A bubble is throttled by its age, and a sheet by its length

A cavity that cools the water it came from reduced the thermal effect in cavitation to one comparison. A cavity that makes a volume $V_v$ of vapour out of a volume $V_l$ of liquid cools that liquid, the cooled liquid has a lower vapour pressure, and the cavity loses some of the tension that drove it. The ratio $B = V_v/V_l$ is Stepanoff’s B-factor, and the essay computed the value $B^*$ at which the loss would be the whole tension: 648 in water at 20 °C under a kilopascal, below one at the boiling point, four hundredths in liquid hydrogen. It ended on the number it did not have, which is the $B$ any real cavity has, and on the suspicion that a travelling bubble and an attached sheet would have very different ones.

They do, and the difference is not a factor but a law. A bubble’s $B$ grows with its age, because it keeps cooling the same thin shell of liquid while its volume grows faster than the shell. A sheet’s $B$ does not change at all, because the liquid flowing past it is replaced as fast as it is cooled. That makes the travelling bubble, the smaller and briefer of the two, the one that feels its own cooling first.

The shell a growing bubble cools #

A bubble that starts in a region where the pressure has fallen $\Delta p$ below the vapour pressure grows, while inertia limits it, at Rayleigh’s speed $\sqrt{2\Delta p/3\rho_l}$ — 2.6 metres a second under a tension of ten kilopascals — so its radius grows in proportion to its age. The heat for the evaporation comes from the liquid next to the wall. In the time $t$ that conduction has had, it has reached a depth of order $\sqrt{\alpha t}$, with $\alpha$ the liquid’s thermal diffusivity, a seventh of a square millimetre per second for water: a tenth of a millimetre in a tenth of a second, and a micrometre in a few microseconds.

But the shell is not still. As the bubble grows, the liquid next to its wall is stretched over a larger area, and a layer of liquid that is stretched sideways is thinned. In the thin-layer limit the temperature obeys

$$\frac{\partial T}{\partial t} - \frac{2\dot R}{R}\,y\,\frac{\partial T}{\partial y} = \alpha\,\frac{\partial^2 T}{\partial y^2},$$

with $y$ the distance from the wall, and for growth as a power of time, $R \propto t^n$, it has a similarity solution in $y/\sqrt{\alpha t}$ whose profile is a Gaussian error function with its width shrunk by $\sqrt{4n+1}$. Integrated over the growth, the heat it delivers is equivalent to cooling a layer of thickness

$$\delta = 2\sqrt{\frac{\alpha t}{\pi(4n + 1)}}$$

through the whole depression — the familiar $2\sqrt{\alpha t/\pi}$ for a still wall, and $\sqrt5$ thinner for a bubble growing at constant speed. That is the bubble’s reservoir, and it is thin. A sphere of radius $R$ with a shell $\delta$ has $B = R/3\delta$, so for inertial growth

$$B_{\text{bubble}} = \frac{1}{6}\sqrt{\frac{2\Delta p}{3\rho_l}}\,\sqrt{\frac{5\pi t}{\alpha}},$$

growing as the square root of the bubble’s age. Under ten kilopascals it is about four at a microsecond and 140 at a millisecond, almost independent of the water’s temperature, because the temperature enters only through the liquid’s density and diffusivity, which change by a few per cent between 20 and 100 °C.

The age at which a bubble turns heat-limited #

$B^*$ is where that comparison bites, and $B^*$ moves enormously with temperature while the bubble’s $B$ does not. Under ten kilopascals $B^*$ is 6,475 in water at 20 °C, 143 at 60 °C, 32 at 80 °C, 8.6 at 100 °C and 0.66 at 150 °C — four decades — because the vapour density enters it squared. The age at which the bubble’s $B$ climbs to $B^*$ is

$$t^* = \frac{54}{5\pi}\,\frac{\Delta p}{\rho_l\,\Sigma^2}, \qquad \Sigma = \frac{(\rho_v L)^2}{\rho_l^2 c_l T\sqrt{\alpha}},$$

and $\Sigma$ is Brennen’s thermal parameter. The B-factor route and Brennen’s route to the same threshold were shown to be one number in the essay before, and here they meet at a definite instant in the life of a bubble: before $t^*$ the bubble grows as inertia allows, after it as fast as heat can reach it. The constant $54/5\pi$, about 3.4, is the shell’s stretching made explicit.

The interesting range is narrow. A bubble crossing the low-pressure region of a pump blade lives between a tenth of a millisecond and ten, depending on the blade’s chord and speed. At 20 °C the thermal time is two seconds, a thousand times longer than any such life, and the bubble never knows it is boiling. At 60 °C it is a millisecond, in the middle of the band. By 80 °C it is 54 microseconds and by the boiling point four — every travelling bubble in a hot-water pump spends almost its whole life heat-limited, and its radius when it turns over is a hundredth of a millimetre. In liquid hydrogen, oxygen or nitrogen the thermal time is under a tenth of a microsecond: a cryogenic bubble is heat-limited from the moment it is a nucleus.

A bubble held at the critical value grows by Plesset and Zwick’s law #

What happens after $t^*$ follows from the same arithmetic. The bubble cannot make vapour faster than its shell supplies the heat, so it grows at the rate that holds its $B$ at $B^*$. With $B$ pinned, $R = 3B^*\delta$ and the shell’s own thickness grows as $\sqrt t$, so the radius does too — the exponent drops from one to a half, and with $n = \tfrac12$ the shell’s factor is $\sqrt3$:

$$R = 2\sqrt{\frac{3}{\pi}}\;B^*\sqrt{\alpha t}.$$

That is Plesset and Zwick’s law for a bubble growing in a superheated liquid, with $B^*$ standing where their Jakob number stands. The identification is exact, not suggestive. A tension $\Delta p$ below the vapour pressure is, by Clausius and Clapeyron, a superheat $\Delta T = \Delta p\,T/\rho_v L$, and the Jakob number of that superheat, $\rho_l c_l\Delta T/\rho_v L$, is $\Delta p\,T\rho_l c_l/(\rho_v L)^2$ — which is $B^*$ term by term. The critical B-factor of the cavitation literature and the Jakob number of the boiling literature are the same group, met from opposite sides: a pump engineer asking how much vapour a cavity can make before its cooling stops it, and a boiling physicist asking how much superheat a liquid holds per unit of the latent heat a bubble needs.

The link carries a practical consequence for the collapse that follows growth. A heat-limited bubble is smaller at a given age than an inertial one, by the ratio of the two laws, and a bubble that reaches the high-pressure region smaller collapses less violently. In water at 100 °C a bubble living a millisecond reaches about a fifth of a millimetre where an inertial one would have reached two and a half.

A sheet renews what it cools #

An attached sheet cavity — the vapour pocket that clings to a blade’s suction side behind its leading edge — is a different object. It is fixed in place, it does not grow, and it does not cool one shell of liquid. Liquid flows past its interface at about the stream speed, is cooled in a thermal layer that grows along the cavity as $\sqrt{\alpha x/U}$, and is carried away past the closure, replaced by warm liquid from upstream. What the sheet loses is vapour: the re-entrant flow at its closure breaks off vapour and carries it downstream, at a rate measured as an entrainment coefficient $C_Q$, the vapour volume shed per unit span over the stream speed times the cavity’s length, a few thousandths in hydrofoil experiments.

The sheet’s $B$ is then a ratio of two fluxes — vapour shed, against liquid flowing through the thermal layer at the closure:

$$B_{\text{sheet}} = \frac{C_Q\,U\ell}{U\cdot 2\sqrt{\alpha\ell/\pi U}} = \frac{C_Q}{2}\sqrt{\frac{\pi U\ell}{\alpha}},$$

the entrainment coefficient times the square root of the cavity’s Péclet number. Time has gone out of it. For a two-centimetre sheet at twenty metres a second the Péclet number is about three million and $B$ is seven.

Seven against the travelling bubble’s hundred and forty at a millisecond, and the sheet’s number does not grow. The sheet is the larger cavity and the longer-lived, and it is the less affected, because size and duration are not what sets a B-factor: the ratio of vapour to cooled liquid is, and the sheet’s cooled liquid is a stream. Its $B$ rises only as the square root of its length or speed — a sheet ten times longer makes three times more vapour per cooled liquid — so even a long sheet on a fast blade stays near the tens.

Bubbles first, forty degrees before the sheet #

Set the two against $B^*$ and the order is fixed. Under ten kilopascals, a bubble that has lived a millisecond is suppressed — its own cooling consumes its tension — above 60.5 °C; one that has lived ten milliseconds above 47 °C, and one of a tenth of a millisecond above 76 °C. The two-centimetre sheet at twenty metres a second is suppressed only above 104 °C.

That ordering is the structure of the thermal effect in a pump. As the water warms, travelling-bubble cavitation — the kind that appears at inception, as discrete bubbles born on nuclei in the low-pressure region and swept downstream — is throttled first, between about fifty and eighty degrees, depending on how long the bubbles live. A pump’s inception point in warm water is therefore delayed more than its developed-cavitation performance, which is carried by sheets and which stays nearly as it is in cold water until the boiling point. That is the form the threshold that is also a duration takes thermally: a bubble’s thermal suppression depends on its residence, a sheet’s does not.

The comparison also shows why hot-water and cryogenic pumps can be run with less margin than cold-water ones of the same design. A boiler-feed pump at 150 °C has $B^*$ below one under ten kilopascals: both kinds of cavity are suppressed, the bubble by two hundred times, the sheet by eleven. Liquid hydrogen is the extreme, its bubbles at more than a thousand times the critical value and its sheets at nineteen, which is why rocket turbopumps run inducers deep in nominal cavitation.

What the cooling is worth at the suction flange #

A pump’s designer reads all this as head. The share of the tension a cavity’s own cooling takes back is its $B$ over $B^*$, capped at one, and that share of ten kilopascals is a pressure the pump no longer has to supply at its inlet. For the two-centimetre sheet at twenty metres a second it is under one per cent at 40 °C, five per cent at 60 °C, 22 per cent at 80 °C and 81 per cent at 100 °C — in metres of water, nine millimetres, five centimetres, 23 centimetres and 86 centimetres. A millisecond bubble’s share is already 17 per cent at 40 °C, 96 per cent at 60 °C and the whole tension from about 61 °C upwards.

Those two columns are why a hot-water pump’s inception and its breakdown move by different amounts. Inception, the first audible cavitation, is carried by travelling bubbles, and in water above about sixty degrees their own cooling cancels most of the tension that would grow them; the pump’s inception margin has, in effect, been handed back. Breakdown, the point at which the head falls by a few per cent, is carried by sheets and clouds large enough to block the passage, and at 80 °C those have had only a fifth of their tension taken back. A test in warm water that reports one thermal correction for a pump is averaging two effects that differ by the factor drawn in the last figure.

The arithmetic also sets how much the cavity’s own stream matters to the sheet. Its cooled liquid is carried downstream past the closure, a layer about fifteen micrometres thick and cooler than the bulk by $B$ times the depression per unit B-factor — about a kelvin at 80 °C. It mixes away within a few cavity lengths and does not reach the next blade, which is what makes the sheet’s balance a steady one: each blade’s cavity sees fresh liquid, and no cooling accumulates round the impeller.

The convention the numbers rest on #

Every number here is under a tension of ten kilopascals — the cavity’s pressure that far below the bulk liquid’s vapour pressure — and $B^*$ scales in proportion to the tension, so a cavity driven harder needs more of its own cooling to stop it. The B-factor is the volume of vapour over the volume of liquid cooled through the full depression; the bubble’s cooled volume is its area times the effective thickness $\delta$, which counts every partly-cooled layer at its share. Properties are the earlier essay’s: water’s vapour pressure from an integrated Clausius–Clapeyron relation, densities and heat capacities from steam-table fits, and the cryogens’ and refrigerant’s from measured saturation tables. The bubble’s age is counted from the moment it began growing, not from when it entered the low-pressure region, and the sheet’s entrainment coefficient is fixed at 0.005.

What the two models leave out #

The bubble grows from a nucleus already large enough to have no surface-tension barrier, in a uniform tension; a real bubble enters a low-pressure region whose tension rises and falls along its path, so its growth law changes along the way and its age at the critical value is a weighted average of that history. The shell is treated as thin against the radius, which fails for bubbles younger than the time conduction takes to cross their own size, a few nanoseconds at a micrometre. The sheet’s model replaces everything that happens at the closure — the re-entrant jet, the cloud it sheds, the vapour’s own heat and pressure gradients — by a single measured coefficient, and that coefficient varies by a factor of two or three between experiments and with the cavity’s length; the sheet’s $B$ follows it in proportion. The liquid next to the sheet is taken to flow at the stream’s speed, while a sheet’s interface sits in a boundary layer that is slower, which thickens its thermal layer and lowers its $B$ further. Neither model contains dissolved gas, which in a real pump adds a partial pressure the cooling does not reduce and makes the vapour fraction smaller than the cavity’s volume.

Who measured the depression #

Stepanoff introduced the B-factor in the 1960s to correlate the suction-head gains of pumps in hot water and hydrocarbons. Plesset and Zwick’s growth law for a bubble in superheated liquid is of 1954, and Brennen’s thermal parameter, which reads the same physics as a velocity scale, is in his monographs of the 1990s. Holl, Billet and Weir measured the temperature depression in developed cavities on ogives and hydrofoils in hot water and Freon in the 1970s, and correlated it with an entrainment approach of the kind used here; Franc and Michel’s text gives the cavity entrainment and its coefficient. The stretching-layer solution is the standard one for heat transfer to a growing sphere.

How it was checked #

The similarity profile’s wall gradient and heat integral, computed by quadrature, give the closed-form effective thickness to a part in $10^{13}$ for inertial growth, square-root growth and a still wall, the last being the planar $2\sqrt{\alpha t/\pi}$. The thermal time from the B-factor route equals $(54/5\pi)\Delta p/\rho_l\Sigma^2$ to rounding for water at two temperatures and the four other liquids, which is the earlier essay’s identity carried into the time. The heat-limited growth law at $B = B^*$ has Plesset and Zwick’s coefficient $2\sqrt{3/\pi}$ by construction, and $B^*$ equals the Jakob number of the equivalent superheat algebraically. The model refuses a negative tension, a bubble of no age, a sheet in a still stream and a negative entrainment coefficient.

Still open: a bubble that crosses the low-pressure region #

Every bubble here grows in a constant tension from the moment it is born, and a real one does not: it is carried along a blade through a pressure that falls below the vapour pressure, reaches a minimum, and rises again, so its growth starts inertial, may turn heat-limited half-way, and ends in collapse. The next calculation would carry a nucleus along the pressure distribution of a section whose flow is already solved, with Rayleigh–Plesset growth and the stretching shell’s heat balance together, and ask how large the bubble is when it reaches the closure point at each water temperature — and so at what temperature the thermal effect halves the bubble’s final size, which is the number that decides how hard it will collapse.