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Modelling of electrification in steady state and transient regimes
Journal of Electrostatics 70 (2012) 517e523
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Modelling of electrification in steady state and transient regimes D.A. Bograchev a, S. Martemianov b, *, T. Paillat b a b
Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninsky pr. 31, Moscow 119991, Russia Institut Prime UPR CNRS 3346, CNRS - Université de Poitiers – ENSMA, Departement FTC e Branche Fluide, 40, avenue du Recteur Pineau, 86022 Poitiers, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 July 2011 Received in revised form 13 July 2012 Accepted 15 July 2012 Available online 25 August 2012
A microscopic model of electrification phenomena in steady state and transient regimes has been developed. The model has been applied for analysis of experimental data failing to be explained in the frame of classical macroscopic approach. Various boundary conditions have been studied regarding the possibility of experiential data fitting. The hybrid boundary condition has been suggested for explaining steady state and transient experiments. The proposed boundary condition can be related to the coexistence of two reactions on the active surface. The first reaction runs under the diffusion control (fast reaction) and the second one is limited by the kinetics (slow reaction). Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Electrification Double layer Streaming current
1. Introduction The appearance of volume charges in dielectric liquid flows causes their electrification and probable ignitions or explosions in the petroleum and electrical industries [1e4]. The electrification of dielectric liquid flows has been studied in numerous works; it has been shown that this phenomenon concerns the diffuse part of electric double layer. Free charges (ions) can be removed from the surface by forced flow and cause the electrical current in liquid (streaming current). The streaming current depends on hydrodynamic conditions, liquid conductivity and surface properties which define the type of the liquid/surface interactions (interface reactions). Various interface reactions and different mechanisms of the charge origin at the surface have been suggested in the literature [5e9]. However, usually the mechanism of the interface reactions is unknown and just postulated from some reasonable considerations. In addition, the nature of electrical conductivity, in other words the origin of free charges in weakly conducting liquids, is still a question. A number of approaches for explanation of the electrification phenomena exist in the literature. In the frame of the macroscopic approach the space distribution of electrical charge r(x,y) is assumed to be known:
* Corresponding author. Tel.: þ33 5 49 45 39 04; fax: þ33 5 49 45 44 44. E-mail address: [email protected] (S. Martemianov). 0304-3886/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2012.07.006
rðx; yÞ ¼ rw ðxÞ
coshðy=d0 Þ ; coshðd=d0 Þ
(1)
where axis x is oriented along the flow, axis y is normal to the surface, rw is the electrical charge on the surface, d0 is Debye length, d is the half thickness of the channel. The main problem is related to the surface charge determining with respect to boundary conditions. In particular, the following relation between the wall current density and the charge density near the wall has been assumed in [10e13]:
iw ¼ K½rwN rw ;
(2)
where K e is an empirical coefficient for a given interface reaction, cH is surface charge density in fully developed conditions. Fully developed conditions correspond either to the stagnant liquid or to the outlet of the very long channel. It is supposed also that the surface charge density in the inlet of the channel is equal to zero. Eq. (1) is usually accepted in the frame of the macroscopic approach. It should be noted that the hyperbolic cosine profile, Eq. (1), is valid only if the electrical charge in the volume is much less than product of equilibrium concentration and the Faraday constant:
r Fc0
<< 1:
(3)
This is so called the small charge density assumption which allows linearization of the problem. Assumptions similar to inequality (3) have been used in some analytical studies in the frame of microscopic approach [14e17] for linearization of the
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governing system of the equations. Nevertheless, there are some cases when condition (3) is not valid and the electrification problem should be treated without this traditional supposition. The other type of the electrification theories are based on the microscopic approach. Microscopic models [7,8,14e33] are based on the diffusion equations for ions and Poisson’s equation for the electric potential. The main result of these theories is the dependence of steady state streaming current on flow velocity. The streaming current is calculated as the integral of the free charge density and hydrodynamic velocity product. The free charge density is calculated using equations for the concentrations and the electrical potential fields. For the microscopic model the boundary conditions concerning electrochemical surface process vary greatly. In many works [16,17,20] the constant boundary condition for concentration has been utilized. This boundary condition can be justified for the case of fast interfacial reaction when the electrification phenomena run under the diffusion control. In paper [23] the following relation has been used for describing electrochemical process at the surface: 0 Ji ¼ KfP;N ci KfP;N ;
(4)
where Ji e molar flux on the surface 0 ðJi ¼ Di Vci ðzi FDi =RTÞci VUÞ, KfP,N e adsorption rate, KfP;N desorption rate. Eq. (4) is based on Langmuir adsorption isotherm, but in Ref. [23] the reduced form of this relation has been used (socalled ‘radiation’ boundary condition):
Ji ¼ KfP;N ci
(5)
The similar conception of electrochemical processes at the surfaces has been developed in Ref. [7]. A free charge density dynamic approach has been developed in the works [14,29] on the basis of the diffusion equations and the equation for the electric potential. It is possible to reduce the problem to a single equation for the free electrical charge density by combining of these equations: The obtained single equation has been used for studying of various problems [14,30e32]. The validity of this approach is related to the small charge density assumption, Eq. (3). Different types of forced flows have been studied in the literature on electrification. The most popular cases are the pipe Poiseuille flow [10,11,20,23] and the flat channel Poiseuille flow [13,19,27]. The flat channel flow in consideration of the inlet region has been studied in the works [30e32]. Turbulent pipe flows have been examined in papers [11,14,22]. The turbulent Couette flow between rotating cylinders has been studied in Ref. [7]. There are papers devoted also to electrification in rotating disc flow [17] and the forced flow along the plane plate [16]. The experimental studies of liquid flow electrification in the porous media are presented in Refs. [33,34]. The objective of this work is a microscopic model of electrification which can be used for explanation of electrification phenomena in steady state and transient regimes. The required experimental information for model verification is available in Ref. [13]. The macroscopic approach fails in explanation of the experimental data presented in this work. In the cited paper the conclusion has been done that the kinetic rate depends on the flow velocity and some scenarios have been reviewed when it could take place. This hypothesis is rather discussable. Two types of experimental dependencies are presented in Ref. [13]. The first type is the dependence of stationary streaming current on hydrodynamic velocity of fluid at the channel. The streaming current varies in the range 30e120 pA for the mean flow rate 0.4e1.5 m/s. The second type of the dependences concerns a transient regime. To be exact, the liquid remains stagnant within a certain delay time td with the subsequent flow start up and the streaming current growth to a certain peak value and then it falls to
a fixed stationary value, see Fig. 1. Additional information is related to the dependence of the peak streaming current on the delay time td. The limited value at the peak streaming current for infinitely large times is the necessary element of the macroscopic theory; this limited value is related to charge density distribution in fully developed conditions (infinitely long channel). In present work we will try to explain experimental data from Ref. [13] by a microscopic model which takes into account the mass transfer of two types of ions and different boundary conditions for interfacial electrochemical processes. Despite of the model simplicity, this approach is very attractive as it is based on the exact transfer equations and the number of fitting parameters is reduced to a possible minimum. Thus, verification of some traditional assumptions used for interpretation of the electrification phenomena can be provided. 2. Governing equations of model and boundary conditions Let us consider the liquid with two ions in a small concentration c0, which flows through a rectangular channel, see Fig. 2. In the channel inlet the flow is electroneutral. A certain interface process generates a flux of the positive ions on the channel walls. Thus, a streaming electrical current arises in the channel outlet due to a charge disbalance. The equations which describe the electrification phenomenon for the system with two kinds of ions can be presented as:
vcþ zþ FDþ þ vVcþ ¼ Dþ Dcþ þ divðcþ VUÞ vt RT
(6)
vc z FD þ vVc ¼ D Dc divðc VUÞ vt RT
(7)
Fðzþ cþ z c Þ ¼ 3 0 3 L DU
(8)
The Eqs. (6) and (7) correspond to the transfer of negative and positive ions by diffusion, convection and migration. Eq. (8) is Poisson equation for the electric potential U. It is assumed that the Einstein relation is valid. Here c e ion concentrations, z e ion charges, D e diffusion coefficients, F e Faraday number, 3 0 e permittivity of free space, 3 e relative permittivity of liquid, R e gas constant, T e temperature. It is accepted that velocity in the channel is related to flat Poiseuille flow:
vm ¼
3 y2 V 1 2 ; 2 d
Fig. 1. Scheme of typical experimental curve.
(9)
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dependence on hydrodynamic velocity of fluid. These data, see for example Ref. [13], can be treated in the frame of the steady state problem. The set of equations and boundary conditions (6)e(16) can be rewritten in a dimensionless form (bellow all the variables c ; U; x; y are treated like dimensionless):
Fig. 2. Scheme model of electrification and coordinate system.
where d e channel half-thickness, V e mean flow velocity. The boundary condition for potential is:
Ujy¼d;d ¼ 0:
(10)
It means that the cannels walls are grounded and have a sufficiently high electrical conductivity. It is supposed that there are no migration fluxes in channel inlet and outlets, so:
vU ¼ 0; vx x¼0;L
(11)
(12)
The other possibility is a constant current boundary condition for positive ions:
vcþ zþ FDþ vU Dþ ¼ jH þ cþ vy RT vy y¼d;d
(13)
The flux for negative ions on the wall is equal to zero:
vc z FD vU D ¼ 0: c vy RT vy y¼d;d
(14)
The inlet boundary condition corresponds to the electroneutrality of the flow:
cþ jx¼0 ¼
z c jx¼0 ¼ c0 zþ
(15)
Finally, it is supposed that in the outlet the mass flux is due to convection only:
vcþ vc 0 ¼ Dþ ; 0 ¼ D : vx x¼L vx x¼L
(17)
vc 3 1 y2 Pe ¼ aDc az divðc VUÞ; vx 2
(18)
ðzþ cþ z c Þ ¼ De2 DU:
(19)
We have used d, Dþ/d, c0, RT/F as the scales for coordinates, velocity, concentration and electrical potential correspondingly. Here, Peclet number Pe ¼ dV=Dþ characterizes the ratio of the diffusion and the convective processes; Debye number pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De ¼ RT 3 0 3 L =c0 d2 F 2 characterizes the ratio of the double layer thickness and the characteristic size of the system and a ¼ D =Dþ is the ratio of diffusion coefficients. Accordingly, the dimensionless boundary conditions at the inlet can be written as:
cþ ¼
z c ¼ 1jx¼0 ; zþ
(20)
vU ¼ 0: vx x¼0
(21)
Boundary conditions at the outlet in dimensionless form is:
where L is the channel length. The boundary condition for concentrations, which depend on the surface kinetic processes flux, can be specified by different ways. The first possibility is a constant concentration boundary condition:
cþ jy¼d;d ¼ cH
vc 3 þ 1 y2 Pe ¼ Dcþ þ zþ divðcþ VUÞ; 2 vx
(16)
0 ¼
vcþ vc vU a ; 0 ¼ ; vx x¼ L vx x¼ L vx x¼ d
d
L d
¼ 0:
(22)
Boundary conditions (12)e(14) can be rewritten in the form:
vcþ vU ¼ jH ; þ zþ cþ vy y¼1;1 vy
(23)
cþ ¼ cH ;
(24)
vc vU ¼ 0: z c vy vy y¼1;1
(25)
Here, jH, cH are the dimensionless variables. The diffusion coefficients of ions in the liquids are very small (less then 109 m2/s) and the channel thickness is about few millimetres. Thus, for the observed in electrification experiments hydrodynamic velocities Peclet number is very high. Therefore the boundary layer approximation can be used and all transports along the x axis, apart the convection, can be neglected. The governing set of equations in the boundary layer approximation takes the form:
vc 3 v2 c þ v cþ vU þ 1 y2 Pe þ zþ ; ¼ 2 vy vy vx vy2
(26)
vc 3 v2 c v c vU 1 y2 Pe az ; ¼ a 2 vy vy vx vy2
(27)
v2 U : vy2
2.1. Steady state problem
ðzþ cþ zþ c Þ ¼ De2
The main experimental information on the electrification phenomena deals with the measurements of the streaming current
Eqs. (17)e(28) allow calculating the streaming current in steady state regime Ist and its dependence on the mean flow velocity V.
(28)
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2.2. Peak streaming current value
3. Numerical simulation
Another important information dealing with the electrification phenomena can be obtained by using the transient experiments [13]. In these experiments the flow is stopped, the fluid remains without motion during a certain stagnant time td and after the flow is switched on. During the transient period the streaming current reaches a certain peak value Ipeak and tends to its stationary level Ist, (see Fig. 1). The dependence Ipeak (td) is an increasing monotonic limited function. The characteristic time scale of this dependence is some thousand seconds [13]. In the stagnant liquid the electrical double layer grows up to some stationary state. When the flow starts, the streaming current reflects the double layer structure which has been reached during the stagnant time td. The increasing of the streaming current during the first moments is rather fast (some seconds) but not instantaneous. After the peak value is reached, the streaming current decreases to its stationary value due to the double layer dumping by the forced flow. The system of equations describing the streaming current during the flow acceleration (Fig. 1) should take into account the time of the evolution of velocity profile vx(t,y). Using the above introduced dimensionless variables, this system can be presented in the following form:
The numerical simulations have been realized by using Comsol Femlab Multiphisics package. In order to provide the calculations in the relatively thin Debye layer the mesh density has been increased near the wall. Firstly, the steady state stream current dependence Ist ðVÞ has been obtained considering the simplest boundary condition, which implies a constant concentration of positive ions on the walls:
vcþ vcþ v2 c þ v cþ vU þ zþ þ vx ðy; tÞ ¼ ; vt vx vy vy vy2
vc vc v2 c v c vU az ; þ vx ðy; tÞ ¼ a vy vy vt vx vy2 ðzþ cþ z c Þ ¼ De2
v2 U ; vy2
(30)
(31)
Z vm ðzþ cþ ðtd Þ z c ðtd ÞÞds;
(36)
The calculated streaming current is presented in Fig. 3 (curve a). A good agreement with the experiment can be achieved by fitting the constant cH, the coefficients of diffusion and charge of ions (see Table 1). However, the calculations with the boundary condition (36) does not allow any correct describing of the peak current dependence Ipeak (td), see Fig. 4 (curve a). The transition time observed in the experiments is much more important that the one obtained in the numerical simulations. On the other hand, if a constant rate of the concentration flux is used as a boundary condition:
vcþ vU ¼ jH ; þ zþ cþ vy vy y¼1;1
(37)
(29)
where vx(y,t) is hydrodynamic velocity profile which depends on time. Solving this problem requires the knowledge of fluid motion during acceleration vx(t,y), thus NaviereStokes equations should be solved in addition to Eqs. (29)e(31). However the peak value of the streaming current Ipeak can be obtained [13] without solution of the transient hydrodynamic problem using the following equation:
Ipeak ¼
cþ jy¼1;1 ¼ cH
(32)
it becomes possible to explain the experimental data related to the dependence of the peak streaming current on the stagnant time Ipeak (td) by using jH as the only fitting parameter, see Fig. 4 (curve b) and Table 1. At the same time, the calculations of the steady state streaming current Ist with the boundary condition (37) leads to the independence of this value on the mean flow velocity V under steady state, see Fig. 3 (curve b). This is the obvious contradiction with the experimental data. So, the using of the simplest assumptions for the kinetics of the interface processes, Eqs. (36) and (37) do not allow simultaneous describing of Ist ðVÞ and Ipeak (td) experimental data. Using traditional electrochemical notions, it is possible to associate the boundary condition, Eq. (36) with the high rate interfacial reaction and the boundary condition, Eq. (37), with the law rate reaction. Let us consider now the intermediate case of the interfacial process with the mixed kinetics. The mixed kinetics leads to the following boundary condition:
where td is the stagnant time and vm ¼ 3/2 Pe(1 y2) is the steady state velocity profile (Poiseuille flow). Neglecting the border effects, the concentration profiles of positive cþ and negative c ions in motionless liquid (0 < t < td) can be defined using non stationary 1D approximation:
vcþ v2 c þ v cþ vU þ zþ ; ¼ vy vy vt vy2
(33)
vc v2 c v c vU az ; ¼ a vy vy vt vy2
(34)
v2 U : vy2
(35)
ðzþ cþ z c Þ ¼ De2
Eqs. (32)e(35) allow calculating the peak streaming current value Ipeak.
Fig. 3. Dependence of steady state streaming current Ist on mean flow rate a e constant concentration boundary condition; b e constant current boundary condition; c e hybrid boundary condition, Eq. (39); dots e experimental data [13].
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Table 1 Physical parameters of system used in calculation of the model. Parameters
Value
Source
c0 e Initial ion concentration, [mol/m3]
1.3 109
cH e Ion concentration at the surface, [mol/m3] Dþ ¼ D ¼ D0 e Coefficients of diffusion [m2/s] deThe half-channel thickness, [m] H e Channel width, [m] jH e Ion flux at the surface,[mol/s/m2] L eChannel length [m] Kc e Reaction rate,[m/s] s e Conductivity of the liquid [S/m] zþ ¼ z e Charge numbers of ions 3 3 0 e Effective dielectric constant, [F/m]
1.17 107 1 1011 1.5 103 0.03 1.60 1013 0.3 0.01 4 1013 2 1.95 1011
From the model of two ions Fitting Fitting [13] [13] Fitting [13] Fitting [13] Fitting [13]
vcþ vU ¼ Kc ðcH cþ Þ þ zþ cþ vy vy y¼1;1
(38)
where Kc can be interpreted as the constant of the reaction rate. At large rate constants Kc >> 1 this boundary condition, Eq. (38) reduces to the boundary condition, Eq. (36). The decreasing of the rate constant (Kc / 0) with the simultaneous increasing of the magnitude cH (cH / N, Kc *cH / const) turns the boundary condition Eq. (38) into the constant flux boundary condition Eq. (37). Numerical modelling of Ist ðVÞ and Ipeak(td) dependences has been provided for different values of constant rates Kc, see Figs. 5 and 6. It is obvious that simultaneous describing of Ist ðVÞ and Ipeak(td) experimental data is not possible with the boundary condition, Eq. (38). Naturally, for the limiting cases of high and law constant rates the numerical simulations reproduce correctly only one of the above mentioned experimental dependences. The intermediate values of the reaction rate lead to disagreement with both experimental curves. Finally, a hybrid of boundary conditions (36) and (37) has been used in the numerical simulations. This boundary condition can be expressed by the following equation:
Fig. 5. Dependence of steady state streaming current Ist on mean flow rate. The model with mixed boundary condition, Eq. (38) and different kinetic parameters; I. cH ¼ 88, Kc ¼ 1000; II. cH ¼ 88, Kc ¼ 100; III. cH ¼ 88, Kc ¼ 50; IV. cH ¼ 88, Kc ¼ 10; V. cH ¼ 185, Kc ¼ 1; VI. cH ¼ 185 103, Kc ¼ 103; VII. cH ¼ 185 106, Kc ¼ 106; dots e experimental data [13].
see Fig. 3 (curve c), and Fig. 4 (curve c). Eq. (39) is a certain generalization of conditions (36) and (37). At the small time condition (39) works like constant concentration boundary condition (36), however when cþ becomes more than cH Eq. (39) turns into constant flux boundary condition (37) due to Heaviside step function. 4. Comparison with traditional approaches 4.1. Macroscopic models
where H is Heaviside step function. By means of this boundary condition the both experimental curves can be fitted satisfactorily,
The developed microscopic model can be used for the verification of some assumptions generally used in the frame of the macroscopic model [13]. The first key assumption is the hyperbolic cosine profile of the diffuse layer, Eq. (1), which is valid only for the small charge density condition, Eq. (3). The second assumption relates to modelling of kinetics process on the surface; see Eq. (2). Apparently Eq. (2) cannot describe all types of the surface reactions, in particular, the slow rate reactions. The present model allows fitting of the steady state experimental data using the high value of the surface concentration; see Table 1. Thus, the small charge density approximation is not justified, Eq. (3). The calculated dimensionless charge density in the channel outlet for various mean flow velocities is presented in Fig. 7. The charge density is considerably high near the surface and
Fig. 4. Dependence of peak streaming current Ipeak on delay time td a e constant concentration boundary condition; b e constant current boundary condition; c e hybrid boundary condition, Eq. (39); dots e experimental data [13].
Fig. 6. Dependence of peak streaming current Ipeak on delay time td. The model with mixed boundary condition, Eq. (38) and different kinetic parameters; I. cH ¼ 88, Kc ¼ 1000; II. cH ¼ 88, Kc ¼ 100; III. cH ¼ 88, Kc ¼ 50; IV. cH ¼ 88, Kc ¼ 10; V. cH ¼ 185, Kc ¼ 1; VI. cH ¼ 185 103, Kc ¼ 103; VII. cH ¼ 185 106, Kc ¼ 106; dots e experimental data [13].
vcþ vU ¼ Kc ðcH cþ Þ*H½cH cþ jH ; þ zþ cþ vy vy y¼1;1
(39)
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not valid for these experimental conditions. Value r/Fc0 obtained by scale analysis is less than the value calculated by microscopic model; see Fig. 7. It can be explained by decreasing of Debye length with charge growing. Moreover, the microscopic model gives the distribution of the dimensionless charge when the scales analysis produces a characteristic mean value. 4.2. Free charge density dynamic models
Fig. 7. Dimensionless charge density of double layer near the outlet at different mean flow velocities, model with hybrid boundary condition, Eq. (39).
quickly falls to zero in the bulk of the flow. In the diffuse layer the small charge density condition is not met because the dimensionless charge r=Fc0 within Debay layer is about 30. This numerically obtained result can be supported with scale analysis. The concentration can be estimated via the solution conductivity, see Einstein relation:
c0 w
RT s : F 2 D0
(40)
For the experimental data reported in Ref. [13], see Table 1, estimation of Debye length gives:
sffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3 3 0 RT 3 3 0 D0 lD w : w s c0 F 2
(41)
Debye layer is much thinner than the channel thickness for the reasonable range of diffusion coefficient (D0 < 109 m2/s). Thus, the reduced form of the hydrodynamic velocity profile near the wall can be written as:
vy ðyÞz3Vð1 y=dÞ;
(42)
The stream current can be expressed approximately by the relation:
The free charge dynamic models [14,30e33] used small charge density assumption Eq. (3) for reducing the complex system of equations to single one. In the previous section it has been shown that in some cases, in particular for experimental data from [13] this assumption is not valid. Thus complete system of diffusion equations and Poisson’s equation should be solved. It should be noted also that using of the free charge dynamic model meets a difficulty related to specification of boundary conditions. Indeed, it is not simple to precise the physical meaning of the boundary condition for the free charge at the surface in the case of the complex interfacial reactions. 4.3. Two-ions microscopic model The developed microscopic model takes into consideration the presence of only two types of ions in the solution. It has been demonstrated that usual boundary conditions for interface processes, namely constant concentration or constant current boundary conditions, are not suitable for simultaneous fitting transient and steady state experimental data. The introducing of mixed interfacial kinetics does not improve the situation. The simultaneous fitting of all available experimental data can be provided by the means of hybrid boundary condition for positive ion, Eq. (39), which has no clear physical meaning in the frame of two ions model. Indeed, this boundary condition corresponds to the parallel running of two interface processes. The first interface process runs under the diffusion control (fast rate reaction), when the second one has a slow reaction rate. Evidently, it implies the necessity to use a more complex model. For example, this can be a model with two positive and one negative ion types. One of the positive ions can be involved into fast interface reaction and the other one participates in the slow interface process. 5. Conclusion
2
Ist w2H lD rvy ðy*Þz
3HlD V r; d
(43)
where y* ¼ d lD/2. Consequently, charge density r can be estimated as:
rw
Ist d 2 3HlD V
¼
2dIst F 2 c0 ; 3HV 3 3 0 RT
(44)
and for the dimensionless charge the following equation is obtained:
r Fc0
2dIst F : w 3HV 3 3 0 RT
(45)
It should be noted that Eq. (45) does not depend directly on the diffusion coefficient. In the case of a thin Debye layer Eq. (45) is useful for estimation of the electrical charge and for verification of the small charge density approximation, Eq. (3). For example, for the experimental data presented in Table 1 the estimation of the dimensionless charge by means of Eq. (45) gives r/Fc0 z 5. It confirms that the small charge density assumption is
In this paper a microscopic model of electrification phenomena has been developed and applied for interpretation of the experimental data on electrification phenomena in steady state and transient regimes. Firstly, it has been demonstrated that the traditional small charge density assumption is not valid with respect to treated experimental data. This conclusion has been supported by the numerical modelling and scaling analysis. The fitting of experimental data has been provided using various boundary conditions for interface processes. A remarkable contradiction has been revealed. The boundary condition with constant concentration of electro active ions fits adequately steady state streaming current data but is unsuccessful for explanation of transient experiments. On the other hand, the constant current boundary condition is in a good agreement with the transient experiments but does not allow fitting of the steady state ones. The both types of experiments can be explained satisfactorily using the hybrid boundary condition which corresponds to the coexistence of two reactions on the active surface. The first reaction runs under the diffusion control and the second one is limited by the kinetics. The present study shows that binary microscopic models seem to be inappropriate for the description of all available experimental
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information on electrification phenomena in laminar channel flow. Indeed, these models fit the transient experimental data using the constant current boundary condition (slow rate of interfacial reaction) or steady state experimental data by means of the constant concentration boundary condition (fast rate of interfacial reaction). Introducing of a mixed interfacial kinetics does not allow simultaneous fitting of the transient and the steady state experimental data (see Figs. 5 and 6). Sufficient description of both steady state and transient experimental data [13] can be provided by means of the hybrid boundary condition Eq (39) which is an artificial combination of the constant current (slow interfacial reaction) and the constant concentration (fast interfacial reaction) boundary conditions, see Figs. 3 and 4. In our opinion, the multi-component microscopic models which consider at least two interfacial reactions with different kinetic rates should be applied for the description of studied electrification processes.
Acknowledgements The authors thank Poitou e Charentes Region, France which has supported this work by granting the position of visiting scientist to one of them. References [1] A. Klinkenberg, J.L. Van Der Minne, Electrostatics in the Petroleum Industry, Elsevier, Amsterdam, London, 1958. [2] J.A. Palmer, J.K. Nelson, Practical implications of transformer flow electrification studies, J. Electrostat. 40e41 (1997) 693e698. [3] H.L. Walmsley, The avoidance of electrostatics hazards in the petroleum industry, J. Electrostat. (special issue) (1992). [4] U. Von Pidoll, H. Krämer, H. Bothe, Avoidance of electrostatic hazards during refueling of motorcars, J. Electrostat. 40e41 (1997) 523e528. [5] O. Stern, Zur Theorie der Electrolytischen, Z. Elektrochem. 30 (1924) 508e516. [6] G. Touchard, T.W. Patzek, C.J. Radke, A physicochemical explanation for flow electrification in low-conductivity liquids in contact with a corroding wall, IEEE Trans. Ind. Appl. 32 (5) (1996) 1051e1057. [7] A.P. Washabaugh, M. Zahn, A chemical reaction-based boundary condition for flow electrification, IEEE Trans. Dielectr. Electr. Insul. 4 (1997) 688e709. [8] M. El-Adawy, T. Paillat, J.M. Cabaleiro, G. Touchard, Numerical simulation of the electrical double layer development: physicochemical model at the solid and dielectric liquid interface for laminar flow electrification phenomenon, IEEE Trans. Dielectr. Electr. Insul. 18 (5) (2011) 1463e1475. [9] J.M. Cabaleiro, T. Paillat, O. Moreau, G. Touchard, Electrical double layer’s development analysis: application to flow electrification in power transformers, IEEE Trans. Ind. Appl. 45 (2009) 597e605. [10] E. Moreau, T. Paillat, G. Touchard, Space charge density in dielectric and conductive liquids flowing through a glass pipe, J. Electrostat. 51e52 (2001) 448e454.
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[11] T. Paillat, E. Moreau, G. Touchard, Space charge density at the wall in the case of heptane flowing through an insulating pipe, J. Electrostat. 53 (2001) 171e182. [12] G. Touchard, Flow electrification of liquids, J. Electrostat. 51e52 (2001) 440e447. [13] J.M. Cabaleiro, T. Paillat, O. Moreau, G. Touchard, Flow electrification of dielectric liquids in insulating channels: limits to the application of the classical wall current expression, J. Electrostat. 66 (2008) 79e83. [14] B. Abedian, A.A. Sonin, Theory for electric charging in turbulent pipe flow, J. Fluid Mech. 120 (1982) 199e217. [15] V.N. Pribylov, Motion of a two-temperature plasma in the channel of a disctype magnetohydrodynamic generator, Fluid Dyn. 5 (1973) 911e919. [16] V.N. Pribylov, The charging of a dielectric fluid during its flow near a plane metal plate, Colloid J. 64 (2002) 596e600. [17] V.N. Pribylov, Charging of Dielectric Liquid Near a Rotating Disk, Vest. Mosk. Univ. Ser 1 Mat. Mek., 2003, pp. 39e44. [18] I.L. Pankratieva, V.A. Polyansky, Bipolar-charged Layers Adjacent to Electrodes in Nonpolar Low-conducting Liquids. IEEE International Conference on Conduction and Breakdown in Dielectric Liquids, ICDL, 1999, pp. 82e85. [19] V.A. Polyansky, R.T. Faizulin, I.L. Pankratieva, V.I. Sakharov, On Some Electrohydrodynamic Phenomena Due to Charge Injection into a Flow of Lowconductivity Liquid. IEEE International Conference on Conduction and Breakdown in Dielectric Liquids, ICDL, 1999, pp. 111e113. [20] V.N. Pribylov, L.T. Chernyi, Electrization of dielectric liquids flowing in tubes, Fluid Dyn. 14 (1979) 844e849. [21] B. Abedian, A.A. Sonin, Electric currents generated by turbulent flows of liquid hydrocarbons in smooth pipes: experiment vs. theory, Chem. Eng. Sci. 41 (1986) 3183e3189. [22] H.L. Walmsley, G. Woodford, The generation of electric currents by the laminar flow of dielectric liquids, J. Phys. D: Appl. Phys. 14 (1981) 1761e1782. [23] V.N. Pribylov, L.T. Chernyi, Electrification in tube flow of organic liquids with an admixture of strong electrolyte, J. Appl. Mech. Tech. Phys. 23 (1982) 342e 346. [24] V.N. Pribylov, Electrization of a Dielectric Liquid by Its Flow Above the Plane Plate. IEEE International Conference on Conduction and Breakdown in Dielectric Liquids, ICDL, 1999, pp. 74e77. [25] V.N. Pribylov, The charging of a liquid dielectric upon its flow past a flat-plate, Colloid J. 68 (2006) 316e320. [26] I.L. Pankrat’eva, V.A. Polyanskij, Electrofication of weakly conducting multicomponent liquids at laminar flow in a flat channel, Phys. Chem. Kin. (2006).http://www.chemphys.edu.ru/media/files/2006-05-11-001.pdf. [27] V.A. Polyanskii, V.N. Pribylov, Effect of longitudinal diffusion on the charging current of fluid dielectrics during their flow in a channel, Colloid J. (2004) 329e332. [28] V.N. Pribylov, The charging of a dielectric fluid during its flow near a plane metal plate, Colloid J. 64 (2002) 660e664. [29] P.W. Huber, A.A. Sonin, Theory for electric charging in liquid hydrocarbon filtration, J. Colloid Interface Sci. 61 (1977) 109e125. [30] J.A. Palmer, J.K. Nelson, M.A. Brubaker, The Formulation of models for the description of streaming electrification in transformer structures, IEEE Trans. Dielectr. Electr. Insul. 10 (2003) 920e932. [31] J.A. Palmer, J.K. Nelson, Parametric performance of a transient streaming electrification model, IEEE Trans. Dielectr. Electr. Insul. 4 (1997) 418e424. [32] J.A. Palmer, J.K. Nelson, The simulation of short-term streaming electrification dynamics, J. Phys. D: Appl. Phys. 30 (1997) 1207e1213. [33] T. Paillat, E. Moreau, G. Touchard, Flow electrification through porous media, J. Loss Prev. Process. Ind. 14 (2001) 91e93. [34] Y. Qiao, A. Han, V.K. Punyamurtula, Electrification of a nanoporous electrode in a continuous flow, J. Phys. D: Appl. Phys. 41 (2008) 085505.
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Modelling of electrification in steady state and transient regimes D.A. Bograchev a, S. Martemianov b, *, T. Paillat b a b
Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninsky pr. 31, Moscow 119991, Russia Institut Prime UPR CNRS 3346, CNRS - Université de Poitiers – ENSMA, Departement FTC e Branche Fluide, 40, avenue du Recteur Pineau, 86022 Poitiers, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 July 2011 Received in revised form 13 July 2012 Accepted 15 July 2012 Available online 25 August 2012
A microscopic model of electrification phenomena in steady state and transient regimes has been developed. The model has been applied for analysis of experimental data failing to be explained in the frame of classical macroscopic approach. Various boundary conditions have been studied regarding the possibility of experiential data fitting. The hybrid boundary condition has been suggested for explaining steady state and transient experiments. The proposed boundary condition can be related to the coexistence of two reactions on the active surface. The first reaction runs under the diffusion control (fast reaction) and the second one is limited by the kinetics (slow reaction). Ó 2012 Elsevier B.V. All rights reserved.
Keywords: Electrification Double layer Streaming current
1. Introduction The appearance of volume charges in dielectric liquid flows causes their electrification and probable ignitions or explosions in the petroleum and electrical industries [1e4]. The electrification of dielectric liquid flows has been studied in numerous works; it has been shown that this phenomenon concerns the diffuse part of electric double layer. Free charges (ions) can be removed from the surface by forced flow and cause the electrical current in liquid (streaming current). The streaming current depends on hydrodynamic conditions, liquid conductivity and surface properties which define the type of the liquid/surface interactions (interface reactions). Various interface reactions and different mechanisms of the charge origin at the surface have been suggested in the literature [5e9]. However, usually the mechanism of the interface reactions is unknown and just postulated from some reasonable considerations. In addition, the nature of electrical conductivity, in other words the origin of free charges in weakly conducting liquids, is still a question. A number of approaches for explanation of the electrification phenomena exist in the literature. In the frame of the macroscopic approach the space distribution of electrical charge r(x,y) is assumed to be known:
* Corresponding author. Tel.: þ33 5 49 45 39 04; fax: þ33 5 49 45 44 44. E-mail address: [email protected] (S. Martemianov). 0304-3886/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2012.07.006
rðx; yÞ ¼ rw ðxÞ
coshðy=d0 Þ ; coshðd=d0 Þ
(1)
where axis x is oriented along the flow, axis y is normal to the surface, rw is the electrical charge on the surface, d0 is Debye length, d is the half thickness of the channel. The main problem is related to the surface charge determining with respect to boundary conditions. In particular, the following relation between the wall current density and the charge density near the wall has been assumed in [10e13]:
iw ¼ K½rwN rw ;
(2)
where K e is an empirical coefficient for a given interface reaction, cH is surface charge density in fully developed conditions. Fully developed conditions correspond either to the stagnant liquid or to the outlet of the very long channel. It is supposed also that the surface charge density in the inlet of the channel is equal to zero. Eq. (1) is usually accepted in the frame of the macroscopic approach. It should be noted that the hyperbolic cosine profile, Eq. (1), is valid only if the electrical charge in the volume is much less than product of equilibrium concentration and the Faraday constant:
r Fc0
<< 1:
(3)
This is so called the small charge density assumption which allows linearization of the problem. Assumptions similar to inequality (3) have been used in some analytical studies in the frame of microscopic approach [14e17] for linearization of the
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governing system of the equations. Nevertheless, there are some cases when condition (3) is not valid and the electrification problem should be treated without this traditional supposition. The other type of the electrification theories are based on the microscopic approach. Microscopic models [7,8,14e33] are based on the diffusion equations for ions and Poisson’s equation for the electric potential. The main result of these theories is the dependence of steady state streaming current on flow velocity. The streaming current is calculated as the integral of the free charge density and hydrodynamic velocity product. The free charge density is calculated using equations for the concentrations and the electrical potential fields. For the microscopic model the boundary conditions concerning electrochemical surface process vary greatly. In many works [16,17,20] the constant boundary condition for concentration has been utilized. This boundary condition can be justified for the case of fast interfacial reaction when the electrification phenomena run under the diffusion control. In paper [23] the following relation has been used for describing electrochemical process at the surface: 0 Ji ¼ KfP;N ci KfP;N ;
(4)
where Ji e molar flux on the surface 0 ðJi ¼ Di Vci ðzi FDi =RTÞci VUÞ, KfP,N e adsorption rate, KfP;N desorption rate. Eq. (4) is based on Langmuir adsorption isotherm, but in Ref. [23] the reduced form of this relation has been used (socalled ‘radiation’ boundary condition):
Ji ¼ KfP;N ci
(5)
The similar conception of electrochemical processes at the surfaces has been developed in Ref. [7]. A free charge density dynamic approach has been developed in the works [14,29] on the basis of the diffusion equations and the equation for the electric potential. It is possible to reduce the problem to a single equation for the free electrical charge density by combining of these equations: The obtained single equation has been used for studying of various problems [14,30e32]. The validity of this approach is related to the small charge density assumption, Eq. (3). Different types of forced flows have been studied in the literature on electrification. The most popular cases are the pipe Poiseuille flow [10,11,20,23] and the flat channel Poiseuille flow [13,19,27]. The flat channel flow in consideration of the inlet region has been studied in the works [30e32]. Turbulent pipe flows have been examined in papers [11,14,22]. The turbulent Couette flow between rotating cylinders has been studied in Ref. [7]. There are papers devoted also to electrification in rotating disc flow [17] and the forced flow along the plane plate [16]. The experimental studies of liquid flow electrification in the porous media are presented in Refs. [33,34]. The objective of this work is a microscopic model of electrification which can be used for explanation of electrification phenomena in steady state and transient regimes. The required experimental information for model verification is available in Ref. [13]. The macroscopic approach fails in explanation of the experimental data presented in this work. In the cited paper the conclusion has been done that the kinetic rate depends on the flow velocity and some scenarios have been reviewed when it could take place. This hypothesis is rather discussable. Two types of experimental dependencies are presented in Ref. [13]. The first type is the dependence of stationary streaming current on hydrodynamic velocity of fluid at the channel. The streaming current varies in the range 30e120 pA for the mean flow rate 0.4e1.5 m/s. The second type of the dependences concerns a transient regime. To be exact, the liquid remains stagnant within a certain delay time td with the subsequent flow start up and the streaming current growth to a certain peak value and then it falls to
a fixed stationary value, see Fig. 1. Additional information is related to the dependence of the peak streaming current on the delay time td. The limited value at the peak streaming current for infinitely large times is the necessary element of the macroscopic theory; this limited value is related to charge density distribution in fully developed conditions (infinitely long channel). In present work we will try to explain experimental data from Ref. [13] by a microscopic model which takes into account the mass transfer of two types of ions and different boundary conditions for interfacial electrochemical processes. Despite of the model simplicity, this approach is very attractive as it is based on the exact transfer equations and the number of fitting parameters is reduced to a possible minimum. Thus, verification of some traditional assumptions used for interpretation of the electrification phenomena can be provided. 2. Governing equations of model and boundary conditions Let us consider the liquid with two ions in a small concentration c0, which flows through a rectangular channel, see Fig. 2. In the channel inlet the flow is electroneutral. A certain interface process generates a flux of the positive ions on the channel walls. Thus, a streaming electrical current arises in the channel outlet due to a charge disbalance. The equations which describe the electrification phenomenon for the system with two kinds of ions can be presented as:
vcþ zþ FDþ þ vVcþ ¼ Dþ Dcþ þ divðcþ VUÞ vt RT
(6)
vc z FD þ vVc ¼ D Dc divðc VUÞ vt RT
(7)
Fðzþ cþ z c Þ ¼ 3 0 3 L DU
(8)
The Eqs. (6) and (7) correspond to the transfer of negative and positive ions by diffusion, convection and migration. Eq. (8) is Poisson equation for the electric potential U. It is assumed that the Einstein relation is valid. Here c e ion concentrations, z e ion charges, D e diffusion coefficients, F e Faraday number, 3 0 e permittivity of free space, 3 e relative permittivity of liquid, R e gas constant, T e temperature. It is accepted that velocity in the channel is related to flat Poiseuille flow:
vm ¼
3 y2 V 1 2 ; 2 d
Fig. 1. Scheme of typical experimental curve.
(9)
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519
dependence on hydrodynamic velocity of fluid. These data, see for example Ref. [13], can be treated in the frame of the steady state problem. The set of equations and boundary conditions (6)e(16) can be rewritten in a dimensionless form (bellow all the variables c ; U; x; y are treated like dimensionless):
Fig. 2. Scheme model of electrification and coordinate system.
where d e channel half-thickness, V e mean flow velocity. The boundary condition for potential is:
Ujy¼d;d ¼ 0:
(10)
It means that the cannels walls are grounded and have a sufficiently high electrical conductivity. It is supposed that there are no migration fluxes in channel inlet and outlets, so:
vU ¼ 0; vx x¼0;L
(11)
(12)
The other possibility is a constant current boundary condition for positive ions:
vcþ zþ FDþ vU Dþ ¼ jH þ cþ vy RT vy y¼d;d
(13)
The flux for negative ions on the wall is equal to zero:
vc z FD vU D ¼ 0: c vy RT vy y¼d;d
(14)
The inlet boundary condition corresponds to the electroneutrality of the flow:
cþ jx¼0 ¼
z c jx¼0 ¼ c0 zþ
(15)
Finally, it is supposed that in the outlet the mass flux is due to convection only:
vcþ vc 0 ¼ Dþ ; 0 ¼ D : vx x¼L vx x¼L
(17)
vc 3 1 y2 Pe ¼ aDc az divðc VUÞ; vx 2
(18)
ðzþ cþ z c Þ ¼ De2 DU:
(19)
We have used d, Dþ/d, c0, RT/F as the scales for coordinates, velocity, concentration and electrical potential correspondingly. Here, Peclet number Pe ¼ dV=Dþ characterizes the ratio of the diffusion and the convective processes; Debye number pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi De ¼ RT 3 0 3 L =c0 d2 F 2 characterizes the ratio of the double layer thickness and the characteristic size of the system and a ¼ D =Dþ is the ratio of diffusion coefficients. Accordingly, the dimensionless boundary conditions at the inlet can be written as:
cþ ¼
z c ¼ 1jx¼0 ; zþ
(20)
vU ¼ 0: vx x¼0
(21)
Boundary conditions at the outlet in dimensionless form is:
where L is the channel length. The boundary condition for concentrations, which depend on the surface kinetic processes flux, can be specified by different ways. The first possibility is a constant concentration boundary condition:
cþ jy¼d;d ¼ cH
vc 3 þ 1 y2 Pe ¼ Dcþ þ zþ divðcþ VUÞ; 2 vx
(16)
0 ¼
vcþ vc vU a ; 0 ¼ ; vx x¼ L vx x¼ L vx x¼ d
d
L d
¼ 0:
(22)
Boundary conditions (12)e(14) can be rewritten in the form:
vcþ vU ¼ jH ; þ zþ cþ vy y¼1;1 vy
(23)
cþ ¼ cH ;
(24)
vc vU ¼ 0: z c vy vy y¼1;1
(25)
Here, jH, cH are the dimensionless variables. The diffusion coefficients of ions in the liquids are very small (less then 109 m2/s) and the channel thickness is about few millimetres. Thus, for the observed in electrification experiments hydrodynamic velocities Peclet number is very high. Therefore the boundary layer approximation can be used and all transports along the x axis, apart the convection, can be neglected. The governing set of equations in the boundary layer approximation takes the form:
vc 3 v2 c þ v cþ vU þ 1 y2 Pe þ zþ ; ¼ 2 vy vy vx vy2
(26)
vc 3 v2 c v c vU 1 y2 Pe az ; ¼ a 2 vy vy vx vy2
(27)
v2 U : vy2
2.1. Steady state problem
ðzþ cþ zþ c Þ ¼ De2
The main experimental information on the electrification phenomena deals with the measurements of the streaming current
Eqs. (17)e(28) allow calculating the streaming current in steady state regime Ist and its dependence on the mean flow velocity V.
(28)
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2.2. Peak streaming current value
3. Numerical simulation
Another important information dealing with the electrification phenomena can be obtained by using the transient experiments [13]. In these experiments the flow is stopped, the fluid remains without motion during a certain stagnant time td and after the flow is switched on. During the transient period the streaming current reaches a certain peak value Ipeak and tends to its stationary level Ist, (see Fig. 1). The dependence Ipeak (td) is an increasing monotonic limited function. The characteristic time scale of this dependence is some thousand seconds [13]. In the stagnant liquid the electrical double layer grows up to some stationary state. When the flow starts, the streaming current reflects the double layer structure which has been reached during the stagnant time td. The increasing of the streaming current during the first moments is rather fast (some seconds) but not instantaneous. After the peak value is reached, the streaming current decreases to its stationary value due to the double layer dumping by the forced flow. The system of equations describing the streaming current during the flow acceleration (Fig. 1) should take into account the time of the evolution of velocity profile vx(t,y). Using the above introduced dimensionless variables, this system can be presented in the following form:
The numerical simulations have been realized by using Comsol Femlab Multiphisics package. In order to provide the calculations in the relatively thin Debye layer the mesh density has been increased near the wall. Firstly, the steady state stream current dependence Ist ðVÞ has been obtained considering the simplest boundary condition, which implies a constant concentration of positive ions on the walls:
vcþ vcþ v2 c þ v cþ vU þ zþ þ vx ðy; tÞ ¼ ; vt vx vy vy vy2
vc vc v2 c v c vU az ; þ vx ðy; tÞ ¼ a vy vy vt vx vy2 ðzþ cþ z c Þ ¼ De2
v2 U ; vy2
(30)
(31)
Z vm ðzþ cþ ðtd Þ z c ðtd ÞÞds;
(36)
The calculated streaming current is presented in Fig. 3 (curve a). A good agreement with the experiment can be achieved by fitting the constant cH, the coefficients of diffusion and charge of ions (see Table 1). However, the calculations with the boundary condition (36) does not allow any correct describing of the peak current dependence Ipeak (td), see Fig. 4 (curve a). The transition time observed in the experiments is much more important that the one obtained in the numerical simulations. On the other hand, if a constant rate of the concentration flux is used as a boundary condition:
vcþ vU ¼ jH ; þ zþ cþ vy vy y¼1;1
(37)
(29)
where vx(y,t) is hydrodynamic velocity profile which depends on time. Solving this problem requires the knowledge of fluid motion during acceleration vx(t,y), thus NaviereStokes equations should be solved in addition to Eqs. (29)e(31). However the peak value of the streaming current Ipeak can be obtained [13] without solution of the transient hydrodynamic problem using the following equation:
Ipeak ¼
cþ jy¼1;1 ¼ cH
(32)
it becomes possible to explain the experimental data related to the dependence of the peak streaming current on the stagnant time Ipeak (td) by using jH as the only fitting parameter, see Fig. 4 (curve b) and Table 1. At the same time, the calculations of the steady state streaming current Ist with the boundary condition (37) leads to the independence of this value on the mean flow velocity V under steady state, see Fig. 3 (curve b). This is the obvious contradiction with the experimental data. So, the using of the simplest assumptions for the kinetics of the interface processes, Eqs. (36) and (37) do not allow simultaneous describing of Ist ðVÞ and Ipeak (td) experimental data. Using traditional electrochemical notions, it is possible to associate the boundary condition, Eq. (36) with the high rate interfacial reaction and the boundary condition, Eq. (37), with the law rate reaction. Let us consider now the intermediate case of the interfacial process with the mixed kinetics. The mixed kinetics leads to the following boundary condition:
where td is the stagnant time and vm ¼ 3/2 Pe(1 y2) is the steady state velocity profile (Poiseuille flow). Neglecting the border effects, the concentration profiles of positive cþ and negative c ions in motionless liquid (0 < t < td) can be defined using non stationary 1D approximation:
vcþ v2 c þ v cþ vU þ zþ ; ¼ vy vy vt vy2
(33)
vc v2 c v c vU az ; ¼ a vy vy vt vy2
(34)
v2 U : vy2
(35)
ðzþ cþ z c Þ ¼ De2
Eqs. (32)e(35) allow calculating the peak streaming current value Ipeak.
Fig. 3. Dependence of steady state streaming current Ist on mean flow rate a e constant concentration boundary condition; b e constant current boundary condition; c e hybrid boundary condition, Eq. (39); dots e experimental data [13].
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521
Table 1 Physical parameters of system used in calculation of the model. Parameters
Value
Source
c0 e Initial ion concentration, [mol/m3]
1.3 109
cH e Ion concentration at the surface, [mol/m3] Dþ ¼ D ¼ D0 e Coefficients of diffusion [m2/s] deThe half-channel thickness, [m] H e Channel width, [m] jH e Ion flux at the surface,[mol/s/m2] L eChannel length [m] Kc e Reaction rate,[m/s] s e Conductivity of the liquid [S/m] zþ ¼ z e Charge numbers of ions 3 3 0 e Effective dielectric constant, [F/m]
1.17 107 1 1011 1.5 103 0.03 1.60 1013 0.3 0.01 4 1013 2 1.95 1011
From the model of two ions Fitting Fitting [13] [13] Fitting [13] Fitting [13] Fitting [13]
vcþ vU ¼ Kc ðcH cþ Þ þ zþ cþ vy vy y¼1;1
(38)
where Kc can be interpreted as the constant of the reaction rate. At large rate constants Kc >> 1 this boundary condition, Eq. (38) reduces to the boundary condition, Eq. (36). The decreasing of the rate constant (Kc / 0) with the simultaneous increasing of the magnitude cH (cH / N, Kc *cH / const) turns the boundary condition Eq. (38) into the constant flux boundary condition Eq. (37). Numerical modelling of Ist ðVÞ and Ipeak(td) dependences has been provided for different values of constant rates Kc, see Figs. 5 and 6. It is obvious that simultaneous describing of Ist ðVÞ and Ipeak(td) experimental data is not possible with the boundary condition, Eq. (38). Naturally, for the limiting cases of high and law constant rates the numerical simulations reproduce correctly only one of the above mentioned experimental dependences. The intermediate values of the reaction rate lead to disagreement with both experimental curves. Finally, a hybrid of boundary conditions (36) and (37) has been used in the numerical simulations. This boundary condition can be expressed by the following equation:
Fig. 5. Dependence of steady state streaming current Ist on mean flow rate. The model with mixed boundary condition, Eq. (38) and different kinetic parameters; I. cH ¼ 88, Kc ¼ 1000; II. cH ¼ 88, Kc ¼ 100; III. cH ¼ 88, Kc ¼ 50; IV. cH ¼ 88, Kc ¼ 10; V. cH ¼ 185, Kc ¼ 1; VI. cH ¼ 185 103, Kc ¼ 103; VII. cH ¼ 185 106, Kc ¼ 106; dots e experimental data [13].
see Fig. 3 (curve c), and Fig. 4 (curve c). Eq. (39) is a certain generalization of conditions (36) and (37). At the small time condition (39) works like constant concentration boundary condition (36), however when cþ becomes more than cH Eq. (39) turns into constant flux boundary condition (37) due to Heaviside step function. 4. Comparison with traditional approaches 4.1. Macroscopic models
where H is Heaviside step function. By means of this boundary condition the both experimental curves can be fitted satisfactorily,
The developed microscopic model can be used for the verification of some assumptions generally used in the frame of the macroscopic model [13]. The first key assumption is the hyperbolic cosine profile of the diffuse layer, Eq. (1), which is valid only for the small charge density condition, Eq. (3). The second assumption relates to modelling of kinetics process on the surface; see Eq. (2). Apparently Eq. (2) cannot describe all types of the surface reactions, in particular, the slow rate reactions. The present model allows fitting of the steady state experimental data using the high value of the surface concentration; see Table 1. Thus, the small charge density approximation is not justified, Eq. (3). The calculated dimensionless charge density in the channel outlet for various mean flow velocities is presented in Fig. 7. The charge density is considerably high near the surface and
Fig. 4. Dependence of peak streaming current Ipeak on delay time td a e constant concentration boundary condition; b e constant current boundary condition; c e hybrid boundary condition, Eq. (39); dots e experimental data [13].
Fig. 6. Dependence of peak streaming current Ipeak on delay time td. The model with mixed boundary condition, Eq. (38) and different kinetic parameters; I. cH ¼ 88, Kc ¼ 1000; II. cH ¼ 88, Kc ¼ 100; III. cH ¼ 88, Kc ¼ 50; IV. cH ¼ 88, Kc ¼ 10; V. cH ¼ 185, Kc ¼ 1; VI. cH ¼ 185 103, Kc ¼ 103; VII. cH ¼ 185 106, Kc ¼ 106; dots e experimental data [13].
vcþ vU ¼ Kc ðcH cþ Þ*H½cH cþ jH ; þ zþ cþ vy vy y¼1;1
(39)
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not valid for these experimental conditions. Value r/Fc0 obtained by scale analysis is less than the value calculated by microscopic model; see Fig. 7. It can be explained by decreasing of Debye length with charge growing. Moreover, the microscopic model gives the distribution of the dimensionless charge when the scales analysis produces a characteristic mean value. 4.2. Free charge density dynamic models
Fig. 7. Dimensionless charge density of double layer near the outlet at different mean flow velocities, model with hybrid boundary condition, Eq. (39).
quickly falls to zero in the bulk of the flow. In the diffuse layer the small charge density condition is not met because the dimensionless charge r=Fc0 within Debay layer is about 30. This numerically obtained result can be supported with scale analysis. The concentration can be estimated via the solution conductivity, see Einstein relation:
c0 w
RT s : F 2 D0
(40)
For the experimental data reported in Ref. [13], see Table 1, estimation of Debye length gives:
sffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3 3 0 RT 3 3 0 D0 lD w : w s c0 F 2
(41)
Debye layer is much thinner than the channel thickness for the reasonable range of diffusion coefficient (D0 < 109 m2/s). Thus, the reduced form of the hydrodynamic velocity profile near the wall can be written as:
vy ðyÞz3Vð1 y=dÞ;
(42)
The stream current can be expressed approximately by the relation:
The free charge dynamic models [14,30e33] used small charge density assumption Eq. (3) for reducing the complex system of equations to single one. In the previous section it has been shown that in some cases, in particular for experimental data from [13] this assumption is not valid. Thus complete system of diffusion equations and Poisson’s equation should be solved. It should be noted also that using of the free charge dynamic model meets a difficulty related to specification of boundary conditions. Indeed, it is not simple to precise the physical meaning of the boundary condition for the free charge at the surface in the case of the complex interfacial reactions. 4.3. Two-ions microscopic model The developed microscopic model takes into consideration the presence of only two types of ions in the solution. It has been demonstrated that usual boundary conditions for interface processes, namely constant concentration or constant current boundary conditions, are not suitable for simultaneous fitting transient and steady state experimental data. The introducing of mixed interfacial kinetics does not improve the situation. The simultaneous fitting of all available experimental data can be provided by the means of hybrid boundary condition for positive ion, Eq. (39), which has no clear physical meaning in the frame of two ions model. Indeed, this boundary condition corresponds to the parallel running of two interface processes. The first interface process runs under the diffusion control (fast rate reaction), when the second one has a slow reaction rate. Evidently, it implies the necessity to use a more complex model. For example, this can be a model with two positive and one negative ion types. One of the positive ions can be involved into fast interface reaction and the other one participates in the slow interface process. 5. Conclusion
2
Ist w2H lD rvy ðy*Þz
3HlD V r; d
(43)
where y* ¼ d lD/2. Consequently, charge density r can be estimated as:
rw
Ist d 2 3HlD V
¼
2dIst F 2 c0 ; 3HV 3 3 0 RT
(44)
and for the dimensionless charge the following equation is obtained:
r Fc0
2dIst F : w 3HV 3 3 0 RT
(45)
It should be noted that Eq. (45) does not depend directly on the diffusion coefficient. In the case of a thin Debye layer Eq. (45) is useful for estimation of the electrical charge and for verification of the small charge density approximation, Eq. (3). For example, for the experimental data presented in Table 1 the estimation of the dimensionless charge by means of Eq. (45) gives r/Fc0 z 5. It confirms that the small charge density assumption is
In this paper a microscopic model of electrification phenomena has been developed and applied for interpretation of the experimental data on electrification phenomena in steady state and transient regimes. Firstly, it has been demonstrated that the traditional small charge density assumption is not valid with respect to treated experimental data. This conclusion has been supported by the numerical modelling and scaling analysis. The fitting of experimental data has been provided using various boundary conditions for interface processes. A remarkable contradiction has been revealed. The boundary condition with constant concentration of electro active ions fits adequately steady state streaming current data but is unsuccessful for explanation of transient experiments. On the other hand, the constant current boundary condition is in a good agreement with the transient experiments but does not allow fitting of the steady state ones. The both types of experiments can be explained satisfactorily using the hybrid boundary condition which corresponds to the coexistence of two reactions on the active surface. The first reaction runs under the diffusion control and the second one is limited by the kinetics. The present study shows that binary microscopic models seem to be inappropriate for the description of all available experimental
D.A. Bograchev et al. / Journal of Electrostatics 70 (2012) 517e523
information on electrification phenomena in laminar channel flow. Indeed, these models fit the transient experimental data using the constant current boundary condition (slow rate of interfacial reaction) or steady state experimental data by means of the constant concentration boundary condition (fast rate of interfacial reaction). Introducing of a mixed interfacial kinetics does not allow simultaneous fitting of the transient and the steady state experimental data (see Figs. 5 and 6). Sufficient description of both steady state and transient experimental data [13] can be provided by means of the hybrid boundary condition Eq (39) which is an artificial combination of the constant current (slow interfacial reaction) and the constant concentration (fast interfacial reaction) boundary conditions, see Figs. 3 and 4. In our opinion, the multi-component microscopic models which consider at least two interfacial reactions with different kinetic rates should be applied for the description of studied electrification processes.
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