22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Examining interfacial gradients in plasma-liquid systems through a coupled
momentum, heat, and mass transport model
A. Lindsay1,2, C. Anderson2, S. Shannon1 and D.B. Graves2
1
2
North Carolina State University, 27695 Raleigh, NC, U.S.A.
University of California-Berkeley, 94720 Berkeley, CA, U.S.A.
Abstract: Transport phenomena at a plasma-liquid interface are investigated with a finiteelement model. Convection-induced evaporative cooling leads to factor of two changes in
aqueous-phase concentrations relative to the case where convection-induced temperature
effects are not considered. Additionally, aqueous concentrations of short-lived species like
OH drop by as many as 9 orders of magnitude within 50 µm of the interface.
Keywords: plasma-liquid interface, coupled transport, convective discharges
1. Introduction
Understanding the physical and chemical interactions
that occur at the interface between plasmas and liquids is
important for a variety of applications, including
biomedicine, bio- and chemical disinfection, and
agriculture [1-3]. The work presented here applies the
concepts of coupled momentum, heat, and mass transport
in order to explore and understand different interfacial
phenomena. In particular, through fluid modelling of a
pulsed streamer-water system, we explore the
interconnectivity of convection, evaporative cooling, and
reaction rates. Interesting results include the significant
divergence of bulk gas and liquid temperatures over the
course of the simulation because of convection-induced
evaporative cooling. More particularly, we find that the
bulk liquid temperature is close to 10 K cooler after
17 minutes than the bulk gas temperature. This has the
effect of reducing the rates of many liquid phase
reactions, resulting in final concentrations that differ by as
much as a factor of 2 from the case where convectioninduced temperature effects are not included.
Additionally, in the aqueous phase we find that the
concentrations of short-lived species like OH drop by as
many as 9 orders of magnitude within 50 µm of the water
surface.
2. Model Description
We model the momentum, heat, and mass transport of
neutral species for a typical pulsed streamer-over-water
geometry without explicitly including the plasma physics
that occur over much shorter time scales. The ionic wind
is included by calculating the maximum gas velocity
using the results found in [4] combined with our
experimental discharge voltage and gap distance, and
using that as a boundary condition. Input conditions for
reactive plasma-generated species are based on the results
reported in [5]. Continuity (Eq. 1), momentum (Eq. 2),
heat (Eq. 3), and mass transport equations (Eq. 4) are
solved using the finite element method implemented in
Comsol Multiphysics 4.4. A copy of the model can be
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found at [6].

∇⋅u = 0


 ∂u   
ρ  + u ⋅ ∇u  = −∇p + µ∇ 2u
 ∂t

C
ρC pu ⋅ ∇T = ∇ ⋅ (k∇T )
C
∇ ⋅ (− Di∇Ci ) + u ⋅ ∇Ci = Ri
(1)
(2)
(3)
(4)
In the above equations, u represents velocity, ρ density,
p pressure, µ viscosity, C p constant pressure heat
capacity, T temperature, k thermal conductivity, D i
diffusivity of species i, C i concentration of species i, and
R i represents the gain or loss of species i through
chemical reactions.
Heat transport is coupled to
evaporation of water through a boundary source condition
at the interface given by equation 5:
Qb = J z , H 2 O ⋅ H vap = − DH 2 O , g ⋅
∂CH 2 O , g
∂z
|z =0 ⋅H vap
(5)
The water vapour mass transport is then selfconsistently coupled back to heat transport through
Antoine’s equation at the interface:
log10 pb = A −
B
C + Tb
(6)
where A, B, and C are constants and p b and T b are the
interface water vapour pressure and temperature
respectively.
Eqs. 1-3 and Eq. 4 for water vapour are solved together
and then the resulting velocity, temperature, and water
vapour profiles are used as inputs for solving the
remaining mass transport equations.
The species
considered in the model are summarized in Table 1.
1
Table 1. Model species.
Liquid phase
OH, H 2 O 2 , NO, NO 2 , N 2 O 4 , HNO 2 ,
HNO 3 , H 2 O
OH, H 2 O 2 , NO, NO 2 , N 2 O 4 , HNO 2 ,
NO 2 -, NO 3 -, ONOOH, H+, OH-
3. Results & Discussion
The ionic wind from the streamer induces re-circulating
convective currents in both the gas and liquid phases, as
shown in Fig. 1.
300
Liquid phase
298
296
Gas phase
294
292
Temperature (K)
Gas phase
302
290
-5
0
5
Axial coordinate (mm)
10
Inlet boundary
Flow development
zone (inert)
Needle tip
Gas-liquid interface
Fig. 1. Fluid flow patterns in the gas and liquid phases.
Arrows in liquid phase scaled ~100x greater than in gas.
The gas-phase convective currents remove water
vapour from the interface. In order to maintain an
equilibrium vapour pressure, liquid water is evaporated
and consequently the temperature near the interface drops
as energy is consumed in latent heating. This coupling of
fluid flow, heat, and mass transfer is analogous to the
classic wet-bulb, dry-bulb problem found in chemical
engineering texts [7]. As the interface cools, heat is
transferred from the liquid bulk to the surface, leading to
an overall cooling of the liquid relative to the impinging
gas as seen in Fig. 2. The relationship between
convective flow and heat transfer is not just one way.
Over the course of the simulation, as the water cools from
300 to 291 K, its viscosity increases by 20%. This
increased liquid viscosity inhibits momentum transfer
between the gas and liquid phases. From a peak value of
8.15 mm/s which occurs at the beginning of the
simulation, the average interfacial velocity drops by 18%
to 6.67 mm/s at the termination of the simulation
(t = 17 minutes).
To the authors’ knowledge, this
illustration of the close coupling between momentum,
heat, and mass transport has not yet been shown in the
plasma-liquid literature.
When analysing the generation of different chemical
species, accounting for convection-induced changes in
temperature is critical because of the strong dependence
of reaction rate coefficients on temperature. To illustrate
this, one study was performed in which a uniform
2
Fig. 2. Temperature profile along axis of symmetry after
17 minutes. Interface is at z = 0. Sharp temperature
gradient in gas boundary layer is evident.
temperature distribution of 300 K was assumed for the
duration of the simulation.
When this study was
compared with a simulation that included the convectioninduced temperature variations, the final concentrations of
aqueous-phase terminal species like H 2 O 2 , NO 3 -, and
NO 2 - differed by factors as large as 2.
In addition to showing the close coupling between
momentum, heat, and mass transport, this model
illustrates vividly the sharp gradients in reactive species
concentrations that arise in the interfacial liquid boundary
layer. A plot of the base-10 log of aqueous hydroxyl
radical concentration is shown in Fig. 3. Even on a
logarithmic scale, the gradients near the interface are
pronounced. Along the z-axis (r = 0), [OH(aq)] drops by
9 orders of magnitudes within 50 µm of the surface. In
addition to the gradients near the interface, Fig. 3 also
illustrates that the bulk OH(aq) concentration is enhanced
within the re-circulating convective loop relative to other
areas of the solution.
Fig. 3. 3-D plot of base-10 log of OH(aq). Sharp
gradients along z = 0, particularly in region of streamer
impingement. Effect of circulating convective loop also
evident.
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4. Conclusions
This work shows the close coupling between
momentum, heat, and mass transport in convective
plasma systems.
Convection produces evaporative
cooling at the interface which leads to a sharp drop in
temperature in the interfacial gas boundary layer. The
result is a significant difference between gas and liquid
bulk temperatures: in this model 9 K and still dropping
after 17 minutes of simulation. Whether this evaporative
cooling effect is included in the model has a significant
impact on system chemistry. The concentrations of longlived aqueous species like H 2 O 2 , NO 2 -, and NO 3 - can
change by as much as a factor of 2 when the convectioninduced changes in temperature are allowed to influence
reaction rate coefficients. In addition to this coupling, the
model reveals strong gradients in short-lived chemical
species concentrations in the interfacial liquid boundary
layer. In the case of OH, concentrations drop by as high
as 9 orders of magnitude within 50 µm of the surface.
5. References
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T. Shimizu, J. van Dijk and J. Zimmerman.
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“Electrohydraulic Discharge and
Nonthermal Plasma for Water Treatment”. Ind.
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Friedman,
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Fridman,
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plasma increase plant growth rate and nutritional
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[4] L. Zhao and K. Adamiak. “EHD flow in air
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J. Electrostatics, 63, 337-350
(2005)
[5] W. Tian and M. Kushner. “Atmospheric pressure
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165201 (2014)
[6] https://drive.google.com/drive/u/1/#folders/0B0XNX7
i2mXeHUzY3SEk3QzExQ0U
[7] B. Bird, W. Stewart and E. Lightfoot. Transport
Phenomena; 2nd edition. (Chichester: J. Wiley &
Sons) 683 (2007)
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