Theme

The thread: What is conserved — page 37

Page 37 of 43, continuing through the 384 essays this motif runs through.

384 essays carry this thread — page 37 of 43.

Where a carburettor's cold comes from. The temperature changes in a carburettor's throat, in kelvin: the cooling of the air as it expands to a hundred metres a second, the far smaller cooling of the metal it flows over, the cooling as petrol evaporates into it, and the warming as the water in humid air condenses back out. The fuel does five times what the venturi does to the air and fifty times what it does to the metal.

Carburettor ice is made by the fuel, not the venturi

Pilots are taught that a carburettor ices because air cools as it expands through the venturi. The expansion does cool the air, by five kelvin at a hundred metres a second, but the metal the ice grows on barely feels it. The cold comes from the petrol evaporating, and whether it makes ice is decided by the water in the air, which warms the mixture as it condenses. The worst day is not a cold one.

Time per litre rises in a straight line with the litres already filtered. The elapsed time divided by the filtrate collected, against the filtrate, for a constant-pressure filtration of a silica slurry through a cloth: integrated step by step, and Ruth's closed form. The line's slope is the cake's resistance, which grows with every litre, and its intercept is the cloth's, which does not. After a quarter of a cubic metre the cake is already the larger.

Fluids at work

A filter is slowed by what it has caught

Push a slurry through a cloth and within minutes the cloth no longer matters: the particles it has caught form a cake, the cake is a packed bed, and every litre filtered makes the bed thicker for the next. The filtrate grows only as the square root of time. And if the cake is soft, it packs tighter against the cloth the harder it is pushed, until more pressure buys almost nothing.

A slow current divides round the fluid over a hill (δ = 4). A uniform stream, left to right, over a Gaussian hill on the floor of a rotating layer, at δ = h₀/(H·Ro) = 4, where h₀/H is the hill's height against the depth and Ro the Rossby number U/fa; the onset for this shape is δ = 3.134. The dashed circle is the hill's e-folding radius. Water crossing the hill is squashed and, keeping its potential vorticity, spins clockwise; that anticyclone adds to the stream on one side and opposes it on the other. Here it stops the stream: the outlined region, 1.54 square radii, holds fluid that circles for ever and never leaves, and it sits beside the summit rather than on it, on the side where the swirl runs against the current.

Regimes and numbers

The hill a slow current will not climb

The Taylor–Proudman theorem says a rotating fluid goes round an obstacle rather than over it, as if a solid column stood above it. Conserving potential vorticity turns that limit into a threshold. A current is stopped over a hill once the hill's height against the depth exceeds a fixed multiple of the Rossby number — 2 for a flat-topped hill, 3.13 for a Gaussian one, 16/3 for a cone — and the fluid it holds sits beside the summit rather than on it.

A washed filter's best cycle is shorter, and still has one. The filtrate a batch filter produces per hour, averaged over filtering, washing and twenty minutes of emptying and refilling, against the filtrate collected in each cycle, for a slurry of 400 kilograms of silica per cubic metre and wash ratios of 0, 1, 2 and 5 pore volumes. Each curve has a best cycle, marked, and the wash both lowers it and moves it to a thinner cake: from 0.307 to 0.217 m³ per m² of cloth.

Fluids at work

A washed filter works as long as it rests

A batch filter that has to be emptied between runs has a best cycle, and the rule for it is exact: the time the cake is responsible for equals the time the filter stands idle. Washing the cake does not break the rule, because a wash through a finished cake costs time in proportion to the square of its thickness, just as filtering it did. What a wash changes is the price, and the price is set by the slurry's concentration and by how far the wash front smears, not by the cake's softness.

Two cylinders side by side pull together. The ideal stream past two cylinders side by side with a gap of a fifth of a radius between them. The stream squeezes through the gap faster than round the outsides, the pressure in the gap is lower, and each cylinder is pulled towards the other with a force of 3.6 ρU²a — while the pair together feels nothing at all.

Ideal flow

The paradox is for the pair, not for each body

D'Alembert's paradox says a body in a steady ideal stream feels no force. Put two bodies in the stream and the theorem still holds — for the pair. Each cylinder on its own is pushed: side by side they pull together, with a force that grows without limit as the gap closes, and one behind the other they push apart, so that the front cylinder is driven upstream into the stream that is flowing past it. The forces are equal and opposite at every spacing and every angle, and their sum is exactly the zero the paradox promised.

Circulation and lift

A fair V is a curved V

In a straight V of birds somebody always pays more than somebody else: at the fairest angle the leader and the two birds at the tips pay half again what the birds between them pay. The V in which every bird pays exactly the same can be solved for, and it is not straight. Its arms leave the leader almost abreast and bend back ever more steeply, to thirty-two degrees at the tips of a flock of nine. The share every bird then pays is the flock's average, fixed by Munk's theorem before any position is chosen — so fairness costs nothing, and the only thing that can make the flock cheaper is flying closer together sideways.

In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

Flows and fields

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

A fuel-wetted plate sits far below the stream that cools it. The temperature of a throttle plate wetted by petrol, against the day's temperature at 80 per cent humidity, beside the temperature of the stream flowing over it — the mixture after half the fuel has evaporated into it, at the plate's recovery temperature. A plate kept wet with fresh fuel sits nine to twenty-three kelvin below the stream, more on warmer days, and below freezing on any day up to about 34 °C. A plate whose film has lost its light ends sits one to six kelvin below it. The dashed line is freezing.

What is taught wrongly

The throttle plate is a wet bulb for petrol

A carburettor's icing is usually worked out for the mixture — the fuel's latent heat spread through all the air — and that puts the warmest icing day near 15 °C, well short of the 30 °C the charts pilots use allow for. The ice does not grow in the mixture. It grows on the throttle plate, and a plate wet with evaporating petrol is a wet-bulb thermometer for fuel: nine to twenty-three kelvin colder than the stream flowing over it. Kept wet with fresh fuel it ices on a 30 °C day at full humidity and a 40 °C day at half — which is what the training handbook warns of. Once its light ends have gone it does not, and that difference is most of the story.

Fluids at work

One fast pump or two slow ones

A pumping station with several identical pumps in parallel has a control that a single pump does not: how many of them are running. With no static lift the answer is all of them, always, slowed together. With lift it is a number that falls with the flow and is almost never whole — the flow times the number of pumps, over the square root of the head the system asks. Staging rounds it, and the right moment to round up comes before the running pumps have run out of speed.

All themes