Cavitation-induced ignition of cryogenic hydrogen-oxygen fluids
V. V. Osipov, C. B. Muratov, E. Ponizovskya-Devine, M.Foygel, V. N. Smelyanskiy
CCavitation-induced ignition of cryogenic hydrogen-oxygen fluids
V. V. Osipov , , C. B. Muratov , E. Ponizovskya-Devine , , M.Foygel , V. N. Smelyanskiy Intelligent Systems Division, D&SH Branch, NASA Ames Research Center, MS 269-1, Moffett Field, CA 94035 Mission Critical Technologies, Inc., 2041 Rosecrans Avenue, Suite 225, El Segundo, CA 90245 and Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 South Dakota School of Mines and Technology, Rapid City, SD, 57701 (Dated: November 9, 2018)The Challenger disaster and purposeful experiments with liquid hydrogen (H2) and oxygen (Ox)tanks demonstrated that cryogenic H2/Ox fluids always self-ignite in the process of their mixing.Here we propose a cavitation-induced self-ignition mechanism that may be realized under theseconditions. In one possible scenario, self-ignition is caused by the strong shock waves generated bythe collapse of pure Ox vapor bubble near the surface of the Ox liquid that may initiate detonationof the gaseous H2/Ox mixture adjacent to the gas-liquid interface. This effect is further enhancedby H2/Ox combustion inside the collapsing bubble in the presence of admixed H2 gas.
The source for the formation of flames in the cryo-genic hydrogen/oxygen (H2/Ox) fuel mixture during theChallenger disaster in 1986 still remains a mystery. Thefireball which caused the orbiter’s destruction appearednear the ruptured intertank section between the liquidH2 (LH2) and liquid Ox (LOx) tanks, but not near thehot jets from the nozzles . Purposeful experimentswith LOx and LH2 tanks carried out by NASA showedthat cryogenic H2/Ox mixtures always self-ignite whenthe flows of cryogenic fluids containing gaseous hydro-gen (GH2), gaseous oxygen (GOx) and LH2 mix witha turbulent LOx stream. Since this effect can lead tocatastrophic events, understanding its mechanisms is aproblem of great importance.In this Letter we propose a cavitation-induced self-ignition mechanism of cryogenic H2/Ox fluids. Cavita-tion is the formation and compression of vapor bubblesin flowing liquids driven by abrupt pressure variations.Due to inertial motion of the liquid this process leads toa rapid collapse of the bubbles and spiking of the gastemperature and pressure inside the bubbles, producingstrong shock waves . Here we discuss possible scenariosof cavitation-induced ignition in cryogenic Ox/H2 fluids.We concentrate on the most transparent scenario relatedto the collapse of a vapor bubble in the Ox liquid nearthe interface between LOx and the GH2/GOx mixture.Vapor bubbles, most likely with admixed GH2, can becreated in the falling LOx blobs as a result of mixingof gaseous H2 and Ox with a turbulent stream of liquidOx. A pressure jump between LOx and the bubbles mayarise, for example, due to shock waves arising as a re-sult of an impact of a LOx blob against a solid object(Fig.1a). The overpressure in such a shock wave is oforder ∆ p (cid:39) ρ L v (cid:38) v (cid:38)
20 m/s of the liquid. Such a “weak” shock wave can-not induce ignition of the GH2/GOx mixture directly,but it can initiate cavitation collapse of the vapor bub-bles inside LOx. The below computations show that suchweak initiating shock waves can lead to the formation ofbubbles of a small radius R min (cid:39) . p (cid:38) T (cid:38) . We treated the liquid as incom-pressible and inviscid, neglected surface tension at theinterface, treated the gas phase as a mixture of idealgases and took into account diffusion and thermodiffu-sion of the admixed GH2 (see supplementary materialfor more detail). We also modeled combustion inside thebubble, using a simplified model based on the assumptionthat the burning rate is limited by the initiation reactionsH +O → OH+OH and H +O → HO +H, which havethe lowest rates. Thus, we modeled the GH2/GOx com-bustion by a brutto reaction H + O → H O + O withthe rate G comb ( T ) = c H c Ox [1 . · exp( − /T ) +1 . · T . exp( − /T )] m /(mol · sec), where c H and c Ox are the molar concentrations of GH2 and GOx, re-spectively, and T is in degrees Kelvin . We note that thisapproximation is close to the one-step mechanism of Mi-tani and Williams . Also, the model predicts the samesteady detonation wave parameters as those obtained with the help of the model taking into account 17 mainchain reactions of GOx/GH2 mixture combustion .Our simulations show that under the action of initi-ating shock wave with overpressure ∆ p ≥ .
15 atm themaximum pressure and temperature in pure vapor bub-bles of initial radius R (cid:38) mm collapsing in LOx ex-ceed 1500 atm and 800K, respectively, when the bub-ble radius reaches its minimum value of R min (cid:39) . p the values of p max and T max increase and the value of R min decreases( R min ≤ .
05 mm at ∆ p ≥ . R min , the harder it is to achieve ignition. Thepresence in the bubble of even a small amount of non-condensable GH2 sharply decreases the values of p max a r X i v : . [ phy s i c s . f l u - dyn ] J a n and T max . However, at large enough ∆ p ≥ . R (cid:38) p max and T max in the bubble reach gigantic values p max ≥ T max ≥ insidethe bubble may be ejected into the space above the LOxsurface and easily ignite the GH2/GOx mixture nearby.The same equations of gas dynamics used in the cavi-tation simulations were also employed to analyze the ig-nition of GH2/GOx mixtures by a localized strong shockwave generated by the collapsing bubble. We envision ahemispherical shock wave propagating in the unconfinedGH2/GOx mixture above the LOx-gas interface (see Fig.1b), initiated by a local increase in gas pressure and tem-perature within the radius R ∼ R min . We see that the FIG. 1: A scenario of cavitation-induced ignition ofGOx/GH2 mixture: (a) an initiating “weak” shock wave in aLOx blob (with a bubble) forms due to its impact with a solidobject; (b) collapse of the bubble near the liquid-gas interfaceunder the action of the initiating shock wave and generationof a strong secondary shock wave inducing detonation of theGH2/GOx mixture. FIG. 2: Bubble collapse dynamics for the initial total gas pres-sure p = 1 atm. Other initial parameters are: radius R = 2mm, overpressure of the initiating shock wave ∆ p = 0 . p H = 0 (curve 1); R = 3 mm,∆ p = 1 . p H = 0 .
01 atm (curve 2); R = 3 mm,∆ p = 2 . p H = 0 .
015 atm (curve 3). The dashed linesand the values in parentheses are obtained without consider-ing combustion in the bubble. local jump of the pressure p (cid:38)
200 atm and tempera-ture T (cid:38) R (cid:38) . FIG. 3: Formation of cavitation-induced hemispheric deto-nation wave in stoichiometric gaseous H2/Ox mixture withtemperature T = 100K and pressure p = 1 atm. The initialconditions are: temperature T = 1000K and pressure p = 250atm in the central area with radius r ≤ .
15 mm (dashedcurves). p cj - steady-state pressure in the detonation wave(Chapman-Jouguet pressure ) liquid pressure. In scenario (ii) bubbles may form as aresult of an impact of two large LOx blobs whose sur-faces were chilled by contact with very cold ( T (cid:39) p L in the liquid bulk. As a result, the bubbleradius will increase and, due to the inertial motion of theliquid, the pressure in the bubble can become much lessthan p L . As a consequence, the bubble will start to col-lapse, and the gas temperature and pressure inside thebubble may achieve very high values, initiating a local ex-plosion and a strong shock wave. Finally, in scenario (iv)heavy droplets of LOx may penetrate deeply into LH2 (alight fluid), causing intense evaporation of LH2 and for-mation of a GH2/GOx bubble inside LH2 that will grow in size and then collapse due to inertial motion of the liq-uid. Since the critical temperature T c = 33 .
2K of H2 issignificantly below the freezing temperature T m (cid:39)
54K ofLOx, the evaporation of LH2 in contact with LOx mayacquire an explosive character, resulting in even moredramatic outcomes.To summarize, we have identified a possible mecha-nism of ignition in cryogenic H2/Ox fluids which relieson the generation of strong shock waves by the cavita-tional collapse of vapor bubbles close to the liquid-gasinterface in the process of cryogenic H2/Ox mixing. Weshowed that the presence of LOx blobs surrounded byGH2/GOx mixture may be sufficient to initiate H2/Oxignition, including strong detonation waves. We furtherproposed several other scenarios that include mixing ofLH2 with LOx and resulting in even more dramatic con-sequences. More detailed studies of these mechanismsare currently underway. Finally, we note that the pro-posed self-ignition mechanisms should be very importantfor understanding conditions and risks of explosion incryogenic H2/Ox-based liquid rockets and other spacevehicles.The work of CBM was supported by NASA via grantNNX10AC65G.
APPENDIX: GOVERNING EQUATIONS
Neglecting the surface tension and treating the liquidas incompressible and inviscid, the equations for the liq-uid phase may be reduced (see e.g. ) to a single ordinarydifferential equation for the bubble radius R : R d Rdt + 32 (cid:18) dRdt (cid:19) + jρ L dRdt + Rρ L djdt = p m − p L ρ L + (2 ρ L − ρ v ) j ρ v ρ L , (1)and the advection-diffusion equation for the liquid tem-perature T L : ∂T L ∂t + (cid:18) Rr (cid:19) (cid:18) dRdt + jρ L (cid:19) ∂T L ∂r = κ L c L ρ L r ∂∂r (cid:18) r ∂T L ∂r (cid:19) . (2)Here r ≥ R ( t ) is the radial coordinate, ρ L , c L and κ L arethe liquid density, specific heat and thermal conductiv-ity, respectively, p m and ρ v are the pressure and the va-por mass density, respectively, in the gas mixture at theliquid-gas interface, p L is the liquid pressure far fromthe bubble. The vapor condensation flux j is given bythe well-known Hertz-Knudsen equation : j = α ( p v − p s ( T s )) √ πR v T s , p s ( T ) = p c ( T s /T c ) λ , (3)where T s and p v are the vapor temperature and pressure,respectively, at the liquid-gas interface, R v is the vaporgas constant, α is the accommodation coefficient. Herewe used a simple approximation for the dependence ofthe saturated vapor pressure p s ( T s ) on the liquid-gas in-terface temperature T s , where p c and T c are critical pres-sure and temperature, respectively, of the vapor, and λ is a dimensionless parameter (see ). Note that p s de-pends strongly on T s , which significantly affects the bub-ble dynamics .In the gas phase, we have the conservation of momen-tum and energy (again neglecting viscosity effects): ∂u m ∂t + u m ∂u m ∂r = − ρ m ∂p m ∂r , (4) ∂E m ∂t + 1 r ∂∂r (cid:16) r u m ( p m + E m ) (cid:17) = 1 r ∂∂r (cid:18) κ m r ∂T m ∂r (cid:19) + Q h G comb . (5)Here ρ m , u m , p m , T m , E m are the density, radial ve-locity, pressure, temperature, and energy density, re-spectively, of the gas mixture located at r ≤ R ( t ), and κ m = (cid:16) T m T (cid:17) / (cid:80) i c i κ i / ( (cid:80) i c i ) is the thermal conduc-tivity of the gas mixture as a function of T m , the molarconcentrations c i = { c H , c Ox , c H O } of different molec-ular components and thermal conductivities of pure gasspecies κ i at some reference temperature T . Finally, G comb is the combustion rate and Q h is the combustionheat. Treating all the gas species as diatomic ideal gasesfor simplicity, we have p m = R g T m c m , E = 52 R g T m c m + 12 ρ m u m , (6)where c m = (cid:80) i c i and ρ m = (cid:80) i c i M i are the total molarconcentration and mass density, respectively, of the gasmixture, with R g the universal gas constant and M i themolar masses of the gas species. Note that we kept thekinetic energy term in the expression for E m in order tobe able to account for possible rapid onset of combustioninside the bubble.The dynamics of the GH2/GO2 mixture combustionare described by the continuity equations for the molarconcentration of the mixture components c i : ∂c H O ∂t + 1 r ∂∂r ( r c H O u m ) = G comb , (7) ∂c Ox ∂t + 1 r ∂∂r ( r c Ox u m ) = − G comb , (8) ∂c H ∂t + 1 r ∂∂r ( r c H u m ) + G comb = 1 r ∂∂r (cid:26) r D H (cid:18) ∂c H ∂r − c H T m ∂T m ∂r (cid:19)(cid:27) , (9) where D H = ( T m /T ) / ( p /p m ) D H ( T , p ) (see ).Note that in the last equation we included the effect ofGH2 diffusion, which may be significant due to high dif-fusivity of hydrogen and its role as a non-condensable gasduring the bubble collapse.Finally, the boundary and the initial conditions for theequations above are: ∂T m ∂r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = 0 , (cid:18) κ L ∂T L ∂r − κ m ∂T m ∂r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = R = jq h ,T m | t =0 = T L | t =0 = T L , T L | r = R = T m | r = R = T s ,T L | r (cid:29) R = T L , ∂c i ∂r (cid:12)(cid:12)(cid:12)(cid:12) r =0 = 0 , c i | t =0 = c i ,u m | r =0 = 0 , u m | r = R = dRdt − jM Ox c Ox , u m | t =0 = 0 ,jc H M Ox c Ox (cid:12)(cid:12)(cid:12)(cid:12) r = R = D H (cid:18) ∂c H ∂r − c H T m ∂T m ∂r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = R ,R (0) = R , dR (0) dt = 0 , (10)where q h is the latent heat of LOx vaporization. Addi-tional conditions are presented in (Fig. 2).The simulations were done using Godunov’s schemewith variable time-step for stiff problem. The time stepwas varied depending on the maximum of the time deriva-tives of bubble radius, temperature and pressure. In par-ticular we use Monotone Upstream-centered Schemes forConservation Laws (MUSCL) based numerical schemethat extends the Godunov’s scheme idea of linear piece-wise approximation to each cell by using slope limitedleft and right extrapolated states.List of all constants used in the simulations are repre-sented in Table I. Cole, M.D., “Challenger: America’s space tragedy”,Springfield, N.J., Enslow Publishers, 1995. Diane, V., “
The Challenger Launch Decision: Risky Cul-ture, Technology, and Deviance at NASA ”, Chicago: Uni-
TABLE I: Parameters of oxygen and hydrogen in SI unitsused in the simulationsParameter Oxygen Hydrogen Meaning c L ρ L V L . · − - kinematic viscousity ofliquid C T c .
58 - critical temperature p c . · - critical pressure λ q h . · - specific heat of evaporation C v
653 10130 specific heat of vaporat V=const,T=300K R v
264 4124 gas constant C p
917 14270 specific heat of vaporat p=const,T=300K γ . . κ i . . κ L .
17 - thermal conductivityof liquid (T=80K) Q h . · - heat of combustion σ . · − - liquid surface tension D i - 6 . · − diffusion constant(T=270K, p=1atm)versity Of Chicago Press, 1996. Osipov, V.V., Muratov, C. B., , Hafiychuk, H, Ponizovskaya-Devine, K, Smelyanskiy, V, Mathias, D,Lawrence, S, and Werkheiser, M, ”
Hazards Induced byBreach of Liquid Rocket Fuel Tanks: Risks of CryogenicH2/Ox Fluid Explosions ”, (arXiv:1012.5135v1). Brennen, C. E., “
Fundamentals of Multiphase Flow ”,Cambridge University Press, 2005; Brennen, C. E., “Cav-itation and Bubble Dynamics”, Oxford University Press,1995. Fujikawa, S. and Akamatsu, T., “
Effects of the non-equilibrium condensation of vapor on the pressure waveproduced by the collapse of the bubble in a liquid ”, J. FluidMech., , part 3, 481-512 (1980). Landau,L. D. and Lifshits, E. M., ”
Course of TheoreticalPhysics: Hydrodynamics ”, v. 6, Pergamon Press, London,1987. Ripley D.L, Gardiner W.C. Jr, ”Shock tube study of thehydrogen-oxygen reaction. II. Role of exchange initiation”,J. Chem. Phys., 44, 2285 (1966); Michael, J.V., Suther-land, J.W., Harding, L.B., Wagner, A.F., “Initiation inH2 /O2: rate constants for H2 + O2 → H+HO2 at hightemperature”, Proceedings of the Combustion Institute,28, 1471-1478 (2000). Mitani, T. and Williams, F. A., ”Studies of CellularFlames in Hydrogen–Oxygen–Nitrogen Mixtures”, Com-bustion and Flame, 39, 169 (1980). Kao, S. and Shepherd, J. E., ”Numerical solution methodsfor control volume explosions and ZND detonation struc-ture”, GALCIT Report FM2006.007 (2008). Clark, J. A., ”
Universal Equations for Saturation VaporPressure ”, 40th AIAA/ASME/SAE/ASEE Joint Propul-sion Conference and Exhibit, Fort Lauderdale, Florida,July 2004. Osipov, V. V., Muratov, C. B., ”
Dynamic condensationblocking in cryogenic refueling ”, APL, , 224105 (2008). Lifshits, E. M. and Pitaevskii, L. P. , ”