Viscous dissipation effects on thermal transport characteristics of combined pressure and electroosmotically driven flow in microchannels

International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Viscous dissipation effects on thermal transport characteristics of combined pressure and electroosmotically driven flow in microchannels Arman Sadeghi, Mohammad Hassan Saidi * Center of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 2 September 2009 Received in revised form 15 March 2010 Accepted 26 March 2010 Available online 8 May 2010 Keywords: Electroosmotic flow Laminar Convection Microchannel Viscous dissipation Joule heating

a b s t r a c t This study investigates the influence of viscous dissipation on thermal transport characteristics of the fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels subject to uniform wall heat flux. Closed form expressions are obtained for the transverse distributions of electrical potential, velocity and temperature and also for Nusselt number. From the results it is realized that the Brinkman number has a significant effect on Nusselt number. Generally speaking, to increase Brinkman number is to decrease Nusselt number. Although the magnitude of Joule heating can affect Brinkman number dependency of Nusselt number, however the general trend remains unchanged. Depending on the value of flow parameters, a singularity may occur in Nusselt number values even in the absence of viscous heating, especially at great values of dimensionless Joule heating term. For a given value of Brinkman number, as dimensionless Debye–Huckel parameter increases, the effect of viscous heating increases. In this condition, as dimensionless Debye–Huckel parameter goes to infinity, the Nusselt number approaches zero, regardless of the magnitude of Joule heating. Furthermore, it is realized that the effect of Brinkman number on Nusselt number for pressure opposed flow is more notable than purely electroosmotic flow, while the opposite is true for pressure assisted flow. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Due to the rapid development of microdevices such as MEMS sensors, micropumps and microvalves, research efforts in fluid flow and heat transfer in these devices have become attractive research fields. Transport phenomena at the microscale reveal many features that are not observed in the macroscale devices. Consequently, fundamental issues related to fluid flow and heat transfer in microchannels need to be resolved for efficient design of microfluidic devices. Fluid delivery is crucial in the microfluidic systems since the operating pressure is substantially high. Although micropumps which are capable of delivering such pressures exist [1], however their moving components are complicated to design and fabricate and they are prone to mechanical failure due to fatigue and fabrication defects which consequently make them unsuitable for microfluidic applications. To meet the pumping requirements of microdevices, various techniques have been proposed for fluid pumping in which the electroosmotic micropump has been favored due to its many advantages over other types of micropumps. Electroosmotic pumps need no moving parts and have much simpler design and easier fabrication. It is applicable to a wide range of fluid conductivity, which is essential for biomedical applications. * Corresponding author. Tel.: +98 21 66165522; fax: +98 21 66000021. E-mail address: [email protected] (M.H. Saidi). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.04.028

Also, precise flow control can be easily achieved by controlling the external electric field. The study of liquid flow in microchannels with consideration of electrokinetic effects can be traced to 1960s. The early analytical works on electroosmotic flow report the electrokinetically driven fully developed hydrodynamics of microchannels [2–4]. Hydrodynamically developing flow between two parallel plates for electroosmotically generated flow has been reported in a numerical study by Yang et al. [5]. Also several researches have been performed to study heat transfer characteristics of electroosmotic flow in microchannels. Maynes and Webb [6] analytically have studied fully developed electroosmotically generated convective transport for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Yang et al. [7] investigated forced convection in rectangular ducts with electrokinetic effects for both hydrodynamically and thermally fully developed flow. They investigated the effects of streaming potential on flow and heat transfer. The flow rate induced by electroosmotic force is usually small and therefore even a small pressure gradient applied along a microchannel may cause velocity distributions and corresponding flow rates that deviate from the purely electroosmotic flow. The pressure gradient may arise from several reasons such as the presence of alternative pumping mechanism, placement of mechanical valve in the flow path and the existence of variations in the wall zeta potential [8]. When electroosmotic and traditional pressure

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

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Nomenclature Br cp Dh e Ex G1 G2 h H ie k kB n0 Nu p q s S T u u* U x y

Brinkman number [=lU2/qH] specific heat at constant pressure [kJ kg1 K1] hydraulic diameter of channel [=4H] electron charge [C] electric field in the axial direction [V m1] dimensionless pressure gradient [Eq. (14)] dimensionless electrical potential gradient [Eq. (14)] heat transfer coefficient [W m2 K1] half channel height [m] current density [A m2] thermal conductivity [W m1 K1] Boltzmann constant [J K1] ion density [m3] Nusselt number [=hDh/k] pressure [Pa] wall heat flux [W m2] volumetric heat generation due to Joule heating [W m3] dimensionless form of s [=sH/q] temperature [K] axial velocity [m s1] dimensionless axial velocity [=u/U] mean velocity [m s1] axial coordinate [m] transverse coordinate [m]

forces are present simultaneously, the resulting velocity profile is a superposition of the electroosmotic and pressure driven flows [9]. Fully developed thermal transport of combined pressure and electroosmotically driven flow in circular microtubes has been analyzed by Maynes and Webb [10]. The two classical thermal boundary conditions of constant wall heat flux and constant wall temperature were considered. Chakraborty [11] and Zade et al. [12] developed closed form solutions for hydrodynamically and thermally fully developed heat transfer in circular ducts and parallel plate channels, respectively, considering isoflux boundary conditions at the walls for combined pressure and electroosmotically driven flow. However, Zade et al. [12] represented the effect of the EDL by a slip velocity at the wall (Helmholtz–Smoluchowsky velocity), which limits their analysis to thin EDL limit only. Jain and Jensen [13] considered fully developed isoflux heat transfer in microchannels formed by parallel plates, analyzing the flow and heat transfer within the EDL but did not consider the effect of Joule heating. Recently, Chen [14] investigated the thermal transport characteristics of fully developed mixed pressure and electroosmotically driven flow in parallel plate micro- and nanochannels subject to uniform wall heat flux considering Joule heating effects. Analytical solutions were obtained for constant fluid properties, while numerical solutions were presented for variable fluid properties. Analytical solutions for thermally developing combined pressure and electroosmotically driven flow in microchannels have been obtained by Dutta et al. [15–17] for a variety of wall boundary conditions. Viscous dissipation effects are typically only significant for high viscous flows or in presence of high gradients in velocity distribution. In macroscale, such high gradients occur in high velocity flows. However, in microscale devices such as microchannels, because of small dimensions, such high gradients may occur even for low velocity flows. So, for microchannels the viscous dissipation should be taken into consideration. Koo and Kleinstreuer [18,19] investigated the effects of viscous dissipation on the temperature field and ultimately on the friction factor using dimensional analysis and experimentally validated computer simulations. It was

y* z

dimensionless transverse coordinate [=y/H] valence number of ions in solution

Greek symbols e fluid permittivity [C V1 m1] f wall zeta potential [V] f* dimensionless wall zeta potential h dimensionless temperature [Eq. (21)] j Debye–Huckel parameter [m1] K dimensionless Debye–Huckel parameter [=jH] kD Debye length [m] l dynamic viscosity [kg m1 s1] q density [kg m3] qe net electric charge density [C m3] r liquid electrical resistivity [Xm] u electrostatic potential [V] U externally imposed electrostatic potential [V] w EDL potential [V] w* dimensionless EDL potential [=ezw/kBTav] Subscripts av average b bulk c critical w wall

found that ignoring viscous dissipation could affect accurate flow simulations and measurements in microconduits. Although there are numerous works considering viscous dissipation effects in pressure driven flows, unfortunately the open literature shows a limited number of papers that deal with viscous heating effects in electroosmotic flow through microchannels. The effect of viscous dissipation in fully developed electroosmotic heat transfer for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions has been analyzed by Maynes and Webb [20].They concluded that the influence of viscous dissipation is only important at low values of the relative duct radius. Sharma and Chakraborty [21] have obtained semi analytical solutions for the temperature and Nusselt number distribution in the thermal entrance region of parallel plate microchannels under the combined action of pressure driven and electroosmotic transport mechanisms, by taking into account the effects of viscous dissipation in the framework of an extended Graetz problem. They considered the constant wall temperature case and represented the effect of the EDL by Helmholtz–Smoluchowsky velocity. The main theme of the present work is to analytically investigate viscous dissipation effects on thermal transport characteristics of both hydrodynamically and thermally fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels subject to uniform wall heat flux. Closed form expressions are obtained for the transverse distributions of electrical potential, velocity and temperature and also for Nusselt number. The results of this investigation will give valuable insight to the effect of viscous dissipation on the flow and heat transfer characteristics that can be useful for active control of electroosmotically driven flow and heat transfer.

2. Problem formulation We consider flow through a microchannel formed between two parallel plates with channel half width of H. Geometry of the

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Fig. 1. Geometry of the physical problem, coordinate system and electric double layer.

physical problem is depicted in Fig. 1. The flow is driven by both pressure gradient and external voltage gradient. In the analysis the following assumptions are considered:  The flow is laminar and both thermally and hydrodynamically fully developed.  Thermophysical properties are constant. This assumption, which has successfully been used by Maynes and Webb [20], is valid for temperature variations less than 10 K.  The channel walls are subjected to a constant heat flux.  The liquid contains an ideal solution of fully dissociated symmetric salt.  The charge in the EDL follows Boltzmann distribution.  In calculating the charge density, it is assumed that the temperature variation over the channel cross section is negligible compared to the absolute temperature. Therefore, the charge density field is calculated on the basis of an average temperature.  Wall potentials are considered low enough for Debye–Huckel linearization to be valid.  The external voltage is significantly higher than the flow induced streaming potential.

  2 d w 2n0 ez ezw ¼ sinh : dy2 kB T av e

ð4Þ

The above non-linear second order one dimensional equation is known as the Poisson–Boltzmann equation. Eq. (4) in the dimensionless form becomes 2

d w 2n0 e2 z2 2 ¼ H sinh w dy2 ekB T av

ð5Þ

in which w* = ezw/kBTav and y* = y/H. The quantity (2n0e2z2/ ekBTav)1/2 is known as Debye length kD. Defining Debye–Huckel parameter as k = 1/kD, we come up with 2

d w 2  k H2 sinh w ¼ 0: dy2

ð6Þ

2.1. Electrical potential distribution

If w* is small enough, namely w* 6 1, the term sinh w* can be approximated by w*. This linearization is known as Debye–Huckel linearization. It is noted that for typical values of e = 1.6  1019 C and Tav = 298 K and using the values of 1 and 1.38  1023 J/K for valence number and Boltzmann constant, respectively, this approximation is valid for w 6 25.7 mV. Defining dimensionless Debye– Huckel parameter K = kH and invoking Debye–Huckel linearization, Eq. (6) becomes

The electrical potential distribution is obtained from solution of the Poisson equation:

d w  K 2 w ¼ 0: dy2

r2 u ¼ 

qe ; e

ð1Þ

where e is the fluid permittivity, and qe is the net electric charge density. The potential u is due to combination of externally imposed field U and EDL potential w, namely:

u ¼ U þ w:

ð2Þ

For an ideal solution of fully dissociated symmetric salt, the electric charge density is given by [9]

 ezw ; qe ¼ 2n0 ez sinh kB T av

2

The boundary conditions for the above equation are

wð1Þ ¼ f ;

where n0 is the ion density, e is the electron charge, z is the valence number of ions in solution, kB is the Boltzmann constant, and Tav is the average absolute temperature over the channel cross section. For fully developed flow, w = w(y) and the external potential gradient is in the axial direction only, i.e., U = U(x). For a constant voltage gradient in the x-direction, Eq. (1) becomes

  dw ¼0 dy ð0Þ

ð8Þ

in which f* is the dimensionless wall zeta potential, i.e., f* = ezf/ kBTav. Using Eq. (7) and applying boundary conditions (8), the dimensionless potential distribution is obtained as follows:

w ¼ f



ð3Þ

ð7Þ

coshðKy Þ : cosh K

ð9Þ

2.2. Velocity distribution Using coordinates shown in Fig. 1, the momentum equation in x-direction under combined action of pressure and electric potential gradient is as follows: 2

l

d u dp ¼  qe Ex ; dy2 dx

ð10Þ

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

  @T ¼0 @y ðx;0Þ

where the electric field in the x-direction Ex is given by

Ex ¼ 

du dU ¼ : dx dx

ð11Þ

Substitution of Ex from the above equation and using Eq. (1), the momentum equation becomes 2

2

d u dp d w dU l 2 ¼ e 2 : dy dx dy dx

  @T T ðx;HÞ ¼ T W ðxÞ and k ¼q @y ðx;HÞ

ð12Þ

T  Tw

hðyÞ ¼

qH k

ð13Þ

*

in which u = u/U, where U is the mean velocity and G1 and G2 which refer to dimensionless gradients for pressure and external electrical potential, respectively, are given below:

G2 ¼

2n0 ezH2 dU : lU dx

  G1 G2 coshðKy Þ : u ¼  ð1  y2 Þ þ 2 f 1  cosh K 2 K

K

1 þ G31 : K 1  tanh K

ð15Þ

ð16Þ

Using the above correlation, the velocity distribution becomes

u ¼ 

G1 ð1  y2 Þ þ 2

1 þ G31 K 1  tanh K

  coshðKy Þ : 1 cosh K

ð17Þ

2.3. Temperature distribution The conservation of energy including the effects of Joule heating and viscous dissipation requires

!

 2 du : þsþl dy

ð18Þ

In the above equation, s and l(du/dy)2 denote the rate of volumetric heat generation due to Joule heating and viscous dissipation, respectively. Joule heating is due to the electrical current and the 2 resistivity of the fluid and equals to s ¼ ie r. The term ie is the current density established by the applied potential and r is the liquid electrical resistivity. The current density is related to the charge potential by the relation [22]

ie ¼

Ex

r

cosh w :

in which Tb is the bulk temperature. The energy balance for an element of duct with the dimensions of H and dx may be written as

qcp UHdT b ¼ q þ sH þ l

Z

H

 2 # du dy dx dy

ð23Þ

After required manipulations, the following expression is obtained for dTb/dx

! dT b 1 lU 2 ¼ q þ sH þ b ; dx qcp UH H

ð24Þ

where b is given by

! !2 1 þ G31 1 þ G31 G21 K2  b¼  2G1 2 K tanh K 3 1  tanh 2cosh K 1  K K 2 3 !2 ! 1 þ G31 1 þ G31 5 K 2 4 tanh K: þ þ G1 K K 2 1  tanh K 1  tanh K K

ð25Þ

Since oT/ox is constant, the axial conduction term in the energy equation will be zero. So the energy equation in dimensionless form will be as follows: 2

Note that G1 = 0 corresponds to purely electroosmotic flow, G1 = 3 corresponds to purely pressure driven or Poiseuille flow, 3 < G1<0 correspond to pressure assisted flow and other values of G1 correspond to pressure opposed flow.

@T @2T @2T qcp u ¼ k þ @x @x2 @y2

ð22Þ

0

It should be noted that since in the definition of u*, G1 and G2 we have used the mean velocity, the two latest parameters are not independent. Their dependency can be obtained invoking the fact that average dimensionless velocity over the cross section of the channel is equal to unity. The dependency of G1 and G2 then will be as:

f ¼ 2

@T dT w dT b ¼ ¼ @x dx dx

" ð14Þ

With zero velocity at each wall, the dimensionless velocity profile becomes

G2

ð21Þ

:

Taking differentiation of Eq. (21) with respect to x gives

2

d u ¼ G1  G2 w dy2

H2 dp ; lU dx

ð20Þ

The dimensionless temperature h is introduced in the following, which depends only on y for fully developed flow:

The momentum equation in dimensionless form becomes

G1 ¼

3785

ð19Þ

For low wall zeta potentials, which is the case in this study, cosh w* ? 1 and the Joule heating term may be considered as the constant value of s ¼ E2x =r [22]. The relevant boundary conditions for the energy equation are as follows:

d h 2 2 ¼ a  by  c coshðKy Þ  dsinh ðKy Þ  ey sinhðKy Þ; dy2

ð26Þ

where a, b, c, d and e are given by

! 1 þ G31 G1 ð1 þ S þ bBrÞ  S; a¼  þ K 2 1  tanh K   G1 ð1 þ S þ bBrÞ þ G21 Br; b¼  2 ! 1 þ G31 ð1 þ S þ bBrÞ ; c¼ K cosh K 1  tanh K !2 1 þ G31 K2 d¼ Br; 2 K cosh K 1  tanh K !   1 þ G31 K G1 e¼4  Br; K cosh K 2 1  tanh K

ð27Þ ð28Þ ð29Þ

ð30Þ ð31Þ

in which Br = lU2/qH is the Brinkman number and S = sH/q is the dimensionless volumetric heat generation due to Joule heating. The thermal boundary conditions in the dimensionless form are written as



dh dy

 ¼ 0;

hð1Þ ¼ 0:

ð32Þ

ð0Þ

Using Eq. (26) and applying boundary conditions (32), the dimensionless temperature distribution is obtained as:

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h¼

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

  a 2 b 4 c coshð2Ky Þ y2  y  y  2 coshðKy Þ  d 2 12 4 K 8K 2    y 2    e 2 sinhðKy Þ  3 coshðKy Þ þ f ; K K

ð33Þ

where

  a b cosh K 1 coshð2KÞ d f ¼ þ c  þ  2 12 4 K2 8K 2   sinh K 2 þ  3 cosh K e: K2 K

3. Results and discussion

ð34Þ

To obtain the Nusselt number, first the dimensionless bulk temperature hb must be calculated, which is given by

R1

u hdy hb ¼ R0 1 ¼  u dy 0

Z

1

u hdy

0

! ! 1 þ G31 1 þ G31 G1 G1 g3 g1 þ g2   þ þ f; K K 2 1  tanh 2 cosh K 1  tanh K K

¼

ð35Þ

where g1, g2 and g3 are as follows:

g1 ¼

g2 ¼

g3 ¼

  a b sinh K 1 sinhð2KÞ cþ d    3 3 6 60 12 K 16K   cosh K sinh K e; 3  3 K K4   a b sinh K cosh K sinh K c    2 þ 2 10 84 K5 K3 K4   1 sinhð2KÞ coshð2KÞ sinhð2KÞ þ d 8  10 þ 10  5 160 K5 K3 K4   cosh K sinh K cosh K sinh K  e; 5 þ 10  10 K5 K3 K4 K6   sinh K cosh K sinh K a þ  2K K2 K3   sinh K cosh K sinh K cosh K sinh K b   þ  2 þ 2 12K K5 3K 2 K3 K4   1 sinhð2KÞ c  þ 2 3 2K 4K   1 sinh K cosh K sinh K sinhð3KÞ þ d 12  24 þ 21  2 3 3 48 K K K K   1 8 coshð2KÞ sinhð2KÞ e:   3þ2 5 3 8 K K K4

constant wall heat flux as well, where q gives a particular value which is a function of the total Joule heating and viscous dissipation and may be obtained from q + sH + (lU2/H)b = 0. The asymptotic solution without internal heating requires an independent procedure and cannot be obtained from the solution presented here.

ð36Þ

ð37Þ

ð38Þ

The dimensionless pressure gradient, dimensionless Debye– Huckel parameter, dimensionless Joule heating term and Brinkman number are the main parameters governing heat and fluid flow in fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels. Here, their interactive effects on the transverse distributions of velocity and temperature and finally on Nusselt number are analyzed. The transverse distribution of dimensionless velocity at different values of dimensionless pressure gradient for K = 50 is presented in Fig. 2. The value of Debye–Huckel parameter, equal to 50, implies that EDL is limited to a small region close to the walls and a significant portion of the channel width is outside the EDL, so, the velocity distribution is nearly a slug flow profile for purely electroosmotic flow (G1 = 0), while as expected, it is a Poiseuille flow for purely pressure driven flow (G1 = 3). The velocity profile for other values of dimensionless pressure gradient is the superposition of both purely electroosmotic and Poiseuille flows. So, the velocity distribution for pressure assisted flow (G1 = 1) shows both a maximum value at the centerline which is related to Poiseuille flow and a sharp gradient at the wall which is inherited from electroosmotic flow. For pressure opposed flow which corresponds to G1 = 1 and G1 = 3, the velocity distribution attains its maximum value at a point close to the wall and reaches a local minimum at the centerline as a result of opposed pressure. Note that for sufficiently large values of G1, reverse flow may occur at the centerline. Fig. 3 exhibits transverse distribution of dimensionless temperature at different values of S in the absence of viscous heating and K = 10. As seen, increasing values of S lead to smaller values of dimensionless temperature for purely electroosmotic flow which implies that Joule heating increases the wall temperature rather than the bulk temperature. The reason is that although the distribution of energy generated by Joule heating is uniform throughout the channel cross section but the energy transferred by convection decreases near the wall and it equals zero at the wall. For S = 10, h is approximately constant over much of the duct cross section rising to a maximum at the wall. In this region at which the velocity

Therefore, the Nusselt number which is based on the definition given by

Nu ¼

hDh qDh 4 ¼ ¼ hb k kðT w  T b Þ

ð39Þ

can be obtained. It is noteworthy that for purely electroosmotic flow, the Nusselt number obtained in the present study and the one given by Chen [14] are in quite agreement. Unfortunately, in the presence of pressure gradient the use of different reference velocities prevents any comparison. For the case that the flow is purely driven by pressure gradient the present results are in excellent agreement with those given by Jeong and Jeong [23]. Although the analysis presented in this section assumed a uniform heat flux at the walls, however, the asymptotic solution of a constant wall temperature boundary condition for which oT/ ox = 0 may also be explored. This corresponds to the condition where all volumetric heating in the fluid is dissipated at the walls, yielding q < 0. Thus, the fully developed condition for imposed constant wall temperature in presence of internal heating is one of

Fig. 2. Transverse distribution of dimensionless velocity at different values of G1.

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

Fig. 3. Transverse distribution of dimensionless temperature at different values of S in the absence of viscous heating: (a) purely electroosmotic flow and (b) pressure opposed flow.

profile is uniform, all the energy generated by Joule heating is transferred by the flow. For pressure opposed flow, the temperature distribution shows a quite different behavior. Although a larger S leads to smaller h in the region adjacent to the wall, it leads to greater values of dimensionless temperature in the region close to the centerline. The interpretation can be expressed is that since as a result of opposed pressure the centerline velocity is a local minimum, so the energy transferred by the flow in this region is smaller compared to the near wall region which attains large velocities. This fact consequently leads to greater dimensionless temperatures for the core region. The transverse distribution of dimensionless temperature at different values of Br for wall cooling case, at which heat is transferred from the walls to the fluid, is shown in Fig. 4. Note that since the wall heat flux is positive, the Brinkman number cannot take negative values. The viscous dissipation behaves like an energy source increasing the temperature of the fluid especially near the wall, since the highest shear rate occurs at this region while it is zero at the centerline. So, for both cases, increasing values of Br lead to smaller dimensionless temperatures. The effect of viscous heating is more notable for pressure opposed flow, as a result of greater velocity gradient existence at the wall. The transverse distribution of dimensionless temperature at different values of Br for wall heating case is presented in Fig. 5. For this case, as Brinkman number increases with negative sign, the dimensionless temperature increases which is an expected behavior. For sufficiently great

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Fig. 4. Transverse distribution of dimensionless temperature at different values of Br for wall cooling case: (a) purely electroosmotic flow and (b) pressure opposed flow.

values of Br with negative sign, the maximum dimensionless temperature occurs at the duct centerline rather than the wall. From the figure, one can see that as Br increases with negative sign, the sign of dimensionless bulk temperature is changed from negative to positive. So, for a value of Brinkman number called Brc, which its value depends on flow parameters, the value of dimensionless bulk temperature will be zero, which this, according to Eq. (39) causes a singularity in Nusselt number values. Fig. 6 depicts Nusselt number versus 1/K at different values of S in the absence of viscous heating for purely electroosmotic flow. Generally speaking, to increase S is to decrease the fully developed Nusselt number. This behavior may be explained by means of Fig. 3a. As seen, increasing values of S lead to greater dimensionless bulk temperatures with negative sign, which this according to Eq. (39) leads to smaller values of Nusselt number. For sufficiently great values of S with negative sign such as S = 10, the behavior is quite different. For these cases, a singularity occurs in Nusselt number values. At the singularity point the wall and the bulk temperatures are the same, so heat transfer cannot be expressed in terms of Nusselt number. Note that after singularity point the Nusselt number takes negative values (not shown in the figure). This phenomenon takes place as a result of the bulk temperature being smaller than the wall temperature and it does not mean that the heat transfer takes place from the wall to the fluid (note that this is a wall heating case). Except for S = 10, a greater value of K causes a greater Nusselt number. By increasing K, EDL will be

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The Brinkman number dependency of the Nusselt number at different values of S for K = 10 is shown in Fig. 7. As seen, increasing values of Brinkman number lead to smaller values of Nusselt number, regardless of the magnitude of dimensionless Joule heating term. The effect of Brinkman number on Nusselt number for pressure opposed flow is more notable than purely electroosmotic flow, as a result of existing greater velocity gradients especially at the

Fig. 5. Transverse distribution of dimensionless temperature at different values of Br for wall heating case: (a) purely electroosmotic flow and (b) pressure opposed flow.

Fig. 6. Nusselt number vs. 1/K at different values of S in the absence of viscous heating for purely electroosmotic flow.

limited to smaller regions close to the walls, resulting in a more plug-like velocity profile. Therefore, as K goes to infinity, for all values of S, the Nusselt number approaches 12 which is the classical solution for slug flow [24].

Fig. 7. Brinkman number dependency of the Nusselt number at different values of S: (a) purely electroosmotic flow, (b) pressure assisted flow and (c) pressure opposed flow.

A. Sadeghi, M.H. Saidi / International Journal of Heat and Mass Transfer 53 (2010) 3782–3791

solid surface, while the opposite is true for pressure assisted flow. Although increasing values of S lead to smaller Nusselt numbers, but for greater values of Br, the effect of S on Nusselt number becomes slighter. This is due to the fact that for great Brinkman numbers, viscous dissipation term dominates heat transfer characteristics. The effect of S on Nusselt number for pressure assisted flow is more pronounced than purely electroosmotic flow, while its effect on Nusselt number for pressure opposed flow is negligible. Note that this is a special case that although the magnitude of dimensionless Joule heating term affects dimensionless temperature distribution, but the dimensionless bulk temperature remains unchanged. Fig. 8 presents Nusselt number values versus G1 at different values of S in the absence of viscous heating. Generally speaking, a greater G1 corresponds to a greater Nu, except for S = 10 which shows a different trend including a singularity point at G1 = 1.46. At G1 = 0.32 the Nusselt number of all values of S coincide with each other. Before this point, the effect of increasing values of S is to decrease Nusselt number, while for greater values of dimensionless pressure gradient it is vice versa. The effect of viscous dissipation on Nusselt number is shown in Fig. 9 which depicts Nusselt number values versus G1 in the absence of Joule heating. As seen, the Brinkman number has a significant effect on Nusselt number especially for Great values of dimensionless pressure gradient. So for the channels having small values of wall heat flux and small height and also for high velocity or high viscous flows, the viscous heating term cannot be discarded from energy equation. Although this phenomenon was observed in the absence of Joule heating, but from Fig. 7 it is clear that the magnitude of Joule heating does not affect the overall behavior of Nusselt number. So it can be expressed that neglecting viscous dissipation may affect accurate flow measurement in flows having great Brinkman number. Fig. 10 illustrates Nusselt number values versus 1/K at different values of Br in the absence of Joule heating. Generally speaking, to increase Brinkman number is to decrease Nusselt number. The effect of Brinkman number on Nusselt number is more significant at higher values of dimensionless Debye–Huckel parameter. At small values of K which correspond to great values of 1/K, the effect of dimensionless Debye–Huckel parameter on Nusselt number becomes insignificant. This is due to the fact that at small values of K the velocity profile is nearly similar to that for Poiseuille flow and variation of dimensionless Debye–Huckel parameter does

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Fig. 9. Nusselt number vs. G1 at different values of Br in the absence of Joule heating.

Fig. 10. Nusselt number vs. 1/K at different values of Br in the absence of Joule heating: (a) pressure assisted flow and (b) pressure opposed flow.

Fig. 8. Nusselt number vs. G1 at different values of S in the absence of viscous heating.

not notably affect profile shape and consequently the energy generated by viscous heating or transferred by the flow. Contrary to small values of K, at great values of K the magnitude of dimensionless Debye–Huckel parameter drastically affects Nusselt number.

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At great values of K, the effect of increasing values of K is to increase Nusselt number for wall heating case and also for the case without viscous dissipation, while the opposite is true for wall cooling case. The effect of K on Nusselt number is more pronounced in the presence of viscous heating and also for pressure opposed flow which takes greater velocity gradients. The reason is that although for great values of K the velocity distribution is nearly uniform over much of the duct cross section, there is a huge velocity gradient at the solid surface. So, although the effect of viscous heating on the bulk temperature is not significant, the wall temperature severely increases and consequently the Nusselt number will be much different. It should be noted that for all non zero values of Brinkman number as K goes to infinity the Nusselt number value approaches zero, regardless of the magnitude of Joule heating. The reason is that as a result of the enormous value of viscous heating, the wall temperature is much greater than the bulk temperature. This behavior of Nusselt number is accompanied by occurrence of a singularity in Nusselt number values of the wall heating cases as a result of changing the sign of dimensionless bulk temperature from negative for small values of K to positive for great values of dimensionless Debye–Huckel parameter. Here, it will be useful to perform a typical dimensional analysis and study the effects of channel size on viscous heating effects. It is assumed that a pressure gradient of 1000 Pa m1 at favorable and opposed directions exists and the mean velocity is fixed at U = 1 cm s1. The relative permittivity of medium, viscosity and the Debye length are assumed to be 80, 103 kg m1 s1 and 1 nm, respectively. The liquid electrical resistivity is set to 104 Xm and the wall heat flux is specified to be 1 kW m2. The zeta potential is considered 25 mV and other parameters have the same values which were considered in Section 2.1. As the channel height is increased the applied electric field is changed accordingly in order to keep the mean velocity constant. The dependence of Nusselt number on the half channel height is shown in Fig. 11, where the cases when viscous heating is neglected are shown by symbols. At smaller values of channel height, the flow is actually driven by electrokinetic effects. Therefore, the Nusselt number for both pressure assisted and pressure opposed flows are the same. As the channel height increases the electrokinetic effects become more and more limited to the region near the wall and the pressure influences become more important. For pressure opposed flow, as channel height increases, due to increasing pressure effects, the velocity decreases in the core region, while it increases near

Fig. 11. Nusselt number vs. half channel height where symbols show the cases when viscous heating is neglected.

the wall. As a result, the energy convected by the flow decreases in the core of channel, while it decreases near the wall, leading to a smaller difference between the temperatures of the wall and the bulk flow and ultimately a higher Nusselt number. As observed, the effect of increasing H is to decrease Nusselt number for pressure assisted flow. This is because the influence of pressure on velocity profile is the opposite of that for pressure opposed flow. For H = 1 lm, neglecting viscous heating leads to about 5% overestimating the Nusselt number for both cases. The influence of viscous heating is decreased with increasing H for pressure assisted flow and neglecting viscous heating for H = 100 lm only results in an overestimating of about 2%. This is due to the fact that as channel height increases, as a result of increasing pressure effects, the velocity gradient at the wall becomes smaller. For pressure opposed flow, the effect of viscous dissipation on Nusselt number increases with increasing channel height and reaches about 15% at H = 100 lm. As mentioned previously, the temperature difference between the wall and the bulk flow decreases with channel size for pressure opposed flow and may reach zero for larger values of H, resulting in a singularity in Nusselt number values. Near the singularity point, the value of Nusselt number becomes severely sensitive to temperature variations and even a little temperature variation caused by viscous heating may notably affect Nusselt number.

4. Conclusions In the present study, the fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels has been studied. The classical boundary condition of uniform wall heat flux was considered in the analysis and the effects of viscous heating as well as Joule heating were taken into account. Closed form expressions were obtained for the transverse distributions of electrical potential, velocity and temperature and also for Nusselt number. The problem was found to be governed by four parameters: dimensionless pressure gradient, dimensionless Debye–Huckel parameter, dimensionless Joule heating term and Brinkman number. The main results of this study can be summarized as follows:  The Brinkman number has a significant effect on Nusselt number. Generally speaking, to increase Brinkman number is to decrease Nusselt number. Although the magnitude of Joule heating can affect Brinkman number dependency of Nusselt number, but the general trend remains unchanged.  Depending on the value of flow parameters, a singularity may occur in Nusselt number values even in the absence of viscous heating, especially at great values of dimensionless Joule heating term.  For a given value of Brinkman number, as dimensionless Debye–Huckel parameter increases, the effect of viscous heating increases. The reason is that although viscous heating effect on the bulk temperature is not significant, the wall temperature drastically increases as a result of the huge velocity gradient attainment at the solid surface and consequently the Nusselt number will be much different.  For a given value of Brinkman number, as dimensionless Debye–Huckel parameter goes to infinity, the Nusselt number approaches zero, regardless of the magnitude of Joule heating.  The effect of Brinkman number on Nusselt number for pressure opposed flow is more notable than purely electroosmotic flow, as a result of existing greater velocity gradients especially at the solid surface, while the opposite is true for pressure assisted flow.

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