LOW MACH (microsecond) STAGE

LOW MACH (microsecond) STAGE

SONOLUMINESCENCE AND INDUCED FUSION WORKSHOP Deuterium-Deuterium Thermonuclear Fusion due to Acoustical Cavitation (Theoretical Analysis) Robert I. NIGMATULIN Ufa-Bashkortostan Branch of Russian Academy of Sciences - President [email protected] Richard T. Lahey, Jr Rensslear Polytechnic Institute Troy, NY, 12180 [email protected] 19 June, 2003 Arlington, VA THE TEAM RUSSIA Ufa Robert I. NIGMATULIN Iskander Sh. AKHATOV Naila K. VAKHITOVA Raisa Kh. BOLOTNOVA Andrew S. TOPOLNIKOV Marat A. ILGAMOV Kazan Alexander A. AGANIN USA RPI Richard LAHEY, Jr. Robert BLOCK Francisco MORAGA ORNL Rusi TALEYARKHAN Colin D. WEST Jeing S. CHO SPHERICAL SHOCK WAVE CONVERGENCE AND CUMULATION Initiation of a Spherical Shock Wave by the Convergent Interface Selfsimilar Cumulation of the Spherical or Cylindrical Shock Wave from the Infinity Guderley, 1942; Landau & Stanyukovich, 1955; Nigmatulin, 1967 Focusing of the Spherical Shock Wave at the Center of the Bubble The Spherical Shock Wave after the Reflection from the Center of the Bubble Specific Features of Single Bubble Sonoluminescence Radius of the bubble a Equilibrium bubble size a0 ~ 3 5 m Adiabatic bulk compression gas temperature Tmax ~ 5000 K (?!)

t Cold water effect Noble gas effect Extremely short light flashes tF ~ 50 ps = 510-11s a0 amin tC ~ 10-8s t t Light Radiation 3 1 T max a min Tmax ~ 5000 K (adiabatic compression) T0 a0 tF ~ 10 s -11 t~ 30s tC ~ 30 ns tF ~ 50 ps t 6 days 7 min 0,7 s Supercompression by Convergent Spherical Shock Wave Moss et al (Livermore National Laboratory, 1994) Radius of the Hot Plasma Core: 109 m = 1 nm Density: 10 g/cm3 = 104 kg/m3 Temperature: 106 K Time Duration: 1011 s = 10 ps No Thermonuclear Fusion HOW TO AMPLIFY THE SUPERCOMPRESSION? AMPLIFING THE ACOUSTIC WAVE (pI 15-20 bar) GAS IN THE BUBBLE: CONDENSING VAPOR (VAPOR CAVITATION) - Minimizing Effect of Gas Cushioning - Higher Kinetic Energy of Convergent Liquid COLD LIQUID LARGE MOLECULES (ORGANIC) LIQUID Low Sound Speed in Vapor CLUSTER of the Bubbles Kinetic Energy of Convergent Flow around the Bubble (CFAB) 3 K p Rmax (in SBSL p 1.5 bar) Rmax 500 800 mcm (in SBSL Rmax 50 80 mcm) p 15 bar In our experiments: the Kinetic Energy K of CFAB is 104 times higher the maximum mass of the gas 103 times higher BUT the final mass of

the gas in the Bubble m is only 50-100 times higher (because of the condensation) 10 4 K/m and Tmax is = 100 200 times higher (50 100) than in SBSL It means that in our experiment we may get Tmax (100-200)106 K Mass, Momentum, Energy Conservation Differential Equations Liquid a(t) 1 ur2 0, 2 t r r Mass Gas 1 p u 2 u2r 2 0, t r r r Momentum e 1 1 T 2 ur 2 e p 2 r 2 , t r r r r r Energy 2 e u , 2 n Tg p p , T , , T , g g 0 . Tg 0 INTERFACIAL BOUNDARY CONDITIONS Mass: g a u g a u j (r = a(t))

- intensity of phase transition 2 4 u a a Tg T g jl r r p g p Momentum: Energy: Kinetics of phase transition (HertzKnudsen-Langmuir Eqn): j 2 Rg T Tg [T ] 0.45 pS(T) saturation pressure, ps T pg T T g j TS 2 Rg TS g l evaporation heat - accommodation (condensation) coefficient - (Labuntsov, 1968) MI-GRUNEIZEN EQUATIONS OF STATE p T c p p , T cV T , p and pp cold or potential p pp pT pp 2 d p d pT cVT internal energy and pressure due to intermolecular interaction T and pT thermal internal energy and thermal pressure

c - chemical internal energy cV and - averaged heat capacity and Gruneizen Coefficient LENNARD-JONES POTENTIAL pp = R n A m p = pp p V 1 V0 R Q n 1 m 1 0 n 1 m 1 BORN-MAYER POTENTIAL 2/3 1 1 1/ 3 p p A exp b 1 0 C K , 0 0 0 3A p exp b 1 0b 0 1/ 3 C 0

0 K 0 0 LIQUID PHASE (NONDISSOCIATED ) p p d p dV 0 858 kg/m3 , C0 1189 m/s cV 1516.8 m 2 /(s 2 K), l(ch ) 6.048 105 m 2 /s 2 , Al 9.757 107 Pa , Cl 0, K l 4.535 108 Pa , bl 19.07 (), 1 SHOCK ADIABAT (D-u) FOR LIQUID ACETONE (Trunin, 1992) Shock Wave Speed, D, km/s 10 40 Non-dissociated 8 Non-dissociated 30 Dissociated 6 Dissociated 20 4 10 Trunin, 1992 2 Cl 0 0 0 2 4 6

MASS VELOCITY, U, km/s D Shock Wave Speed U Mass Velocity after the Shock Wave 8 10 0 10 20 30 MASS VELOCITY, U, km/s U D 40 SHOCK ADIABAT & ISOTHERMS (P-V) for D-Acetone (C3D6O) Shock adiabat of Liquid 0 .5 10 12 0 .0 6 6000 K NDis P 10 11 Trunin, 1992 10 10 5000 K Dis 10 9 0 .0 0 4000 K 0 .4 0 .6 0 .8 10 8 1 .0 0 .3 PRESSURE p, bar PRESSURE p, Mbar p 0 .0 3 0 .4

3000 K Dis 2000 K 0 .2 1000 K 0 .1 NDis 0 .0 0 .2 0 .4 108 K 10 7 107 K 10 6 106 K 10 5 105 K 10 4 104 K 10 3 RELATIVE VOLUME, 0/ 0 .8 1 .0 103 K 508 K 273 K 10 0 0 0 .6 D is 103 K 10 2 10 1 pp 0 .0 Isotherms of Vapor 10 13 N D is P p -2 1 0 3 4 0 D = (D U) p p0 = 0 D U

-4 1 0 L iq D is 3 10 -2 10 -1 N D is 10 0 10 1 10 RELATIVE VOLUME, / 2 0 .6 6 7 0 .1 1 3 10 3 ISOTHERMS (P-V) & SATURATION LINE for D-Acetone Tcrit 508 K , crit 309 kg/m3 , M 64 kg/kmol, Internal Energy and Evaporation heat Isotherms 12 60 Rg 129.9 J/(kg K) C 40 Vapor ENERGY , 105 m2 /s2 PRESSURE p, bar C 8 508 K 1000 K Liquid 4 20

Evaporation Heat (ig-il) 400 K 0 0 10 0 10 1 10 2 RELATIVE VOLUME, / 10 3 200 300 400 TEMPERATURE, K 500 600 DISSOCIATION of GAS 1 .0 0.9 g mg d md , mg md 1, mg mg (T ), 5(T Tk ) mk 0.5tanh tanh ( 5 ) , T k Td 4.01 eV, k g , d md p p g mg pd md , 0 .5 0.1 0 .0 1000 10000

g gp gT gch d dp dT dch p g p g p p g T pd pd p pd T p p 2 d g p g d gT cVg T p g T g gT cVg 1516.8 m 2 /(s 2 K) gch 6.048105 m2/s2 g 0.113, g 0.9394, g 0.9000, Ag 4.0 107 Pa , C g 1.7435 109 Pa , K g 1.784 109 Pa , bg 24.028 Td T, K 100000 p p 2 dd p d d dT cVd T pd T d dT cVd 1940.0 m 2 /(s 2 K) ch d0 27.6106 m2/s2 d 0.667, d 0.333, Ad 2.403 10 7 Pa , Cd 0, K d 3.585 108 Pa , bd 25.207 1000000 IONIZATION of DISSOCIATED GAS (dch ) (dch0 ) i(ch ) ( i(ch) ionization energy) 6 i(ch) k mk Rk Tk 36 R 64 C k 7 16 TCj mCj 12 R

T m R 64 D D1 D1 64 O j 1 3 D6 O : TOjmOj j 1 M 64 molecular weight M C 12 3 36 ( 36 ) - carbon, 64 M D 2 6 12 ( D 12 ) - deuterium, 64 M O 16 1 16 ( 16 ) 64 - oxygen. k C1, C2, C3, C4, C5, C6; D1; O1, O2, O3, O4, O5, O6, O7; 5(T Tk ) mk 0.5 tanh tanh (5), Tk IONIZATION CONSTANTS RD 4157 m 2 /(s 2 K), TD1 13.60 eV RC 692.8 m 2 /(s 2 K) , RO 519.6 m 2 /(s 2 K), TC1 11.26 eV TO1 13.69 eV, TC2 23.38 eV TO2 35.19 eV, TC3 47.89 eV TO3 54.94 eV, TC4 64.49 eV TO4 77.41 eV, TC5 392.0 eV TO5 113.9 eV, TC6 490.0 eV TO6 138.1 eV, TO7 739.3 eV THERMAL CONDUCTIVITY for acetone 0.08

Gas 3 g , kg m/(s K) / g0 , kg m/(s3 K) g 0.06 0.04 0.02 400 600 T, K 800 1000 6 10 5 10 4 10 3 10 2 10 1 10 0 Gas T 1 , g g 0 1 g 0 Tg 0 g 0 75 g 0 7.5 3 10

5 7 10 10 T, K Tg 0 273 K , 0.5 9 10 0 .2 0 0 .1 6 T 1 , l l 0 1 l 0 Tl 0 0 .1 2 0 .0 8 l 0 0.169 kg m/ s 3 K , l 0 0.609, 0 .0 4 Liquid 0 .0 0 200 300 T , K 400 500 g 0 8.23 10 3 kg m/ s 3 K , g0 0 l , k g m /(s 3 K ) 0.00 200 10 Tl 0 273 K , 1.0 3 He n DD

TH < J 1 2 n2 v , 12 12 v > m 3 /s KINETICS OF FUSION N J dV dt , tV J neutron emission intensity, N number of emitted neutrons, n 6g g 10 -2 1 10 -2 4 10 -2 7 10 -3 0 10 -3 3 10 -3 6 D -T D -D 10 concentration of D atoms CO CD3 2 , 6 10 7 8 10 T , K 10 9 v averaged product of the cross section times the deuterium nucleus thermal velocity, N A 6.02 1026 kmol-1 Avogadro number,

N D 6 N A g 0.56 1026 kg-1 g 64 kmol-1 molecular weight 10 10 Different Stages for Bubble Expansion and Compression Low Mach Regime (M << 1) Rayleigh-Plesset + Thermal Conductivity Eqn Middle & High Mach Regime (M ~ 1, and M >> 1) Hydro Code a M Mach Number Cg a,m 500 BF Tg=Tg(t, r) pg=pg(t) Heat conducting, homobaric gas (M < 10 -1) M>1 SBSL Tg=Tg(t, r) pg=pg(t, r) t, s 30 Low Mach regime a M 1 Cl For GAS (vapor): cVg T , p R g gT R g 129.9 J/(kg K), cVg R g cVg 1.1 For LIQUID: cVl T , cVl 2000 m 2 /(s 2 K) l 858 kg/m 3 const Rayleigh-Plesset equation 2 2 pl d a 3 da a 2 2 dt dt r a l

pI THERMAL CONDUCTIVITY EQUATIONS FOR HOMOBARIC BUBBLE (pg = pg(t)) IN INCOMPRESSIBLE LIQUID (l = const) Tg 1 dp g Tg 2 Tg 2 g r r a : c gp g ug r r r r dt t dp g 3p g u ga 3( 1) Tg , p g (t ) Rg g (r , t )Tg (r , t ) dt a a r r a 1 Tg r dp g ug p g r 3p g dt r a : r a : Tl 1 T 2 Tl cl l l ul r l , 2 r r r r t (l const) ul ula a2 r2 g a u g a u j Tg T g j l, r r p T

s j 2 Rg T p g Tg Cluster Amplification Effect a, R = 0.05 Void fraction 20 Number of bubbles N = 50 Maximum microbubble radius a 0 = a max = 400m 100 Radius of the cluster R 0 = 4 mm 10 r=0 r = 2 mm r = 4 mm 1 12 450 17 22 27 32 37 42 r = 4 mm r = 2 mm t, s p,bar 500 450 400 350 100 300 50 250 r=0 p, bar 0 200

150 -50 t = 32 s 100 -100 50 r, mm 0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 t, s -150 12 17 22 27 32 37 42 LOW MACH (microsecond) STAGE 40 1 -3 0 t* t0 0 .4 0 10 20 t ,s 30 40 , k g /m

10 0 -4 0 10 -1 -2 -4 0 -8 0 10 -3 0 .1 4 300 0 .1 2 290 0 .1 0 280 p , bar 0 .0 10 1 10 3 0 .1 10 2 -6 0 * * -2 0 3 0 0 .3 0 .2 , ng 9 -1 5 4 200 80 0

g 5 400 a ,m 8 120 10 m 6 20 T , K 600 40 d a /d t , m /s 7 160 p I, bar 800 * 0 .0 8 260 0 .0 4 250 10 20 t,s 30 40 5 7 8 -1 5 1 -3 270 0 .0 6 0 4 6 0 p 15 bar, p 50 bar, 2 19.3 kHz, TL 0 273 K, 1.0 10 20

t,s 30 40 LOW MACH (microsecond) STAGE 0 .5 40 8 30 20 0 .3 1 0 .2 2 3 p , bar , k g /m 3 0 .4 7 4 0 .1 6 5 1 2 3 0 5 4 8 7 6 -1 0 0 .0 -2 0 0 200 1 2 40 3 400 600 r , m 4 5 6 800

0 200 400 r , m 300 20 600 800 290 0 280 7 -2 0 T ,K u , m /s 10 -4 0 8 -6 0 270 4 260 -8 0 8 1 -3 5 6 7 250 0 200 400 600 r , m p 15 bar, p 50 bar, 2 19.3 kHz, TL 0 273 K , 1.0 800 0 200 400 600 800

r, m t1 0.77 s, t 2 1.67 s, t3 3.41 s, t 4 6.89 s, t5 9.86 s, t 6 14.46 s, t 7 22.76 s, t8 28.05 s Transition from LOW MACH to HIGH MACH STAGE (microsecond stage) 0 .0 100 15 0 10 2 10 1 10 0 10 -1 - 0 .8 60 15 - 1 .2 20 - 2 .0 0 - 1 .0 - 0 .8 - 0 .6 - 0 .4 t - t* , s - 0 .2 0 .0 - 1 .0 0 .0 14 - 0 .4 13 9 -1 2 - 0 .6 200 r ,m 400 600 p 15 bar, p 50 bar, 2 19.3 kHz, TL 0 273 K , 1.0 0 .0

14 400 13 9 -1 2 300 15 - 1 .2 0 - 0 .2 500 - 1 .0 9 -1 2 - 0 .6 - 0 .4 t - t* , s 15 600 - 0 .8 13 - 0 .8 700 - 0 .2 14 15 40 - 1 .6 0 .0 15 14 , K 10 3 13 80 T 4 - 0 .2 u , k m /s p , bar 10 - 0 .6 - 0 .4

t - t* , s 12 , ng 14 - 0 .8 14 g 200 d a /d t , k m /s a ,m 13 - 1 .0 100 13 - 0 .4 12 300 12 m 400 200 0 200 r ,m 400 600 0 200 r , m 400 600 t* 30.01 s, t9 t * 1.67 s, t10 t * 1.28 s, t11 t * 1.10 s, t12 t * 0.81 s, t13 t * 0.52 s, t14 t * 0.23 s, t15 t * 0.03 s 40 10 , kg/m3 a,m 30 20 10 4

19 18 20 102 * t16 - t 20 16 17 0 p , bar 4 0 -4 -8 - 5.0 * t 16 - t 20 0. 0 t - t*, ns 1012 1010 108 106 104 102 1 19 18 108 19 18 HIGH MACH (nanosecond) STAGE * t * 17.41 s EOS 1, 0 16 2 20.5 kHz, 20 17 4 102 - 5. 0 p 1000 bar, 17 106 10

p 40 bar, 20 5.0 T ,K da/dt, km/s 8 TL 0 273 K, 1.0 16 0.0 t - t *, n s 5.0 10 5 10 4 10 3 10 2 10 1 10 0 100 19 18 10 10 7 10 5 50 17 16 1 .0 r ,m 2 .0 3 10

1 0 16 17 -5 0 3 .0 18 0 .0 19 20 10 20 -1 5 0 0 .0 9 19 -1 0 0 1 0 11 10 9 10 7 1 .0 r ,m 2 .0 3 .0 19 18 T ,K p , bar 20 u , k m /s , k g /m 3 HIGH MACH (nanosecond) STAGE 18 0 .0 17 16 1 .0 r,

m 2 .0 3 .0 10 5 10 3 20 17 0 .0 1 .0 16 r , m 2 .0 p 15 bar, p 50 bar, 2 19.3 kHz, TL 0 273 K, 1.0 3 .0 10 8 10 6 10 4 10 10 6 10 5 10 4 10 3 10 2 2 10 1 10 0 0

10 -1 19 21 3 18 17 * 0 .0 t18 t 0.02 ps , 0 .2 200 t19 t 0.06 ps , 0 20 17 0 .4 0 .6 21 T ,K * -4 0 0 18 -6 0 0 -8 0 0 0 .0 0 .2 0 .4 t - t* * , p s 0 .6 20 19 10 9 10 8 10 7 10 6 10 5

10 4 10 3 10 2 21 18 17 0 .0 19 -2 0 0 u * , k m /s t 21 t 0.17 ps 10 10 10 20 , k g /m * 0.04 ps , t 20 t 0.10 ps, 12 0 .2 0 .6 0 .4 0 .6 0 .2 0 .4 t - t**, ps 0 .6 20 19 21 18 17 0 .4 0 .0 0 .2

7 .0 21 6 .0 5 .0 20 4 .0 N t17 t p , bar PARAMETERS IN THE CENTER OF THE CORE 10 3 .0 2 .0 1 .0 0 .0 19 17 0 .0 18 LIQUID DISSOCIATION IMPACT 50 Cold dissociation because of the super high pressure (105 bar) in liquid needs 102 ns; Super high pressure in liquid (near the bubble interface) takes place 1 ns RADIUS [mkm] 40 30 20 dissociated liquid 10 non-dissociated liquid 0 -10 -5 TIME [ns] 0 5 ubble radius evolution for deuterated acetone C3D6O; COLD ELECTRONS Te << Ti (during 10-13 s) CV = 2000 m2/c2K, not 8000 m2/c2K

Neutron production distribution and maximum density, temperature and velocity 1 f N dt J dV dt 4r 2 J dr dr 4r 2 J (r , t ) dt 0 V 0 V 0 V 1010 10 a 0.16 Tmax 9 max , kg/m3 & Tmax , K a Nr 0.12 108 107 106 0.08 max 105 104 0.04 103 10-2 10-1 100 101 102 r , nm 0 0.00 103 Nr , nm-1 u max, km/s -400 -800 -1200 Nr -1600 10-2 10-1 100 101 102 r , nm 10

0.16 0.12 0 0.00 r=5.29 nm r=13.2 nm r=26.5 nm 0 0.12 3.0 20 r* 40 60 80 100 r , nm rF 2.0 0.04 1.0 0.00 0.0 10-1 100 f N r 4r 2 J (r , t ) dt 0 rF 55 nm - Radius of the fusion core r=2.65 nm 0.04 1 r 11 nm - Radius of the maximum neutron production , r=1.32 nm 0.08 4.0 3 N N r (r ) dr r=0.132 nm r=0.256 nm 0.16 0.08 umax

a N r , nm -1 f N 1 f N r , nm -1 1 101 r, nm 102 INTERNAL GAS ENERGY AS THE SUM OF COMPONENTS 3 k g / m 1 .0 3 T 0 .8 d 0 .6 0 .4 i 0 .2 0 .0 p , T , d , i p , T , d , i 1 .0 p 10 4 10 5 6 10 T, K 10 7 10

8 4 k g / m 3 T 0 .8 p 0 .6 0 .4 i 0 .2 0 .0 , T p , T T T d T i T , p d 10 p 4 10 , T 5 106 T,K 107 T , d d , i i 108 Acetone 1 .2 1 =103 kg/m3 p T/p 0 .8 0 .6 0 .4 =104 kg/m3 0 .2 0 1E+2 1E+3

1E+4 1E+5 1E+6 TEMPERATURE, K 1E+7 1E+8 LOW TEMPERATURE (condensation) EFFECT Normalized neutron production, N/N273 250 MINIMUM MASS, mg min, ng 20 0 15 0 = 0.1 10 0 50 = 1.0 0 250 2 60 2 70 2 80 2 90 LIQUID TEMPERATURE, Tl0, K 3 00 3 = 1.0 2 1 = 0.1 0 250 260 270 280 290 LIQUID TEMPERATURE, Tl0, K Minimum bubble mass and total number of emitted neutrons vs liquid temperature, T0 300 Fig.1. Temporal dependence of the air bubble radius R and some bubble shapes in the course of a single-period harmonic pressure oscillation in water with p = 3 bar, /2 = 26.5 kHz, for a20/R0 = 2.510-2, R0 = 4.5 m . While plotting the shapes, the bubble radius was taken to be R0[1 + 0.3{3.5lg(R/R0) + 1.5|lg(R/R0)|}]. Incopmpressible viscous liquid, homobaric Van-der-Waals gas. a20/R0 = 0.03 Incompressible viscous Liquid Homobaric Van der Waals Gas a20/R0 = 0.001

Temporal dependences of the radius R of an air bubble in water, the sphericity distortion a2 /R and some bubble shapes just before the time of the collapse tc under harmonic forcing with p=5bar, /2=26,5 kHz for two values of the initial distortion. Convergent and divergent shock waves in the bubble are shown in figure (b). SUMMARY OF THE ANALYSIS Bubble Fusion (ORNL+RPI+RAS) Sonoluminescence (LLNL) Density: 20 - 80 g/cm3 10 g/cm3 Temperature: 108 K = 10 KeV 106 K = 10-1 KeV Pressure: 1011 bar Velocity: 900 km/s Time Duration: 10131012 s = 101-100 ps 10 ps Radius of the Fusion Core: 50 nm 1-3 nm Number of nucleus: 20 109 Fast Neutron & Tritium Production 10-1 - 10 per collapse FINDINGS COLD LIQUID Effect CLUSTER effect NON-DISSOCIATION of Liquid COLD Electrons SHARPENNING: Node size for Fusion Core r 0.1 nm << a 10 nm << a 10 000 nm

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