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:
(1)
at initial conditions:
(2)
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):
(3)
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
(4)
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:
(5)
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
|
Density
at 20 ºC,
kg/m3
|
ρc2,
109(Joule
/m3) |
WSL,
In relative units
|
|
Mercury |
1453 |
13600 |
28,71 |
- |
|
Glycerin |
1923 |
1270 |
4,70 |
28 |
|
Sulphuric
acid |
1440 |
1830 |
3,79 |
- |
|
Ethylene
glycol |
1658 |
1115 |
3,07 |
22 |
|
Aniline |
1656 |
1023 |
2,81 |
- |
|
Nitrobenzene
|
1460 |
1200 |
2,56 |
- |
|
Sea-water |
1531 |
1030 |
2,41 |
- |
|
Water
|
1484 |
1000 |
2,20 |
6 |
|
Castor
oil |
1477 |
850 |
1,85 |
- |
|
Disulfide of carbon
|
1149 |
1290 |
1,70 |
- |
|
Benzol
|
1324 |
900 |
1,58 |
2 |
|
Toluol
|
1328 |
866 |
1,53 |
3 |
|
Kerosene |
1324 |
800 |
1,40 |
- |
|
Carbon
tetrachloride |
920 |
1595 |
1,35 |
- |
|
Diesel
oil |
1250 |
850 |
1,33 |
- |
|
Ethanol |
1207 |
790 |
1,15 |
2 |
|
Acetone |
1174 |
810 |
1,12 |
1 |
|
Methanol |
1103 |
792 |
0,96 |
- |
|
Ethylic
ether |
985 |
714 |
0,69 |
-
|
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 = 210-5 s.
2 - R(t) at μ=0,001 Pas
3 - R(t) at μ=1 Pas
4 - R(t) at μ=10 Pas
5 - R(t) at μ=50 Pas
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:
|
R0,
m |
ρ,
kg/m3 |
P∞,
Pa
|
Pm,
Pa
|
μ,
Pas |
P0,
Pa
|
γ |
F,
hertz
|
Δt,
s |
|
0,002 |
1200 |
100000 |
100000 |
1,0 |
100000 |
1,4
|
1100 |
2,010-7
|
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
(4)
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
|
Temperature,
ºC |
Dynamic viscosity of glycerin,
mPas |
Dynamic viscosity of water,
mPas
|
|
0 |
12100 |
1, 792 |
|
5 |
7050 |
1, 519 |
|
10 |
3950 |
1, 308 |
|
15 |
2350 |
1, 140 |
|
20 |
1480 |
1, 005 |
|
30 |
600 |
0, 8007 |
|
100 |
13 |
0, 2838 |
|
160 |
1 |
-
|
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.
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