A tank that turns a hammer into a swing

Stopping water costs more than moving it found the price of shutting a valve on a moving column: the wave speed times the velocity destroyed, over gravity, and nothing about the pipe’s strength or the valve’s design can lower it. The part of the closure a pipe cannot see found the only escape, which is time — and found that “slowly” has to mean slowly against one number, the time a pressure signal takes to reach the far end of the line and come back.

A hydroelectric scheme is the case where that escape is closed. Its water arrives down a long tunnel from a reservoir, and its turbine has a governor whose whole job is to cut the flow in seconds when the electrical load drops, because a turbine that keeps its water after losing its load runs away. The tunnel’s round trip is measured in seconds too. A valve that must shut in two seconds on a line whose round trip is four and a half is, by the argument of both earlier essays, a valve shut instantaneously.

The engineering answer is a vertical shaft, open to the air, standing on the tunnel just upstream of the turbine: a surge tank. What it does is easy to describe wrongly — it does not absorb the pressure, and it does not cushion anything — and the rest of this essay computes what it does instead.

The scheme used throughout is one of ordinary proportions: two kilometres of tunnel three metres across carrying water at two metres a second from a reservoir a hundred metres above the turbine, five metres of that head lost to friction at full flow, a tank ten metres across, and two hundred metres of penstock 2.2 metres across between the tank and the valve. The wave speed is a thousand metres a second in both pipes.

A free surface sends the wave back upside down #

The two traces in that figure are the same valve closing in the same two seconds, and the difference between them is where the first reflection happens.

A pressure wave travelling up a pipe meets the pipe’s far end and is reflected. What it does there depends on what the end can hold. A closed end cannot let water move, so the wave comes back with its own sign and the pressure doubles; that is what the valve does. An open end into a large body of water cannot hold a pressure different from its own depth, so the wave comes back with its sign reversed, cancelling the rise it arrived with; that is what the reservoir does, and it is why the earlier essays’ traces invert after one round trip.

A surge tank is a reservoir moved to within two hundred metres of the valve. Its surface is large and open to the air, so for the few milliseconds a wave takes to pass, it is a place where the head cannot change. The closure’s wave runs up the penstock, meets the tank, and returns inverted after 0.4 seconds rather than after 4.4. The tunnel beyond the tank is fed only by whatever small part of the wave crosses the junction, and that part is small because the tank’s surface is so much larger than either pipe.

So the valve is no longer shutting a two-kilometre line. It is shutting a two-hundred-metre one, whose round trip is a tenth as long, and a two-second closure on that line is five round trips — slow, in the only sense the earlier essay allows the word.

The trace also shows what the tank does not do. Once the valve is shut, the penstock is a pipe closed at one end and open at the other, and a pipe like that rings at the period in which a wave makes the journey four times: 0.8 seconds here, 47 metres either side of the old head. Quasi-steady friction over two hundred metres takes almost nothing out of that ringing, which is the calculation being honest about its own friction model rather than a prediction — the friction a pipe actually has depends on the history of its flow, and a real penstock rings down in a few periods.

The knee moves from the tunnel to the penstock #

The previous figure is one closure time. The question a specification asks is the whole curve, and it exposes a belief about surge tanks that is common and wrong.

The belief is that a tank cushions a hammer, and that a valve behind one can therefore be shut as fast as its actuator allows. The left end of the figure says otherwise. Shut in a fifth of a second or in 0.4, the valve takes 381 metres of head with the tank and without it — the penstock’s own Joukowsky rise of 379 metres, and the little the friction packs on top. Nothing else is possible, because the peak is made before the wave has reached the tank, and a surface two hundred metres away cannot act on a pressure that has not arrived.

What the tank moves is the knee — the closure time at which a slower closure starts to help. On the bare line the knee is at the whole line’s round trip, so closing in 1, 2 and 4 seconds gives 300, 256 and 233 metres, all within a quarter of the instantaneous answer. Behind the tank the knee is at the penstock’s round trip, and the same three closures give 123, 51 and 26 metres. A four-second closure behind the tank makes a rise the bare line cannot match until it is given somewhere between twenty and forty seconds.

That is the whole of the engineering value, and it is a statement about time rather than about pressure. The tank buys the governor the right to act fast. It does not buy the right to act instantly, and a turbine inlet valve that trips shut in a tenth of a second is designed against the full penstock rise whether a tank stands upstream of it or not.

The marked points on the bare line are the other thing the tank prevents. After every closure up to six seconds, the reservoir’s inverted reflection arrives at the valve end while the line is still packed and drives the pressure below what water can hold. The column separates, a cavity opens, and the collapse that follows is the case the first water-hammer essay drew as the worst one. The characteristics used here do not follow a column once it has parted, so those rises are the largest before separation rather than the largest overall, which makes the bare line’s curve an understatement.

Where the tunnel’s momentum goes #

The tank has kept the wave out of the tunnel, but it has not made the tunnel’s water stop. Fourteen cubic metres a second of it are still arriving at a turbine that has shut, and they have nowhere to go but up the shaft.

On the time scale of that, the water is incompressible. The tunnel is a single slug of fluid, and two equations describe it. The level in the tank, $z$, measured above the reservoir’s, is the only thing pushing back on the slug, together with its own friction, so its velocity obeys

$$\frac{L}{g}\frac{dV}{dt} = -z - cV|V|$$

which is the unsteady energy equation with the term that is usually dropped kept in — $L/g$ times an acceleration is the head it takes to change the velocity of a column $L$ long. And the tank fills with whatever the tunnel delivers and the turbine does not take:

$$A_s\frac{dz}{dt} = A_tV - Q_T$$

With the friction dropped and the turbine shut, those are a mass on a spring. The mass is the tunnel’s water, and the spring is gravity acting on the column of water standing in the tank: every metre the level rises is a metre of head pushing the slug back. So the level rises, stops, falls below the reservoir’s, and swings.

Its height comes out of an energy balance with nothing else in it. The slug carries a kinetic energy of $\tfrac12\rho L A_t V_0^2$ — twenty-eight megajoules here. At the top of the swing all of it is potential energy of the water lifted into the tank, $\tfrac12\rho g A_s z_{\max}^2$, so

$$z_{\max} = V_0\sqrt{\frac{L A_t}{g A_s}}, \qquad T = 2\pi\sqrt{\frac{L A_s}{g A_t}}$$

For this scheme that is 8.57 metres and 299.1 seconds. The hammer that would have been 204 metres of head on the bare tunnel, delivered at once, has become 8.57 metres delivered over a minute and a quarter. Nothing has been dissipated to make that trade. The momentum has been given a free surface to push against, and a free surface gives way slowly.

Friction starts the swing lower and ends it sooner #

The frictionless swing is the closed form’s; the real one differs in two ways that are both visible in the figure, and both are to the tank’s advantage.

The first is where it starts. A tunnel losing five metres of head to friction at full flow delivers its water to the tank five metres below the reservoir’s level, because that is the head it takes to push the flow through. The swing begins from a level already five metres down, so a rise of ten and a half metres brings it only to 5.61 metres above the reservoir.

The second is what happens during the rise. The friction term acts against the slug’s velocity whichever way it is moving, so it helps the tank decelerate the column on the way up and it opposes the column’s return on the way down. Each swing loses a share of its energy, and the troughs and peaks close in: 5.61, then −3.70, then smaller.

Both effects come from the same five metres, and there is a small irony in that. The earlier essays treated friction as a nuisance the solver had to carry. Here it is the thing that stops the tank from ringing for ever, and a siphon’s friction turned out to be a margin rather than a loss for a different reason with the same shape. The next essay finds a case where that shape becomes the whole of the story: a turbine governor that removes damping, and a friction head that is the only thing left to supply it.

Height against width #

The closed form has the tank’s area under a square root in both of its answers, with opposite signs, and that is the designer’s trade written as algebra.

A tank twice as wide has four times the area, so the frictionless peak halves and the period doubles. A shaft must be taller than the highest level it will ever see — or it overflows, which some designs allow and most avoid — so a narrow tank is a tall one and a wide tank is a short one, and the rock or concrete it takes to build either is a question of how much of each is cheap at that site.

The frictional curve falls faster than the frictionless one, and for a reason worth stating. The friction drawdown is the same five metres whatever the tank’s size, so it is a larger share of a small swing than of a large one; and a wide tank’s slower swing gives friction more seconds to act on each cycle. From ten metres to twenty the frictional peak falls to a third rather than a half.

The cost is the period. A swing of ten minutes is a slow disturbance to the head at the turbine, and the governor, which is trying to hold the machine’s speed and power steady, sees the head changing under it throughout. That interaction is harmless for fixed guide vanes and is not harmless for a governor, and it is the reason the tank’s area has a lower limit that has nothing to do with its height.

The floor is sized by taking load on #

A tank has two design cases, and the one that sizes its bottom is the opposite of a shutdown.

When the turbine opens, it demands fourteen cubic metres a second at once, and the tunnel cannot supply them: its water is at rest, and accelerating two kilometres of it takes the head of a falling level in the tank. So the tank drains into the turbine while the tunnel catches up, and the level overshoots the new steady level of five metres down before it swings back.

That downsurge is what fixes how far below the reservoir’s level the tank must extend. If the level falls below the top of the tunnel where it enters the shaft, air is drawn into the tunnel — and air in a pressure tunnel is the failure a surge tank exists to prevent, for the reason air carried into a siphon’s crown breaks it: a pocket of gas in a line that relies on being full does not leave on its own.

The figure also shows the downsurge’s own version of the closure-time curve. Taking load on over a minute buys only 45 centimetres, because a minute is a fifth of the swing’s period; taking it on over five minutes, longer than a period, lets the tunnel keep pace and the overshoot nearly vanishes. Slow is again measured against the system’s own clock, and the tank’s clock is minutes long.

Two calculations that share no equations #

Every result above has come from one of two models, and the second can be used to check the first.

The rigid column assumes the water is incompressible. That is plainly false — the whole of the water-hammer argument is that it is not — and the justification for it is a ratio of times: the swing takes 299 seconds, and a pressure wave crosses the whole line and back in 4.4. Sixty-eight round trips happen during one swing, and over that many of them the waves have averaged into a uniform pressure along the tunnel. A memory number is always one time over another, and this one is large.

The characteristics calculation assumes nothing of the kind. It carries every pressure wave through a compressible tunnel and penstock, reflecting at the reservoir, at the valve and at the tank, and it integrates the tank’s level from the difference between the flows arriving from both pipes. It knows nothing about masses, springs or closed forms. Run for fifteen minutes of simulated time, forty-five thousand steps, it produces the slow swing anyway, and its peak agrees with the rigid column’s to four millimetres.

That agreement is the useful sentence in this section. The rigid column is not a separate physics from the wave; it is the wave’s own long-time limit, once the round trips are short against the swing. The rigid column can be trusted for the tank’s level for that reason, and not for the valve’s head, which is decided inside a single round trip — which is why the two earlier figures were drawn with characteristics and the three after them were not.

What the tank model leaves out #

The tank’s connection. The model joins tank and tunnel with no loss. Many tanks deliberately put a restriction there, so that water entering or leaving the tank pays a loss that needs no viscosity to compute. A throttled tank damps its swing much faster, at the price of passing more of the hammer into the tunnel, and choosing the orifice is a trade between the two curves this essay drew separately.

The shape of the tank. A shaft of one diameter is the simplest case. Real tanks widen at the top and bottom to take the extremes cheaply, add internal risers that make the tank respond quickly at first and slowly afterwards, or trap air under a closed roof so that a short tank behaves like a tall one. Each changes the spring, and none changes the argument.

Friction. The Darcy term is applied quasi-steadily, as if each instant’s flow had been running for ever. For the swing, whose period is minutes, that is accurate: the tunnel’s velocity profile has time to follow, and a pulsating flow only stops having a steady profile when its period is short against the time viscosity takes to cross the pipe. For the penstock’s ringing it is not, and the undamped ringing in the first figure is that assumption showing.

The column that separates. The bare-line curves stop at separation because the calculation has no cavity model. Every figure with the tank in place stays well above vapour pressure throughout, so the omission affects only the comparison and not the tank.

The governor. Every closure here is a prescribed motion of the valve. A real turbine’s gate is moved by a governor that is watching the machine’s speed, and the head at the turbine is changing under it the whole time the tank swings. That is the next essay’s subject.

A castle for water #

The German name for a surge tank is Wasserschloss — a water castle — and the tall concrete shafts on Alpine hillsides above hydro stations are the reason. By the early twentieth century the rigid-column equations above were standard, the tank’s height and floor were being sized from them, and the design question had moved on to the thing they did not settle: whether a tank large enough to hold the surge was large enough to be stable once the turbine was being governed. That question was answered in 1910, and the answer is a lower limit on the tank’s area that grows as the tunnel gets better.

Still open: what a governor does to the swing #

Every closure in this essay was imposed on the turbine from outside. A governed turbine does something different when the head at it falls: to hold its power, it opens further and draws more water, and the tank level falls further still. A flow that increases as the pressure driving it decreases is a negative resistance, and a swing fed by a negative resistance is damped only by whatever positive resistance the system has — which, in a tunnel, is its friction.

That is the better tunnel needs the bigger tank: Thoma’s criterion for the smallest stable tank, why a smoother tunnel needs a larger one, and the eigenvalue that changes sign at exactly the area the criterion names. Beside it sits the same instability in a different place — a roll that damps itself only until the lift curve turns over is a negative damping too, and so is every constant-power load on an electrical network.