The snorkel that cannot be made longer

Leonardo da Vinci’s notebooks include a design for an underwater breathing apparatus: a diver under water, a tube running up to a float at the surface, the diver breathing through it. Similar sketches go back to antiquity, and the idea is revived regularly by children with garden hoses. It does not work, and a person who tries it with a hose more than about half a metre long discovers the fact quickly and alarmingly. Every snorkel on sale is short — safety standards settle on lengths of around thirty-five to forty centimetres for adults — and the reason is not timidity on the part of the standard-setters.

Two separate pieces of physics close on the long snorkel, and either alone would be enough. The first is the weight of water on the chest. The second is the tube itself, which has to be breathed through twice for every breath. Neither is about how hard it is to suck air through a narrow pipe, which is the first thing most people suspect, and which turns out to matter least.

The water on the chest #

The pressure that only knows depth set out the rule: the pressure in a still fluid depends on nothing but depth, rising by $\rho g$ per metre, 9.8 kilopascals in fresh water. A swimmer’s chest a metre below the surface is squeezed on all sides by that extra pressure. The lungs inside are connected by the snorkel to the air at the surface, at atmospheric pressure. To breathe in, the chest has to expand, and to expand it has to push the water back against a pressure difference of $\rho g d$ across the chest wall.

The breathing muscles are not built for that. The strongest suction a healthy adult can generate, measured by having them inhale with all their strength against a closed valve, is typically eight to twelve kilopascals, a metre of water, and it can be held for a second or two. The load that can be sustained, breath after breath, without the muscles tiring, is far smaller: of the order of a kilopascal or two, the equivalent of fifteen or twenty centimetres of water. So a person with their chest a metre down can take perhaps one breath with all their strength, and then no more. Even at half a metre the effort is exhausting within minutes.

That is the snorkel’s real depth limit, and it would remain even if the tube had no volume and no resistance at all. A swimmer floating face down at the surface, held up by the weight of the water the body displaces as the weight of the water that is not there described, has their chest a few tens of centimetres below the water, and a snorkel of about thirty-five centimetres reaches the surface from there with a little to spare. A longer tube is pointless, because the swimmer cannot use the depth it would allow.

The same arithmetic explains why breath-hold divers can go far deeper than snorkellers. A breath-holding diver does not draw air from the surface; the air in their lungs is compressed along with their chest, and the pressure inside and outside the chest stays nearly equal. It is only connecting the lungs to a different pressure — the surface’s, through a tube — that makes depth unbearable. Force multiplied and nothing gained found the same principle in a hydraulic press: pressure differences, not pressures, do work, and a tube to the surface creates a pressure difference across the chest wall that would otherwise not exist.

Two columns and a wall between them #

The geometry is worth stating carefully, because it is a general one. There are two fluid columns, the water around the body and the air in the tube and lungs, and a flexible wall between them, the chest. Within each column the pressure follows its own hydrostatic rule. In the air, density is so small that the pressure in the lungs is the surface’s pressure to within a few pascals. In the water, it rises by 9.8 kilopascals per metre. The wall feels the difference between the two columns at its own depth, and it is that difference, not the pressure of either, that the breathing muscles must work against. The push that has no direction stressed that pressure acts equally in every direction at a point; the chest is squeezed from all sides, and only the connection to a different pressure through the tube makes the squeeze into a load.

The building that breathes through its doors followed the same structure on a larger scale. A heated building in winter is a column of warm air inside a column of cold air, and the difference between their hydrostatic gradients makes a pressure difference across the walls that grows with height, pushing air in at the bottom and out at the top. A swimmer with a snorkel is a column of air inside a column of water, with a much larger difference in density and so a much larger pressure difference per metre, and the chest wall is the building’s envelope. The bottle that pours at one rate however full it is used the same idea deliberately: a tube open to the air, dipped into a sealed bottle, fixes the pressure at its tip to atmospheric whatever the level above, and the snorkel’s mouthpiece does exactly that to the lungs.

The comparison with breath-holding makes the point sharp. A breath-holding swimmer at a metre has lungs at the water’s pressure, because nothing connects them elsewhere; the chest is compressed slightly and the air inside follows, the compression that makes a free diver lose buoyancy with depth and, past a certain depth, sink, as the depth past which it must sink found for any compressible body. Nothing has to be pushed against. Connect the same lungs to the surface with a tube and the pressure inside drops by 9.8 kilopascals relative to outside, the blood in the chest’s vessels is pushed towards the lungs by the higher pressure around the body, and the breathing muscles face the whole difference. Remove the tube again and the load vanishes.

The animal that snorkels deep #

One animal does routinely what the snorkel’s arithmetic forbids. An elephant crossing a river can walk along the bottom with its trunk raised above the surface, breathing with its chest two metres or more below the water. The pressure difference across its chest wall is then around twenty kilopascals, far beyond what a human could breathe against, and in most mammals a difference of that size would rupture the small blood vessels in the thin pleural space between lung and chest wall.

The elephant’s lungs are unusual: in place of the fluid-filled pleural space that lets the lungs of other mammals slide freely inside the chest, the elephant’s lungs are bound to the chest wall by dense connective tissue. The physiologist John West proposed in 2001 that this anatomy is an adaptation to exactly this load, making the chest and lungs a single structure that the pressure difference cannot tear apart, and suggested that it reflects an aquatic stage in the elephant’s ancestry. Whether that is the reason the anatomy evolved remains debated, but the mechanics are clear: the elephant has strengthened the wall between the two columns, and its breathing muscles, scaled to a body of several tonnes, are strong enough to work against the difference.

A tube that is breathed twice #

The second limit is the tube’s volume. At the end of each breath out, the snorkel is full of exhaled air, low in oxygen and high in carbon dioxide. The next breath in draws that air back into the lungs before any fresh air arrives from the surface. The same thing happens in the body’s own airways — the mouth, the throat, the windpipe and bronchi hold about 150 millilitres that never reaches the gas-exchanging surfaces of the lungs, the anatomical dead space. A snorkel adds its volume to that.

A quiet breath is about half a litre. Subtract the 0.15 litres of the body’s own dead space and 0.35 litres of fresh air reaches the lungs. A snorkel of 20 millimetres bore and 35 centimetres length holds 0.11 litres, which cuts the fresh air to about 0.24 litres per breath, a reduction of a third. A tube of the same bore a metre long holds 0.31 litres and cuts it to 0.04 litres, almost nothing. A 30-millimetre tube half a metre long holds the whole useful volume of a quiet breath: the swimmer would shuttle the same air up and down, breathing in what they breathed out, and suffocate while breathing normally.

There is no way around this by breathing harder through the same tube, only by breathing deeper. A deeper breath moves more air past the fixed volume of tube and airways, and the excess is fresh. That is what the next picture measures.

The body regulates its breathing by the carbon dioxide in its blood, and the rate of carbon dioxide production at rest, about a quarter of a litre a minute, sets how much fresh air the lungs need. With a snorkel the body has to breathe the tube’s volume in addition, every breath. A swimmer with an ordinary snorkel breathes about a fifth more deeply than without one, which is noticeable but harmless. A swimmer with a metre of hose would need breaths nearly twice the normal size just to stand still, and would need them against the water’s weight on the chest. People who try long hoses report breathlessness that builds over a minute or two, which is carbon dioxide accumulating faster than their breathing can clear it.

The tube’s resistance is small #

The intuitive objection to a long snorkel — that air will not flow easily through a long thin pipe — is the one that matters least.

Air is light and not very viscous, and at the flows of breathing — half a litre a second at rest, a litre or two in hard swimming — a tube two centimetres across offers almost no resistance. The pressure lost in a 35 centimetre snorkel is a few tens of pascals, a few millimetres of water. Only when the bore falls below about a centimetre does the resistance grow to matter, because it rises very steeply as the bore shrinks: once the flow is turbulent, roughly as the inverse fifth power of the diameter. A designer who tried to fix the dead-space problem by narrowing the tube would meet the resistance problem at around ten millimetres, and could not reduce the volume much further without choking the swimmer.

So the snorkel is boxed in by three numbers. Its bore must be large enough, about 15 to 25 millimetres, to keep the resistance negligible. Its length must be short enough that its volume, at that bore, adds only modestly to the body’s own dead space, which means a few tens of centimetres. And its depth below the surface cannot exceed what the chest can breathe against, which is also a few tens of centimetres. All three limits arrive at about the same length, which is why every snorkel looks the same.

The cost in watts #

The water’s weight on the chest can be put in terms of power.

Each breath expands the chest by its tidal volume against the pressure $\rho g d$, doing work $\rho g d \, V_T$: about five joules per breath at a metre, sixty joules a minute, a watt. Quiet breathing at the surface needs about a twentieth of a watt of mechanical work, so a metre of depth multiplies the work of breathing twenty-fold. The breathing muscles convert metabolic energy to mechanical work at a few per cent efficiency, so that watt of mechanical work costs tens of watts of metabolism, and the muscles doing it — the diaphragm and the muscles between the ribs — are endurance muscles tuned to small loads, not to lifting.

There is a pleasing comparison with the height a siphon cannot pass, where atmospheric pressure limits how high water can be lifted by suction to about ten metres. The snorkel is the inverse problem. A siphon fails when the pressure difference the atmosphere can supply runs out; a long snorkel fails when the pressure difference the chest can supply runs out, at about a tenth of the siphon’s height, because a chest is a much weaker pump than the atmosphere.

How divers solved it #

The solution, when it came, was to stop fighting the pressure difference and supply air at the pressure of the water around the chest. Standard diving dress in the nineteenth century had pumps at the surface pushing air down to the diver at the pressure of the depth, so that the chest no longer had to work against the water. The modern regulator, introduced with the aqualung in the 1940s, does the same thing on the diver’s back: it delivers air from a high-pressure tank at exactly the ambient pressure, sensed through a diaphragm open to the water, so that the lungs breathe as easily at thirty metres as at the surface. A regulator is a device for abolishing $\rho g d$ across the chest wall.

The regulator also solves the dead-space problem. It sits at the mouth, so the only extra volume is in the mouthpiece; exhaled air is vented to the water through a one-way valve and never breathed again. Long hoses survived in surface-supplied diving, where the air flows one way down the hose and exhaled air escapes at the diver — the same principle of separating the inward and outward paths that a snorkel cannot use, since it has only one tube.

What the picture cannot show #

The breathing figures are representative of a healthy resting adult. Tidal volume, dead space, carbon dioxide production and the strength of the breathing muscles vary widely between people, with size, age, fitness and illness, and the figures should be read as orders of magnitude. The 1.5 kilopascal figure for a sustainable load is an estimate from studies of breathing against resistance, and some trained people can sustain considerably more for a while. The maximum inspiratory pressures quoted are for a brief effort against a closed valve, not for breathing.

The dead-space treatment assumes that the air in the tube is perfectly mixed exhaled air at the start of each breath, and that the breath in draws it all back before any fresh air. Real flow in a tube is not plug-like: the fastest air at the tube’s centre arrives first and some mixing occurs, so a little fresh air arrives slightly earlier than the simple model says. The figures also leave out the swimmer’s position, which affects chest depth: a snorkeller floating face down has the chest only twenty or thirty centimetres below the surface, while one swimming slightly downward has it deeper.

Resistance was computed for a smooth straight tube with a simple entrance loss. Real snorkels have bends, valves and mouthpieces that add resistance, and some have purge valves or splash guards at the top. These matter more than the straight tube does, but they do not change the conclusion that resistance is not what limits the length.

Still open: how breathing fails under load #

The point at which breathing muscles fatigue under a load, and what happens then, is studied in respiratory physiology because it matters for diving, for high-performance flight, and for patients being weaned from ventilators. The diaphragm fatigues when the load it carries, as a fraction of its maximum, multiplied by the fraction of each breath spent working, exceeds a threshold. That index, measured in the laboratory, predicts when a patient will fail to breathe on their own. How well it carries over to the very different conditions of a swimmer — horizontal, cold, exercising, with water pressure on the abdomen as well as the chest — has been studied much less, and the safe limits for diving equipment that adds breathing resistance are still argued over. A snorkel is the simplest such device, and its length was fixed by experience long before the physiology was understood.

A snorkel connects lungs at depth to air at the surface, so the chest must expand against ρgd — 9.8 kPa at a metre, about the most an adult can make once, against roughly 1.5 kPa that can be sustained, reached 15 cm down — while the tube’s volume adds to the dead space and is breathed back each time, so a 30 mm tube half a metre long delivers no fresh air; the tube’s resistance, 22 Pa for an ordinary snorkel, is the smallest of the three limits. The habit worth carrying is to ask where a pressure difference comes from, because the weight of water on a body does no harm until a tube connects the inside to somewhere else.