expansion to half pressure gives. A real wall is not adiabatic, and by Reynolds’ analogy the same turbulence that makes the friction carries heat from the wall into the gas, at a rate set by the same friction factor.

On a short line the blowdown is too quick for the wall to do much: at 30 friction lengths the gas still reaches 0.82 of the wall’s temperature at half pressure. On a long line the blowdown is slow — it scales as $\sqrt{F}$ — while the heat exchange runs at a rate set by the friction and the gas speed, and the wall keeps up: 0.94 at 300 friction lengths, 0.99 at 3,000. The long line’s blowdown is isothermal, which is the assumption the isothermal pipeline makes for steady flow, arrived at here from the other side.

The consequence is in the third figure. Gas held warm keeps its pressure up for a given amount left in the line, and passes less mass for a given pressure at the break, and both slow the decay: on every line longer than about a hundred friction lengths the wall’s heat lengthens the half-time by 27 to 30 per cent. In the steady line the heat changed the capacity by under half a per cent. In the unsteady one it is the second-largest effect after the friction itself.

A rule for a line of any length #

Put together, the figures give a rule with three regimes. A line shorter than about ten friction lengths empties as a frictionless one would: a rarefaction back and forth, a half-time a little over one transit time, the break choked throughout. Between ten and a hundred friction lengths the half-time grows but the break stays choked, and the choked-outlet estimate, applied to the gas near the break rather than to the whole line, is still the right tool. Beyond a hundred, friction sets the pace, the half-time is close to $0.37\sqrt{F}$ transit times, and the break is choked for a shrinking first part of it.

For the line of the example — 10 km of 0.5 m pipe, two hundred friction lengths, a transit time of about twenty-three seconds — the march gives a half-time of 5.6 transit times with an adiabatic wall, a little over two minutes, and 7.3 with the wall’s heat, nearly three. Its break unchokes after 4.7 transit times, most of the way to half pressure. A line five times as long, with a transit time five times as long and a friction length five times as great, would take $5\sqrt{5} \approx 11$ times as long to halve — about twenty-five minutes — and its break would be choked for only the first three.

The same line with water in it #

A liquid line that breaks does something quite different, and the contrast shows what the gas’s compressibility is doing. In water the pressure wave travels at the pipe’s wave speed, and the drop it carries is the Joukowsky rise run in reverse: the break’s pressure falls to the air’s at once, the column behind it starts to move, and within a round trip or two the whole column is accelerating as a rigid slug against its friction. There is almost nothing stored in the liquid to come out, so a broken water main empties at the rate its reservoir can drive a column through the pipe. A gas line has its whole contents stored in its compressibility, and every kilogram of it must travel the line’s length against friction to leave. That is why the friction length, and not the break, is the number that decides a gas line’s blowdown.

What was checked #

The sixth figure is the ledger. With the closed end replaced by a reservoir held at the initial state, the march settles on steady Fanno flow, and between two stations clear of the choke the friction length it implies matches Fanno’s closed form to a quarter of a per cent; at the choke itself, where the steady solution’s gradient is infinite, any grid rounds the answer off, and the check is deliberately made away from it. With no friction, and before the rarefaction has come back, the break passes exactly the sonic state of a centred rarefaction, $p/p_0 = (2/(\gamma+1))^{2\gamma/(\gamma-1)} = 0.279$, to five parts in a hundred thousand. The mass that leaves through the break equals the mass the line loses to $10^{-15}$. The half-time moves by half a per cent when the grid is doubled. And the square-root collapse holds to two and a half per cent between 1,000 and 3,000 friction lengths.

What a one-dimensional line leaves out #

The jet outside. A choked break discharges an underexpanded jet into the air, and a real rupture has a crater, a torn pipe end and a flow that is not one-dimensional near the break. The model’s break is an ideal open end.

Real gas. Natural gas at pipeline pressures is not a perfect gas, and its speed of sound and its cooling on expansion both differ; the Joule–Thomson cooling of a real gas adds to the expansion cooling the figure shows.

Friction that is not steady. The friction factor is held constant, as a steady turbulent pipe’s would be. In the first transit time the flow accelerates from rest and its wall shear is larger than the steady value; in the last stages it is slow enough for the friction factor to rise. Both affect the details and neither the scaling.

One closed end. A line broken in the middle empties from both halves, each a closed-end line of half the length, and a line fed from a live compressor station does not empty at all; the case computed is the simplest one.

The convention the numbers depend on #

$F = fL/D$ with $f$ the Darcy friction factor; time is in transit times $L/c_0$, with $c_0$ the gas’s initial speed of sound; pressures are fractions of the line’s initial pressure, and the air outside is at a twentieth of it. The half-time is the time for the pressure at the closed end to fall to half. The wall exchanges heat with a Stanton number of $f/8$, Reynolds’ analogy for a turbulent pipe, and is held at the gas’s initial temperature.

Who found it, and when #

Fanno’s steady friction-choked flow is from 1904. The unsteady blowdown of long gas lines became an engineering question with the growth of high-pressure transmission pipelines after the Second World War, and was taken up by the method of characteristics from the 1960s, among others by Fannelöp and Ryhming, whose analyses of massive releases from long lines in the 1980s treated the friction-dominated stage of a rupture; the square-root law is what its similarity variable gives. Reynolds’ analogy between wall friction and wall heat transfer is from 1874.

Still open: a line whose gas is not a perfect gas #

Every number here is for air-like gas with a ratio of specific heats of 1.4 and no real-gas effects, and the lines that break with the most at stake do not carry that. Natural gas at seventy or a hundred bar has a compressibility factor well below one and a strong Joule–Thomson effect, and carbon dioxide lines for capture and storage cross into two phases as they depressurise — the rarefaction then carries a boiling front behind it, and the speed of sound in the mixture drops by an order of magnitude. The next calculation replaces the perfect gas with a real equation of state and asks how much the square-root law and the one-third heat effect survive, and whether a carbon dioxide line’s two-phase front becomes the thing that sets the half-time in place of the friction.