st
21 International Symposium on Plasma Chemistry (ISPC 21)
Sunday 4 August – Friday 9 August 2013
Cairns Convention Centre, Queensland, Australia
Plasma Bubbles in a Water Jet Excited by Microwave Radiation
E. Gidalevich, R. L. Boxman.
Electrical Discharge and Plasma Laboratory, Tel Aviv University
Tel Aviv 69978, Israel
Abstract: Plasma excitation is considered in flowing water. The basic difference between this and
the static cases is that the heat transfer will primarily be by forced convection. Calculations were
performed using a one-dimensional approximation. For fields E0 > 5.5 kV/m, the plasma is near
thermodynamic equilibrium. The plasma is displaced downstream relative to the central field point.
Keywords: Microwave plasma, Submerged discharge, Water vapor ionization.
1. Introduction
Purifying water from chemical and biological
contaminants is a challenging scientific and technological
problem. One purification method uses an electrical
discharge within the water [1]. It was demonstrated that
pulsed plasma arc discharges between electrodes in water
effectively kill bacteria and decompose various organic
pollutants in water [2]. However electrode erosion products
remain in the water. In contrast, a microwave electrodeless
discharge should not have this shortcoming. As a first step,
a static stationary model of a spherical plasma bubble at
rest relative to the surrounding water was considered [3].
The basic difference between this static case and flowing
water is that in the latter the heat transfer will primarily be
by forced convection. Under these conditions, the plasma
bubble is not spherical but rather stretched in the water
flow direction. The flowing case has not heretofore been
considered. The objective of this paper is to present a
theoretical model of such a plasma bubble in a focused
microwave field.
2. Model description.
We consider in this work a stationary distribution of the
temperature and all other plasma parameters, which
generally depend on the temperature. The physical
processes in the plasma bubble will be described
considerably differently for "weak" and "strong" applied
fields -- the boundary between "weak" and "strong" field
will be explained below. A schematic diagram showing the
arrangement for coupling microwave power into a
submerged discharge is presented in Fig. 1. An
electromagnetic field is focused in the center of a spherical
water vessel, after passing through a hemispherical quarterwave dielectric layer, which reduces reflection. The
focusing is far from ideal, due to the relatively large
microwave wavelength, even in water, and
the strong field region will have a characteristic radius on
the order of magnitude of the microwave wavelength in
water, R   / 2   0.006748 m, where   81 is the
dimensionless dielectric constant for water, and =0.122 m
is the wavelength in free space at f=2.45 GHz.
Fig.1
setup for plasma bubble
excitation.
The strength will be approximated by
 r 2  z2
E  E 0 exp  
R2




,
(1)
where z and r are axial and radial distances from the central
point of the focused field, i.e. in the direction along the
water jet and normal to that direction, respectively.
3. Plasma bubble in a strong electromagnetic field.
The thermal energy produced by Joule dissipation is
balanced by energy loss through convection and thermal
conduction.
The outgoing energy flux F is given by:
F  - T  w v
(2)
where  and T are the thermal conductivity and
temperature of the plasma, v is the plasma velocity and
1
w
nkT is the internal energy, where  is the
 1
adiabatic index, assumed to be 5/3, n is the particle
concentration, and k is Boltzmann's constant. Thermal
plasma conductivity is given by

1  kT 


  T 
3 e
3 2  ma 
2 k 
2
1
2
k a
 aa   1
(3)
where ma and a are the mass and the atomic number of the
atoms or ions in the plasma (hereafter referred to as heay
particles),, which are assumed equal for all heavy particles,
aa is the neural-neutral collision cross-section, e is the
electron charge and  is the electrical conductivity:
 
nee 2  e
me  e2   2
(4)
where ne, me are the electron concentration and mass
respectively,  is the electrical field frequency and e is the
electron collision frequency:
e 
kT  1  e 2


me  2   0 kT

2


 ln   ne   ea na 




(5)
where 0 is the vacuum permittivity, and ln is the
Coulomb logarithm. The electron concentration is
determined by ne=xn, where n is the total heavy particle
concentration and the ionization degree x is determined
from the Saha's formula (modified for constant pressure P):
x2
 2 me kT 
 2

1  x2
h2


3
2

kT
 

exp 
P
 kT 
(6)
where h is the Planck's constant, and  is the ionization
energy of the neutral particles. In equation (3), the first
term describes the electron part of the thermal conductivity
and second term describes the heavy particle part of the
thermal conductivity.
The energy balance equation including the convective
term is
T 

T 2  k nv  T   E 2  0
T
 1
(7)
This equation takes into account mass conservation, i.e.
nv   0 . It was assumed that: 1) the electrical field is
sufficiently strong so that the plasma temperature is
relatively high and the plasma will be near to local
thermodynamic equilibrium, and therefore the electron and
the ion/neutral temperatures are the same, and 2) both
the water and plasma motion are one-dimensional, in the zdirection. The equation set (1) - (7) in principle describes
the properties of the plasma as it passes through the
microwave field.
4. Plasma bubble in a weak electromagnetic field.
In contrast to the strong field case, the weak
electromagnetic field produces a lower plasma temperature,
and the heavy particle temperatures will not necessarily
equal the electron temperature. Instead of equation (7),
equations for the electron and heavy plasma components
must be written. The electron component equation is:
e Te 
 e
Te 2  k ne v  Te   E 2   Te  Ta   0
 Te
 1
(8)
where Te and Ta are the temperatures of the electron and
heavy plasma components, respectively, e is the thermal
conductivity of the electrons, and  is the coefficient of
energy transfer from the electrons to the heavy plasma
components [3]:
   e ne k
me
ma
(9)
It is important to note that in the convective term, instead
of n (the total particle concentration), the electron
concentration ne appears. Furthermore, the temperature
relaxation is much slower in the weak field case than in the
strong field case. Another equation must be formulated for
the heavy plasma component heating:
a Ta 
 a
Ta 2  k na v  Ta   Te  Ta   0
 Ta
 1
(10)
where index "a" indicates heavy particle temperature,
concentration etc. One dimensional water/vapor/plasma
motion is assumed also in this case.
5. Solution.
First, we consider plasma motion through the strong
electromagnetic field. Let us estimate the relative part of
the convective and conductive terms in the thermal flux
expression (2). Comparing these terms, and taking into
account
that
in
the
temperature
interval
1.0×103 K <T<1.8×103 K, the coefficient of the thermal
conductivity has a value 2-5 W/(m K), we obtain that the
conductive thermal transfer exceeds the convective thermal
transfer only if temperature changes significantly over an
interval L
L

k nv
(11)
.
With a water velocity of v=1 m/s and a total particle
concentration n  
where  is the water density,
mH 2O
-6
mH 2O is the water molecule mass, we have L  10 m. As
so short intervals are not relevant, we conclude that the
energy transfer is primarily convective. Thus in the zeroorder approximation we can write
k
 1
nv   T   E 2  0
(12)
Equation (12) with assumption (1) may be transformed to
the form:
dT
  (T ) 
  1 E 02 R
k
where   z
vn
R


exp( 2 2 ) 1  erf ( 2 )
;  r
R
Fig. 3. Longitudinal plasma temperature distributions
for different radial distances.
(13)
.
In this equation the right side contains only spatial
variation, while the left side contains all the physical
conditions.
Similarly, the longitudinal temperature distribution
depended on the value of the field, E(0,0) (figure (4)). We
see that the plasma temperature on the axis strongly
increased when E(0,0) increased 0.9105 V/m to
1.0105 V/m.
Equation (13) was solved numerically for v = 1 m/s,
Eo = 105 V/m, and R = 0.005 m. In the weak field case,
equations (8) and (10) was solved numerically, neglecting
the thermal conductivity terms.
6. Results
The strong field results are presented in figures 2 - 8. The
temperature distribution is shown in figure 2. The field
E(,) maximum was at (,)=(0,0), while the maximum
temperature was at (2,0).Thus the plasma was displaced
downstream due to the convection.
Fig. 4. Plasma temperature distribution along the
axis with different applied field; E0 = 105 V/m.
Figures 2-4 were obtained by considering only the
dominant convective heat transfer, and neglecting thermal
conduction. A more accurate solution was obtained by
considering also conduction, substituting the temperature
distribution obtained from equation (12) into the thermal
conductance terms of equation (2). The resulting axial
temperature distribution is shown in figure 5.
Fig. 2. Isotherms from 2-14 kK in strong field example.
Longitudinal temperature distributions for different
normalized radial positions  are presented on figure 3. The
plasma temperature increased as the water vapor plasma
passed through the field spot. The maximum temperature
decreased with increasing radial distance, This is a direct
consequence of the assumed field profile (1).
Fig. 5. Axial temperature distribution after taking into
account the plasma thermal conductance.
We can see that the downstream temperature decreased
with longitudinal distance. This was due to radial
conduction to the surrounding environment. However the
maximum temperature value was only slightly reduced by
the thermal conductance (compare with figure 3)
result was obtained using the one dimensional motion
approximation, in the real case considerable velocity should
likewise be observed.
Results for a weak field were obtained by solving
equations (8) and (10). The electron and heavy particle
temperature distributions for E0 = 2 kV/m and 5.5 kV/m are
presented in figures 9 and 10, respectively, and are unequal
in the weak field case.
Fig. 6. Power density distribution.
The power density distribution is shown in figure 6. The
power density maximum was displaced downstream from
the field maximum (0,0). Total power P was calculated by
 
P  2 R 3
   (T ) E( , )
2
(14)
 d d
 0
and is presented on figure (7) as a function of the electrical
field in the center of the focus.
Fig. 9. Electron and heavy particle temperature
distributions along the axis with E0 = 2000 V/m.
Fig. 7. Total power vs applied field. E0=105 V/m
Fig. 10 Electron and heavy particle temperature
distributions along the axis with E0 = 5500 V/m.
The location of the strong rise of Te in figures 9 and 10
near =2 determines the upstream plasma bubble boundary.
Note that the maximum value of T e is considerably greater
than the maximum value of Ta in figure 9, while
downstream the two temperatures are identical in figure 10.
This indicates that the boundary between the weak and
strong electric field cases is someplace between 2 000 and
5 500 V/.
Fig. 8.
a
It should be noted that is the plasma bubble has a
hydrodynamic interaction with the water jet. As nv = const
in one-dimensional motion, while n 
P0
kT
, where P0 is
atmospheric pressure, we have that after evaporation and
ionization of the water vapor, the plasma velocity is a
function of the temperature, i.e. of the r – coordinate. The
velocity distribution for E0=105 V/m is shown in figure 8.
We can see that the strong field electrical discharge in the
water stream a supersonic plasma flow. Although this
7. References.
[1] Khlustova A V, Maximov A I, Subbotka I N, High
Temperature Material Processes, 14, No. 1-2, pp. 185 –
191, 2010.
[2] Parkansky N, Alterkop B, Boxman R L, Mamane H,
Avisar D, Plasma Chem Plasma Process, 28, pp. 583
– 592, 2008.
[3] E. Gidalevich and R. L. Boxman, J. Phys. D: Appl.
Phys., 45, 245204, 2012.