When the pore fluid is mechanically forced to flow through a porous media, some of the excess charges are dragged to move, therefore causing streaming electric current in[r]
Trang 1ELECTROKINETICS IN A CYLINDRICAL CAPILLARY
Luong Duy Thanh 1,* , Phan Van Do 1 , Pham Thi Thanh Nga 1 , Nguyen Trong Tam 2 , Pham Thi Na 3 , Phan Thi Ngoc 3
1 Thuyloi University, 2 Vietnam Maritime University,
3 University of Science - TNU
ABSTRACT
Electrokinetic phenomena are induced by the relative motion between a fluid and a solid surface and are directly related to the existence of an electrical double layer with excess charges In this work, we use a theoretical study of electrokinetics in a narrow cylindrical capillary to obtain the streaming potential and electroosmosis coefficients under the thin double layer assumption We
use the obtained theoretical coefficients to compare with experimental data available in literature
The results show a good agreement between the theory and the experimental data and that validates the obtained model The model for a narrow cylindrical capillary is a basis to understand electrokinetics in porous media
Keywords: electrokinetics, zeta potential, porous media, electric double layer,
Electrokinetic phenomena consist of different
effects such as streaming potential,
electroosmosis etc When the pore fluid is
mechanically forced to flow through a porous
media, some of the excess charges are
dragged to move, therefore causing streaming
electric current in porous media, which is
referred to as the streaming potential effect
(SP) Conversely, an applied electric field
forces the excess charges to move, therefore
driving pore fluid flow, which is referred to as
the electroosmosis effect (EO)
Electrokinetics plays an important role in
geophysical applications, environmental
applications, medical applications and other
applications For example, SP measurement is
used to detect subsurface flow in oil
reservoirs or to monitor subsurface flow in
geothermal areas and volcanoes It is also
used to detect seepage of water through
retention structures such as dams, dikes, and
canals etc [1] SP has been utilized to
generate electric power by pumping liquids
such as tap water through tiny micro channels
[2,3] EO is one of the promising technologies
for cleaning up low permeable soil in
*
Email: thanh_lud@tlu.edu.vn
environmental applications In this process, the contaminants are separated by the application of an electric field between two electrodes inserted in contaminated masses Therefore, it has been used for the removal of organic contaminants, heavy metals, petroleum hydrocarbons etc in soils, sludge and sediments Additionally, EO has been used to produce microfluidic devices such as
EO pumps with several outstanding features: ability of generating constant and pulse-free flows, facility of controlling the flow magnitude and direction of EO Pumps, no moving parts EO Pumps have been used in microelectronic equipment for drug delivery etc [4]
Figure 1 Porous media as a bundle of parallel
capillaries taken from [5]
Porous media can be simply approximated as
an array of parallel capillaries as shown in Fig 1 Therefore, having knowledge of electrokinetics in a single capillary is a basis for understanding electrokinetics in porous media In this report, the theoretical
Trang 2background of streaming potential and
electroosmosis is presented for a cylindrical
capillary The electrokinetic coefficients are
then obtained and then compared with
experimental data available in literature
THEORETICAL DEVELOPMENT
Surfaces of the minerals of porous media are
generally electrically charged, creating an
electric double layer (EDL) containing an
excess of charge that counterbalances the
charge deficiency of the mineral surface [6]
Fig 2 shows structure of the EDL: a Stern
layer that contains only counterions coating
the mineral with a very limited thickness and
a diffuse layer that contains both counterions
and coions but with a net excess charge The
shear plane that can be approximated as the
limit between the Stern layer and diffuse layer
separates the mobile and immobile part of the
water molecules when subjected to a fluid
pressure difference The electrical potential at
the shear plane is called the zeta potential (ζ)
[6] The zeta potential is a complicated
function of many parameters such as mineral
composition of porous media, ionic species
present in the fluid, the pH of fluid, fluid
electrical conductivity and temperature etc In
the bulk liquid, the number of cations and
anions is equal so that it is electrically neutral
Most reservoir rocks have a negative surface
charge and a negative zeta potential when in
contact with ground water The characteristic
length over which the EDL exponentially
decays is known as the Debye length λ and is
on the order of a few nanometers
The distribution of the excess charges in the
diffuse layer of a capillary is governed by the
Poisson-Boltzmann equation:
0
) ( )
( 1
r
r dr
r d r
dr
d
where ψ(r) and ρ(r) is the electric potential (in
V) and the volumetric charge density (in C m
-3
) in the liquid at the distance r from the axis
of the capillary, respectively; ε r is the relative
permittivity of the fluid (78.5 at 25oC for
water) and ε o is the dielectric permittivity in vacuum (8.854×10−12 C2 J−1 m−1)
For symmetric electrolytes such as NaCl or CaSO4 in the liquid, ρ(r) is given by [7]
T k r eZ eZC
N r
b f
A
where C f is the electrolyte concentration in the bulk fluid representing the number of ions (anion or cation) (mol m−3), e is the
elementary charge (e = 1.6×10−19 C), Z is the
valence of the ions under consideration
(dimensionless); k b is the Boltzmann’s constant (1.38×10-23 J/K), T is the kelvin temperature (in K) and N A is the Avogadro’s number (6.022 ×1023 /mol)
Figure 2 Schematic view of the EDL (a) Charge
distribution (b) Electric potential distribution
Putting Eq (2) into Eq (1), one obtains
) ) ( sinh(
2 ) ( 1
r eZ eZC
N dr
r d r dr
d
b
The boundary conditions to be satisfied for the cylindrical capillary surface are: (1) the
potential at the surface r = a (a is the radius of
the capillary), (a); (2) the potential at
the center of the capillary r = 0,
0 /
) (
0
r dr r
By solving Eq (2) and Eq (3) with the linear
approximation, the analytical solution ρ(r) are
obtained as [7]
) (
) ( )
a I
r I r
o
o r
Trang 3where Io is the zero-order modified Bessel
function of the first kind and is the Debye
length characterizing EDL thickness given by
f A
b r o
C e Z N
T k
2 2 2
Figure 3 Development of streaming potential
when an electrolyte is pumped through a capillary
Streaming potential
The streaming current is created by the drag
of the excess charges in the EDL due to the
fluid flow in the capillary (Fig 3) The
streaming current is given by
a
I
0
2 )
( )
(6)
where ρ(r) is charge density and v(r) is the
velocity profile in the capillary that is given
by [8]
4 )
L
P r
where ΔP is the pressure difference across the
capillary, η is the dynamic viscosity of the
fluid and L is the length of the capillary
Putting Eq (4), Eq (7) into Eq (6) and
evaluating the integral, one obtains:
) (
) ( 2
1 0
2
a I
a I a L
a
P
I
o
r
where I1 is the first-order modified Bessel
functions of the first kind
The streaming current is responsible for the
streaming potential As a consequence of the
streaming current, a potential difference
called streaming potential (ΔV) will be set up between the ends of the capillary This streaming potential in turn will cause an electric conduction current opposite in direction with the streaming current (Fig 3) The conduction current when taking into account only bulk conduction of the capillary
is given by
R
V
I c
(9)
where R is the resistance of the capillary that
is related to the conductivity of fluid σ w by
L
a R
w
2
Eq (9) is now written as
L
a V
c
2
At steady state, the sum of the streaming current and the conduction current in the capillary needs to be zero Therefore, one has
) (
) ( 2 1
1 0
a I
a I a P
V
o w
Ratio of ΔV/ΔP is referred to as the streaming potential coefficient Ksp Consequently, the following is obtained
) (
) ( 2 1
1 0
a I
a I a K
o w
r
The streaming potential coupling coefficient
is defined as [9]
) (
) ( 2 1
1 0
a I
a I a K
L
o
r w sp
Electroomosis
Electroosmosis is the opposite effect of the streaming potential Namely, when an electric field is applied parallel to the wall of a capillary, ions in the diffuse layers experience
a Coulomb force and move toward the
Trang 4electrode of opposite polarity, which creates a
motion of the fluid near the wall and transfers
momentum via viscous forces into the bulk
liquid So a net motion of bulk liquid along
the wall is created and is called
electroosmotic flow (see Fig 4)
Figure 4 Electroosmosis flow in a capillary
The velocity profile in the capillary under
application of a voltage ΔV is given by [7]
) (
) ( )
a I a
r I L
V r
v
o
o
Therefore, the volumetric flow rate due to the
electroosmosis in the capillary is given by
Q eoa v r rdr
0 2 )
Combining Eq (15) and Eq (16), the
following is obtained
) (
) ( 2
1 2
0
a I
a I a L
a V
Q
o
r
The pressure necessary to counterbalance
electroosmotic flow is termed the
electroosmotic pressure (P eo) Under that
pressure, the counter volumetric flow rate is
given by [10]
L
P a
cou
8
4
At the steady state, the sum of the
electroosmotic flow and by the flow caused
by the pressure is zero
0
cou
Consequently, one obtains
) (
) ( 2 1
2 0
a I
a I a a
V
P K
o
r eo
Ratio of ΔP eo /ΔV is referred to as the electroosmosis coefficient Keo
The electroosmosis coupling coefficient is defined as [9]
E eo
K
where is the permeability of the capillary and is given by [10]
8
2
a
Eq (21) is now rewritten as
) (
) ( 2 1
1 0
a I
a I a L
o
r
By comparison, it is seen that Eq (14) and Eq
(23) are identical, that is Lsp = Leo This result
is what we expected because the coupling coefficients must comply with the Onsager’s reciprocal equation in the steady state [1] Eq (13) and Eq (20) show the dependence of the streaming potential coefficient and the electroosmosis coefficient on the capillary radius and electrokinetic parameters such as ionic concentration, valence of ions, temperature and the zeta potential
RESULTS AND DISCUSSION
In this part, a system of 1:1 symmetric electrolytes such as NaCl, KNO3 (Z = 1) and
silica-based surfaces are considered at room
temperature (T = 295 K) for the modeling
because of the availability of input parameters For silica-based rocks saturated
by 1:1 symmetric electrolytes, the C f -
relation is found to follow [11]:
ζ = a + blog10(C f) (24)
where a = -9.67 mV, b = 19.02 mV (ζ in mV) The C f -w relation for monovalent electrolytes of concentration ranging from 10
Trang 5-6M to 1 M and temperature ranging from 15
to 25°C is found to be [13]
f
w10C
From Eq (13), Eq (24) and Eq (25), the
variation of the K sp with electrolyte
concentration is shown in Fig 5 for two
values of capillary radius
Figure 5 Streaming potential coefficient as a
function of electrolyte concentration for two
values of the capillary radius (0.1 μm and 1.0 μm)
It is seen that the Ksp decreases with
increasing electrolyte concentration as
reported in [1, 11, 12] For ground water
saturating rocks or soils, the Debye length λ is
about few nm and a typical pore radius of
rocks is around in order of µm Therefore, the
thickness of the EDL is normally much
smaller than the capillary radius (thin EDL
assumption) In this case the ratio
2I1(a/λ)/I0(a/λ) can be neglected Under these
conditions, Eq (13) may be simplified as
w
r sp K
0
Eq (25) becomes the well-known
Helmholtz-Smoluchowski (HS) equation Based on the
HS equation, one can explain the behavior in
Fig 5 at high electrolyte concentration where
K sp is independent of the capillary radius Eq
(14) is also valid for porous media as reported
[12] Therefore, we use it to predict the
dependence of the K sp on the electrolyte
concentration for silica-based rocks saturated
by NaCl electrolyte (see the dashed line in
Fig 6) The experimental data available in
literature [1, 14] for K sp is also shown in Fig
6 (see symbols) It is seen that the HS
equation is in good agreement with the experimental data
Figure 6 Comparison between the HS equation
and experimental data available in literature
Similarly, for the thin EDL assumption the electroosmotic pressure P eo in the porous media is simplified as
V a
Figure 7 The comparison between Eq (26) (see
the solid line) and experimental data obtained
from [15] (see symbols)
Fig 7 shows the variation of P eo with the applied voltage obtained from measured data
in [15] for a sand pack of 10 μm diameter particles (symbols)
The relationship between particle diameter and the capillary radius is given by [16]
2
d
where d = 10 μm and θ is the theta transform
function depending on parameters of the porous media such as porosity, cementation exponent, and formation factor For the porous sample made of the monodisperse
spherical particles arranged randomly, θ is
taken to be 3.3 [16] Therefore, the capillary
radius a is found to be 1.52 μm To model the
observed result in Fig 7, Eq (26) is used with
Trang 6the knowledge of zeta potential magnitude of
17 mV for sand packs [1] and a = 1.52 μm
The theoretical prediction is shown by the
solid line in Fig 7 It is seen that the theory
can reproduce the main trend of the measured
data available in literature
CONCLUSIONS
In this report, we present the theoretical
background of streaming potential and
electroosmosis for a cylindrical capillary
Then we obtain the electrokinetic coefficients
The theoretical predictions are performed and
compared with experimental data in literature
for both the streaming potential coefficient
and the electroosmotic pressure The results
show a good agreement between them and
that validates the models derived in this work
ACKNOWLEDGMENTS
This research is funded by Vietnam National
Foundation for Science and Technology
Development (NAFOSTED) under grant
number 103.99-2016.29
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Trang 7TÓM TẮT
HIỆN TƯỢNG ĐIỆN ĐỘNG HỌC TRONG ỐNG MAO DẪN HÌNH TRỤ
Lương Duy Thành 1* , Phan Văn Độ 1 , Phạm Thị Thanh Nga 1 , Nguyễn Trọng Tâm 2 , Phạm Thi Na 3 , Phan Thị Ngọc 3
1 Đại học Thủy lợi, 2 Đại học Hàng Hải Việt nam,
3 Trường Đại học Khoa học - ĐH Thái Nguyên
Hiện tượng điện động học được gây ra bởi chuyển động tương đối giữa chất lỏng và bề mặt rắn và
nó có liên hệ trực tiếp với sự tồn tại của lớp điện tích kép tại mặt phân cách giữa chất lỏng-bề mặt rắn Trong báo cáo này, chúng tôi trình bày cơ sở lý thuyết của hiện tượng điện động học trong một ống mao dẫn hình trụ Trên cơ sở đó, chúng tôi thu nhận được hệ số điện thế chảy và hệ số thẩm điện Các biểu thức lý thuyết sau đó được so sánh với kết quả thực nghiệm ở các tài liệu đã được công bố trong trường hợp bề dày của lớp điện tích kép rất nhỏ so với bán kính của ống mao dẫn Kết quả cho thấy có sự phù hợp tốt giữa lý thuyết và thực nghiệm Kết quả trong báo cáo này
sẽ là cơ sở để nghiên cứu hiện tượng điện động học trong môi trường xốp
Từ khóa: hiện tượng điện động học, thế zeta, môi trường xốp, lớp điện tích kép
Ngày nhận bài: 14/11/2018; Ngày hoàn thiện: 05/12/2018; Ngày duyệt đăng: 15/12/2018
*
Email: thanh_lud@tlu.edu.vn