Joule Heating Effects in Electroosmotically Driven Microchannel Flows

International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 www.elsevier.com/locate/ijhmt

Joule heating eﬀects in electroosmotically driven microchannel ﬂows Keisuke Horiuchi, Prashanta Dutta * Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920, USA Received 26 September 2003; received in revised form 23 February 2004

Abstract Analytical solutions for temperature distributions, heat transfer coefficients, and Nusselt numbers of steady electroosmotic flows are obtained for two-dimensional straight microchannels. This analysis is based on an infinitesimal electric double layer in which flow velocity becomes ‘‘plug-like’’, except very close to the wall. Both constant surface temperature and constant surface heat flux conditions are considered in this study. Separation of variables technique is applied to obtain analytical solutions of temperature distributions from the energy equation of electroosmotically driven flows. The thermal analysis considers interaction among inertial, diffusive and Joule heating terms in order to obtain thermally developing behavior of electroosmotic flows. Heat transfer characteristics are presented for low Reynolds number microflows where the viscous and electric field terms are very dominant. For the parameter range studied here (Re 6 0:7), the Nusselt number is independent of the thermal Peclet number, except in the thermally developing region. Analytical results for cases with and without Joule heating are also compared with the existing heat transfer results, and an excellent agreement is obtained between them. 2004 Elsevier Ltd. All rights reserved.

Keywords: Microﬂuidics; Electroosmotic ﬂow; Joule heating

1. Introduction electro-mechanical systems (MEMS) have made a breakthrough progress toward utilizing electroosmotic Electroosmosis, one of the main electrokinetic effects, flows in complex microfluidic networks for ‘‘lab-on-a- is the process by which an ionized liquid moves with chip’’ applications. Lately, scientists at Sandia National respect to a stationary charged surface under the action Lab [3] and Stanford University [4,5] have successfully of an external electric field. The electroosmotic phe- developed electrokinetic micropumps capable of creat- nomenon was first observed by Ruess in 1809 from an ing in excess of 20 atm by packing micron sized non- experimental study on clay diaphragms [1]. In the mid- porous silica particles in a 500–900 lm diameter fused 19th century, Wiedemann repeated the experiment and silica capillary. Due to the ability of pumping a wide formally introduced the mathematical theory behind it range of working fluid, this non-mechanical pumping [2]. Though the electroosmotic phenomenon has been has become an attractive fluid handling method in the known for more than a century, the application of emerging ‘‘lab-on-a-chip’’ microfluidic devices for sam- electroosmotic flow was only limited in the field of ple loading, mixing, flushing and reagent transporting analytical chemistry to transport samples in capillaries. [6,7]. So far, most of the studies have been focused only But, recent developments in microelectronics and micro- on the fluid flow behavior where the velocity distribution is plug like in most of the channel [8–12], and not much attention has been paid on the thermal behavior. * Corresponding author. Tel.: +1-509-335-7989; fax: +1-509- One of the main objectives of electroosmotic action is 335-4662. to generate pressure head for pumping liquid in micro- E-mail address: [email protected] (P. Dutta). fluidic devices for bioanalytical applications. Here the

0017-9310/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2004.02.020 3086 K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095

Nomenclature

2 AC cross-sectional area of a microchannel [m ] b0 Fourier coefficient cp specific heat at constant pressure [J/(kg K)] bn Fourier coefficient D half channel height [m] d99 effective Debye layer thickness [m] DH hydraulic diameter [m] e permittivity of the medium [C/(V m)] ~E applied electric field [V/m] U viscous dissipation function

Ex electric field component in the streamwise cn exponentialpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decay parameter, 2 direction [V/m] ð PeT þ 4kn PeTÞ=2 e electron charge [C] g non-dimensional cross-stream distance, G non-dimensional generation term y=D h local heat transfer coefficient [W/(m2 K)] k Debye length [m] k thermal conductivity [W/(m K)] kn eigenvalues 2 kB Boltzmann constant [J/K] l dynamic viscosity coefficient [N s/m ] L channel length [m] h non-dimensional temperature m_ mass-flow rate [kg/s] ha auxiliary temperature field n0 average number of positive/negative ions in hp particular type of function the buffer [1/m3] Dh non-dimensional temperature difference Nu Nusselt number, Dh=k hs h 3 PeT thermal Peclet number, Re Pr qf fluid density [kg/m ] 3 Pr Prandtl number, m=a qe electric charge density [C/m ] 00 2 qs surface heat flux [W/m ] r electrical conductivity of the buffer fluid Re Reynolds number, umD=m [S/m] 2 Rev ratio of Joule heating to viscous dissipation t kinetic viscosity [m /s] T temperature [K] n non-dimensional streamwise distance u velocity component in the streamwise x=D direction [m/s] w electrokinetic potential [V] uHS Helmholtz–Smoluchowski velocity [m/s] f zeta potential [V] V~ velocity field [m/s] Subscripts v velocity component in the cross-stream c centerline direction [m/s] h homogenous W channel width [m] i inlet z valence m mean Greek symbols o non-homogenous 2 a thermal diffusivity, k=ðqcpÞ [m /s] s surface an Fourier coefficient pumping action can be achieved by applying a very high little attention, especially for electroosmotic microflows. external electric field along the channel. This high However, there are several analytical and experimental external electric field creates both conduction and con- works that appeared in literature describing the effect of vection current in the liquid, where convection current Joule heating in capillary electrophoresis for identical contributes to net flow in the system and conduction electric field and ion concentration [14,15]. Those works current generates volumetric Joule heating in the system identified thermal band broadening due to Joule heating. [8]. This Joule heating effect is very significant for high Explicit analytical solutions have been reported in electric field strength and/or highly conductive buffer. the past for both developing and fully developed thermal Since Joule heating is a volumetric phenomenon, its transport problems with or without volumetric heat magnitude is directly related to the channel volume. In generation. In a seminal work, Sparrow and co-workers typical electroosmotic flows with a fluid of 103 S/m [16] studied the effect of an arbitrary generation term in electrical conductivity and an applied electric field of 10 pressure driven thermally developing flows, and they V/mm, the Joule heating term contributes 105 W/m3. obtained analytical solutions for heat transfer charac- Depending on the surface (thermal) conditions, this teristics. Their framework is very similar to this study, thermal energy may elevate the system enthalpy signifi- but in our case the hydraulically fully developed elec- cantly [13]. The thermal energy generation and the troosmotic (plug like) velocity is used instead of pressure associated dissipation mechanisms have received very driven parabolic velocity profiles. In literature there also K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 3087 exist numerous heat transfer studies in the slug flows by the anode. This results in the net movement of ion- both in parallel plates and in circular channels [17,18], ized fluid in the direction of the electric field. where they have not considered any Joule heating or generation terms. Recently, Maynes and Webb have 2.1. Governing equations presented thermal analysis of electroosmotic flows for circular tubes and parallel plates under Debye Huckel The steady electroosmotic flow is governed by the linearization [19]. However, that study is limited only to modified Navier–Stokes equations as [20] the fully developed thermal region. q ðV~ rÞV~ lr2V~ þrP q ~E ¼ 0 ð1Þ In this article, we obtain analytical solutions for f e temperature distributions and heat transfer characteris- ~ where qf is the fluid density, l is the dynamic viscosity, E tics for thermally developing steady electroosmotic flow is the applied electric field, and V~ ¼ðu; vÞ is the velocity in two-dimensional straight microchannels. Our analysis field. The incompressibility condition requires a diver- takes care of the interaction among convection, viscous, gence free velocity field (rV~ ¼ 0), subjected to no slip and Joule heating terms to obtain the temperature dis- and no penetration boundary conditions at the surface. tribution of the fluid in microchannels. This analysis The first and second terms of Eq. (1) indicate the inertia identifies the effects of Joule heating during electroos- and viscous forces, respectively. While the third term motic pumping in designing electroosmotically driven ~ shows the pressure force, and the final term, qeE, rep- micropumps, valves, and mixers where surface can be resents the electrokinetic body force due to the forma- maintained either at constant temperature or at constant tion of the EDL next to the surface. Here qe is the heat flux with the surrounding. electric charge density, and for a symmetric univalent dilute electrolyte it can be found as [20] 2. Theory ezw qe ¼2n0ez sinh ð2Þ kBT The formation of electroosmotic flow in a straight microchannel is shown in Fig. 1. Here the channel walls where w is the electrokinetic potential, n0 is the average attain net negative charges due to ionization, ion number of positive or negative ions in the buffer, e is the absorption, or ion adsorption from the polar liquid next electron charge, z is the valence, kB is the Boltzmann to the solid surface. These trapped surface charges sig- constant, and T is the absolute temperature. At steady nificantly influence the distribution of ions in the liquid state, the electrokinetic potential, w, can be obtained side. Due to the presence of negative charges on the from the Poisson–Boltzmann equation as [21] surface, positive ions from the solution get attracted to q r2w ¼ e ð3Þ the channel surface, and negative ions get repelled from e the surface, forming an electric double layer (EDL) very close to the channel wall. The EDL is typically 1–30 nm where e is the permittivity of the medium. thick depending on the concentration of the solutions. For steady electroosmotic microflow, the governing For example, ion concentrations of 1 and 100 mM equation for thermal energy transport can be presented correspond to EDL thickness of 10 and 1 nm, respec- as tively. In the EDL, the ion density of counter-ions is ~ ~ ~ qf cpðV rÞT rðkrT ÞlU rðE EÞ¼0 ð4Þ greater than that of co-ions. The electroosmotic micro-

flow occurs when an EDL interacts with the externally where cp is the specific heat capacity, T is the tempera- applied electric fields. In Fig. 1, the positively charged ture, k is the thermal conductivity, U is the viscous dis- ions of EDL are attracted towards cathode and repelled sipation, and r is the electrical conductivity of the buffer fluid. The first and second terms of Eq. (4) indicate the thermal energy transfer due to convection and thermal qs" diffusion, respectively, while the third and fourth terms ------show thermal energy generation in the system due to y + + + + + + + + - - viscous dissipation and Joule heating, respectively. - + - - Ex x uHS + 2D +

Anode + + + - - + Cathode 2.2. Assumptions and approximations + + + + + + + + ------The main simplifying assumptions and approxima- q" s tions in our analysis are as follows: Fig. 1. Schematic view of electroosmotic flow between two parallel plates subjected to constant heat flux boundary con- • The fluid viscosity is independent of the shear rate. ditions at the top and bottom surfaces. Hence, we assume a Newtonian fluid. 3088 K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095

• The fluid viscosity is independent of the local electric Table 1 field strength. This condition is an approximation. Typical length scale, electrokinetic and flow parameters con- Since the ion concentration and the electric field sidered in this theoretical work strength within the EDL are increased, the viscosity Parameter Notation Range of the fluid may be affected. (Unit) • The Poisson–Boltzmann equation is valid when the Half channel height D (lm) 10–300

ion convection effects are negligible. Hence, our anal- Electrolyte concentration n0 (mM) 1–100 ysis is valid for Stokes flows, or for hydraulically fully Debye length k (nm) 3–30 developed channel flows. Characteristic length ratio D=k 1000–30,000 • The solvent is continuous, and its permittivity is not Axial electric field Ex (V/mm) 1–300 ) ) affected by the overall and the local electric field Zeta potential f (mV) 100 to 25 strength. Reynolds number Re 0.001–0.700 Peclet number Pe 0.001–5.000 • The ions are point charges. T • The zeta potential is uniform throughout the channel wall. Hence relatively low temperature variation in effective Debye layer thickness, d99 ðd99 6 5kÞ. For more the system is assumed. information about the EDL, d99, electrokinetic poten- • Joule heating takes place uniformly throughout the tial, and electroosmotic velocities, readers are advised to channel. consult classical electroosmotic flow studies by Hunter • Fluid properties are independent of temperature [21]. change. This can be justified for smaller temperature It is clear from Eqs. (5) and (6) that electroosmotic change (less than 10 K). flow velocity remains plug-like, except very close to the • No pressure driven component is present in the veloc- wall. Therefore in Eq. (4), the viscous dissipation term is ity distribution. only active within the effective EDL, d99, where local velocity changes from uHS to zero. But, Joule heating 2.3. Analysis of pure electroosmotic flow due to electric current will take place uniformly over the entire volume. For pure electroosmotic flow in a 2-D Electroosmotic flow in microchannels has been straight channel, the ratio of Joule heating to viscous extensively studied in the past, and detailed analysis of dissipation can be expressed as electroosmotic flow can be obtained from earlier flow rE2ð2DWLÞ Dd r l R x 99 studies [8–11]. In this study, we review the necessary ev ¼ 2 ffi 2 2 ð7Þ ou f e parameters for the complete understanding of the elec- l oy ð2d99WLÞ troosmotic phenomena. From experimental studies it has been found that microchannel electroosmotic flow is where W is the channel width and L is the channel generally very slow in nature with very low Reynolds length. From Eq. (7), it is clear that the ratio of gener- number, Re 1. Therefore, the inertial term in Eq. (1) is ation terms, Rev, is independent of the externally applied negligible compared to viscous term. Moreover, for pure electric field, but it depends on characteristic dimen- electroosmotic flow in a 2-D straight microchannel there sions, effective Debye layer thickness, and conductivity. is no pressure gradient, and hence the electroosmotic Therefore, for typical electroosmotic flow with d99 ¼ body force term is counterbalanced by the viscous term. 108 m, r ¼ 103 S/m, l ¼ 103 N s/m2 and f ¼100 In a two-dimensional straight channel, the steady elec- mV, the Joule heating term is dominating when D is 10 troosmotic velocity distribution becomes [10] lm or higher. Hence we assume that our channel depth w is larger than 20 lm(D P 10 lm), and the viscous dis- uðx; yÞ¼u 1 ð5Þ HS f sipation term was neglected for this analysis. where f is the zeta potential and uHS is the Helmholtz– Smoluchowski electroosmotic velocity, which is given by 3. Normalized energy equation uHS ¼efEx=l. For a straight microchannel (D > 10k), the electroosmotic potential can be found by solving We normalize streamwise and cross-stream coordi- Eq. (3) as [10] nates x; y with the half channel height D n x=D; ð Þ ð ¼ 4k T ezf jyjD g ¼ y=DÞ. Here, D is used as the characteristic length as w ¼ B tanh1 tanh exp ð6Þ opposed to hydraulic diameter, DH ðDH ¼ 4DÞ. Hence ez 4kBT k the flow Reynolds and Nusselt numbers in this study will where k is the Debye length and D is the half channel be one fourth of their conventional values where those height. The electrokinetic and flow parameter ranges numbers are calculated based on hydraulic diameter. studied for this analysis, as presented in Table 1, show Here we assume that the channel half height, D, is much that the electrokinetic potential diminishes within the smaller than the channel width W ðW DÞ. Therefore, K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 3089 the flow can be treated as two-dimensional as shown in h ¼ hh þ ho ð10Þ Fig. 1. For convenience we use two different schemes to normalize the temperature: one for constant surface where hh and ho are the homogenous and non-homogeneous part of the solution, respectively. temperature as h ¼ðT T Þ=ðT T Þ and the other one s i s The homogeneous solution is obtained by using the for constant wall heat flux as h ¼ kðT T Þ=Dq00, where i s separation of variables technique as [22] Ts is the channel surface temperature, Ti is the inlet 00 X1 pffiffiffiffiffi temperature, and qs , is the surface heat flux [17]. Therefore, for both boundary conditions, the normal- hh ¼ an expfcnng cos kng ð11Þ ized governing equation becomes n¼0 2 2 2 1 oh o h o h where the eigenvalues, kn, are given by kn ¼fðn þ 2Þpg , Pe G 8 n T ¼ 2 þ 2 þ ð Þ 2ðffiffiffiffi1Þ on on og the Fourier coefficients, an, are calculated as an ¼ p , kn where G is the normalized generation term that repre- and the exponential decay parameter, cn, is defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sents the ratio of Joule heating to surface heat flux. Since Pe2 þ 4k Pe we used two different expressions to normalize the k ¼ T n T : n 2 temperature, the normalized source term, G, also has two different forms. For constant wall temperature, G On the other hand, the non-homogeneous solution has 2 can be expressed as rð~E ~EÞD =kðTi TsÞ, and for con- been found as [23] ~ ~ 00 stant wall heat flux G is presented as rðE EÞD=qs .In X1 pffiffiffiffiffi an G Eq. (8), PeT is the thermal Peclet number and can be h ¼ ½1 expfc ng cos k g ð12Þ o k n n expressed as n¼0 n

uHSD m uHSD PeT ¼ Re Pr ¼ ¼ Therefore, the total solution for the non-dimensional m a a temperature distribution becomes where a is the thermal diffusivity and Pr is the Prandtl h ¼ hh þ ho number. In this study, we only deal with thermal Peclet G X1 G pffiffiffiffiffi number. ¼ ð1 g2Þþ a 1 expfc ng cos k g 2 n k n n n¼0 n 4. Thermal analysis ð13Þ P pffiffiffiffiffi 1 an 1 2 where ð Þ cosð kngÞ can be replaced as ð1 g Þ. 4.1. Isothermal wall condition n¼0 kn 2 At any section of the channel, the normalized bulk mean temperature can be obtained from following expression In this case, the normalized energy transport equa- Z tion is subjected to the following boundary conditions T T 1 h ¼ m s ¼ ðhuÞdA ð14Þ m T T u A hðn ¼ 0; gÞ¼1 ð9aÞ i s m AC hðn !1; gÞ < 1ð9bÞ where Tm is the mean temperature, um is the mean

velocity and AC is the cross-sectional area of the channel oh ¼ 0 ð9cÞ (AC ¼ 2WD). Since the flow is plug like in most part of og ðn;g¼0Þ the channel, we can assume u ¼ um ffi uHS, and the normalized bulk mean temperature becomes hðn; g ¼ 1Þ¼0 ð9dÞ Z Here, the normalized generation term (G) can be positive 1 D h ¼ hdy or negative depending on the relative magnitude of the m 2D D X1 inlet temperature, Ti, over the surface temperature, Ts. G 2 G G ¼ þ 1 expfcnngð15Þ The generation term, ¼ 0 corresponds to Stokes flow 3 k k with no Joule heating. If the difference between surface n¼0 n n and inlet temperature reaches to zero, the generation term goes to infinity. Therefore, this analysis is partic- Hence, for the isothermal boundary condition, the local ularly valid when inlet temperature is different from heat transfer coefficient can be found as surface temperature. We obtain the solution of the 00 qs normalized temperature from governing equation (8) hn ¼ Ts Tm based on boundary conditions presented by Eqs. (9). P G þ 1 21 G expfc ng Since Eq. (8) is a non-homogenous partial differential k n¼0 kn n ¼ P ð16Þ equation, we divide it into two components such that the D G 1 2 G þ 1 expfcnng total solution, h, becomes 3 n¼0 kn kn 3090 K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095

o and the corresponding local Nusselt number becomes ha ¼ 0 ð21dÞ P og 1 G ðn;g¼1Þ G þ 21 expfcnng Dhn n¼0 kn Nun ¼ ¼ P ð17Þ k G 1 2 G Like the constant surface temperature case, now we þ 1 expfcnng 3 n¼0 kn kn decompose the governing equation of the auxiliary temperature into two components: one contains the homogeneous part and the other carries the non- 4.2. Constant wall ﬂux condition homogeneous part so that

Here the normalized governing equation (8) is sub- ha ¼ hh þ ho ð22Þ jected to the following boundary conditions: Using the separation of variables technique, we find the hðn ¼ 0; gÞ¼0 ð18aÞ homogeneous solution as hðn !1; gÞ < 1ð18bÞ X1 pffiffiffiffiffi hh ¼b b expfc ng cos kng ð23Þ 0 n n oh n¼1 ¼ 0 ð18cÞ pffiffiffiffiffiffiffiffiffiffiffiffi o Pe2 þ4k Pe n g ðn;g¼0Þ 2 T n T 1 2ð1Þ where kn ¼ðnpÞ , c , b ¼ and b ¼ 2 . n 2 0 6 n ðnpÞ o h The non-homogeneous solution becomes, ho ¼ ¼ 1 ð18dÞ o ð1þGÞn. Therefore the total solution for the normalized g ðn;g¼1Þ PeT temperature distribution in constant wall heat flux can 00 In this analysis, heat flux qs is positive when thermal be written as energy enters the control volume. Hence the normalized h ¼ h þ h þ h generation term, G, is positive for heat addition into the p h o X1 1 þ G pffiffiffiffiffi channel, and G is negative for heat rejection. We varied ¼ n þ b ½1 expfc ng cos k g ) Pe n n n the normalized generation term, G, between 1 and 1, T n¼1 where G ¼1, 0, 1 corresponds to Joule heating equal ð24Þ to surface heat removal, no Joule heating, and Joule P pffiffiffiffiffi heating equal to surface heat addition, respectively. For 1 2 1 where hp ¼ 2 g ¼ b0 þ n¼1 bn cosð kngÞ. Hence the flow of de-ionized (DI) water (r ¼ 103 S/m) in a 40 lm normalized surface temperature becomes deep channel with an electric field, Ex ¼ 250 V/mm and X1 00 2 1 þ G 2 heat flux, qs ¼ 1200 W/m will result in G ¼ 1. h ¼ n þ ½1 expfc ng ð25Þ s Pe k n In constant heat flux case, the boundary conditions n¼1 n presented by Eqs. (18) are not homogeneous. In order to obtain homogeneous boundary conditions at the wall, For plug like electroosmotic flow velocity, the normalized bulk mean temperature can be found as we decompose the normalized temperature as [23] Z Z D h h h 19 1 1 1 þ G ¼ a þ p ð Þ hm ¼ ðhuÞdA ¼ hdy ¼ n ð26Þ umA AC 2D D Pe where ha is the auxiliary temperature field and hp is a particular type of function that provides homogeneous Therefore, the local heat transfer coefficient can be boundary condition for the auxiliary temperature field. obtained as 2 For a particular function, hp ¼ð1=2Þg , the resulting 00 qs k P 1 governing equation for the auxiliary temperature field hn ¼ ¼ 1 2 ð27Þ Ts Tm D ½1 expfcnng becomes n¼1 kn and the corresponding local Nusselt number becomes oh o2h o2h Pe a ¼ a þ a þ½1 þ Gð20Þ h D 1 T 2 2 n P on on og Nun ¼ ¼ ð28Þ k 1 2 ½1 expfc ng n¼1 kn n and the corresponding boundary conditions for the auxiliary temperature are 1 5. Discussion of results h ðn ¼ 0; gÞ¼ g2 ð21aÞ a 2 For both isothermal and constant surface heat flux haðn !1; gÞ < 1ð21bÞ channel surface thermal conditions, the analytical solutions for local temperature, wall temperature, bulk mean oh a temperature, local heat transfer coefficient, and local o ¼ 0 ð21cÞ g ðn;g¼0Þ Nusselt number are obtained in terms of infinite series K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 3091 solutions as presented by Eqs. (13)–(17) and (24)–(27). (a) 1 Numerical techniques are used to find the convergence results for the above-mentioned parameters with an 0.8 ξ = 0.01 accuracy of 99.99% by summing at least 1000 terms of 0.6 ξ = 0.10 the infinite series. In the absence of any special func- ξ = 1.00 tions, convergence of the series was straightforward 0.4 ξ = 10.00 without any numerical difficulty. Analytical solutions ξ = 100.00

θ ξ = 1000.00 are presented for DI water for Re < 1, Pr ¼ 7, and 0.2

D > 1000k. Therefore, the hydrodynamic entry length is 0 negligible, and the uniform ‘‘plug-like’’ velocity is justi- ﬁed from the very beginning of the channel. -0.2

-0.4 5.1. Isothermal wall condition

0 0.25 0.5 0.75 1 The normalized fluid temperature distribution, h, η across the channel is shown in Fig. 2 at different axial (b) 1 locations for Peclet number, PeT ¼ 1. Due to the symmetric hydrodynamic and thermal boundary conditions, we have plotted temperature distributions for the upper 0.8 ξ = 0.01 half of the channel. Although the analytical result ob- ξ = 0.10 ξ = 1.00 tained in Eq. (13) is valid for any finite value of the ξ = 10.00 0.6 normalized source term (G), we have presented the ξ = 100.00 ξ = 1000.00 temperature distribution for only three different source θ term values (G ¼1, 0, 1). 0.4 Fig. 2(a) shows the temperature development along the channel for a dimensionless source term G ¼1. 0.2 For the negative value of the non-dimensional source term, the channel surface temperature is higher than the 0 inlet temperature of fluid (Ts > Ti). The normalized 0 0.25 0.5 0.75 1 temperature becomes zero at the surface (g ¼ 1) to η conserve the isothermal boundary condition at the wall. At the entry, the fluid temperature is very close to the (c) 1 inlet temperature, and the normalized temperature becomes unity across the channel. In the entry region, since the surface temperature is higher than the fluid 0.8 temperature, the heat flux enters into the channel for

G ¼1. However, as the fluid travels along the channel, 0.6 the normalized fluid temperature (h) and the wall heat θ flux decrease, but the dimensional fluid temperature (T ) 0.4 increase due to Joule heating. ξ = 0.01 Further down the channel, both normalized fluid ξ = 0.10 ξ = 1.00 temperature and wall heat flux become negative, and the 0.2 ξ = 10.00 temperature profile does not change anymore on or after ξ = 100.00 n ¼ 10. Therefore, we obtain thermally fully developed ξ = 1000.00 region within 10 characteristic dimensions from the en- 0 0 0.25 0.5 0.75 1 try. In the fully developed region, the thermal energy η leaving through the boundary is equal to the heat gen- erated by the Joule heating. Fig. 2. Non-dimensional temperature, h, distribution across the channel in pure electroosmotic flow at different downstream Fig. 2(b) shows the normalized temperature distri- locations, n, for both walls subjected to isothermal conditions at bution for the negligible or no Joule heating case. This is PeT ¼ 1:0 for (a) G ¼1, (b) G ¼ 0, and (c) G ¼ 1. a classical heat transfer situation where a fully developed flow with uniform (slug flow) velocity passes through a 2-D isothermal microchannel. Although Joule heating the channel as fluid travels towards the downstream always exists in the electroosmotic flow, we presented direction. However, the non-dimensional temperature that in order to compare with the existing literature. In never goes beyond zero, and both fluid and surface this case, the normalized temperature decreases along temperature reach the same value by the 10 channel 3092 K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 characteristic dimension from the inlet, n ¼ 10. Without (a) 70 internal heat generation, the heat transfer between the channel and environment takes place only in the entry 60 region. In the entry region, the net wall heat flux is po- 50 Pe = 0.001 sitive or negative, depending on the relative magnitude T PeT = 0.010 of fluid inlet temperature, Ti, over surface temperature, PeT = 0.100 40 PeT = 1.000 T . On the other hand, the rest of the channel has no Pe = 5.000 s ξ T temperature change. Nu 30 The normalized temperature distribution across the channel for the positive source term, G ¼ 1, is presented 20 in Fig. 2(c). In this case, the fluid inlet temperature is higher than the channel surface temperature (Ti > Ts). 10 Unlike the negative source term (G ¼1), the normal- 0 ized temperature ðhÞ is always positive. Therefore, fluid -2 10 10-1 100 101 temperature for this case is always higher than the ξ channel surface temperature. Here both local fluid temperature and heat flux decrease as the flow pro- 100 (b) gresses, but they do not change after reaching n ¼ 10. 80 60 Pe =0.001 After that, the amount of heat rejected from the channel T PeT =0.010 Pe =0.100 is balanced with the volumetric thermal energy genera- 40 T Pe =1.000 tion due to Joule heating. T PeT =5.000 Pe =30.00 Fig. 3 shows the local Nusselt number distribution 20 T Pe =100.0

ξ T along the channel for all three cases (G ¼1, 0, and 1) PeT =100.0 (Burmeister, 1994) Nu presented in Fig. 2 for PeT 6 5. Since the local Nusselt number is the same for each case after reaching the fully developed condition, they have been presented up to n ¼ 10. For isothermal channel surface condition, the Nusselt number is very high at the entry point, but it decays abruptly to reach a flat value within one characteristic length (D). The local Nusselt number distri- 0 10 20 bution along the channel also identifies the exact ξ location where the flow is thermally fully developed. The (c) 70 fully developed Nusselt number is independent of the Peclet number in the electroosmotically driven slug flow 60 region, but it depends on the magnitude of the source 50 term as shown in Fig. 3. For both positive and negative PeT = 0.001 Pe = 0.010 ) T source terms considered in this study (G ¼ 1 and 1), PeT = 0.100 40 Pe = 1.000 the Nusselt number reaches a value of 3.0 in the fully T PeT = 5.000 developed region. On the other hand, without Joule ξ 30 heating the heat transfer characteristics are identical to Nu the classical isothermal heat transfer in a slug flow, for 20 which the fully developed Nusselt number is 2.467 [18]. 10 5.2. Constant wall flux condition 0 -2 ξ Analytical solution for normalized temperature, h, 10 10-1 100 101 distribution within the channel is presented in Eq. (24), Fig. 3. Local Nusselt number distribution for both walls sub- and the corresponding wall temperature distribution is jected to isothermal boundary conditions under various Peclet given in Eq. (25). The first linear term of Eq. (24) shows number for (a) G ¼1, (b) G ¼ 0, and (c) G ¼ 1. The fully the temperature variation only on the streamwise developed Nusselt number for case (a) or (c) is 3.0. For no Joule direction, while the second term has contribution from heating (Case b), our analytical solutions exactly match existing both streamwise and cross-stream directions. In Fig. 4, results [17,18], and the fully developed Nusselt number for this case is 2.467. we show the normalized wall temperature (hs) distribution along the channel for three different values of generation term, though our analytical results are valid for the linear first term, in normalized wall temperature any value of non-dimensional generation term, G. Here distribution (Eq. (25)), causes a monotonic temperature K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 3093

2000 1 0.5 ξ = 0.01 1750 0.8 ξ = 0.10 ξ = 1.00 0.6 s ξ = 10.00 1500 θ 0.4 0.4 ξ = 100.00 0.2 ξ = 1000.00 1250 0 0 0.1 0.2ξ 0.3 0.4 0.5

s 0.3 1000 G=-1 θ G= 0 ∆θ G= 1 750 0.2 500

250 0.1

0 0 0 200 400 600 800 1000 ξ 0 0.2 0.4 0.6 0.8 1 η

Fig. 4. Normalized wall temperature, hs, distribution along the channel in pure electroosmotic flow for both walls subjected Fig. 5. Non-dimensional temperature difference, Dh ¼ hs h, across the channel in pure electroosmotic flow at different constant heat flux boundary conditions at PeT ¼ 1:0 for different normalized source terms. Onset figure shows the variation downstream locations, n, for both walls subjected to constant of wall temperature at the entry region. wall heat flux at PeT ¼ 1:0. change with axial location for any value of G, except for For G ¼1 case, the heat is taken away from the 00 G ¼1 case. For G ¼1, the amount of Joule heating channel by external means (qs < 0), and there is no is equal to the surface heat rejection, and hence the change in the wall temperature after the thermally dimensional wall temperature (Ts) reaches a constant developing region (refer to Fig. 4). Therefore the value after the thermal entry region. Although non- dimensional fluid temperature (T ) is higher than the dimensional wall temperature can change a couple of channel surface temperature (Ts), although the non- orders of magnitude in the downstream direction (for dimensional temperature difference, Dh, is positive G 6¼1), the dimensional temperature change, (Ts Ti) within the fluid. On the other hand, in case of positive 00 is less than 5 K due to the conversion factor qs D= source term (G ¼ 1), both heat generation and heat k ¼ 5 104 K, even for an external electric field, addition take place simultaneously. As expected, both ~E ¼ 100 V/mm and electrical conductivity, 300 lS/m in a surface and centerline temperature increase as the flow 20 lm deep channel. Therefore, temperature indepen- proceeds along the downstream direction, and the sur- dence on fluid properties can be justified at any section face temperature is always higher than centerline tem- of the channel. perature. The normalized temperature difference, Dh ¼ hs h, The no Joule heating case (G ¼ 0) is very similar to distribution across the channel is plotted in Fig. 5 at classical heat transfer study for constant surface heat different axial locations for PeT ¼ 1. The general trends of flux without any generation. There could be two possible normalized temperature difference, Dh, distribution re- scenarios: heat rejection from the channel and heat main same for any value of G. The temperature distri- addition to the channel. For heat rejection from the bution is almost uniform at the beginning of the channel channel, both dimensional surface and centerline tem- (n ! 0:0), and the corresponding fluid temperature is peratures decrease. On the other hand, for heat addition very close to inlet temperature, Ti. As the flow proceeds both dimensional surface and centerline temperature along the channel, the non-dimensional temperature increase. From Fig. 5, it is clear that the dimensional difference (Dh) increases, especially at the centerline, until surface temperature is higher than the fluid temperature, it reaches a constant value of 0.5 at the centerline. Fig. 5 if heat is added in the system, and vice versa. We also indicates that there is no change of Dh distribution across obtained temperature distribution (not presented here) the channel after ten characteristic dimensions from the at high Peclet number flows for G ¼ 0, and our results entrance, n ¼ 10. However, the dimensional temperature, replicate the existing analytical work [17,18]. T ðx; yÞ, distribution changes along the channel since the The mean temperature, which is presented by Eq. wall temperature, TsðxÞ, varies in the downstream direc- (26), shows only linear dependence on axial positions. tion. In following section, we discuss the dimensional However, the difference between non-dimensional sur- temperature distribution for three different normalized face and mean temperature, (hs hm), is independent of generation terms (G ¼1, 0, and 1). the source term G. Therefore, in the fully developed 3094 K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095

100 • In electroosmotically driven microchannel flows, the 80 Joule heating contribution is important if the channel 60 thickness is higher than 20 lm. PeT =0.001 PeT =0.010 • For isothermal channel surface condition, the local 40 Pe =0.100 T fluid temperature decreases for positive source term Pe =1.000 T (G > 0) and increases for negative source term PeT =5.000 PeT =30.00 (G < 0), along the channel, before it reaches an iden- 20 PeT =100.0 ξ tical profile. Although both local and mean (normal- PeT =100.0 (Burmeister, 1994)

Nu ized) temperature is a function of the source term, the normalized heat transfer coefficient reaches the same value for both positive and negative source terms as long as the magnitude of G is the same. • In constant surface heat flux condition, the local fluid temperature increases monotonically along the channel for positive and no source term (G P 0). But for the negative source term (G < 0), the local fluid temperature may increase or decrease depending on the 02010 ξ relative values of G. Fig. 6. Local Nusselt number distribution under various Peclet • For surface heat flux case, the Nusselt number does number for both walls subjected to constant heat flux boundary not depend on the magnitude of the source term. conditions. At high Peclet numbers our results exactly match • Under no Joule heating case, our results verify the the existing results [17,18]. The fully developed Nusselt number Nusselt number of classical slug flows in 2-D chan- for constant wall heat flux case is 3.01. nels. • Within the thermally fully developed region, the local Nusselt number is independent of the Peclet number region we obtain identical heat transfer coefficient and for the electroosmotic microflow region considered in Nusselt number for any value of Joule heating and this study. external heat flux. Fig. 6 shows the local Nusselt number distribution along the channel for different values of the Peclet number. Within the microflow regime considered Acknowledgements in this study, the heat transfer characteristics are independent of the flow Reynolds number, except in the The authors thank Dr. Hong-Ming Yin for his developing region where the Nusselt number is very valuable suggestions. This investigation was supported high. From Fig. 6, it is clear that the flow is thermally in part by funds provided by the Washington State fully developed within five characteristic dimension from University Office of Research. the entrance, n ¼ 5, and the corresponding Nusselt number for this case is 3.01. Therefore the Nusselt number in the fully developed region is within 1% of that References of fully developed uniform flow with no Joule heating case reported earlier [17,18]. [1] F.F. Reuss, Charge-induced flow, Proceedings of the Imperial Society of Naturalists of Moscow 3 (1809) 327– 344. 6. Summary and conclusions [2] G. Wiedemann, Pogg. Ann. 87 (1852) 321. [3] P.H. Paul, M.G. Garguilo, D.J. Rakestraw, Imaging of We obtained analytical solutions for heat transfer pressure and electrokinetically driven flows through open characteristics of electroosmotically driven Newtonian capillaries, Anal. Chem. 70 (1998) 2459–2467. fluid in two-dimensional straight microchannels for [4] S. Zeng, C.H. Chen, J.C. Mikkelsen, J.G. Santiago, constant zeta potential, buffer concentration, and exter- Fabrication and characterization of electroosmotic micro- nal electric field. The energy equation for electro- pumps, Sensor. Actuator. B Chem. 79 (2001) 107– osmotically originated flows have been analyzed for 114. [5] S. Zeng, C.H. Chen, J.G. Santiago, J.R. Chen, R.N. Zare, both constant surface temperature and constant wall J.A. Tripp, F. Svec, J.M.J. Frechet, Electroosmotic flow heat flux thermal boundary conditions in the presence of pumps with polymer frits, Sensor. Actuator. B Chem. 82 significant Joule heating. This analysis is based on (2002) 209–212. infinitesimal Debye length (k=D P 1000), and hence a [6] A.M. Leach, A.R. Wheeler, R.N. Zare, Flow injection plug like uniform velocity profile is assumed throughout analysis in a microfluidic format, Anal. Chem. 75 (2003) the channel. Our analysis resulted in the following: 967–972. K. Horiuchi, P. Dutta / International Journal of Heat and Mass Transfer 47 (2004) 3085–3095 3095

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