The Electrokinetic Microfluidic Flow In Mu Lti-Channels With Em Ergent .
Korean J. Chem. Eng., 22(4), 528-535 (2005)The Electrokinetic Microfluidic Flow in Multi-Channels with Emergent ApplicabilityToward Micro Power GenerationTae Seok Lee, Myung-Suk Chun , Dae Ki Choi*, Suk Woo Nam* and Tae-Hoon Lim*†Complex Fluids Research Laboratory, *Environment and Process Technology Div.,Korea Institute of Science and Technology (KIST), PO Box 131, Cheongryang, Seoul 130-650, Korea()Received 17 January 2005 accepted 9 June 2005Abstract In order to elaborate the possible applicability of microfluidic power generation from conceptualizationto system validation, we adopt a theoretical model of the electrokinetic streaming potential previously developed forthe single channel problem. The ion transport in the microchannel is described on the basis of the Nernst-Planck equation, and a monovalent symmetric electrolyte of LiClO4 is considered. Simulation results provide that the flow-inducedstreaming potential increases with increasing the surface potential of the microchannel wall as well as decreasing thesurface conductivity. The streaming potential is also changed with variations of the electric double layer thickness normalized by the channel radius. From the electric circuit model with an array of microchannels, it is of interest toevaluate that a higher surface potential leads to increasing the power density as well as the energy density. Both thepower density and the conversion efficiency tend to enhance with increasing either external resistance or number ofchannels. If a single microchannel is assembled in parallel with the order of 103, the power density of the system employing large external resistance is estimated to be above 1 W/m3 even at low pressure difference less than 1 bar.Key words: Microfluidics, Electrokinetics, Streaming Potential, Microchannel, Electrolyte Solution, Power DensityINTRODUCTIONnetic micro battery consisting of an array of microchannels [Yanget al., 2003]. Yang et al. conducted analytical solutions on the timedependent microflow, and found a good agreement between the predicted results and those from experiments with microporous glassfilter. Their results suggest the hydrostatic pressure can be converted into electrical work in the order of W/m , depending on theproperties of the electrolyte solution and channel wall.Current MEMS and fabrication technologies allow us to predictthat the streaming potential technique will become an establishedprocess in the field of electrokinetic microfluidics. The streamingpotential occurs due to the charge displacement in the EDL causedElectrokinetic phenomena have been necessarily of concern inthe design of diagnostic micro devices and micro-chips [Harrisonet al., 1993], particle manipulation techniques [Effenhauser et al.,1997], and micro flow control in micro-electro mechanical system(MEMS) devices [Polson and Hayes, 2000; Yang and Kwok, 2003;Chang and Yang, 2004]. Especially, electrokinetic techniques havethe advantage of being easily adaptable into microfluidic systemswhen compared to external systems [Karniadakis and Beskok, 2003].Among the electrokinetic phenomena, both electro-osmosis andelectrophoresis use an applied electric field to induce motion. Onthe other hand, both the streaming potential and the sedimentationpotential have the opposite electrokinetic coupling in that they usemotion to produce an electric field. Basically, such electrokineticphenomena are present due to the electric double layer (EDL), whichforms as a result of the distribution of electric charges near a dielectric charged surface [Hunter, 1981; Probstein, 1994].Conventional works in the area of electrokinetic transport phenomena mainly focus on electro-osmotic flow. It should also be recognized that the streaming potential has been an emergent technique for surface characterization of materials with solid-liquid interfaces. From the relations involving a quantitative expression of theforce that creates the observed electrokinetic phenomena, the measurement of the streaming potential permits the determination ofthe zeta potential of the hydrodynamic phase boundary [Bowen andJenner, 1995; Szymczyk et al., 1999; Ren et al., 2001; Sung et al.,2003; Chun et al., 2003]. A relevant work is a new insight recentlyrevealed that the streaming potential is applicable to an electroki-3Fig. 1. Schematic view of the development of flow-induced streaming potential along a charged microchannel filled with electrolyte solution.To whom correspondence should be addressed.E-mail: mschun@kist.re.kr†528
Electrokinetic Microfluidics for Micro Power Generationby an external force shifting the liquid phase tangentially againstthe solid. As illustrated in Fig. 1, the counter-ions in the diffusive(or mobile) part of the EDL are carried toward the downstream endfor the applied pressure p. Then the streaming current results in thepressure-driven flow direction of electrolyte solution, and the streaming potential φ generates corresponding to this streaming current.This flow-induced streaming potential acts to drive the counter-ionsin the mobile part of the EDL to move in the direction opposite tothe streaming current. This opposite-directional flow of ions willgenerate a conduction current in the Stern layer of the EDL. Theoverall result is a reduced flow rate in the direction of pressure dropreferred to as the electroviscous effect [Yang and Li, 1997; Vainshtein and Gutfinger, 2002; Chun, 2002]. The convective transport ofhydrodynamically mobile ions can be detected by measuring thestreaming potential between the two electrodes positioned at upand downstream in the liquid flow.It is the purpose of this paper to evaluate the required performance for applicability toward electrokinetic micro power generation. A theoretical part associated with the single microchannel isthat of Chun and co-workers [Chun et al., 2005], who dealt withindepth analysis on microfluidics in a microchannel encompassingelectrokinetic phenomena. They developed the momentum equation for an incompressible ionic fluid by verifying the external bodyforce and the relevant flow-induced electric field, from the theoretical analysis of the Navier-Stokes (N-S) equation coupled with thePoisson-Boltzmann (P-B) equation. Basic principle of the net current conservation was faithfully applied in the microchannel takinginto account the Nernst-Planck (N-P) equation. In the present study,the flow-induced streaming potential in a multi-channel array is estimated with variations of surface potential, surface conductivity ofthe microchannel, and bulk electrolyte concentration. As the 1 : 1type electrolytes, LiClO having lower mobility are considered. Weestimate the power density as well as the energy density of the electric circuit with variations of design parameters (e.g., pressure difference, the number of channels, and external resistance) in orderto address a performance comparison for practical applications. Theelectric conversion efficiency is also discussed.4ELECTROKINETIC MICROFLOWWe consider a model for pressure-driven and steady-state electrokinetic flow through a uniformly charged straight cylindrical microchannel. Cylindrical coordinates (r, θ, z) are introduced, where rdenotes the radial distance from the center axis and z is the distancealong the axis of a microchannel. The development of the electrokinetic flow equation extends those of previous works [Rice andWhitehead, 1965; Levine et al., 1975] to symmetric electrolytes inwhich the mobilities of anions and cations may individually be specified.For one-dimensional incompressible laminar flow, the velocityof ionic fluid is expressed as v [0, 0, vz(r)], the pressure p p(z),and the flow-induced electric field E [0, 0, Ez(z)]. Neglecting gravitational forces, the body force per unit volume ubiquitously causedby the z-directional action of flow-induced electric field Ez on thenet charge density ρe can be written as Fz ρeEz [Yang and Li, 1997;Chun, 2002]. The Ez is defined by the flow-induced streaming potential φ as Ez dφ (z)/dz. With these identities, the N-S equation529reduces todv-z dpd- r-----µ 1--r---------- ρ Edr dr dz e z(1)where µ is the fluid viscosity. This momentum equation correspondsto the Stokes equation for the flow situation of this study. The boundary conditions applied to vz(r) arevz(r) 0dvz( r)------------- 0drat r a,(2a)at r 0.(2b)Once the charged surface is in contact with an electrolyte, theelectrostatic charge would influence the distribution of nearby ions.Then, an electric field is established and the positions of the individual ions in solution are replaced by the mean concentration of ions.It is well-known that the nonlinear P-B equation governing the electric potential field is given as [Hunter, 1981; Probstein, 1994] Ψ κ sinhΨ.(3)Here, the dimensionless electric potential Ψ denotes Zeψ/kT andthe inverse EDL thickness (namely, inverse Debye radial thickness)κ is defined by κ 2nbZi e /ε kT , where nb is the electrolyte ionicconcentration in the bulk solution at the electroneutral state, Zi thevalence of type i ions, e the elementary charge, ε the dielectric constant, and kT the Boltzmann thermal energy. The nb (1/m ) equalsto the product of the Avogadro’s number (1/mol) and bulk electrolyte concentration (mM). The Boltzmann distribution of the ionicconcentration of type i (i.e., ni nb exp( Zieψ/kT)) provides a localcharge density Zieni.The finite difference method evolved in the previous work is applied here and detailed procedure is analogous to the scheme [Chunet al., 2005]. To obtain the solution of the nonlinear P-B equationwith the boundary conditions imposed as Ψ Ψs at r a and dΨ/dr 0 at r 0, the five-point central difference method is taken onthe left-hand side of Eq. (3). The sinh Ψ on the right-hand side canbe linearized as sinhΨ jk (Ψ jk Ψ jk)coshΨ jk, where k means theiteration index and the grid index j 1, 2, N [Chun, 2002]. Thefinite difference form of the nonlinear P-B equation becomes22223 1kkkkkΨj 2Ψ j Ψ jj Ψ ------------------ Ψ 1 1 1 1 1 1 1 1 12rj r rkk κ [ sinh Ψ j (Ψ j Ψ kj )cosh Ψ kj ].(4)kEq. (4) can be solved for Ψ j by successive iterative calculation,using the value of Ψ obtained in the k-th iteration. From a relationofΨ k Ψ k encountered in the boundary condition at the center,one can derive as2Ψ k [2 (κ r) cosh Ψ k ]Ψ k (κ r) ( sinhΨ k Ψ k coshΨ k ).(5)From a series of algebraic equations expressed in a matrix form,the electric potential Ψ is obtained and then we can determine thenet charge density ρe ( Σi Zieni Ze(n n )), as followsρe Zenb[exp( Ψ ) exp(Ψ )] 2ZenbsinhΨ.(6)For a high surface potential of Ψs 1 (i.e., above kT/e 25.69 mV),substituting Eq. (6) into Eq. (1) yieldsKorean J. Chem. Eng.(Vol. 22, No. 4)221 1 1 111 1 121002000
530T. S. Lee et al.1--- ---d- rdv-------z --1- dp--------------b sinh Ψ d-----φ- .------ 2Zenr dr dr µ dz µdz(7)Eq. (7) takes an integral from 0 to L with respect to z together withsetting p p pL and φ φ φL, then we derive a formula for thevelocity profile subject to the above boundary conditions as0Rs Rf - .1 - -------------R --------------------R11 ----- ----- s Rf Rs R f 0 p 2Zen r- -------------------b ------φ a 1---( r r sinh Ψdr)drvz(r) a-----------4µ L µ L r r 2resistance R along the microchannel consists of the surface resistance Rs and the fluid resistance Rf in parallel [Yang et al., 2003],so that20(8)where a and L indicate the radius and the length of channel, respectively.Subsequently, one obtains the volumetric flow rate q by 2πav(r)rdr. zIn the case where the surface of the microchannel wall satisfiesa condition of low potential (Ψ s 1) with a 1 : 1 type electrolyte system, the P-B equation may be linearized corresponding the DebyeHückel (D-H) approximation [Hunter, 1981; Chun et al., 2003].For a cylindrical channel, the linearized P-B equation leads to (1/r)(d/dr)(rdΨ/dr) κ Ψ, and its solution can be obtained with boundaryconditions as Ψ (r) ΨsI (κr)/I (κa), where I is the modified Besselfunction of the first kind of zeroth order. The net charge density isthen determined byas000I (κ r)-.ρe ε ψ εκ ψsI------------(κ a)22(9)00The analytical solution for velocity profile vz(r) is derived asεψ-s 1 ------------I (κ r) p ------ r- -----vz(r) a-----------4µ L µI (κa)2200 ------φ . L (10)ELECTROKINETIC FLOW-INDUCED STREAMINGPOTENTIAL1. Mathematical Formulation for Single MicrochannelThe N-P equation describes the transport of ions in terms of convection and migration resulting from the pressure difference andelectric potential gradient, respectively. Diffusion is not consideredhere due to an assumption of no axial concentration gradient. Ionsin the mobile region of the EDL are transported through the singlechannel, commonly causing the electric convection current (i.e.,streaming current) IS. The accumulation of ions provides the streaming potential difference φ ( EzL). This field causes the conductioncurrent IC to flow back in the opposite direction. In this case, thenet current I consists of IS and IC, and it should be zero at the steadystate, viz. I IS IC 0 [Werner et al., 1998; Szymczyk et al., 1999;Chun et al., 2003].The streaming current is defined as an integration of the productof the velocity profile and the net charge density, yieldingIS 2π a ρe(r)vz(r)rdr.(11)0For a low surface potential, Eq. (11) becomesπa εψ-s 1 -----2-------------I (κ a) p IS --------------µκaI (κa)- -----L a ε κ ψ-s 1 ----I (κ a-)2 I (κ a-) ------------ π---------------------I (κ a) I (κ a)µκa-------------2211200,IC, f A Zienivd, idA 2πZ e NA ------φ a [K n (r) K n (r)]rdr (16)Li220where the subscripts and of the ionic number concentration nidenote the cations and the anions, respectively. Consequently, weobtain the fluid resistance that is addressed as an inverse quantityof the integration of the local fluid conductivity λf over the crosssectional area per channel length:L - ----------------------------L-.Rf ---------------------------2π a λf(r)rdr 2πZ e NA a [K n (r) K n (r)]rdr220(17)0The total resistance can be identified asLR --------------------------------.2π [aλs Z e NA a (K n K n )rdr ]22(18)0Applying Eqs. (12) and (14) gives an analytic formula of thestreaming potential for a low surface potential:πa εψ-s 1 -----2-------------I (κ a-) p--------------µκa I (κ a) φ ------------------------------------.I (κ a)a ε κ ψ-s 1 ----2 I (κ a)- ---------------L- π---------------------µI ( κ a) I ( κ a)κa-------------R ------φ L 2(12)where I is the modified Bessel function of the first kind of first order.Details for deriving Eq. (12) are provided in Appendix A. The totalJuly, 20051,,002where IC s ( φ/Rs) is the conduction current through the wall andIC f ( φ/Rf) is the conduction current through the electrolyte solution. The surface resistance is readily defined as Rs L/2πaλs, where2πa equals the wetted perimeter, and λs indicates the specific surface conductivity depending on the material property of microchannel.The fluid resistance certainly varies with the radial position bymeans of the contribution of radial concentration gradients to theelectric current. The influence of an axial concentration gradientupon the conduction current vanishes reasonably. As in the previous study, we invoke again the drift velocity of ion species i, givenas [Chun et al., 2005]vd i ZieNAKi φ(15)where NA is Avogadro’s number and Ki the mobility of ion speciesi defined as its velocity in the direction of an electric field of unitstrength [Bowen and Jenner, 1995; Szymczyk, 1999; Chun et al.,2005]. Now the conduction current through the electrolyte solutionfor arbitrary channel cross-section can be expressed as follows:112(14)222Using Ohm’s rule, the conduction current IC can then be expressedIC IC, s IC, f ------φR02(13)22221(19)1200The streaming current for a high surface potential is derived fromEqs. (6), (8), and (11):
Electrokinetic Microfluidics for Micro Power GenerationπZenb ------p a r(a r )sinhΨdrIS -------------µ L a1 rπ(Zenb)- ----- φ a 8----------------------µ L rsinhΨ r --r-( rsinhΨdr)dr dr.220(20)0We combine the conduction current IC defined as in Eq. (14), andthen finally write the streaming potential as the following expression:πZenb[ a r(a r )sinhΨdr] p-------------µ-. φ ---------π(Zenb)- a rsinhΨ a 1---( r rsinhΨdr)dr drL---- 8----------------------r rRµ22(21)20In Eq. (21), the electric potential profile is determined by performing the finite difference method, for which the grids of 5 10 werebuilt within the channel and the convergence criterion is given as10 to satisfy the accuracy requirement.382. Illustrative ComputationsLet us consider a fully developed flow of the aqueous solutionin a cylindrical microchannel made of inorganic materials such aseither glass or fused silica. The dielectric constant and the viscosityof the fluid are taken as 80 (8.854 10 ) Coul /J·m and 1.0 10 kg/m·sec, at room temperature. The ionic concentration of 1 : 1 typeelectrolyte equals the solution ionic strength, in which the EDL thickness κ (nm) is given by [solution ionic strength (mol)] /3.278.The EDL thicknesses correspond to 3.1, 9.7, and 96.5 for bulk electrolyte concentrations of 10, 1.0, and 10 mM, where thinning of122131/22Table 1. Each mobility of ions in aqueous solutions at 298.15 KMonovalent electrolyteLiClO4(lithium perchlorate)2 2 100the EDL means the decrease of electrostatic interaction. The adoption of LiClO electrolyte having lower ion mobility was shown toresult in a higher streaming potential [Chun et al., 2005]. Its individual mobilities provided in Table 1 can be estimated from a molarconductance of infinite dilution based on the Kohlrausch rule ofthe migration of ions [Dean, 1999].Fig. 2 shows the profile of fluid conductivity, i.e., λf(r) Z e NA[K n (r) K n (r)]. The position where the fluid conductivity λf reachesthe bulk value moves more closely towards the channel wall as κaincreases. Here, a κa value of 10 denotes the EDL thickness κ isequal to 1 µm. The bulk fluid conductivity has about 1 µS/m at κa 10, which relatively corresponds to the literature value. A ten timesincrease in κa results from changes of either a ten times increase inthe channel radius for the constant bulk electrolyte concentration ora hundred times increase in the bulk concentration for the constantradius. Fig. 2 allows one to figure out the significance of both thechannel radius and the bulk electrolyte concentration as design parameters regarding an efficient tool for electrokinetic micro powergeneration.Fig. 3 demonstrates how the streaming potential changes withvariations of the surface conductivity λs as well as the bulk electrolyte concentration for microchannel radius of 10 µm. Plotting thestreaming potential coefficient ( φ/ p) can verify that simulationresults are related to the performance evaluation. The optimum flowrate should be considered to keep up with the pressure drop. Theincreasing trend in φ/ p with decreasing surface conductivity ismore considerable for higher surface potential. If p of 1 bar is appliedthe value of streaming potential gains up to about 6 V, although thatis a theoretical prediction.We note that there exists a maximum value of φ/ p in Fig. 3.In the low κa region, the rate of an increase in the streaming current is much higher than that of the total resistance R. In fact, thetotal resistance maintains almost constant compared with the streaming current IS, appearing as a rise of φ/ p with increasing bulk electrolyte concentration. The amount of mobile ions becomes satu420531IonsMobility(Cation, Anion) (10 13 mol s/kg)Li 4.16ClO4 7.29 Fig. 2. Fluid conductivity profiles for several dimensionless inverseEDL thicknesses for monovalent electrolyte of LiClO4,where Ψs 1 and a 10 µm.Fig. 3. Effect of bulk concentration of LiClO4 electrolyte on φ/ pwith different surface conductivities λs for Ψs 0.5 and 4,where L 10 2 m.Korean J. Chem. Eng.(Vol. 22, No. 4)
532T. S. Lee et al.rated after a rapid increase in the streaming current; therefore, thestreaming current approaches the plateau regime around a maximumpoint. As the bulk electrolyte concentration continues to increase,the total resistance decreases, whereas the streaming current remainsconstant. As κa becomes large enough, the streaming potential isalmost constant for Ψs 0.5. Further, the streaming potential increaseswith decreasing surface conductivity of the microchannel. Oncethe surface conductivity becomes lower, a deficiency of the conduction current occurs. Based on the principle of net current conservation, higher streaming potential should necessarily be generatedto make up the deficiency.ELECTRIC CIRCUIT ANALYSIS AND RESULTSNow, let us hypothesize that the external resistance RL is appliedin the multi-channel array assembled with number of N in parallel,as shown in Fig. 4. Then, the net current equals N(IS IC) IL, whereIL means the external current. It is taken to be zero at the steady state,viz. I N(IS IC) IL 0. In a similar way of deriving Eq. (19) for alow surface potential, the streaming potential in the multi-channelarray circuit with external resistance can be obtained:πa εψ-s 1 -----2-I------------(κa-) p--------------µκaI (κ a) φL --.(22)I (κ a)I (κ a)- ------------L - ---------------------πa ε κ ψ-s 1 -----2-------------L---- --------µκ a I ( κ a) I ( κ a)R NRL2102222211200The external current passing the external resistance is determinedby IL φL/RL. By corresponding analogy for a high potential case,Fig. 5. Effect of external resistance RL on φL/ φ with different surface conductivities λs, where Ψs 4, κa 50, N 2,000, andL 2 10 3 m.we again derive the streaming potential in this circuit, expressed asπ-------------Zenb[ a r(a r ) sinhΨdr] pµ - (23) φL -------------------------π(Zenb)- a rsinhΨ a 1---( r rsinhΨdr)dr drL - 8-------------------------L- --------r rR NRµ22020L0from which one can determine the external current IL φL/RL. In Fig.5, an enhancement of the streaming potential φL in the electriccircuit is shown with increasing either the external resistance RL orthe surface conductivity. The enhancement is developed rapidly forsmaller RL, however, a gentle slope is taken with increasing the RL.The power density of our system can be evaluated from the electric power divided by the overall volume of microchannels for theimposed external resistance, and the electric power divided by theoverall flow rate becomes the energy density, given as, respectively, φLIL - ------------------------ φL -,power density ------------------N ( π a L ) N ( π a L ) RLφLIL ----------- φ L - .energy density -----------Nq NqRL22223July, 2005(24b)We present Fig. 6 for the case of low potential, where higher pressure difference causes the enhancements of both the power densityand the energy density as expected. They also increase with increasing either the number of channels or the external resistance. Althoughthe overall trend of Fig. 7 is the same as Fig. 6, the channel wallinvolving a high surface potential is shown to result in increasingabout 70 times the low potential case. Fig. 7 illustrates that, depending on the electrokinetic properties, our theoretical prediction resultsin the higher order of magnitude for power densities reported in theliterature [Koeneman et al., 1997; Yang et al., 2003]. For the number of channels above the order of 10 , one can observe that the valueof power density takes above 1 W/m even at low p (i.e., less than1 bar). Hence, a proper choice of design condition should be considered to obtain the improved performance.3Fig. 4. Schematic diagram of multi-channel array circuit with external resistance.(24a)
Electrokinetic Microfluidics for Micro Power Generation533Fig. 6. The predicted power density (a) and energy density (b) as afunction of p with different external resistances and number of channels for Ψs 0.5, a 10 µm, κa 50, L 2 10 3 m,and λs 5 10 9 S; solid curve: N 2,000, dashed curve: N 1,000, dotted curve: N 200.Fig. 7. The predicted power density (a) and energy density (b) as afunction of p with different external resistances and number of channels for Ψs 4, a 10 µm, κa 50, L 2 10 3 m,and λs 5 10 9 S; solid curve: N 2,000, dashed curve: N 1,000, dotted curve: N 200.We define the electric conversion efficiency as the ratio of theelectric work generation to the flow-induced work without the external resistance, as follows:creased, as the RL increases. Further studies remain ahead in orderto access the present analysis on a practical system. Instead of theporous glass filter used in the previous work [Yang et al., 2003],we suggest more advantageous MEMS devices such as the microfluidic-chip with multi-channel type. φLIL- ------------------- φ L - .ηeff ------------- φNI φ NI R2SSL(25)Fig. 8 shows the effect of the number of channels N as well as theexternal resistance RL upon the conversion efficiency, where theefficiency rises entirely with increasing the N. The efficiency is almost zero for RL 0, because a large IL can be obtained and theelectric work becomes zero. It is evident that the efficiency is in- CONCLUSIONSThis study motivates research on emerging MEMS technologiesand relevant micromachining techniques, which have been increasingly employed in the electrokinetic microfluidic system. We emKorean J. Chem. Eng.(Vol. 22, No. 4)
534T. S. Lee et al.εψ-s 1 ------------I (κ r) φ rdr. ------µ I (κa-) -----L (A1)00It is rearranged asψs ------p a (a r )------------I (κ r)-rdr---------------IS πεκ2µ L I (κa)πε κ ψ-s ------φ a ------------I (κr) I (κr)- rdr. 2--------------------I (κ a)µ L I (κa-) 1 ------------2220022020000(A2)0The first term in the right-hand side of Eq. (A2) is identified asψs -----I (κ r)-rdr p a (a r )------------IS, πεκ---------------I ( κ a)2µ L πεκ ψs- ------p [a a rI (κr)dr a r I (κr)dr], -------------------2µI ( κ a ) L 22120002230000(A3)0and the second one isπε κ ψ-s ------φ a ------------I (κr) I (κr)- rdrIS, 2--------------------I ( κ a)µ L I (κa-) 1 ------------2πε κ ψ-s ------φ a rI (κ r)dr ------------1 - a rI (κ r)dr . --------------------µI ( κ a) L I (κ a) 2Fig. 8. Effect of external resistance RL on the conversion efficiencywith different number of channels for Ψs 4, a 10 µm, κa 50, L 2 10 3 m, and λs 5 10 9 S.ployed the previously developed analysis concerning the streamingpotential due to the electrokinetic microflow in cylindrical microchannels.A profile of fluid conductivity has been estimated in terms ofboth the concentration profile and the mobilities of anions and cations. In order to get the design methodology in the system of electrokinetic micro power generation, one should consider the parameter variations of the surface potential of a channel wall, the surfaceconductivity λs, the electrolyte concentration of bulk solution, andthe pressure drop across channel p. These variations emphasizethe system performance.With increasing surface potential of the wall, the flow-inducedstreaming potential certainly increases. Both the making up mechanism for the deficiency of conduction current and the principle ofnet current conservation result in a higher streaming potential withdecreasing the surface conductivity. Simulation results show that amaximum value of the streaming potential could be obtained withvariations of bulk electrolyte concentration when the microchannelradius is constant. We have shown the feasibility by consideringthe electric circuit model consisting of an array of circular microchannels. An enhancement of the power density and corresponding energy density as a function of the pressure drop has been observed with increasing either the number of channels or the external resistance.APPENDIX A: ANALYTICAL SOLUTION FOR LOWPOTENTIAL CASEThe relevant equation for streaming current given in Eq. (12) isderived; starting with substitution of Eqs. (9) and (10) into Eq. (11),we write asI (κ r)- a----------- r- ------p IS 2π a εκ ψs------------I (κ a) 4µ L 22000July, 2005222200000222200000(A4)0The integral by parts is able to perform using the following:a 0rI (κ r)dr --a-I (κ a)a 0r I (κr)dr a----I (κ a) 2a-------I (κ a) 4a-----I (κ a)(A6)a 0rI (κ r)dr a----[I (κ a) I (κ g Eqs. (A5) and (A6) into Eq. (A3) givesπa εψ-s ------p 1 -----2-I------------(κ a)IS, --------------µ L κaI (κa-) ,2(A8)110and substituting Eqs. (A5) and (A7) into Eq. (A4) givesa ε κ ψ-s -----I (κ a) I (κ a-) . φ 2-------------IS, π---------------------κaI (κa-) ------------µ L 1 ----I (κ a)2222212(A9)1200Then, the streaming current can be derived as Eq. (12).ACKNOWLEDGMENTThis work was supported by the Basic Research Fund (Grant No.R01-2004-000-10944-0) from the Korea Science and EngineeringFoundations (KOSEF) provided to M.-S.C. We acknowledge financial support as the Future-Oriented Battery Research Fund (2E18270)from the Korea Institute of Science and Technology (KIST).NOMENCLATUREAaEz: cross-sectional area of microchannel [m ]: radius of microchannel [m]: flow-induced electric field in axial direction [V/m]2
Electrokinetic Microfluidics for Micro Power GenerationeFzII,I01ICIC f,IC s,ILISKkLNnpqRRfRLRsrtTvvdZz: elementar
hte left-hand side of Eq. (3). The sinh on the right-hand side can be linearized as sinh k j (j k 1 j k h os )c j k, where k means the iteration index and the grid index j 1, 2, N [Chun, 2002]. The mo formsnoer P-B equnlineecof theaecon nereb f ti a f di nite fi (4) Eq. (4) can be solved for j k 1 by successive iterative calculation,
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