The better tunnel needs the bigger tank

A tank that turns a hammer into a swing sized a surge tank against things done to the turbine from outside: a valve shut in a given time, a load taken on over a given time. Every one of those swings decayed, because the tunnel’s friction took a share of each cycle’s energy and nothing put any back.

A real turbine is not driven from outside. It has a governor, and the governor’s job is to hold the machine’s output steady while the conditions around it change. One of the conditions that changes is the head at the turbine, which rises and falls with the tank’s level through the whole of every swing. What the governor does about that turns out to decide whether the swing decays at all.

The scheme is the same one: two kilometres of tunnel three metres across, two metres a second, a hundred metres of gross head of which five are lost to friction at full flow.

A turbine that insists on its power #

The two tanks in that figure differ in area by less than a factor of two. Both are given the same small change, a two per cent cut in the power the turbine is asked for. One swing dies away; the other grows until, after thirteen minutes, the calculation refuses to go on, because the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine can pass.

Nothing about the tanks’ heights is involved. The difference is in how the turbine’s flow responds to the head, and it is clearest drawn directly.

A turbine with its guide vanes fixed behaves like an orifice. If the head across it falls, less water passes, at a slope of 0.074 cubic metres a second per metre about this operating point. That is an ordinary resistance: push harder and more flows, push less and less flows. In the tank equations it acts as damping, because a falling level reduces the outflow and so slows its own fall.

A governed turbine does the opposite. Its power is the flow times the head times a constant, and the governor holds the power. If the head falls, the flow must rise to compensate, at a slope of −0.149 cubic metres a second per metre — twice as steep as the orifice’s, and in the other direction. A flow that increases when the pressure driving it decreases is a negative resistance. In the tank equations it acts as negative damping: a falling level draws more water out of the tank, which makes the level fall faster.

The same object is familiar elsewhere. Any electrical load that regulates its own power — a switching power supply, a motor drive — draws more current as its supply voltage sags, and an input filter ahead of it that has too little resistance of its own oscillates. Power electronics sizes those filters with a criterion of the same shape as the one this essay derives, and an aircraft whose roll damping changes sign past the peak of the lift curve has the same thing in a third form.

Two terms, and only one grows with the tank #

The rigid-column equations from the essay before carry over unchanged, and they are linearised about the operating point: the level $z_0 + \zeta$, the tunnel velocity $V_0 + v$. The governor makes the turbine flow change by $-(Q_0/H_n)\,\zeta$ when the level changes by $\zeta$, since holding $QH$ constant makes the fractional changes in $Q$ and $H$ equal and opposite.

Eliminating the velocity leaves one equation for the level, a damped oscillator,

$$m\,\zeta'' + b\,\zeta' + k\,\zeta = 0,\qquad m = \frac{L A_s}{g A_t},\qquad b = \frac{2 c V_0 A_s}{A_t} - \frac{L Q_0}{g A_t H_n}$$

and the whole criterion is in $b$. It has two terms with opposite signs.

The first is the tunnel’s friction. Its size is proportional to the tank’s area, and the reason is worth having in words: a wide tank changes level slowly for a given imbalance of flow, so for a given swing of level the tunnel’s velocity has to swing further, and friction acts on velocity.

The second is the governor. It does not contain the tank’s area at all. The governor sees the level, the level’s effect on the flow is fixed by the operating point, and nothing about the tank’s size changes that.

So there is an area at which the two are equal, below which the governor wins and the swing grows, and above which friction wins and it decays. Setting $b = 0$ and writing the friction coefficient $c$ in terms of the friction head at full flow, $h_f = cV_0^2$, gives

$$A_{\text{Th}} = \frac{L A_t V_0^2}{2 g\, h_f\, H_n}$$

which is Thoma’s criterion. For this scheme it is 6.07 square metres: a tank 2.78 metres across.

The curve says two things beyond the criterion. The first is that without the governor there is no limit: a fixed-flow turbine adds no negative damping, friction is unopposed, and a tank of any size is stable. Thoma’s area is entirely the governor’s doing.

The second is how slowly a margin is bought above it. At twice Thoma’s area a swing still keeps nearly half of itself each cycle, and at five times it keeps an eighth. A tank at 1.3 times the limit takes ten and a half cycles — nearly eighteen minutes — to bring a disturbance down to a twentieth. The criterion marks where the damping changes sign, and damping that has only just changed sign is not much damping.

Where a growing swing gets its energy #

A swing that grows is gaining energy, and a surge tank has no pump. The energy comes from the reservoir, and the governor is the valve that lets it in.

The swing keeps its energy in two places: the kinetic energy of the tunnel’s water, $\tfrac12\rho L A_t v^2$, and the potential energy of the water lifted above or drawn below the tank’s mean level, $\tfrac12\rho g A_s \zeta^2$. Differentiating their sum and using the two linearised equations gives, to leading order,

$$\frac{dE}{dt} = -\,2\rho g A_t\,c V_0\,v^2 \;+\; \rho g\,\frac{Q_0}{H_n}\,\zeta^2$$

The first term is the friction, and it always removes energy. The second is the governor, and it always adds it. When the level is below its mean the governor draws extra water, which lowers the level further; when the level is above its mean the governor draws less, which lets the tunnel raise it further. Both halves of the cycle push the level away from its mean, so the governor feeds the swing at a rate set by the square of the level’s own excursion, with no moment in the cycle at which it takes anything back.

The criterion is the statement that, averaged over one cycle, the two terms are equal — and the reason the tank’s area decides it is plainer here than in the damping coefficient. The governor’s input depends only on how far the level swings. The friction’s drain depends on how fast the tunnel’s water moves, and for a given rate of change of level that velocity is the tank’s area over the tunnel’s. A tank of twice the area, swinging through the same height, needs twice the tunnel velocity at a given rate of change of level, and swings more slowly by the square root of two, so its level changes more slowly. The net is a velocity swing larger by the square root of two and a friction drain larger by two, while the governor’s input has not moved. The balance falls at Thoma’s area, which is a further derivation of the same number from a quantity — energy — that the eigenvalue never mentions.

The sign change is the criterion, not an estimate of it #

A criterion derived by setting a coefficient to zero is easy to get subtly wrong — a factor of two in the governor term, a sign in the linearisation — and nothing about the formula’s appearance would reveal it. So the claim is checked twice more, by routes that do not pass through $b$.

The first check is the eigenvalue itself. The damped oscillator’s two roots have a real part $-b/2m$, and computing it over a range of areas and locating the zero numerically puts the zero at 1.000 times the closed-form area. That is not a coincidence waiting to be explained: it is the statement that the closed form is the eigenvalue’s sign change, rather than an approximation to it.

The second check shares nothing with the linearisation. The rigid-column equations are integrated in full, with the governor’s $1/H$ law and the friction’s $V|V|$ intact, after a half-per-cent load change small enough to stay near linear. Through four cycles a tank at 0.7 times Thoma’s area grows its swing 4.63-fold, against 4.70 from the eigenvalue; a tank at 1.3 times it shrinks the swing to 0.323 of itself, against 0.321. The first discrepancy is a percentage point and a half, and it is the nonlinearity starting to show in a swing that has grown fivefold.

Why the better tunnel needs the bigger tank #

The criterion has the tunnel’s friction head in its denominator, and that single fact contradicts an instinct almost everyone has about pipes.

Friction in a hydro tunnel is lost energy — five metres of head out of a hundred here, five per cent of the plant’s output for its whole life — and there is every reason to reduce it. A concrete lining instead of bare rock, a larger bore, a smoother finish: each lowers the friction head, and each is good engineering by every measure except this one.

Cut the friction from five metres to one and the smallest stable tank’s area grows by a factor of 4.8, from a shaft 2.78 metres across to one 6.09 metres across. The governor’s negative damping has not changed, the friction’s positive damping has fallen to a fifth, and the tank must grow until its area makes up the difference. A better tunnel needs a bigger tank, and the statement is not a paradox once friction is seen as the only thing holding the swing down.

The same reading explains a device that looks perverse. Many surge tanks are built with a deliberate restriction where they join the tunnel, so that water entering or leaving the tank pays a loss that comes from momentum rather than viscosity. That loss costs nothing in steady running, because no water flows into a tank whose level is steady; it acts only on the swing. It is friction placed exactly where the damping is wanted and nowhere else, and it lets a tank be smaller than Thoma’s area for the tunnel alone.

It is also the same shape a draining siphon’s friction takes, where the loss that slows the flow is the loss that keeps the crown’s pressure up. In both, friction is doing a second job that a design reduces it at its peril.

A small-swing answer #

The criterion came from linearising about one operating point, and a governed plant does not stay at one operating point.

At the smallest load changes each curve sits on its linearised value — 0.979, 0.949 and 0.902 per cycle — which is the four-cycle check again, from a different direction. As the change grows, the two directions separate.

Taking load off moves the plant to a lower flow, a smaller friction loss and a higher net head. Thoma’s area goes as one over the net head, so the new operating point needs a slightly smaller tank than the old one, and every tank drawn stays stable.

Putting load on does the reverse. The plant moves to a higher flow and a lower net head, where Thoma’s area is larger; a tank sized with two per cent to spare at the old point has less than none at the new one once the increase passes about thirteen per cent. The nonlinear terms add to that, since a large swing spends half its time at heads lower still, where the governor’s $1/H$ is steeper.

So a tank sized exactly at Thoma’s area is stable only for the swings the linearisation was made about, and a margin above it is not optional. It is the same lesson in a different setting as a flow whose every mode decays and which grows anyway: an eigenvalue answers the question it was asked, about infinitesimal disturbances, and a real disturbance is not infinitesimal.

Which of the two sizes a tank #

A tank now has two lower limits on its area. The essay before sized it for the swing a full load rejection makes; this one sizes it for stability. Which one decides is a question with a remarkably clean answer.

A tank sized to limit the frictionless rise to $\Delta z$ needs an area $LA_tV_0^2/g\,\Delta z^2$, from the closed form in the essay before. Thoma’s area is $LA_tV_0^2/2g\,h_f H_n$. Their ratio is

$$\frac{A_{\text{surge}}}{A_{\text{Th}}} = \frac{2\,h_f\,H_n}{\Delta z^2}$$

and the tunnel’s length, its bore and its velocity have all cancelled. What is left is two heads against the square of a third.

For this scheme — five metres of friction, 95 of net head, a rise of 8.57 metres — the ratio is 12.9. The tank that holds the surge is thirteen times as large as stability requires, and the governor is irrelevant to its size. That is the ordinary case for a high-head scheme, where the net head is large and a swing of a few metres is a small fraction of it.

A low-head scheme is the opposite. With one metre of friction and twenty of net head, the same allowed rise gives a ratio of 0.54: a tank sized for the swing would be about half as large as stability requires, and it would oscillate. High heads are sized by the swing and low heads by the governor, and the boundary between them is $\Delta z = \sqrt{2 h_f H_n}$ — 30.8 metres for this scheme, a swing larger than a tank on a scheme of this size is built to allow.

What the criterion leaves out #

The governor has no dynamics. The constant-power law is applied instantly, as if the governor measured the head and reset the guide vanes in no time. Against a swing whose period is a minute or more, a governor that responds in seconds is effectively instantaneous — the swing is slow enough to be steady from the governor’s point of view. Real governors add their own lag and a deliberate droop, and both change the effective slope of the flow against head.

The plant is isolated. A turbine holding its own power is what a machine does when it is the only one feeding its load. A machine on a large grid does not see its speed change when its head does, and many are dispatched to a set vane opening or a set power by an operator rather than by their own governor. The criterion is the isolated case, which is the one a designer has to survive.

The turbine’s efficiency is constant. Power is flow times head times efficiency, and a real turbine’s efficiency changes with head. If efficiency falls as head falls, the flow must rise faster to hold the power, and the negative resistance is steeper than $1/H$; the correction multiplies Thoma’s area by a factor that comes from the turbine’s hill chart and not from the tunnel.

The water is incompressible. The swing is minutes long and the pressure waves in the penstock are fractions of a second, so the rigid column is a sound model of the level. A governor that acts fast enough to interact with the penstock’s waves has a second, much faster stability problem of its own, which this criterion does not see.

Thoma, 1910 #

The criterion is Dieter Thoma’s, published in 1910, when governed turbines at the ends of long tunnels were becoming common and whether a governed tank could oscillate on its own had become a practical question rather than a curiosity. Later work added the governor’s dynamics, the turbine’s efficiency, throttled and differential tanks and the nonlinear margin, and none of it has displaced the first result, which remains the number a designer computes first.

What makes it durable is that it is a statement about signs. It does not predict how large a tank should be for comfort — the essay before does that, for a high-head scheme, far above the limit — but where the damping of a governed system changes sign, and why friction appears in the denominator. That second part is still the one most often met with surprise.

Still open: what a spring of air does to the swing #

The tanks here are shafts open to the air, so their stiffness is gravity acting on the height of water in them. Where the rock above a tunnel is too deep, or the hillside too steep, for a shaft to reach daylight, schemes use a closed chamber with compressed air trapped above the water instead. The air is a spring far stiffer than the water’s weight in a short chamber, the swing is faster, and the gas’s own compression — somewhere between isothermal and adiabatic depending on how fast the swing is — enters the stiffness term $k$, which for an open tank is fixed by gravity and the operating point alone.

That changes Thoma’s criterion in a way the open tank never shows: the governor’s term is unchanged, the friction term is unchanged, and the stiffness now depends on the air’s volume and pressure. Beside it sits the restriction the essay mentioned in passing, where the loss at a sudden expansion becomes a design variable for damping, and the question of how small a tank a throttle can buy before the part of the closure a pipe cannot see starts reaching the tunnel again.