Calculation of installation for cumulation of energy in gas bubbles

The authors: Smorodov E.A., Galiahmetov R.N.
(Translation into the English language has executed A.J. Streltsov.)

Research of a high-temperature status of dense substance represents significant interest for thermonuclear the power engineering. In a some  works of last years  is shown, that similar cumulation of the energy can be achieved at the acoustic cavitation. So in works [1,2,3] affirms, that in a similar way can be achieved of  the  thermonuclear temperatures. In the article [4],  authors by experimentally do investigated of the "one-bubble" sonoluminescence (OS) and then  do  the inference, what the temperature during of the compression of the gas bubble can reach 20 thousand degrees.  

Studying of acoustic methods of the cumulation of energy in gas-filled bubble represents  significant complexities for  production of experiments.  It is caused of the bad controllability of process of a pulsation bubble and his microscopic sizes at the end of compression.   Even in case of steady fluctuations of the single bubble (the mono-bubble cavitation) impossible purposefully do the changing in any parameters of process - equilibrium radius of bubble, frequency of fluctuations, acoustic pressure - because an all this results in infringement of stability of fluctuations.

Therefore in the given work is put the task to investigate the opportunity of cumulation of energy in gas-filled bubble at the strictly controllable parameters. With this purpose instead of acoustic methods of excitation of fluctuations of bubble is offered the method of the percussive compression by the piston of a working liquid with artificial injection of the gas bubble with the established initial radius (image 1). Injection  of the bubble with the established volume (for example, with the help of a syringe) imply,  what his radius have of the size  approximately R0 = 0,5 ... 10 mm. From a condition, what the bubble should be single and he should  be situated far from walls of a vessel, i.e. the L >> R0, then the characteristic size of a vessel should be L, d ~100 mm. The high percussive pressure are provided with high mechanical durability and massiveness of walls and the piston of installation.


 Image 1. The scheme and the geometrical sizes of installation of percussive compression.


For definition of parameters of the device and impact of blow we shall use of equation of Nolting-Neppiras for an incompressible liquid and adiabatic compression of gas:



at initial conditions:



where R - the current radius of  bubble, m,

P0 - starting pressure in bubble, Pa,

P - static pressure in a liquid, Pa,

Ps - pressure of the saturated vapor of liquid, Pa,

P (t) - pressure, created by the external device (for example, by the percussive device), Pa,

ρ - density of a liquid, kg/m3,

μ - coefficient of dynamic viscosity of a liquid, Pa ? s,

σ - coefficient of a surface tension of a liquid, N/m,

γ - ratio of specific heats.


In this case we have not such  target how calculating of  the temperature and pressure in gas at the end of compression of the bubble, because this equation will of bad for use. Us interests first of all what function Pm(t) for the most effective cumulation of energy should be at  the given equilibrium radius R0. For this purpose it is necessary to define general time of compression bubble up to the minimal radius.

Is known the analytical decision of the equation (1) for  getting of time of collapse for a case with the vacuum bubble in an ideal liquid  (Reyleigh formula):


In the general case the equation (1) not can be solved in quadratures, however his numerical decision does not represent complexity.

Numerical decisions of the equation for various initial radiuses of the gas bubbles in glycerin are given on image 2.  At calculation we accepted, what the pressure, which  creates the external device, will changed under the law P(t)=Pm ? (1-exp(-t/τ)). How follows from calculations, for bubbles, with small equilibrium radius the speed of increase of pressure at percussive interaction can be  insufficient for effective transfer of the energy to the gas in bubble. Energy of the percussive impact in this case is spent on heating of a liquid at fading pulsations of the bubble in a viscous liquid.

Image 2. Numerical decisions of the equation of pulsations of the bubbles in glycerin at percussive compression.


1 - Pressure P(t)=Pm ? (1-exp(-t/τ)) at  Pm=10 MPa, τ = 2 ? 10-5 s

2 - Change of the relative radius at R0=3 mm

3 - Change of the relative radius at R0=0,5 mm


In the bubble, with the big equilibrium radiuses  the pressure has time for increase up to the maximal value up to the moment of achievement by bubble the minimal size, i.e. the big radius of bubble  be conductive for high efficiency of transfer of energy.

If the impulse  of pressure P(t) be  close to rectangular (τ→
0), then the energy of external forces, transferred to gas in bubble in an ideal liquid, can be accepted equal E= Pm ? ΔV, where ΔV - change of volume of bubble. Considering, that the amplitude of fluctuations is great, i.e. R0>> Rmin we can write 



The mass of gas in bubble at the predetermined equilibrium of pressure also is proportional to the initial volume of bubble, therefore the density of energy (or temperatures), reserved in bubble at the accepted conditions does not depend on his initial radius. In other words, the temperature of gas in the adiabatic  case does not depend on initial radius bubble, but depends only on size of the enclosed pressure.

In a real case, at the account of finiteness of speed of a sound in a liquid and its viscosity, it not so. Finiteness of speed of a sound in a liquid is resulted in expansion of front of a percussive impulse of pressure, therefore, how it is visible from image 2, the big compression ratio (and more the heat) is reached inside bubble with the big initial radius. Besides, if is put into effect  the mechanism of heating of the central part of bubble with help of a spherical converging shock wave then the  radius of bubble should be enough big, because  is necessary the time for shaping  of the shock wave. Speed of increase of pressure at the percussive compression depends on speed of a sound in a liquid. Therefore emerges a task of an optimum choice of a working liquid for installation of the percussive compression of the gas bubble. The duration of front of a impulse of pressure, for example in the thick-walled cylinder with the percussive piston (image 1), it is possible to estimate, knowing the characteristic sizes of a vessel with liquid L and speed of a sound in a liquid c: τp L/c.  Properties of a liquid should select from a condition τp< τ, or, taking into account earlier accepted condition L~10R0 and taking into account of the time of compression (3) we shall receive criterion for a choice of a liquid for the percussive compression:




We can notice, that a condition (5) though and  is received from the other considerations, but he coincide to the exponent  with criterion of the incoercibility liquid

  ,                                                       (6)


what with the physical party means the most full transfer of kinetic energy of a liquid to the gas in bubble  (The minimal losses at reflection of the percussive wave from bubble). Comparison of properties of various liquids by these criteria is given in list 1.


List 1

Properties of some liquids and their comparison by criterion of a choice (6)


The liquid

acoustic speed

at 20 ºC, m/s



at 20 ºC,



109(Joule /m3)


In relative units











Sulphuric acid





Ethylene glycol

























Castor oil





Disulfide of carbon




















Carbon tetrachloride





Diesel oil




















Ethylic ether






In the right column are given the data on intensity of sonoluminescence in these liquids with the air bubbles on the data [5]. Draw the attention on that fact, what the intensity of the sonoluminescence is practically unequivocally connected to size of criterion ρ ?c2, that can be consequence of the mentioned above condition of the incompressible fluid.

Thus, how follows from list 1, the mercury is  the optimal choice of a liquid by the accepted criterion. Her use, nevertheless, is undesirable owing to toxicity and a high pressure of the saturated vapors.

The most real candidate for a role of a working liquid in installation of the percussive  compression is the glycerin - a harmless transparent liquid with insignificantly pressure of the saturated vapors. Draw attention, however, what the uppermost rows (except the first) in list 1 occupy liquids with high viscosity. Therefore it is necessary to find out, how the viscosity influences on cumulation of energy in the gas bubble.

On the image 2 are submitted the numerical decisions of the equation (1) for two initial radiuses bubbles at various viscosity of a liquid. How follows from the analysis of decisions, viscosity most strongly influences on the bubble with small initial radius. So at R0 = 3 mm character of movement and time of compression practically do not differ from designed under Reyleigh formula (3) up to viscosity approximately μ =10 Pa ? s (glycerin at 0 º C) while at R0 = 0,5 mm influence of viscosity starts to have an effect already at μ =1 Pa ? s. For water ( μ = 0,001 Pa ? s at 20º C) and in less viscous liquids of the decision in the accepted conditions already do not depend any more on viscosity.

Example of calculation:

Calculation R0 = 5  mm, glycerin, ρ = 1270 kg/m3, c = 1923 m/s

At percussive influence  Pm=10 MPa the time of compression will τ = 55,4 mks.

The time of increase of pressure at blow is  approximately of the time of double pass of a sound wave on length of the cylinder, i.e. τp 2L/c. Having accepted L=10 sm (condition L>>R0) we shall receive τp=2 ? 0,01/1923=10,4 mks. Thus, under these conditions will satisfied of the condition τp< τ and  can be  effective cumulation of energy for enough big bubbles.

Image 3. Numerical decisions of the equation of pulsations of the bubbles with initial radius 0,5 and 3 mm in liquids with various viscosity at percussive compression.



1 - Pressure P(t)=Pm?(1-exp(-t/τ))  at Pm=10 MPa, τ = 2?10-5 s.

2 - R(t) at μ=0,001 Pa?s

3 - R(t) at μ=1 Pa?s

4 - R(t) at μ=10 Pa?s

5 - R(t) at μ=50 Pa?s


The second way of accumulation of energy in bubble is the method of the resonant pulsations. Advantage of a method is that, are not required of a high pressures and he is widely used for excitation of the cavitation by ultrasound. For the bubbles the millimetric sizes is necessary the frequency acoustic fluctuations 100-1000 hertz. It does not allow to use a free acoustic wave since the sizes of a vessel should be not less than length of a wave, i.e. about several meters. The same problem arises and with the sound oscillators.

This task can be solved at piston influence of fluctuations on a working liquid in the closed (waterproof) cylinder.

Let's accept length of a cylindrical part of installation L ~ 1 of meter. Therefore at frequency of fluctuations 100 ... 1000 hertz we can consider the liquid as incompressible. Diameter of cylinder D is chosen from condition D >> Rmax, where Rmax - the maximal radius of the bubble.


Image 4. Resonant pulsations bubble in glycerin. Numerical calculation from the equation (1). 1 - external pressure, 2 - change of radius of bubble.


Parameters and initial conditions are given in the list:




























Let's accept Rmax= 10 R0 = 1 mm. In this case it is possible to accept D ~ 0,1 m.

The system a liquid - bubble represents nonlinear oscillatory system, and her the resonant frequency depends on amplitude of fluctuations. Therefore for excitation of fluctuations in a liquid the usual generators and acoustic oscillators with the sinusoidal vibrations is of little use. On image 4 is shown the numerical decision of the equation (1) at excitation of fluctuations by a sine wave signal. On the diagram is well visible, what with increase of the amplitude of fluctuations of bubble of his the period of fluctuations is increased, how result is the discontinuance of the further energy pumping into the bubble. Therefore frequency of  the forced oscillations should change (be decreased) in process of increase of amplitude of radial fluctuations of bubble.

On image 5 is given the circuit of experimental installation with positive feedback , the frequency of impulses of pressure in which is regulated by a hydrophone,  which located in the working liquid. In this case already  itself  the system a liquid - bubble  determine of the frequency of oscillator, what automatically results to necessary frequency adjustment of fluctuations for achievement of the maximal amplitude (and energy of bubble).


Image 5. The resonant method of excitation of fluctuations of the bubble.


Draw attention, what the influence of viscosity in a resonant method will be much stronger, than at percussive compression of the bubble. Therefore it is desirable to inform enough the big a quantity of energy to the bubble during only a small number of the periods of fluctuations, then we need in enough the big amplitude of fluctuation of pressure (0,1..1 MPa). Therefore as the electrodynamic converter of fluctuations we used specially  worked out device providing a course of the piston of 5 mm at his diameter of 15 mm, and be able to creating of amplitude of pressure in a liquid up to 1 MPa. Electric capacity of the converter be 1000 watt, therefore his winding was cooled by flowing water.

Gas bubble of the given volume is entered with the help of a syringe through the device of injection in bottom of the cylinder (in the circuit on image 5, he not shown). Use of liquids with high viscosity allows to slow down process of  surfacing of  bubble and  his  speed is calculated by formula of Stokes


where ρ0 and ρ' - density of a liquid and gas accordingly;

μ - dynamic viscosity of liquid (list 2);

R0 - radius of bubble.

List 2

Dependence of viscosity of glycerin and water with temperature



Dynamic viscosity of glycerin, mPa?s

Dynamic viscosity of  water, mPa?s



1, 792



1, 519



1, 308



1, 140



1, 005



0, 8007



0, 2838





From the other physical properties of glycerin, which do influencing on dynamics of the gas bubble, we shall note factor of a superficial tension σ = 63 mN/m (at 20o C) and on very low pressure of the saturated vapors Ps = 0,133 kPa (at 125 oC). (For mercury Ps = 171 kPa at 20o C).

We can enough simply calculate, what at temperature +5º C and the initial radius of the bubble R0 = 3 speed of surfacing of the bubble in glycerin will about 5,3 mm/s, so at height of the cylinder in  1 meter we can do supervision of the bubble within several minutes. The viscosity we can very simply regulate in a wide range, by changing temperature of a liquid.

In the  following publication will be given  the data of experiments with the described installations.


The literature:

1.     Taleyarkhan R. P., West C.D., Lohey R.T., Nigmatulin R.I., Block R.C. Evidence for nuclear emissions during acoustic cavitation, Science 295, pp.1868-1873 (2002).

2.     Nigmatulin R.I., Akhatov I.Sh., Topolnikov A.S. et al. -  Theory of supercompression of vapor bubbles and nanoscale thermonuclear fusion //Physics of Fluids. - 17, 107106 (2005)

3.     Taleyarkhan R. P., West C.D., Lohey R.T., Nigmatulin R.I., Block R.C., Y.Xu. Nuclear Emissions During Self-Nucleated Acoustic Cavitation. Pys. Rev. Lett. 96, 034301 (2006)

4.     Flannigan D. J., Suslick K.S.. Molecular and atomic emission during single-bubble cavitation in concentrated sulfuric acid //Acoustics Research Letters Online, 2005 - Volume 6, Issue 3, pp. 157-161

5.      Margulis M.A. Sound-chemical of reaction and sonoluminescence, Moscow: Chemistry, 1986.-288 pages.