Hydrodynamic effects in liquids subjected to pulsed low ... - Exvacuo

Apr 28, 2004 - discharges in liquid nitrogen [5] and in water [6] were studied experimentally to produce ... rotational energy levels by plasma particle collisions.
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INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 37 (2004) 1509–1514

PII: S0022-3727(04)69979-7

Hydrodynamic effects in liquids subjected to pulsed low current arc discharges E Gidalevich1,2 , R L Boxman1,2 and S Goldsmith1,3 1

Electrical Discharge and Plasma Laboratory, Tel Aviv University, Tel Aviv 69978, Israel Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 3 School of Physics and Astronomy, Faculty of Exact Science, Tel Aviv University, Tel Aviv 69978, Israel 2

E-mail: [email protected]

Received 7 October 2003 Published 28 April 2004 Online at stacks.iop.org/JPhysD/37/1509 DOI: 10.1088/0022-3727/37/10/014

Abstract Hydrodynamic phenomena in liquids associated with low current electrical discharges were analysed theoretically and the radial plasma column expansion was considered as an arbitrary discontinuity propagation. For assumed initial discharge channel radii of R0 = 10−6 and 10−5 m, the initial temperatures in the discharge channel was found to be T0 = 1.4 × 106 and 6.5 × 104 K for a discharge current of I = 1 A, and T0 = 6.5 × 106 and 3 × 105 K for I = 10 A. The temperature decreases during the initial ∼10−8 s, due to radial plasma column expansion. During the initial 10−8 s the velocity of the plasma column expansion exceeds the speed of sound in the undisturbed water (by 5–15 times), creating a shock wave in the water with a pressure discontinuity of ≈105 atm, and a relaxation length of 10−4 m.

1. Introduction

2. Description of the model

Electrical discharges in liquids are important for technological applications, e.g. for underwater cutting and welding of metals [1], and for medical and biological applications [2]. Roi and Frolov [3] considered a spark discharge in water as a source of large amplitude positive pressure pulses, using currents in the range of 1370–2700 A and a voltage of 30 kV. Robinson [4] analysed the plasma channel expansion in discharges with currents of tens of kiloamperes. Lagrangian variables were used for a finite-difference simulation. Recently electrical discharges in liquid nitrogen [5] and in water [6] were studied experimentally to produce carbon nanotubes. However, the hydrodynamic phenomena caused by low current arc pulses in liquids have not been analysed theoretically. The objective of this work is to develop a theoretical model of the postbreakdown plasma expansion and, on the basis of this model, to determine the hydrodynamic behaviour of a liquid subjected to a low current (1–10 A) pulsed electrical discharge, and specifically to obtain the typical pressure discontinuity and the typical temporal and spatial scales of shock fronts produced by the discharge.

In laboratory devices, low current discharges are initiated by breakdown in a liquid between electrodes separated from each other by a short gap. The situation is complicated by both the details of the breakdown process and the interaction with the electrodes. This theoretical analysis will be simplified by neglecting the details of the breakdown process and the electrodes, by assuming that at time t = 0 an infinitely long cylindrical plasma column with known conditions exists, through which the discharge current will flow. For t > 0, the plasma column will expand radially due to the high temperature and pressure caused by Joule heating from the discharge current. The inertia of the surrounding liquid opposes the plasma expansion. It is assumed that the predominant energy loss mechanism is the mechanical work applied against the resistance of the surrounding liquid. The plasma column expansion is determined by conservation of momentum, mass and energy [7]:

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∂ v 1 ∂p ∂v +v + = 0, ∂t ∂r ρ ∂r

(1) 1509

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∂ρ χ(T )T ∂ρ = 0, +v + ρ(∇ · v ) − 2 2 ∂t ∂r R ηliquid

(2)

      p v2 v2 ∂ ρ ε+ + ∇ · ρv ε + + = λ(T )E 2 , (3) ∂t 2 ρ 2 where r is the radial distance, t is time, v , ρ and p are the radial velocity, density and pressure of the plasma, respectively, λ(T ) is the electrical conductivity of the plasma, E is the axial electric field strength in the plasma, ε is the internal energy of the gas (plasma), ε = (1/(γ − 1))(kT /m), where k is Boltzmann’s constant, m is the particle mass, T is the plasma temperature and γ is the heat capacity ratio. We take into account that the plasma column radius increases not only due to hot plasma pressure, but also due to evaporation of the cylindrical liquid wall surrounding the plasma (see figure 1). The last term in equation (2) describes additional vapour penetrating the plasma column due to liquid evaporation (detailed in section 3.2). Hereafter χ(T ) is the thermal conductivity of the plasma, R is the time-varying plasma column radius, ηliquid , is the heat of evaporation of the liquid: ηliquid = 22.6 × 105 J kg−1 for water. The calculation of the exact value of γ for a partially ionized vapour plasma is very difficult and was investigated by Granovskii [8]. Hereafter we assume γ = 4/3, appropriate for the case where the dominant energy storage in the media is excitation of translational and rotational energy levels by plasma particle collisions. After a transformation of equation (3) and using equations (1) and (2), we obtain [7, 9]:     λ(T )E 2 ∂ kT kT ∂ kT +v + (γ − 1) (∇ · v ) − ∂t m ∂r m m ρ   χ(T )T kT γ − 1 2 = 0. (4) + + v m 2 R 2 ρηliquid In addition, ∂R (5) = vf , ∂t where vf is the plasma column front velocity. An expression for vf will be developed in the next section. Directly solving equations (1), (2) and (4) presents serious difficulties. First, the plasma velocity, which is zero at r = 0 (due to cylindrical symmetry), increases with r, and will exceed the local speed of sound at some value of r. Finding this transition point is very difficult [10] and must be repeated

Plasma/water density

Plasma

Water Shock wave Undisturbed water

Expanding plasma column

∂ρ vf χ(T )T = 0, + 2ρ − 2 2 ∂t R R ηliquid ∂ ∂t

 kT kT vf λ(T )E 2 + 2(γ − 1) − m m R ρ   χ(T )T kT γ − 1 2 = 0. + vf + m 4 R 2 ρηliquid

Figure 1. General scheme of the shock wave formation near the plasma bore.

(7)



(8)

R In equation (8), the average value v 2  = ( 0 [v(r)]2 r dr)/ R ( 0 r dr) = vf2 /2 was used instead of v 2 in the last term. Equations (5), (7) and (8) form a closed system of equations for the unknowns R, ρ and T . A concrete expression for vf will be developed in the next section.

3. Plasma column radial expansion The radial plasma column expansion velocity vf consists of two components: the thermal expansion velocity of the plasma vfT and the velocity of the plasma front relative to the liquid, i.e. the phase expansion velocity vph , determined by the rate of liquid evaporation: vf = vfT + vph .

(9)

Each term will be considered in turn in the following sections. 3.1. Simplified description of arbitrary discontinuous propagation If we consider the discharge channel as a uniform plasma column, then the outward plasma boundary with the surrounding liquid is an arbitrary discontinuity [7, 9], illustrated in figure 2. Let us assume that in one-dimensional space a compressed gas (designated 1) is located at the left side (x < 0) of a partition placed at x = 0. Another material, fluid 2 (which may be a vacuum, a gas or a liquid), is located in the semi-infinite space x > 0 (figure 2(a)). At time t = 0, the partition is removed, and thereafter a rarefaction wave propagates to the left in the region x < 0 with velocity v = c0 , where c0 is the speed of sound in the undisturbed gas 1. To the right of the wave front, the gas velocity at an arbitrary point is v=

Radial distance

1510

at every time. Instead, we will assume that the plasma temperature, density and pressure are distributed uniformly at all times. Since the pressure is assumed to be uniform, the radial plasma velocity cannot be obtained from equation (1). In a homogeneous expansion, (∇ · v ) must be a function of time only. Therefore r v = vf , (6) R where vf and R are functions of time. Neglecting the spatial derivatives of ρ, p, T , and taking into account that (∇ · v ) = 2(vf /R), we obtain from equations (2) and (4)

2 (c0 − c), γ −1

(10)

which is the Riemann solution of the hydrodynamic equations [9]. Here c is the speed of sound at the arbitrary point. If fluid 2 is vacuum, c = 0 to the right of the front formed by

Post-breakdown hydrodynamic effects in liquids

density jump by the equation

(a) t=0

p2 − p2∞ = B

Gas 2

M=

Boundary of separation

x=0

(b) t>0

Shock wave

c

Figure 2. An arbitrary discontinuity propagation: (a) at the time t  0, (b) at the arbitrary time t > 0.

the escaping gas, and the front velocity is v = (2/(γ − 1))c0 . If the expansion of gas 1 is impeded by fluid 2, p1 = p2 on the boundary separating the two materials [9] (see figure 2(b)), where p1,2 are the pressures on either side of the boundary. With adiabatic gas expansion, the speed of sound in gas 1 on the boundary is  cboundary = c0

p2 p0

(γ −1)/2γ .

(11)

Substituting this expression into equation (10), we obtain the propagation velocity of the boundary:   (γ −1)/2γ  2c0 p2 vboundary = 1− . (12) γ −1 p0 Pressure p2 also depends on the boundary propagation velocity. It is equal to undisturbed pressure in fluid 2 if the boundary propagation velocity is subsonic relative to fluid 2. But if the boundary propagation is supersonic relative to fluid 2, the pressure p2 is not equal to the undisturbed pressure, but must be calculated from the specific physical conditions. In our case fluid 2 is a liquid, specifically water, in the numerical calculations which will follow. If the boundary velocity is supersonic relative to the undisturbed liquid 2, a shock wave will propagate through the liquid on the right side of the boundary of separation (see figure 2). The basic properties of shock waves in water have been described previously [11]. The density is discontinuous across the shock front, and the ratio between the value to the left of the shock front and the value in the undisturbed fluid, ρ2 /ρ2∞ , is given by 8

 −



7

−1 ,

(14)

ρ2 ρ2∞

 (1 + 7M 2 ) + 7M 2 = 0,

v2 1 , c2∞ 1 − ρ2∞ /ρ2

(15)

which is a direct corollary of mass flux conservation on the shock front [9], we have a system of four algebraic equations (12)–(15) with four unknowns, v2 , p2 , ρ2 and M, where p0 and ρ2∞ are given. All of the unknowns depend on c0 = (γ kT /m)1/2 . As all of the physical parameters in the plasma column are assumed to be uniform and depend only on time, by equating the instantaneous value of the plasma pressure to the pressure in the shock wave (14) and using expressions (13) and (15), we obtain:

v

Undisturbed Gas 1

ρ2 ρ2∞

ρ2 ρ2∞

where B = 3214 atm. For hypersonic velocities, p2∞ may be neglected. At last, using

Gas 1





(13)

where M is the Mach number (here, the ratio of the shock front velocity to the speed of sound in the undisturbed water). The pressure jump across the shock front in water is related to the

vboundary = c2∞ 1/2  1 p1 − p2∞ (1 + (p2 − p2∞ )/B)1/7 − 1 × , 7 B (1 + (p2 − p2∞ )/B)1/7

(16)

where c2∞ = 1.5 × 103 m s−1 is the speed of sound in the undisturbed water. 3.2. Phase front propagation velocity In addition to the thermal expansion of the plasma, the liquid at the plasma–liquid interface is subject to a heat flux which evaporates liquid from the boundary, and thus the plasma– liquid phase boundary propagates to the right with a velocity vph which must be superimposed on the thermal expansion velocity calculated in the preceding section. The vapour molecules thus produced eventually are disassociated and ionized, and mix with the plasma column. The phase velocity vph is estimated by equating the heat flux incident on the liquid by thermal conduction from the plasma to the heat flux removed from the liquid by evaporation, and thus χ(T ) ∂T vph = − . (17) ρliquid ηliquid ∂r As a plasma column model is assumed to be radially uniform, we change −∂T /∂r ≈ T /R; thus vph =

χ(T )T , ρliquid ηliquid R

(18)

while the contribution to the average plasma density change is   χ(T )T ∂ρ =2 2 , (19) ∂t ph R ηliquid which was already used in section 2. Although ρliquid is the density of the liquid compressed by the shock wave, the compression may be neglected. Thus ρliquid = ρH2 O = 103 kg m−3 will be used in the following numerical calculations. Substituting expressions (18) and (16) into equation (5), we close the system of equations (5), (7) and (8) with the dependent variables R, ρ and T in the plasma, the independent variable t, and where E will be treated as a given parameter. 1511

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4. Calculations

Table 1. Input parameters used in calculations.

Since the breakdown stage is not considered in the present work, we start calculations from the assumed initial value of the plasma column radius R0 . The initial temperature T0 was then calculated so that electric current I , would be equal to a pre-selected value (1 or 10 A), according to I = πR02 λ(T0 )E,

(20)

I (A) 1 V = 10 V

R0 R0 V = 100 V R0 R0

where λ(T ) is the electrical conductivity of the partially ionized plasma, given by (21)

where e and me are the electron charge and mass, respectively, x = ni /n is the ionization degree and σei is the cross-section for electron–ion collisions [12]:  2 2 1 e ln , (22) σei = 4π ε0 kT where ε0 is the vacuum permittivity, ln ≈ 10 is the Coulomb logarithm, σea is the cross-section for electron–neutral-particle collisions. σea was calculated numerically as a function of the temperature [13], taking into account the Ramsauer effect [14, 15]. The ionization degree was calculated from Saha’s equation,   x2 gi 2πme kT 3/2 1 −(χ0 /kT ) , (23) e =2 h2 n 1−x gn where gi and gn are the statistical weights of the ionized and neutral states, h is Planck’s constant, χ0 is the energy of ionization and n is the total density of heavy particles. A specific expression for the plasma thermal conductivity (21) was presented and discussed previously [13]. The electric field E is assumed to be homogeneous along the plasma column and determined approximately as E=

Vs arc , L arc + s

(24)

where Vs is the power supply voltage, L is a channel length (distance between the electrodes), arc is the arc resistance and s is the internal resistance of the power source. The internal resistance was assumed to be s = 10 . Substituting the well known formula  = (1/λ(T ))(L/πR 2 ) into expression (24), we have Vs . (25) E= L + λ(T )πR 2 s Equation (7) requires an initial condition for the plasma density, whose determination from first principles is problematic. Given the extreme values of heat flux from the plasma to the liquid, resulting in extreme values of evaporation rates, for the purposes of this preliminary analysis it will be assumed that the initial plasma density is equal to the liquid density. The input parameters of calculations are summarized in table 1. The system of equations (5), (7) and (8), together with expression (16), was solved numerically using the Mathsoft package ‘Mathcad 2001 Professional’. 1512

Temperature, 104 K

e2 x , me kT xσei + (1 − x)σea

= 10 m = 10−5 m = 10−6 m = 10−5 m

T0 T0 T0 T0

10 = 1.4 × 10 K = 6.6 × 104 K = 2.9 × 105 K = 1.5 × 104 K 6

T0 T0 T0 T0

= 6.45 × 106 K = 2.85 × 105 K = 1.38 × 106 K = 6.3 × 104 K

40 30 20

R0 = 10-6

10

R0 = 10-5 m

0 0

0.005

0.01

Time, µs

Figure 3. Temperature versus time graph, for I = 1 A, V = 10 V. 30

Temperature, 104 K

λ(T ) = √

−6

20

R0 = 10-5 m

10 0 0

0.005

0.01

Time, µs

Figure 4. Temperature versus time graph, for I = 10 A, V = 10 V.

5. Results Calculations were done using the input parameters presented in table 1. For electrical currents of I = 1 and 10 A and for two assumed, a priori values for the power supply voltage V = 10 and 100 V with an inter-electrode gap of L = 10 µm, the initial temperatures were calculated from expressions (20)–(23) and (25). The dependence of the plasma temperature on time for I = 1 A and V = 10 V are shown in figure 3. In the R0 = 10−6 m case, the temperature decreases from ≈1.4 × 106 K down to 1.2 × 105 K during the first 10 ns and then continues to decrease, although less rapidly. With R0 = 10−5 m, the plasma temperature decreases from 5×105 K down to ≈4×105 K in the same time. For I = 10 A, we also see considerable decrease of the temperature with time (figure 4). Figure 5 shows that the radial expansion velocity of the plasma column decreases from more than 2 × 104 m s−1 down to 1.5 × 103 m s−1 at 8 ns for R0 = 10−6 m, while for R0 = 10−5 m the expansion velocity decreases from ≈1 × 104 m s−1 down to ≈2 × 103 m s−1 . These velocities are supersonic and indicate that the plasma column radius increases ≈18 times (figure 6) for V = 10 V, I = 1 A and

Post-breakdown hydrodynamic effects in liquids

The usual hydrodynamic criterion must be satisfied:

Velocity, m/s

20000

λj k R(t)

R0 = 10-6 m

R0 = 10-5 m 0 0.002

0.004 0.006 Time, µs

Normalized radius, R/R0

20

R0 = 10-6 m

R0 = 10-5 m

0 0

0.005

0.01

Time, µs

Figure 6. Plasma column radius versus time graph, for I = 1 A, V = 10 V.

Pressure, 105 atm.

10

5

R0 = 10-6 m R = 10-5 m

0 0

0.002

λ= 

0.004

0.006

Time, µs

Figure 7. Pressure jump in the shock wave for I = 1 A, V = 10 V.

initial radius R0 = 10−6 m. But for R0 = 10−5 m, the plasma radius increases only ≈3 times, thus reaching a radius of 3 × 10−5 m, compared to 1.8 × 10−5 m in the R0 = 10−6 m case. From figure 7 we can see that the pressure in a shock wave in water with R0 = 10−6 m reaches 5 × 105 atm during the first 8 ns and after that considerably decreases, while for R0 = 10−5 m the maximum pressure is about 3 × 105 atm. The thickness of the shock wave affected region in the liquid may be estimated using (15) and (16) and the results are seen in figure 5 to be ∼10−4 m at 5 ns for R0 = 10−6 m.

6. Discussion The validity of the hydrodynamic description of the early stages of the post-breakdown discharge in water must be examined because of very short distances and times involved.

(27)

where ni is the density of particles of kind i, and σik is the crosssection for i–k particle collisions. Assuming for simplicity that the collisions can be treated by a hard particle model in a partially ionized plasma, we have, λ=

10 5

1 , n i i σik

0.008

Figure 5. Plasma boundary propagation velocity versus time graph, for I = 1 A, V = 10 V.

15

(26)

where λj k is the mean free path for particles of type j moving in a background gas of particles of type k, which may be ions, electrons or neutrals; τj k is the mean time between collisions of these particles, R is the plasma column radius and t is the time. To examine these inequalities we take into account that

10000

0

τj k t,

1 , n[xσii + (1 − x)σia ]

(28)

where x, σii , σia are described in section 4. Let us turn to the initial conditions for the case with I = 1 A, V = 10 V and R0 = 10−6 m. Using expressions (22) and (23) and values of σia ≈ 10−19 m2 [16], we have from equation (26) λ = 3 × 10−8 m, that is, ≈300 times less than the initial plasma radius, and thus the first of conditions (24) is satisfied. During radial plasma column expansion, the plasma density as well as temperature decreases. According to our calculation, despite the density decrease, the free mean path also decreases, because of the increasing ion–ion collision cross-section (22), ion–ion collisions being more frequent than ion–neutral collisions. This also occurs in all the other cases considered here. Examination of the mean time between two consequent collisions, τ = λ/(kT /m)1/2 , shows that it is about 10−13 s for the case with I = 1 A, V = 10 V and R0 = 10−6 m. That is much less than the time of interest, and hence the second condition (26) is also satisfied. Likewise, for all the cases considered, the calculated results show that criterion (26) is satisfied for all times considered, and thus the hydrodynamic model is appropriate. The role of the ionizing thermal radiation as well as the radiative heat transport from the plasma to water was not taken into account—doing so would be very complicated. The spectral composition of the radiation from a thin plasma layer is very different from Planck’s distribution [13] and, up to now, cannot be estimated accurately under the conditions considered in this work. Equations (1) and (3) take into account plasma pressure p, but neglect magnetic pressure and the pressure associated with the surface tension of the curved liquid surface. Let us compare these two pressures with the calculation for the plasma pressure presented in figure 7. Both these additional pressures reach a maximum when the channel has the minimum radius. Therefore we estimate their values at moment t = 0, e.g. for an initial channel radius of R0 = 10−6 m. The magnetic pressure is pB = B 2 /(2µ0 ), where the magnetic field from a linear current element is B = (µ0 I )/(2π R0 ), where µ0 is the permeability of vacuum. For I = 10 A, we have pB ≈ 106 Pa p (see figure 7). We next consider the surface tension of the curved water surface. For a cylindrical channel, 1513

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pα = α/R0 , where α is a surface tension coefficient. For the water at room temperature α = 0.073 J m−2 . The pressure is thus pα = 7.3 × 104 Pa p. Thus both pB and pα are much less than the calculated plasma pressure, and thus neglecting them is justified here. The consequences of the obtained results are high initial temperatures and thus a very large pressure jump in the shock wave in the water. This may be explained by the assumption that all of the electrical current at the earliest stage flows into a narrow plasma channel with radius R0 . In practice, the plasma starts to expand in the breakdown stage, and the electrical current grows relatively smooth. A model which takes this process into account might result in significantly lower plasma temperatures and pressures compared to the cases considered earlier. A plasma expansion model which includes breakdown will be formulated in the future.

7. Conclusions Post-breakdown hydrodynamic effects in liquids are considered theoretically as a discharge plasma column expansion into the liquid vicinity. The theory of arbitrary discontinuity propagation is used to describe a plasma–liquid boundary motion. In the frame of the simplified model of homogeneous plasma column expansion a numerical solution is found for a low current, pulsed electrical discharge initially in a narrow plasma channel within a liquid. The plasma channel rapidly expands, both pushing against the surrounding liquid, and evaporating liquid from the boundary region. For 1–10 A

1514

discharges in water, pressures ∼105 atm and plasma expansion velocities of ∼104 m s−1 are calculated.

References [1] Kandel C 1947 Mater. Methods 25 78 [2] Jagadeesh G and Takayama K 2002 J. Indian Inst. Sci. 82 49 [3] Roi N A and Frolov D P 1958 Sov. Phys.—Dokl. 3 118 [4] Robinson J W 1973 J. Appl. Phys. 44 76 [5] Munindradasa D A I, Chhowalla M, Amaratunga G A J and Silva S R P 1998 J. Non-Cryst. Solids 230 1106 [6] Sano N, Wang H, Chowalla M, Alexandrou I and Amaratunga G A J 2001 Nature 414 506 [7] Zel’dovich Ya B and Raizer Yu P 1966 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (New York: Academic) [8] Granovskii V L 1971 Electrical Current in Gas (Moscow: Nauka) [9] Stanyukovich K P 1960 Unsteady Motion of Continuous Media (London: Pergamon) [10] Gidalevich E, Goldsmith S and Boxman R L 1999 IEEE Trans. Plasma Sci. 27 1045 [11] Ridah S 1988 J Appl. Phys. 64 152 [12] Spitzer L Jr 1962 Physics of Fully Ionized Gases (New York: Interscience) [13] Gidalevich E, Goldsmith S and Boxman R L 2002 Plasma Sources Sci. Technol. 11 513 [14] Ramsauer C 1921 Ann. Phys. 66 546 [15] Townsend J and Baily V 1921 Phil. Mag. 43 593 [16] Shkarofsky I P, Johnston T W and Bachynski M P 1966 The Particle Kinetics of Plasmas (Reading, MA: Addison-Wesley)