The pipe that carries heat as vapour

The heat that changes no temperature measures latent heat: a kilogram of water at its boiling point absorbs 2.26 megajoules to become a kilogram of steam at the same temperature, more than five times the heat needed to warm it from freezing to boiling. That energy is carried by the vapour and returned when it condenses. A substance changing phase moves a great deal of heat with no change in temperature at all, and a device that boils a liquid in one place and condenses it in another moves heat between the two places without needing a temperature difference to push it.

That is a heat pipe. Proposed in 1942, patented in its modern form by George Grover at Los Alamos in 1963 for cooling nuclear reactors in spacecraft, it is now one of the most widely manufactured thermal devices there is: almost every laptop has two or three, carrying heat from the processor to the fins of a fan, and satellites, solar collectors, permafrost-preserving piles under the Trans-Alaska Pipeline and the cooling systems of LED lamps use them too. A heat pipe is a sealed tube, evacuated, with a small amount of a working fluid inside and a wick lining its wall. There are no moving parts and no pump.

The pipe that out-conducts any solid #

The comparison with solid conduction is the reason for the device.

Heat conducted along a rod needs a temperature gradient: the power is the conductivity times the cross-section times the temperature difference divided by the length. A copper rod six millimetres across has a cross-section of under thirty square millimetres, and carrying thirty watts along thirty centimetres of it would require nearly eight hundred degrees of difference between its ends. Diamond, the best conductor of heat at room temperature, would still need a hundred and sixty. No solid could do the job in a laptop, where the processor must stay below a hundred degrees and the fan’s air is at forty.

The heat pipe does it with less than a degree. Inside, the water at the hot end is boiling, in a wick of copper powder sintered onto the wall, and the vapour fills the tube’s hollow core at a pressure set by the temperature. At the cold end the vapour condenses onto the wick there, which is slightly cooler, and gives up its latent heat. The vapour flows from the hot end to the cold because the hot end’s vapour pressure is very slightly higher. The temperature drop of the whole device is almost all in the two thin layers of wet wick where heat enters and leaves the metal; the vapour core itself is very nearly isothermal.

Why the vapour needs so little push #

How little temperature difference the vapour needs is a consequence of the steepness of the boiling curve.

A boiling point is a pressure shows that the temperature at which a liquid boils is fixed by the pressure above it, along a curve whose slope Clausius and Clapeyron derived from thermodynamics: the change in boiling temperature per unit change in pressure is $RT^2/Lp$ for a vapour behaving as an ideal gas. At 50 °C, water vapour’s pressure is about twelve thousand pascals, and the relation gives 1.6 millikelvin per pascal. The vapour flowing along a heat pipe needs a pressure difference of a few tens of pascals to push it at the speeds involved, a few metres a second, and that corresponds to a temperature difference of a few hundredths of a degree.

The same curve shows where heat pipes struggle. At low temperature the vapour is thin, its pressure falls exponentially, and each pascal of pressure difference costs more temperature: at 5 °C, seventeen millikelvin per pascal, and the vapour must flow faster to carry the same heat, needing more pascals. A water heat pipe started from cold is sluggish until it warms up, and at the lowest temperatures the vapour flow is limited by the speed of sound in it — the vapour cannot be pushed out of the evaporator faster than sound travels, whatever the pressure difference, and the pipe chokes. Heat pipes for cold environments use fluids such as ammonia, whose vapour is much denser at low temperature.

How much a wick can lift #

The liquid has to get back to the hot end, and the wick does it by capillarity. The pores of the wick, a few tens of micrometres across, hold liquid in curved menisci; at the hot end, where liquid evaporates, the menisci retreat into the pores and curve more sharply; at the cold end, where liquid is supplied, they are nearly flat. The difference in curvature is a difference in pressure, which pushes the liquid along the wick from the cold end to the hot.

The greatest pressure a wick can develop is the capillary pressure of its finest menisci, $2\sigma/r$, which for water at 50 °C in pores of fifty micrometres is about 2,700 pascals — enough to hold a column of water twenty-eight centimetres high, the same physics that sets how high water will climb in a tube. That pressure must drive the returning liquid through the wick against its viscosity, at a rate set by the heat being carried, and also against gravity if the hot end is above the cold. When the demand exceeds the supply, the wick at the hot end dries out, the evaporator overheats, and the heat pipe fails abruptly. That is the capillary limit, and it is the one that usually decides a heat pipe’s rating.

The figure shows how strongly it depends on orientation. Level, the model pipe carries forty-four watts. Tilted hot end down, gravity helps the liquid down to the evaporator and the limit rises. Tilted hot end up, gravity opposes the wick, and at a height equal to the capillary column, twenty-eight centimetres, nothing can be returned at all. This is why laptop heat pipes are laid flat in the case and why heat pipes in aircraft and spacecraft are rated for the accelerations they will meet. In orbit, with no gravity, a pipe that could not lift its liquid against gravity on the ground works perfectly.

The wick that has to be fine and open at once #

The wick’s design involves a trade-off that the pore that lifts highest fills slowest describes in general. Fine pores give high capillary pressure, since the pressure goes as one over their radius, but they resist flow, since the permeability of a porous medium goes roughly as the square of its pore size. A wick of fine powder pumps hard and slowly; a wick of coarse grooves pumps gently and freely. The maximum heat a wick can return is proportional to the product of the capillary pressure and the permeability, and so roughly proportional to the pore size — but only while gravity is unimportant. Against gravity, the fine wick wins because only it can develop the pressure at all.

Heat-pipe engineers therefore build composite wicks: fine sintered powder at the evaporator, where the menisci must be curved sharply, and open grooves or coarse mesh along the length, where the liquid must flow freely. Laptop heat pipes use sintered copper throughout for its tolerance of orientation; spacecraft heat pipes, which never fight gravity, often use axial grooves extruded in an aluminium tube. The same balance between suction and permeability governs how water rises through soil and through the vessels of trees, where the column that is pulled, not pushed is held up by menisci in the cell walls of leaves.

The temperature water works best at #

The working fluid’s properties set how much heat a wick of given geometry can return, through a single combination of them.

The heat a wick can return is the mass of liquid it can move times the latent heat each kilogram carries. The mass flow it can drive is proportional to the capillary pressure, and so to the surface tension, divided by the liquid’s viscosity, and multiplied by its density. So the fluid’s merit is the product of density, surface tension and latent heat divided by viscosity. For water it rises from room temperature to a maximum near 150 °C, because viscosity falls faster than surface tension and latent heat do, and then declines towards the critical point, where both surface tension and latent heat go to zero and the distinction between liquid and vapour disappears — the point where the two become one. Water’s merit is higher than that of any other common fluid over its range, by a large factor, thanks to its large latent heat and surface tension, both products of hydrogen bonding; other fluids are used only where water cannot be.

Every fluid has its window #

A heat pipe works only where its fluid can be both liquid and vapour, which is a stretch of the fluid’s phase diagram.

Below its freezing point, or its triple point, a fluid has no liquid to return; above its critical point it has no latent heat to carry. A heat pipe is therefore chosen by its operating temperature: helium for cryogenic instruments at a few kelvin, nitrogen near seventy kelvin for cooling infrared detectors, ammonia for spacecraft at room temperature and below, water from room temperature to two hundred degrees, and liquid metals — sodium, lithium — at the hundreds or thousands of degrees of nuclear reactors and furnaces. Within each window the useful range is narrower than the bar: near the bottom, the vapour is too thin and the pipe chokes; near the top, the latent heat and surface tension have dwindled. The liquid-metal heat pipes that cool the leading edges of some hypersonic vehicles and the cores of proposed space reactors start up from frozen, melting their own fluid as they warm.

The Trans-Alaska Pipeline uses the opposite of what a laptop needs: some sixty thousand vertical heat pipes in its supports, filled with ammonia, carry heat out of the ground in winter, when the air is colder than the soil, and do nothing in summer, when the soil is colder than the air, because their condensers are at the top and the liquid can only return by falling. They are thermal diodes, and they keep the permafrost frozen beneath a pipeline whose oil is warm. A heat pipe with its condenser above its evaporator, relying on gravity rather than a wick to return the liquid, is called a thermosiphon, and its one-way conduction is a direct consequence of where the liquid must go.

The limits that are not the wick’s #

The capillary limit is the one that decides most pipes, but not all. At the evaporator the heat must pass through the wet wick into the liquid, and if the heat flux is too high, the liquid in the wick boils inside the pores rather than evaporating from the menisci. Bubbles form in the wick, block the liquid’s path to the surface, and the evaporator dries out even though the wick as a whole could have returned enough liquid. This boiling limit is the heat-pipe version of the burnout that the boiling curve that turns back traces for a heated surface in a pool: beyond a critical heat flux, vapour blankets the surface and the heat transfer collapses. Engineers design evaporator wicks with fine pores to resist it and use thin wicks, so that the heat crosses little liquid before reaching a meniscus.

At high powers the vapour rushing down the core can move fast enough to tear droplets from the liquid in the wick, which is flowing the other way, and carry them back to the condenser — the entrainment limit, set by the vapour’s speed and the liquid’s surface tension. And at startup, as described, the vapour’s speed is limited by the speed of sound in it. A heat pipe’s rated power at any temperature is the lowest of these limits, and the shape of the rating curve, low when cold, highest in the middle, falling when hot, is the envelope of all of them.

Why every liquid carries about the same heat per molecule #

The latent heat that makes heat pipes possible has a regularity across liquids that explains why so many fluids can be used. The entropy every liquid gains on boiling is Trouton’s rule: at their normal boiling points, most liquids gain about the same entropy on vaporising, about 85 to 90 joules per kelvin per mole, because the vapour’s freedom compared with the liquid’s is roughly the same for all simple liquids. The latent heat per mole is then roughly proportional to the boiling temperature. A fluid that boils at a higher temperature carries more heat per molecule, and heat pipes for higher temperatures are correspondingly more capable, provided their fluid’s surface tension and viscosity cooperate.

Water breaks the rule: its entropy of vaporisation is about 109 joules per kelvin per mole, a quarter higher than Trouton’s value, because liquid water is unusually ordered by its hydrogen bonds, and breaking that order costs extra. That excess, together with a molar mass of only eighteen, is why water carries more heat per kilogram than any other liquid near room temperature — 2.26 megajoules at its boiling point — and why its merit number is unmatched in its range.

Where the heat finally goes #

A heat pipe moves heat; it does not destroy it. In a laptop the heat arrives at the fins of a radiator through which a fan blows air, and the fan’s airflow sets the temperature of the pipe’s condenser end, and with it, through the pipe’s tiny temperature drop, the processor’s. In a satellite the heat pipes carry heat to radiator panels that emit it to space, the sink that the cold reservoir overhead shows is available even on Earth through the atmosphere’s infrared window. The heat pipe’s virtue is that it makes the place where heat is generated and the place where it is rejected thermally almost the same place, even when they are tens of centimetres apart and the space between them is too narrow for any metal to fill.

A straight pipe and one limit drawn #

The water properties are fitted correlations, accurate to a few per cent over the range drawn. The pipe is a straight cylinder with a uniform sintered wick and fixed evaporator and condenser lengths; real laptop pipes are flattened and bent, which reduces their capacity, and their wick structure is graded. The temperature drop of the pipe counts only conduction through the wet wicks at its ends, with an effective wick conductivity of 40 W/m·K, and neglects the vapour’s own drop and the interface resistances at the menisci, which add a few tenths of a degree. The capillary limit uses Darcy’s law for the liquid’s flow and the full pore radius for the capillary pressure, ignoring the vapour’s drag on the liquid, which reduces the limit at high power. The other limits — the boiling limit, at which bubbles form in the evaporator wick and block it, and the entrainment limit, at which fast vapour tears liquid from the wick’s surface — are described in words but not drawn; one of them, rather than the capillary limit, sets the rating of some pipes.

Still open: how thin a heat pipe can be #

As phones and tablets thin, heat pipes have been flattened into vapour chambers less than half a millimetre thick, with wicks of etched copper or woven mesh and vapour spaces a few tenths of a millimetre high. At those dimensions the vapour’s flow is no longer easy: its pressure drop grows as the inverse cube of the gap, the same lubrication physics that makes an air film slow to drain, and the vapour’s temperature drop, negligible in a laptop pipe, becomes a significant part of the total. Where the practical limit lies — how thin a vapour chamber can be before it conducts no better than the copper it is made of — is set by the competition between vapour flow, liquid return and the manufacturability of wicks a few tens of micrometres thick, and is being pushed by every generation of handheld devices.

The habit worth carrying away is to ask how a flow is being carried. Thirty watts along a 6 mm, 300 mm rod needs 796 K of difference through copper and 0.8 K through a water heat pipe, because the heat travels as latent heat in vapour whose flow, by Clausius–Clapeyron, costs only millikelvin per pascal — and the price is a wick whose 2,700 Pa of capillary pressure must return the liquid, which it does level and cannot do stood 28 cm on end. A heat pipe is a boiling curve put to work.