M. Y. ABDOLLAHZADEH JAMALABADI， J. H. PARK， M. M. RASHIDI， J. M. CHEN

1. Department of Ship Engineering ，Chabahar Maritime University， Chabahar， Iran

2. Department of Aerospace and System Engineering ，Gyeongsang National University， Jinju， Korea，

3. Engineering Faculty of Bu-Ali Sina University， Hamedan， Iran

4. Department of Mechanical and Nuclear Engineering， Kansas State University， Manhattan， USA

E-mail: muhammad_yaghoob@yahoo.com

Effects of thermal boundary conditions on the joule heating of electrolyte in a microchannel*

M. Y. ABDOLLAHZADEH JAMALABADI1，2， J. H. PARK2， M. M. RASHIDI3， J. M. CHEN4

1. Department of Ship Engineering ，Chabahar Maritime University， Chabahar， Iran

2. Department of Aerospace and System Engineering ，Gyeongsang National University， Jinju， Korea，

3. Engineering Faculty of Bu-Ali Sina University， Hamedan， Iran

4. Department of Mechanical and Nuclear Engineering， Kansas State University， Manhattan， USA

Joule heating effects on a slit microchannel filled with electrolytes are comprehensively investigated with emphasis on the thermal boundary conditions. An accurate analytical expression is proposed for the electrical field and the temperature distributions due to Joule heating are numerically obtained from the energy balance equation. The results show that a thermal design based on the average electric potential difference between electrodes can cause severe underestimation of Joule heating. In addition， the parametric study of thermal boundary conditions gives us an insight into the best cooling scenario for microfluidic devices. Other significant thermal characteristics， including Nusselt number， thermophoretic force， and entropy generation， are discussed as well. This study will provide useful information for the optimization of a bioMEMS device in relation to the thermal aspect.

microchannel， electrolyte， joule heating， Nusselt number， thermophoretic force， entropy generation， optimized thermal design

Introduction

The application of an external potential field on a net surplus of charged ions can cause bulk liquid motion across a microchannel through the Coulomb force［1］. Recently， Joule heating in microfluic systems has been of great interest because many biological applications use physiological electrolytes with high electric conductivity （≥1S/m） under high electric fields， which could bring a considerable temperature increment to the system by Joule heating［2］. Such undesirable local thermal loading causes a major intemperance of bio-particle catching［3］， manipulation of bacteria in biological samples［4］， magnetohydrodynamic pump［5，6］， alternation of cell phenotypes［7］， damage of living cells， hyperthermia etc.. Thus， a systematic understanding of the physical details of Joule heating effects is essential in designing reliable bioMEMS using an electric field. Even the detailed investigations were performed on thermal effect because of its importance on the thermal control of manipulated particle，but the boundary conditions which can be applied correlates to the realistic situation in microfluidics is not discussed fully. Researchers have exerted intensive efforts to unveil various characteristics of the Joule heating phenomena in microfludic systems. Erickson et al.［8］found a nearly fivefold increase in the maximum buffer temperature in PDMS/PDMS chips over PDMS/Glass systems due to Joule heating in a high electric field. Tang et al.［9］reported that Joule heating can be considerably affected by the solute concentration: With an increase in solute concentration， the Joule heating effects could lead to a 20oC increment in the solution temperature. In addition， De Mello et al.［10］used Joule heating to control the temperature of ionic liquids in a microchannel with high precision and accuracy. Ross et al.［11］employed the Joule heating technique to generate inhomogeneous temperature distribution in microfluidic circuits to test the accuracy of a new temperature measurement technique using thermally active fluorescent dye. The flow rates as wellas the unique plug-like velocity profile in electroosmotic flows can be severely altered in the presence of Joule heating［12］. There are many approaches to controlling the slip velocity close to the boundaries. For instance， one can use micropolar fluid， applying external magnetic and electric fields［13］. Recently， Chaurey et al.［3］observed that dielectrophoretic trapping forces fall steeply with particle size， especially within physiological media of high-conductivity， where the trapping can be dissipated by the electrothermal flow due to localized Joule heating. Burg et al.［13］examined the temperature field in a dielectrophoresis-based deposition device which was induced by Joule heating. Castellanos et al.［14］derived the scaling relations between various forces in microsystems including dielectrophoresis and Joule heating. Xuan and his colleagues show that Joule heating in a microchannel-reservoir system using insulator-based dielectrophoresis creates fluid inhomogeneities in the constriction region［15］. The effects of other applications of Joule heating on electrokinetic transport were carefully reviewed by Cetin and Li［16］.

Considering the previous works， although a lot of researchers have already performed experimental and numerical investigation of the Joule heating of electrolyte in MEMS， none have considered the effects of boundary conditions. Hence， in this study the effects of Joule heating by an electric field in a microchannel under various thermal boundary conditions are comprehensively investigated. As a representative case， a microchannel is considered in which two microelectrodes are placed on the top and bottom walls. It is well known that the use of microelectrodes has important implications for minimizing Joule heating effects when using DEP to characterize and manipulate cells and other biological particles in aqueous media［17］. The channel is filled with a conducting fluid （e.g.， electrolytes）. This system is found in many dielectrophoretic devices that use dielectrophoretic forces， because this type of electrode configuration can generate a nonuniform electric field. The major contributions of this work differ from those of previous similar investigations by including detailed thermal characteristics such as temperature field （transient and steady-state）， thermophoretic velocity， Nusselt number， and entropy generation. The results of the current study are expected to help the thermal design of electric-field-driven microfluidic systems.

1. Problem definition and mathematical model

In this study， a typical parallel electrode configuration is applied to a microchannel filled with a conductive liquid （see Fig.1）. The bottom wall is entirely covered by one electrode whereas only half of the top wall is covered by another electrode and the other half is under electrical insulation. Such parallel electrode configurations are simple but widely used in various applications［16，17］. A planar electrode with a lower voltage ofminφ is placed on the bottom wall. The top wall is divided into two regions: The left part （0）X＜is an electrode with a higher voltage ofmaxφ whereas the right part （0）X＞ is electrically insulated. The X=0 is the middle point of the top wall， where the electrode starts. Both the top and bottom walls are assumed to be infinitely long because the length of the microchannel is usually sufficiently larger than the channel height H.The top electrode is stretched from 0 to =X-∞ while the insulated region is from the middle point to =X∞. The microchannels are of course three-dimensional in reality. However， if the boundary surfaces in the width-dimension （or -Z direction， perpendicular to -XY plane） are usually considered to be perfectly insulated （heat loss is negligible）， the properties of interest （e.g.， T and E） are uniform along that direction， and the two-dimensional model is reasonably acceptable.

Fig.1 Electric boundary conditions for electric potential. Top wall consists of a semi- infinite electrode in negative -x direction and electrically insulated， i.e.， electrode-less，section in positive x-direction （marked in blue）. Bottom wall is an electrode entirely

The dimensionless electric potential in the microchannel， Φ， is obtained by solving the non-dimensionalized Laplace's equation

Fig.2 Thermal boundary conditions considered in this study （the adiabatic part is marked in red）: （a） All the surfaces are thermally insulated i.e.， adiabatic （?T/?Y=0）. （b） Top wall is adiabatic， while bottom wall is in cooling （T=T∞）. （c） Top electrode（X＜0） is adiabatic， top electrode-less section （X＞0） is in cooling， bottom wall is in cooling. （d） Top electrode is in cooling， top electrode-less section is adiabatic， bottom wall is in cooling. （e） All the surfaces are in cooling. （f） Top electrode is adiabatic， top electrode-less section is in cooling， bottom wall is adiabatic. （g） Top electrode is in cooling， top electrode-less section is adiabatic， bottom wall is adiabatic. （h） Top wall is in cooling while bottom wall is adiabatic

On the top insulation wall （=1Y， ＞0X）

On the top electrode wall （=1Y， 0X＜）

The temperature distribution can be computed by solving the stationary fluid energy equation that the bulk fluid flow effect is not considered. This kind of system is common in some of the particle manipulation devices mentioned above. The energy conservation equation with Joule heating is

The assumption for the thermal properties of the electrolyte in this study is that they are temperature independent. In addition， the temperature increase in Joule heating existing in various cases is not too much to leads to a significant changes in the thermal properties that required the consideration of temperature dependent thermal properties in Eq.（6）. As is common insimilar studies on thermal design［18，19］， the equations are solved in non-dimensional form. After non-dimensionalization with flow-less conditions， Eq.（6） is simplified as

where θ=kf（T-Tθ）/σ/（φmax-φmin）2.

In this study， eight thermal boundary conditions as shown in Fig.2 are considered. They are also summarized in Table 1. All the conditions are carefully chosen to represent practical scenarios. For example，in CASE D the top electrode is in cooling as the ambient temperature =TT∞and the other surfaces are adiabatic i.e.， thermal insulation such as used in Ref.［20］. The adiabatic surface can be created in experiments by attaching a polydimethylsiloxane （PDMS）sheet to the electrode from outside.

Table 1 Thermal boundary conditions of a dielectrophoresis trap

2. Results and discussion

2.1Potential and electric field

Since simple geometry is considered， an analyticcal expression of the electric field is available as derived in the following: Without loss of generality， the solution of Eq.（1）， Φ， subject to the boundary conditions of Eqs.（3）， （4）， and （5） can be expressed as

for the left domain， and

for the right domain. In the above equations，=nπ and=（n -0.5）π. Then， the Fourier transformed form of Eq.（1） after applying the BC of （3） to remove the cosh solution is in the form of

where i is the imaginary unit， i=1- and tanh（）κ has the following product form

Thus， the equality of Eq.（11） can be rewritten as

As the left-hand side of Eq.（13） is analytic in the lower half-plane of the κ domain while the right-hand side is analytic in the upper half-plane， both sides of Eq.（13）should be zero according to Liouville's theorem. Then，the coefficients of+A and A-are computed as

After applying the inverse Fourier transform together with the residual theorem to Eq.（10） combined with Eqs.（14） and （15）， the analytic solution of Eq.（1） subjected to BC's of Eqs.（3）， （4）， and （5） is finally obtained as

Matlab software is used to calculate the above series. A convergence test at （=0X， =1Y） indica-

ting that n=2000terms is sufficient to achieve two digit accuracy （See Table 2）. Figure 3 depicts the （1）electric potential contour， （2） electric field vector， and（3） magnitude of electric field. The potential gradually decreases from top （φ=φmax， Φ=1） to bottom（φ=φmin， Φ=0）. Near the electrodes （left part of top wall and entire bottom wall）， the potential line lies parallel with the electrode. Since the right part of the top wall is electrically insulated， the potential lines arrive at the surface vertically （see Fig.3（a））. A large electric field is mainly found across the channel in which a major potential difference exists. In the left half of the microchannel， the electric field is much larger in comparison with the field in the right domain. The electric field rotates by 90oaround the middle point of the top wall （0，1） that is the end point of the electrode （see Fig.3（b））. Throughout this paper， this point is referred to as the turning point. The maximum electric field found at the turning point is=16.5. These results are compared with the finite difference method. A mesh size of ΔX=0.01 or smaller for a uniform grid， is needed to achieve the same results as the analytical solution with 2000 terms （See Table 3）. The magnitude of the electric field norm is mostly contributed by the y-component of the electric field. This means that， for the present electrode configuration， the local electric field in the medium could be much larger than the average electric field computed by the potential difference between top and bottom electrodes over the channel height， Εavg=（φmax-φmin）/H. Thus， the voltage regulation to prevent device failure due to dielectric breakdown should be determined upon the electric field at the turning point. For example， the critical electric field of dielectric breakdown is about 10 V/μm for water and common electrolytes （e.g.， NaCl solution）. If the maximum available potential difference is computed based on the average electric field， it has a value of 10 V when H=1μm . However， the maximum potential based on the electric field near the turning point is estimated to be about 800 mV. Since the Joule heating is proportional to Ε2， the maximum temperature is expected to be found at the turning point. The edge effect near the turning point in all cases is considered in this study.

Table 2 Convergence study of analytical solution （Eq.（16））

Fig.3（a） （Color online） Dimensionless electric potential， Φ

Fig.3（b） （Color online） Dimensionless electric field，=，）

Fig.3（c） （Color online） Magnitude of electric field，=+）1/2

Table 3 Convergence study of numerical solution mesh size for electric field and n=2000

2.2Temperature distribution

In this section， the temperature fields are examined under various thermal boundary conditions. The dimensionless energy conservation equation of Eq.（7）is numerically solved with the electric field of Eq.（16），by using finite difference method.

Fig.4 （Color online） Unsteady isotherms for CASE A where all the surfaces are adiabatic. The contours are normalized by the maximum value of dimensionless temperature

Fig.5 Transient behavior of maximum dimensionless temperature for CASE A

CASE A represents a situation where all the boundaries are thermally insulated. In this case， the medium temperature keeps increasing as time goes by because all the BC's are Neumann and there are no factors that can regulate the temperature. The actual system temperature is anyway saturated after a long time because it is not possible to remove heat loss completely. Nevertheless， the present results for CASE A provide interesting details about thermal development. As revealed in Fig.4， the initial hot spot formed at the turning point （τ=0.01） spreads all over the channel and the temperature gradient inside fades away with rising temperature. It should be noted that the contour levels are different in each plot. At τ= 0.01， the maximum value of θ is about 0.14 （see Fig.4（a）） while at τ=10， θ has the value larger than 5 at any place in the channel （see Fig.4（d））. The temperature rise of the system can be confirmed by presenting the time history of maximum temperature as in Fig.5. Since a steady-state solution does not exist for CASE A， this case is not considered in the discussion hereafter. The typical values for the thermal properties of fluids leads to α=O （10-6）m2/s and the characteristic size of a microchannel H=O（10-6）mleads to the time scale of t=O（10-6）sfor the time evolution of pure conduction through the fluid. The temperature distribution at steady-state （）τ→∞ with the other boundary conditions （CASEs B-H in Table 1） is illustrated in Fig.6. In CASE B， the entire top wall is adiabatic and the bottom wall is kept at the ambient temperature of T∞（see Fig.6（a））. A significant thermal loading is concentrated near the turning point（τmax=0.6551）. Interestingly， cooling through the bottom wall does not influence the thermal behavior near the top wall very much. In CASE C， revealed in Fig.6（b）， only the top wall electrode is thermally insulated， i.e.， adiabatic， while the electrode-less section of the top wall and the bottom wall are being cooled as T=T∞. Under this condition， the effect of Joule heating is localized in the left domain while the temperature of the right half is successfully regulated close to T∞. Themaxθ lowers to 0.4728 and is observed near the upper-left corner. In Fig.6（c）， the electrode-lesspart of the top wall is adiabatic and the temperatures of the top and bottom electrodes are maintained at T∞（CASE D）. In the figure， a distortion in the temperature field is seen because of the discontinuity at the turning point. The temperature becomes further reduced andmaxθ has a value of 0.1678. In CASE E， all the boundaries are being cooled as =TT∞. Then， the high temperature zone due to Joule heating is isolated in the middle left of the microchannel. However， it should be noted that a meaningful temperature increment is still observed at the very vicinity of the turning point where the highest electric field is applied. Now， θmaxhas a value of 0.1250. This is smaller than any θmaxof CASEs A-D in which only the bottom wall is being cooled as T∞. However， the difference between θmaxin CASEs D and E is not that significant. It indicates that cooling the electrodes is the optimal strategy in thermal design unless you can cool all the surfaces together.

Fig.6 （Color online） Various isotherms at steady state isotherms.

In CASEs F-H， the bottom wall is adiabatic and the temperatures are generally higher than those of CASEs B-E （The bottom wall is in cooling with T∞）. In CASE F， the cooling is applied only to the electrode-less top wall， which results in a dramatic increase in temperature at the left side of the channel θmax=2.9984）. Hence， the temperature field is stratified in the -xdirection， which may induce an undesirable heat transfer along the channel axis （see Fig.6（e））. The maximum temperature of CASE F is about 24 times higher than that of CASE E. In CASE G， although the cooling is applied to the left electrode section of the top wall， its effect is not significant in comparison with CASE F because the high electric field there is sufficiently strong to generate heat over cooling （see Fig.6（f））. Although the maximum temperature is not as high as 0.5413， the high temperature zone is widely found in the right domain of the microchannel. Finally，in CASE H the cooling condition is applied to the entire top wall while the bottom wall is adiabatic. This is a counter-scenario to CASE B with respect to cooling strategy. As a result， in contrast to CASE B， a high temperature zone is formed at the lower left corner. Themaxθ becomes slightly lower than in CASE H 0.4701. Summarizing all the observations in this section， the best performance in regulating the medium temperature is of course to cool all the boundary surfaces with θ∞simultaneously.

If this is not possible， cooling on the electrodes near the high electric fields has to be applied. Table 4 lists the ratio of electric conductivity to thermal conductivity， /kσ， for the typical electrolytes of NaCland KCl solutions with different concentrations. Since the temperature increment， TΔ， is given as

σ/k is sufficient to identify the temperature elevation in the dimensionless form as presented in the Table 4. The concentration-dependent thermal conductivity k and the electric conductivity σ of the NaCl or KCl solutions are taken from the data in Refs.［21，22］. For CASE F， with the applied potential difference （=φΔ φmax-φmin） of 800 mV， the medium temperature can be elevated by about 30oC for 1.0 M NaCl or KCl solutions and about 101oC for 3.0 M solutions. Such thermal loading is enough to cause severe adverse effects in biological applications［3，5，9，10］.

Table 4 /kσ with various concentrations for common electrolyes of NaCl and KCl

As stated above Joule heating in the microchannel flow is highly coupled with the electric， thermal and flow fields. In this study， the system is decoupled and the potential field solved without consideration of temperature variation. In addition， the temperature field is solved numerically without accounting for the influences of the flow field. At such high temperature differences， simplification by ignoring the coupling nature of the problem， is not valid and the relation in Eq.（17） for such cases is just an estimate. Although the same relationship is presented in Ref.［6］with validation by the experimental results， the physical properties for high temperature cases， such as fluid viscosity， permittivity， thermal/electric conductivity，mass diffusivity， and heat capacity， are a function of temperature， in particular those properties such as permittivity and viscosity are strongly dependent on local temperature. As we are dealing with Joule heating， it is not unreasonable to ignore the non-isothermal effects on local physical properties. The variable-property model should be employed in the analysis of this class of electro-thermal-flows and the non-linear problem has different solution from the current analytical solution.

2.3Nusselt number along the surface

In a steady state， the heat generation inside the channel should be the same as the heat lost over all the boundary walls. The heat flux thorough wall can be quantified by introducing the Nusselt number， Nu. Rather than the original definition of the Nusselt number as Nu=hL/k where h is the convective heat transfer coefficient and L is the characteristic length，in this study Nu is computed by using the modified formula as

The steady state Nusselt numbers for the boundary conditions of B to E are compared in Fig.7. The red symbols stand for the top wall and the green ones for the bottom wall. In these four cases， the bottom wall is in cooling at T∞while the top wall BC's are varied. The Nusselt number becomes zero at the adiabatic wall，i.e.， there is no heat transfer （see Fig.7（a））. In addition，the electric field at the right end is negligible and subsequently the heat transfer as well as the heat generation is not noticeable there. The Nusselt number at the bottom wall gradually decreases after =0X. When the electrode side of the top wall is adiabatic， the Nusselt number there has the value of unity （=1）Nu because the heat loss occurs through the bottom wall only （see Figs.7（a） and 7（b））. If the heat loss happens at the top and bottom walls simultaneously， Nu at the top wall becomes 0.5 and Nu at the bottom wall has the value of -0.5. The sign indicates the direction of heat flux. The positive sign means that the heat flux is to the （+）-y direction while the negative sign stands for the （-）-y direction. The divergence in the value of Nu is found near the turning point of the cooling side （top wall）， for example， the region with 0X＞for CASE C （Fig.7（b））. The largest Nu appears as 5.3 for CASE C in which the electro-less part is being cooled， because the heat transfer through the top wall is concentrated on the right half （The left half is thermally insulated）. The Nusselt numbers of the cases with a thermally insulated bottom wall are revealed in Fig.7 （CASEs F-H）. Since the entire bottom wall is adiabatic， the heat loss through the top wall is significantly increased. The most significant heat loss is found as Nu=14.2 for CASE F in which the electrode-less part is in cooling.

Fig.7 Nusselt numbers at top and bottom walls

2.4Thermophoresis

Molecules drift along temperature gradients， an effect called thermophoresis， the Soret effect， or thermodiffusion. Since a non-uniform temperature distribution within a device inevitably causes thermophoretic flows with a thermophoretic velocityTU， which is proportional to the temperature gradient: UT=b?T，where b is the thermophoretic mobility［23］. So the thermophoretic velocity is a function of the temperature gradient and the thermophoretic mobility which is dependent on the absolute temperature， the fluid viscosity， and fluid density p， the thermal conductivity of the fluid， and particle. Thermophoresis was modeled in various studies through the hydrodynamic analysis based on Navier-Stokes theory with slip-corrected boundary condition （Stokes-Cunningham expression）， kinetic theory （Boltzmann equation with temperature jump on particle interface）， or irreversible thermodynamics （Onsager's reciprocity relations） in various regimes such as free molecule regime， near-continuum regime， and Transition regime， with considering effect of finite gas volume. Considering this， in this section the potential strength of the thermophoretic flow of the present system is examined in continuum regime with Navier-Stokes theory by introducing the magnitude of temperature gradient，=［（?θ/?X）2+（?θ/?Y ）2］1/2. The assumed linear relationships between fluxes and gradients are valid in the Navier-Stokes level of approximation within the framework of kinetic theory. The results are shown in Fig.8. The analytical solution presented here provides a better derivation of variables in comparison with the numerical solution using the finite difference method， and the steep gradients which appear are obtained by the same number as the series number （n=2000） while the FDM needs a mesh size of ΔX =0.005 or smaller for the uniform grid to achieve the same results as the analytical solution because of the edge effect in =0X （See Table 5）. Interestingly， the order ofis not the same as the order of θmax. In addition， it is not proportional to the percentage of the adiabatic surfaces enclosing the system. The maximum ofis observed at the turning point where a significant change in temperature field occurs and the electric field is maximized. If the temperature distribution is symmetric with respect to the channel center（=0.5）Y， the thermophoretic motion has the minimum value along the centerline and gradually increases as it approaches the boundary surface （e.g.， right side in CASE C， left side in CASE D， and the entire channel in CASE E）. It is noteworthy to mention that here stationary liquid is assumed. If electrolyte liquid particle migration occurs due to the temperature gradient in this electrophoresis device the thermal properties of the electrolyte can change which leads to variations in the temperature field and heat transfer. Furthermore， the current analysis is different from laser induced self-thermophoresis motion which depends on Soret coefficient.

Fig.8 （Color online） Contours of non-dimensionalized temperature gradient at steady state

Table 5 Convergence study of numerical solution for temperature field

Fig.9 （Color online） Contours of non-dimensionalized entropy generation at steady state

2.5Εntropy generation

In recent years， second law analysis has been encouraged in the design of microfluidic systems with the aim of entropy generation minimization［24］. Flow and heat transfer in these devices is mostly irreversible， and， therefore， system performance can be quantitatively measured with entropy generation representing the irreversibility of the processes. The rate of volumetric entropy generation is given as

and its dimensionless form is

whererefθ is a reference dimensionless temperature determined by the ambient temperature: θref=kT∞/Figure 9 comparesfor various thermal boundary conditions. Overall， maximum entropy generation is observed near cooling surfaces （e.g.，bottom surface of CASE B， C， D， and E） where maximum heat transfer to the environment happens. As the heat transfer becomes larger， the irreversibility， i.e.，entropy generation， increases. In all the cases except CASEs B and F， thedistribution resembles that of?θ （see Fig.9） because the temperature distribution is uniform in a considerable part of the channel（CASEs C-E and H） or the region with a large?θcoincides with the region with a small θ （CASE G）-recall entropy generation is proportional to2（/）TT?When the bottom wall is adiabatic （CASEs F-H）， the weak entropy generation region becomes wider andthe strong generation region is isolated to the top cooling section.

3. Conclusions

In this study， the Joule heating of conducting fluids in a slit microchannel is investigated theoretically under different thermal boundary conditions. A constant potential difference is applied across the channel with common electrical boundary conditions. The bottom wall is an electrode （one section） while the top wall is divided into two sections: The left half is covered with an electrode and the right half is electrically insulated. An analytical expression is derived for the electric field while the temperature field is computed by numerically solving the simplified pure conduction energy balance equation to determine the effect of Joule heating in a micro channel. Eight thermal boundary conditions are considered from a combination of adiabatic （?T/?y=0） and cooling （T=T∞） scenarios at the three sections. Each of the conditions was carefully chosen to mimic a real situation. The significant findings from the results for the estimated physical mechanisms in this study are as follows:

（1） The electric field near the turning point increases by about 16 times in comparison with an average field.

（2） When all the surfaces are thermally insulated，i.e.， adiabatic， the system temperature keeps increasing with time due to Joule heating.

（3） Significant thermal loading is usually concentrated near the turning point.

（4） To cool the electrodes is the optimal strategy in thermal design unless one can cool all the surfaces together.

（5） Even with a small potential difference of 800 mV， the medium temperature may increase by 30oC and 101oC for 1.0 M and 3.0 M common electrolytes， respectively. Such thermal loading is enough to cause severe adverse effects in biological applications.

（6） The Nusselt number， Nu， identifies the heat loss through a wall quantitatively. The most significant heat loss occurs when all the electrodes are adiabatic while the electrically insulated section is in cooling.

（7） The orders of thermophoresis strength and entropy generation are different from those of maximum temperature. If one focuses on the flow field and energy availability of a system， one needs to focus on them rather than the temperature field only. However，it should be noted that Joule heating is essential in the analysis of any thermal features.

Finally the new knowledge obtained from the current study results can use to improve the microfludic system， such as design， efficiency and thermal control.

Acknowledgement

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea （NRF） funded by the Ministry of Education， Science and Technology （Grant No. NRF-2012R1A1A1042920）.

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［17］ KUA C. H.， LAM Y. C. and RODRIGUEZ I. et al. Dynamic cell fractionation and transportation using moving dielectrophoresis［J］. Analytical Chemistry， 2007， 79（18）: 6975-6987.

［18］ JAMALABADI M. Y. A.， GHASEMI M. and HAMEDI M. H. Numerical investigation of thermal radiation effects on open cavity with discrete heat sources［J］. International Journal of Numerical Methods for Heat and Fluid Flow， 2013， 23（4）: 649-661.

［19］ JAMALABADI M. Y. A.， GHASEMI M. and HAMEDI M. H. Two-dimensional simulation of thermal loading with horizontal heat sources［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2012， 226（5）: 1302-1308.

［20］ JAMALABADI M. Y. A. Experimental investigation of thermal loading of a horizontal thin plate using infrared camera［J］. Journal of King Saud University Engineering Sciences， 2014， 26（2）: 159-167.

［21］ VALYASHKO V. M. Hydrothermal experimental data［M］. New York， USA: John Wiley and Sons， 2008.

［22］ RAMIRES M. L. V.， De CASTRO C. A. N. Thermal conductivity of aqueous potassium chloride solutions［J］. International Journal of Thermophysics， 2000， 21（3）: 671-679.

［23］ GOLDHIRSCH I.， RONIS D. Theory of thermophoresis. I. General considerations and mode-coupling analysis［J］. Physical Review A， 1983， 27（3）: 1616-1634.

［24］ JAMALABADI M. Y. A.， PARK J. H. and LEE C. Y. Optimal design of mhd mixed convection ow in a vertical channel with slip boundary conditions and thermal radiation effects by using entropy generation minimization method［J］. Entropy， 2015， 17（2）: 866-881.

E-mail: muhammad_yaghoob@yahoo.com

（January 31， 2015， Revised July 28， 2015）

* Biography： M. Y. ABDOLLAHZADEH JAMALABADI（1983-）， Male， Ph. D.， Assistant Professor