Paraffin makes a hybrid rocket burn 3 times faster. It may also be the reason chamber pressure oscillates at 58 percent of the mean. Two published models say why, and both are built on three numbers nobody has ever measured.
Every published stability prediction for a paraffin hybrid rests on 3 numbers that have never been measured for a liquefying fuel, at any condition.
In 2023 Giuseppe Gallo and Carmine Carmicino ran a 100 N hybrid rocket motor twice. Same hardware. Same oxygen mass flow. They changed only the fuel.
With high density polyethylene, the motor burned stably from ignition to shutdown.
With a paraffin grain, chamber pressure built to 6 atm and held. Then, 1.7 seconds into the burn, it entered a limit cycle. Pressure began swinging between 6 and 9.5 atm peak to peak, and the mean pressure climbed from 6 to 8.4.
A 58 percent pressure oscillation and a 40 percent shift in mean chamber pressure, caused by changing nothing but the fuel.
Their explanation points at the thing that makes paraffin attractive in the first place.
Regression rate is how fast the fuel surface recedes. It is the single number that decides whether a hybrid motor is worth building.
Conventional hybrid fuels pyrolyse straight from the solid. Paraffin does something else. It melts into a thin, low viscosity, low surface tension film on the grain surface, and the oxidiser gas tearing across that film destabilises it. Waves grow. Droplets are torn off and carried into the flow, where they vaporise and burn.
That droplet entrainment is an extra mass transfer path stacked on top of ordinary evaporation, and it is worth a factor of 3 to 4 in regression rate. It is the reason anyone puts wax in a rocket.
Karabeyoglu, Altman and Cantwell wrote the governing relation in 2002. Entrainment scales with the gas dynamic pressure raised to a power, times film thickness raised to another power, divided by melt viscosity and surface tension.
Two features of that relation matter more than they look. The mechanism is mechanical, driven by shear, with no heat needing to soak into a solid first. And it has a hard threshold: below a critical value the entrainment mass transfer is not small, it is exactly zero. The mechanism switches on.
The argument above is a reading of the literature. To test it, the closures have to run against each other in one place, on the same chamber, at their own published calibrations. That is what hybriddyn does19.
The architecture is the method. Property sets know no solvers, closures know no chambers, and every solver takes its closure by injection, so a failure can be attributed to one layer rather than to the code as a whole.
Thermal data from reference 1818, critical point from reference 55.
Regression rate as a mass flux law with exponent 0.8.14
The same law with the exponent rederived below 0.8.13
Regression rate against melt viscosity.7
Entrainment implicit in regression rate, solved with fzero.4
Transient one dimensional conduction in the regressing frame. Enthalpy method for the moving front1516. Implicit Arrhenius surface energy balance, solved with fzero at every right hand side call1. Upwind advection at the regression velocity.
The same depth columns at every axial station of a cylindrical grain, Marxman coupled, with pre and post chamber capacitance.
Renewal limited azimuthal drainage of the film in a horizontal port212223.
Zero dimensional delay differential equation with a ring buffer history. Coupled to a lumped inertance and resistance feed line and a first order vaporisation lag12.
against two phase Neumann analytic17
0.54 percent on front positionuniform flux MeltLayer2D against the verified 1D operator
1e-10 relativelinearised solution against the nonlinear march
0.8 percent of the perturbation amplitudeagainst Gallo and Carmicino3
Not reproducedThe Neumann solution describes a semi infinite slab at uniform initial temperature with a fixed surface temperature. It describes no rocket. That is precisely why it is the right test. It is one of the few phase change configurations whose exact answer is known in closed form, so any disagreement between the code and it is the code's error and nothing else. Benchmark against a fired motor instead and a mismatch could be the numerics, the properties, the boundary conditions or the experiment. Against Neumann there is only one candidate. This is verification, which asks whether the equations were solved correctly. It is a separate question from validation, which asks whether they were the right equations.17
The framework ships three campaigns verbatim from their sources, and seven modelling cases that run against them.
The 100 N engine at 25 g/s of oxygen, the motor of the two fuel experiment above and the chamber the limit cycle test runs against.
The measured frequency bands of the Stuttgart test record, the anchor for the acoustic formulas.
The 75 kN flight scale chamber of Finding 5, where the mode separation of the test engines collapses.
Computes regression rate as a simple power law in mass flux, exponent 0.8. Shows the classical boundary layer answer. Present because it is the baseline every other model is measured against, and because it contains no melt layer at all.
Same calculation with the wall temperature allowed to vary, which drops the theoretical exponent below 0.8 and closer to experiment. Shows that lifting one steady simplification improves the answer. Present as the published precedent for what the melt layer solver does in the time domain.
Regression rate against melt viscosity, measured across fuel formulations. Shows that viscosity is a design lever, since polymer additives trade mechanical strength against burn rate. Present because it is the only closure fitted to reacting engine data rather than to cold films, and because it is entirely time averaged, which is the field's blind spot in one model.
Total regression as vaporisation plus droplet entrainment from a shear unstable melt film, with entrainment scaling on dynamic pressure, film thickness, viscosity and surface tension, and a hard onset threshold below which entrainment is exactly zero. Shows the mechanism that makes liquefying fuels burn 3 to 4 times faster. Present because everything downstream inherits it, including its exponents, which were fitted to cold non reacting kerosene and methanol films.
Entrainment written implicitly in the regression rate and solved exactly rather than lagged. Shows that the same physics can be closed differently and still inherits the same exponents, which the authors themselves flag as possibly unrealistic. Present as the most recent formulation, so the negative result cannot be blamed on using an outdated closure.
Transient conduction with a moving phase front, solving for the temperature field and reading melt thickness off it. Shows that h lags the heat flux. Present because it is the only element in the framework that does not assume the quantity the entrainment law depends on most strongly.
Zero dimensional pressure balance with delayed entrained heat release, feed line inertance and a vaporisation lag. Shows whether the closures, run together, produce a limit cycle. Present because it is the only place where a closure's error becomes an engine's behaviour.
2 independent groups have now modelled paraffin hybrid instability. Gallo and Carmicino built an acoustic model of the coupled feed line and chamber and ran a linear stability analysis. A Turin group, publishing with Avio and the Italian Space Agency, built a multiphysics model with a quasi one-dimensional chamber solver and a neural network standing in for the chemistry.
Both rest on the same 3 quantities. A dynamic pressure exponent. A film thickness exponent. And the delay between a droplet leaving the surface and its heat arriving in the flow.
| Quantity | What it controls | Status |
|---|---|---|
| α | How hard fuel production responds to a flow perturbation | Fitted to 1970s wind tunnel data on kerosene films |
| β | Sensitivity to melt film thickness | Same origin, range 1 to 2 |
| τ | Phase between fuel production and chamber pressure | A free parameter in one model. “Arbitrarily chosen” in the other |
The Turin and Avio paper is unusually candid about this. It sets the exponents to 1.5 and 2, notes in the same paragraph that these values have been shown to produce unrealistic regression curves, and states that finding better approximations is future work. Of the delay term it says plainly that the value is arbitrarily chosen.
Written out, their entrainment term scales with oxidiser mass flux cubed. A cubed sensitivity, published, with the authors flagging that the exponent may be wrong.
Every published stability prediction for a paraffin hybrid rests on 3 numbers that have never been measured for a liquefying fuel, at any condition.
Paraffin wax of carbon number around thirty has a critical pressure near 6.5 bar and a critical temperature near 840 K.
Above roughly 6.5 bar the wax goes supercritical. Surface tension and heat of vaporisation approach zero, and there is no clean distinction between the liquid and the gas. The droplet entrainment picture, the one all of this rests on, is a sub-critical picture.
Every flight engine runs an order of magnitude above that line.
Anna Petrarolo filmed paraffin burning at 1, 13 and 40 bar in an optical chamber rated to fifty. Below the critical point, periodic Kelvin-Helmholtz waves at 300 to 340 Hz, wavelengths of a few millimetres, droplets shedding as the theory says. Above it, no periodic wave structure could be recognised in the flame at all. What she saw instead were intermittent, violent blowing events and flame bursts, getting more frequent as pressure rose.
Her thesis closes that thread with one sentence: some of the chamber pressure oscillations could also derive from these unsteady phenomena. And then it stops.
Sorting the studies by chamber pressure has not, as far as I can find, been done. Here is the first attempt. Hover any marker for the campaign behind it.
The picture is not the one I expected. Feed-coupled instability appears well above the critical pressure. Entrainment-driven acoustic instability has so far only been observed with onset in the neighbourhood of it. What nobody has done is look for the second kind, at scale, above the line.
2 branches, 1 root. The place they would meet is the thing nobody has measured.
Every model in current use traces to one correlation published in 2002, which was itself fitted to wind tunnel measurements of non-reacting liquid films made in 1970. From there the field splits. One branch went to the fuel surface and photographed it. The other went to the chamber and modelled its acoustics. Neither branch has come back for the other. Hover any node.
Read the papers separately and each is coherent. Read them together and they contradict each other.
Gallo and Carmicino say instability is caused by the combustion delay of entrained droplets. No entrainment, no instability. That is why the polyethylene test was quiet.
The Turin and Avio paper says, in its conclusions, that an increase in the entrainment phenomenon raises regression rate but has a beneficial effect on instability. Their design recommendation is to cut additive content so that entrainment goes up.
One says entrainment destabilises. The other says more of it stabilises. There is a related tension underneath: the optical work shows higher viscosity gives a more stable film, while the model says lower viscosity gives a more stable engine. Film stability and engine stability are being treated as the same quantity by a literature that has never said they are different.
Every number in this section is published by the engine builders themselves. Only the arithmetic putting them on one axis is new.
The largest paraffin engine programme in Europe grew out of the University of Stuttgart's HyEnD project, whose test campaigns are the best-documented stability record in this literature. In their 500 N class engines an intrinsic low-frequency instability appeared in most tests, at 70 to 140 Hz. A Helmholtz bulk mode sat at 350 to 400 Hz, vortex shedding near 500. The first longitudinal acoustic mode of the chamber was computed at 2000 to 2500 Hz.
It was never measured. Their own campaign report says why: the acoustic modes could not be resolved because the pressure transducer sampled at 1000 Hz. Everything above 500 Hz was invisible to the instrumentation by construction.
In a small chamber that hardly matters, because the acoustic mode sits a full decade above every driving mechanism in the engine. But the first longitudinal frequency falls as the reciprocal of chamber length, and the same group has published the chamber length of their 75 kN flight engine: about 3.5 metres. Run their own formula at their own length and the mode arrives at roughly 145 to 170 Hz.
The acoustic mode that was 10 times above the instability band in the test engines lands on top of it at flight scale, in the band where the instability was measured in most tests.
Move across the chart to slide a chamber length along the mode line.
This is not a claim that the flight engine is unstable. It is a claim about evidence: the separation that made the acoustic band ignorable in the test campaigns does not survive the scale-up, and the instrumentation that produced the existing stability record could not have seen the collision if it had happened. For a sounding rocket, stability can be observed flight by flight. For a licensed launcher it has to be evidenced, at sampling rates chosen for the modes the chamber actually has.
Every model on this page reads melt thickness off the heat flux acting at that instant1245. That is a quasi-steady assumption. The layer is taken to be in equilibrium with the flux at all times, so a change at the surface appears in the melt thickness with no delay.
Heat does not work that way. It has to travel. Solve the same problem in time, with conduction into the fuel and a moving phase front1516, and melt thickness is no longer an input to be read off. It is a result, and it arrives after the flux that caused it.
The panels below share one input. On the left, the profile the published models assume. On the right, the profile computed by the transient solver. Step the flux and the lag is visible as horizontal separation in the bottom trace. Switch to a sinusoid and the same lag reads as attenuation: at the frequency where this class of motor was measured oscillating3, the computed layer moves far less than the assumed one.
Solved in the page from the properties in reference 18, with a surface temperature of 700 K held fixed and a flux of 3 to 6 MW/m2. It is the mechanism, not a measurement, and the exported solver run will replace it.
Run the melt layer at every axial station of a cylindrical grain and couple the stations through the mass flux, and the port stops being a cylinder. The leading edge regresses fastest, because the boundary layer is thinnest where it starts growing, the x to the power minus 0.2 term. Fuel addition raises the mass flux downstream, the G to the power 0.8 term, and the two effects together open the port into a slight cone. The grain below is the Gallo and Carmicino S-P geometry3. Scrub through the burn and read any station.
Solved in the page from the same Marxman coupling the framework uses. The exported MeltLayer2D run will replace it.
A horizontal port hangs its melt film against gravity, so the film should pool at the bottom and thin at the crown. MeltFilmGravity computes how much, as a renewal limited rimming flow212223: drainage acts on a film that the burn keeps replacing. The balance is the drainage number, rho g h cubed over 3 mu R times the regression rate.
At the verified reference state the film is 0.20 percent thicker at 6 o clock and 0.20 percent thinner at 12. At 2 MW/m² the split grows to 13 percent, because drainage scales as h cubed over the regression rate and both move against gravity as the flux falls.
Drawn from the gravityfilm sweep of the framework. The exported profiles will replace the parametric fit.
The findings split the field's problem in two. Findings 1 to 5 are a measurement gap, response numbers that have never been measured. Findings 6 to 8 are a structural one, response physics the published models delete by construction. The framework exists so that closing either can be attributed to the right layer.
Not a published finding. The part of this piece most likely to be wrong.
The two claims fit together if droplet size is the mediating variable.
Lower melt viscosity gives more entrainment. By the same shear mechanism it also gives finer droplets. Finer droplets vaporise and burn sooner, so the delay between entrainment and heat release shortens. A shorter delay moves the phase between fuel production and chamber pressure away from the driving condition.
So more entrainment destabilises through the sheer quantity of delayed heat release, and stabilises through the shortening of the delay. Which effect wins is a question about how that delay depends on droplet diameter, and droplet diameter is set by viscosity, surface tension and dynamic pressure, which is the same group that sets entrainment in the first place.
Supporting evidence, indirect: the optical work reports that droplet size is expected to fall as chamber pressure rises, and that above the critical point droplet surfaces become indistinct altogether.
Nobody measures droplet size in a firing motor. The one dedicated droplet sizing rig in this literature runs cold, with no combustion at all.
The gravity film adds a falsifiable prediction. Port ovalisation should appear only in low flux thick film burns, at the bottom of the port by pooling before the crown by thinning, because drainage scales as h cubed over the regression rate21.
This is a case study, so the method is part of the output.
13 primary sources were read in full or by targeted section: 2 theory papers, 4 experimental campaigns, 2 doctoral and masters theses, 2 modelling papers, and 3 on additives and injector geometry. Each was entered into an extraction table with a fixed column set, including chamber pressure, regime against the wax critical point, instrumentation sampling rate, attributed acoustic cavity, claimed mechanism, and whether the response parameters were measured or assumed.
A second tranche of 5 programme papers from the HyEnD project and its successor company was read after the table was built: the STERN project report, the HEROS launch campaign paper, the LOX transition paper and two systems papers. Finding 4 is built entirely from numbers published in that tranche.
That last column is where Figure 1 and the central claim come from. Sorted, it reads: defined in 1 paper, assumed in 2, modelled from motor data in 1, not addressed in 9, measured in none.
2 database sweeps established the corpus size. The main query returns 57 records since 2015. The intersection with the supercritical question returns 15. A third query, probing for any measured response function or describing function applied to a hybrid, returns effectively nothing. A documented empty search is a finding rather than a failure, and it is why this piece can say zero rather than few.
Gain and phase against forcing frequency, for a liquefying fuel. This is the closure both models assume. In gas turbine and liquid rocket combustion it is routine practice: force the system, sweep the frequency, phase-lock average against a heat release marker, extract the transfer function. It has never been done for a fuel that entrains.
Vary dynamic pressure and film thickness independently and measure the entrainment response. This is the item the Turin and Avio paper explicitly names as future work.
Optical sizing in a firing motor, against viscosity and pressure. The step past the cold-flow rigs, and the test of the hypothesis above.
One post burn grain fired horizontally against one fired vertically at identical conditions. Sliced web averaging structurally cannot see the azimuthal split of Finding 8, and one scan closes that gap.
None of these is exotic. All four are standard practice in adjacent fields. They have simply never been pointed at this fuel, at the pressure that matters.
13 papers is not a systematic review, and no completeness is claimed. 4 sources cited inside these papers remain unread, and one of them, on film instability under supercritical conditions, sits directly under the supercritical section. The supercritical argument rests on 3 tests compared qualitatively in 1 thesis. It is the best evidence available and it is thin. The lineage reading and the hypothesis are inference, labelled as such. None of it has been checked against proprietary test data, because none was available.
The measurement this calls for
I have built and run the forced-response chain this case study says is missing, on a different fuel: Thermoacoustic Response of Hydrogen Micromix Flames →