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9The following types of anomalous phenomena of radio emission from pulsars are considered: variations in residual deviations, changes in the pulse shape, switching on and off radio emission, period fail-ures(glitches), changes in the dispersion measure and scattering. The connection established between them indicates a common mechanism of their generation in the pulsar magnetosphere and the propagation of disturbances in the environment. The considered phenomena do not require the involvement of a starquake model.
The results of long-term sounding of the Crab Nebula at the PRAO ASC FIAN by monitoring the radio emission of giant pulses from the pulsar in the Crab Nebula are presented. A technique for probing active processes in the Crab Nebula by observing giant pulses and measuring their scattering is proposed and tested.
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DYNAMICS OF PLASMA PISTON IN PIPE FILLED BY A GAS-LIQUID MEDIUM
Fedun V.
Associate Professor of the Department of Physics Pryazovsky State Technical University, Ukraine
ABSTRACT
The work simulates the operation of a plasma generator, which creates elastic waves in a pipe filled with a two-phase fluid. Modeling is based on wave models of non-stationary gas dynamics. The nonlinear nature of the properties of such systems was taken into account in the simulation. A model of the expansion of a plasma formation in a waveguide is proposed, which made it possible to study the process of excitation of elastic pressure pulses. The time dependences of the rate of expansion of the cavity, sound pressure, and energy of acoustic radiation were obtained and analyzed at various values of the gas content and static pressure of the fluid.
Keywords: plasma formation, cavity, plasma piston, homogeneous model of a two-phase medium, gas content.
Introduction. A lot of technological processes in cryogenic devices, in the metallurgical, oil-producing and oil-refining industries are accompanied by the formation of vapor-liquid systems or proceed occur in gasliquid media. The intensification of such processes can be carried out using elastic waves [1,2]. One of the effective ways to create acoustic fields in a liquid is based on the pulsations of a vapor-gas cavity, which is formed by an electric discharge in a liquid [3]. A numerical experiment was carried out in [4] on the generation of elastic pulses by powerful plasma bunches in an acoustic waveguide filled with a single-phase liquid. In this
case, the discharge forms a gas-plasma cavity - a plasma piston, which causes translational motion of the gas-liquid interface. Therefore, there is no added mass, and the transformation efficiency of the discharge energy into the energy of elastic vibrations increases. Since liquids in technological processes contain gas bubbles, the previously obtained results [4] require clarification.
The aim of this work is to simulate the generation of elastic waves by a plasma piston - a vapor-gas cavity,
which is forms by an electric discharge in a hydrody-namic waveguide. In our case, the waveguide is a vertical pipe that is filled with a gas-liquid medium.
The modeling is based on the simplest wave models of unsteady gas dynamics [5], which take into account (at least in the first approximation) the features of two-phase systems. Water, which contains methane bubbles, was taken as a two-phase medium. The hydrate formation processes and other phase transitions are not taken into account in this consideration, and the acoustic properties of the fluid are considered within the framework of a homogeneous model [5].
The homogeneous model is a rather visual and convenient way of continuous representation of a two-phase medium, in particular, a gas-liquid mixture in a bubble mode. A mixture of liquid and gas is considered here as a kind of homogeneous medium with average values of temperature T, density p and pressure p. The basic assumption of this model is the rapid exchange of momentum, heat and mass between phases. Therefore, the temperatures and speeds of the phases are equal. It is considered that the pressure in the liquid phase is either equal to the pressure in the gas phase p1, or differs by the value of the Laplace pressure Ap=2c/R, where R is the radius of the bubbles, c is the coefficient of surface tension. (Hereinafter, the index 1 means the liquid phase, 2 means gas or vapor).
The composition of such a mixture is usually characterized by mass X or volumetric y gas content. By definition, the mass gas content X is the ratio of the mass of the gas component to the mass of the entire mixture. The volumetric gas content y is the ratio of the volume of the gas component V to the volume of the entire mixture. These values are related by the ratios:
x = -P2V--(1)
P2<P + PI(1-<P)
It should be noted that, in contrast to the volumetric mass gas content, in the absence of phase transitions, it does not change.
The average density of the mixture p is related to the mass gas content X by the ratio:
1 _ 1-X ^ x P Pl P2
dW =
The speed of sound in a gas-liquid mixture without phase transitions is defined c as the speed of propagation of small perturbations of weak compression waves c2 = ( dp/d p) s.
We will differentiate expression (2) with respect to the variable and take into account that d p/ d p = c-2; dp1/dp = c-2; dp2/dp = c-2. We get:
1 (1-X)p2 1 Xp2 1
7 2 2+22 (3)
c2 P2 c2 p2 c2
In the case when X << 1, a mixture of bubbles with liquid is considered as a kind of gas, the density of which is similar to the density of the liquid, and the compressibility of this medium is determined by the compressibility of the gas or vapor phase.
As in [4], we will use the model of the expansion of the plasma piston, which is formed at the upper end of the cylindrical vertical waveguide. Note that this model does not take into account the processes of dissipation of the energy of the plasma formation. Thus, it is assumed that the energy that is released in the gas cavity according to a certain law 8E(t) is converted into the internal energy of the plasma piston dW(t), and is also expended on working on the mixture SA(t):
SE = dW + SA. (4)
As shown in [3], the internal energy of the plasma formation can be set equal to W = pnVn/(y — 1)^ where pn and Vn are the pressure and volume of the plasma formation, y is the effective adiabatic exponent of the ionized gas.
The current value of the volume Vn can be expressed in terms of the thickness of the cylindrical plasma piston x and dB - the diameter of the waveguide: Vn = nxd2/4. (5)
When the plasma formation expands, the pressure at the fluid boundary pn, on the one hand, is the sum of the static and sound pressures (pn =p0 +pcv), and on the other, is equal to the pressure in the cavity pn, i.e.
(2)
PndVn+Vndpn Y-1
Pn=Po+ pcv = Po + pc
dx dt
(6)
Then the change in internal energy and the elementary work of the piston can be found by the formulas:
4(7-1)
(p0 + pcv)dx + pcxd-xdt
dt2
(7)
SA = PnVn=^(po+pcv)
(8)
For the critical regime of a pulsed discharge, the law of energy input is described quite well by the expressions:
SE(t)=^(l — 0(t — Tp))sin(^)dt (9)
x p c
where E0 - the energy introduced into the cavity over time Tp, 0 (t) - the Heaviside function.
Substituting expressions (7-9) into equation (4), we obtain the equation of motion of the cavity boundary
.-dpcdx dt2 1 dp dt
d2x , / , dx\ dx
+ Y(po+pcTtht
(10)
x(0) = xo,v(0)=dx\ =0,p(0) = po. (11)
the initial conditions for which are:
d x
dt\t=o
In this consideration, the density of the fluid and the speed of sound in the mixture depend on the current value of the pressure in the fluid. In this case, the gas density is first determined from the equation of state,
and then the fluid density and the sound speed are determined from (2) and (3). Below we consider the case when water contains bubbles of methane, the state of which is described by the Peng-Robinson equation [6]:
p2
v^-b vli(vli+b)+b(vli-by
(12)
where
2
a = acf(T), f(T) = [1 + ip(l
JUT^)]
RT
a
$ = 0.37464 + 1.5422M
2rr2/Vcv
0.26992M2
ac = 0.457235R2Tcr
R = 8.314 J/(mol K), b = 0.077796RTcr/pcr.
The critical values of temperature Tcr, pressure pcr, and molar volume V^, as well as the acentric factor m of methane, are respectively 190.56 K, 4.5992 MPa, 99.0 cm3/mol, 0.01142 [7].
Below are the results of a numerical experiment carried out with the following parameters: Eo = 30 kJ,
Tp = 0.6 ms, dn =25 mm, T = 300 K and x0= 2 mm. The static pressure of the fluid p0 could take values of 2, 10, 20 and 40 atm, and the mass gas content X could be 0,
0.01, 0.1 and 0,4%.
The values of the volumetric gas content y in the medium for the indicated parameters are given in table
1.
Table 1.
The volumetric gas content 9
X the static pressure
2 10 20 40
0.01 7.20 1.50 0.74 0.36
0.1 43.72 13.25 6.96 3.47
0.4 75.71 38.00 23.10 12.61
The dependence of the specific acoustic resistance pc of methane and its derivative dpc/dt on pressure are shown in Fig. 1.
Fig. 1. The dependence of the specific acoustic resistance pc of methane (a) and its derivative dpc/dp (b)
on pressure
The time dependences of the sound pressure are shown in Fig. 2 for the values X = 0 (Fig. 2a) and X = 0,4 % (Fig. 2b).
Fig. 2. Time dependences of sound pressure in the cavity at static pressures of2, 10, 20, 40 atm and mass gas
content of 0% (a) and 0,4% (b)
As can be seen from this figure, the static pressure of the fluid has little effect on the amplitude of the elastic pulse at a constant mass gas content X. In this case, the amplitude decreases slightly with increasing X. This is shown in Fig. 3.
As seen from Fig. 2 and 3, the shape and amplitude of the pressure pulse are practically independent of both the static pressure p0 and the mass content of the gas (up to X = 0,4 %).
The properties of the mixture and external pressure affect the expansion rate of the plasma piston. The results of calculating the velocity of the plasma - two-phase mixture interface, which are illustrated in Fig. 4 and 5 provide insight into this influence. With a relatively low static pressure p0 = 2 atm, as can be seen from Fig. 4a, at the initial stage of cavity expansion, differences in the values of the boundary velocities from the mass gas content appear.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 mS 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 mS
Fig. 3. Time dependences of sound pressure in the cavity at mass content ofgas X = 0, 0.01, 0.1, 0,4% and static
pressures a) p0 = 2 atm and b) p0 = 40 atm
An increase in pressure to 40 atm (see Fig.4b) eliminates these differences. However, the difference in speed values is noticeable even at this pressure. This
difference can be explained by the fact that the specific acoustic resistance of the medium strongly depends on the pressure in the two-phase fluid (see Fig. 1).
Fig. 4. Time dependences of the boundary velocity at mass gas contents X = 0, 0.01, 0.1, 0.4% and static pressures a) p0 = 2 atm and b) p0 = 40 atm
Figure 5 shows the time dependences of the piston-medium interface velocity for various values of the static pressure p0. Analysis of Fig. 5b shows that after the plasma pressure significantly exceeds the static pressure of the liquid, the interface velocity ceases to depend on the preliminary compression of the mixture.
In the case of the mass content of gas X = 0.01%, the time dependence of the interface velocity is practically the same for the entire range of static pressures under consideration.
Fig. 5. Time dependences of the boundary velocity at static pressures p0 = 2, 10, 20, 40 atm and mass gas contents a) 0.4% andb) 0.01%
The energy flux density I of acoustic radiation was As calculations show, this value practically does
determined in a numerical experiment by the formula: not depend on either the static pressure p0 (see Fig. 6a)
I(t) = pcv2. (13) or the mass gas content (see Fig. 6b).
Fig. 6. Time dependences of acoustic radiation energy: a) mass content ofgas X = 0,4% at static pressures p0 = 2, 10, 20 and 40 atm; b) static pressure p0 = 2 atm at mass gas contents X = 0, 0.01, 0.1 and 0,4%.
Conclusions. A model of the dynamics of a plasma piston in a cylindrical waveguide filled with a two-phase fluid is proposed, which makes it possible to study the process of excitation of elastic pressure pulses. The properties of the fluid are found according to the definitions of a homogeneous model of a two-phase medium - a gas-liquid mixture in a bubble mode. The state of the gas (methane) was described by the Peng-Robinson equation. During a numerical experiment the following results were obtained.
1. The time dependences of the sound pressure were obtained for different values of the gas content and static pressure in the liquid, which indicate that these values have little effect on the shape and amplitude of the elastic pressure pulse.
2. An increase in the gas content in the mixture promotes an increase in the interface velocity at the initial stage of the compression pulse formation, and an increase in the static pressure eliminates this difference.
3. It is shown that after the plasma pressure significantly exceeds the static pressure of the fluid, the interface velocity ceases to depend on the preliminary compression of the mixture.
4. The energy of acoustic radiation weakly depends on the value of the static pressure and mass gas content, and it is determined only by the energy that is introduced into the cavity.
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