Analytical solutions for thermo-fluidic transport in electroosmotic flow through rough microtubes

International Journal of Heat and Mass Transfer 92 (2016) 244–251

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Analytical solutions for thermo-fluidic transport in electroosmotic flow through rough microtubes Hadi Keramati a, Arman Sadeghi b, Mohammad Hassan Saidi a, Suman Chakraborty c,⇑ a

Center of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University of Technology, P.O. Box: 11155-9567, Tehran, Iran Department of Mechanical Engineering, University of Kurdistan, Sanandaj 66177-15175, Iran c Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India b

a r t i c l e

i n f o

Article history: Received 29 May 2015 Received in revised form 28 July 2015 Accepted 27 August 2015

Keywords: Electroosmotic flow Rough microtube Joule heating Analytical solution

a b s t r a c t The limitations of the microfabrication technology do not allow producing perfectly smooth microchannels. Hence, exploring the influences of roughness on transport phenomena in microtubes is of great importance to the scientific community. In the present work, consideration is given toward the corrugated roughness effects on fully developed electroosmotic flow and heat transfer in circular microtubes. Analytical solutions based on perturbation technique are presented for the problem assuming a low zeta potential under the constant heat flux boundary condition of the first kind. It is revealed that higher values of the corrugation number and relative roughness give rise to smaller Nusselt numbers. Since the same is true for the mean velocity, one may conclude that the roughness effects on the hydrodynamic and thermal features of electroosmotic flow are negative. Further, the Nusselt number is found to be a decreasing function of the Joule heating rate and an increasing function of the dimensionless DebyeHückel parameter. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Electroosmosis is one of the primary electrokinetic phenomena which was discovered more than two centuries ago [1]. It refers to the flow of an ionic solution tangential to an electrified surface under the application of an electric field. Ionic species within a charged interfacial layer (also known as the electrical double layer, EDL) are forced to move by virtue of the external electric field, which in turn actuates the motion of the solvent molecules through viscous interaction, resulting in so-called electroosmotic transport. The trace of the pertinent literature indicates that the first accurate expression for electroosmotic velocity was derived by Smoluchowski [2] in 1903. Two decades later, Debye and Hückel [3] determined the ionic distribution in solutions of low electrical potential, by means of a linear approximation of the Boltzmann distribution. This paved the way for the development of analytical solutions for electroosmotic flow (EOF) in flat and circular channels by Burgreen and Nakache four decades later [4]. They afterward extended the solutions to account for high surface potentials [5]. ⇑ Corresponding author. E-mail addresses: [email protected] (H. Keramati), [email protected]. ac.ir (A. Sadeghi), [email protected] (M.H. Saidi), [email protected] (S. Chakraborty). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.089 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

Rice and Whitehead [6] investigated fully developed EOF in a narrow cylindrical capillary assuming low zeta potentials. Levine et al. [7] extended Rice and Whitehead’s work to high zeta potentials by means of an approximation method. More recently, Tsao [8] examined the hydrodynamic aspects of EOF in a microannulus, using the Debye-Hückel linearization. His work was extended to high zeta potentials by Kang et al. [9], utilizing an approximate method. Closed form solutions for fully developed EOF in rectangular and semicircular microchannels were presented by Yang [10] and Wang et al. [11], respectively. Xuan and Li [12] developed semi-analytical solutions for electrokinetic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge. Despite the hydrodynamic features being well-explored, the study of the thermal aspects of EOF is recent. The pioneering studies in this scope were performed by Maynes and coworkers and include presenting closed form solutions for thermally fully developed EOF in slit and circular channels of small zeta potential with [13] and without [14] considering viscous heating effects. They thereafter extended their works to account for high zeta potentials, albeit numerically [15]. Subsequently, Chakraborty [16] obtained analytical solutions of Nusselt number for thermally fully developed flow in microtubes under a combined action of electroosmotic forces and imposed pressure gradients. Chen [17] performed the same analysis for a slit channel. The viscous heating

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effects on the thermal characteristics of the fully developed mixed flow in slit and circular microducts were analyzed analytically by Sadeghi and Saidi [18] and Yavari et al. [19], respectively. Yavari et al. [20] derived closed form solutions for the hydrodynamic and thermal characteristics of combined electroosmotic and pressure driven flow in a microannulus. Exactly the same, but this time for a rectangular geometry, was done by Sadeghi et al. [21]. The mixed flow characteristics in triangular microchannels were investigated by Liao et al. [22] through a Galerkin-based numerical approach. In a recent study, Vocale et al. [23] paid attention to the problem of EOF heat transfer in elliptical microchannels under axially constant heat flux boundary condition. The walls of any conduit show some degree of roughness depending on the manufacturing process. Since the surface conditions may affect the flow characteristics to some extent, it is crucial to account for roughness effects in any fluid flow analysis. This is more essential in dealing with microflows, because, as the flow cross section gets smaller the influences of wall phenomena grow significantly. This fact has motivated some researchers to investigate the roughness effects on EOF [24–28]. However, no attention has been given toward the influences of the surface conditions on the associated heat transfer physics, despite recent advancements in using electroosmotic-based micro cooling systems [29–31] that dramatically raise the need for such analyses. The successful utilization of EOF for cooling purposes also draws attention to the thermo-fluidic transport in circular microchannels, as an essential part of microchannel heat sinks [32–35]. In this respect, in spite of considerable attention to the electroosmotic flow and heat transfer in circular microducts, this geometry has been ignored in the studies of the roughness effects. In the present work, the effects of roughness on both the hydrodynamic and thermal features of EOF through circular micropipes are being studied. For a better tractability of the problem, the surface roughness is modeled by considering a corrugated channel surface. The flow is assumed to be both hydrodynamically and thermally fully developed and the thermal boundary condition is assumed to be the constant heat flux of the first kind, H1, which refers to a constant heat flux in the axial direction and a constant temperature in each cross section [36]. Analytical solutions are obtained for the electrical potential, velocity, and temperature fields utilizing the perturbation technique, assuming small amounts of the relative roughness. To the best of our knowledge, this is the first report on analytical solutions for heat transfer in EOFs with surface roughness effects taken into consideration. A complete parametric study is then performed by putting emphasize on the heat transfer aspects and it is shown that roughness has unfavorable effects on the transport phenomena of electroosmotic flow. 2. Problem formulation Steady, laminar, hydrodynamically and thermally fully developed electroosmotic flow inside a circular microtube with a rough surface is considered. The geometry of the problem is shown schematically in Fig. 1. The surface roughness is assumed to be approximately modeled by considering a corrugated wall of the form rw ¼ R þ R e sinðMhÞ, where R is the mean radius of the microtube and M stands for the number of corrugations. Moreover, e denotes the relative roughness. Based on the experimental data for the roughness amplitude of glass microchannels [37], this may take values of the order 0.01 assuming a channel of radius 10 lm. The electroosmotic-based microchannel heat sinks usually are made from silicon which may result in significantly rough surfaces; we here assume the values of up to 0.06 for e based on the experimental data of Qu et al. [38]. Following the findings of Sadeghi et al. [39], the thermophysical properties are assumed to

be computed based on the bulk mean temperature for the temperature-dependent influences to remain negligible. The zeta potential of the channel is not only constant and uniform but also is low enough to permit the usage of the Debye-Hückel linearization. In addition, it is assumed that the effect of temperature variations on the potential distribution within the EDL may be neglected [40]. The thermal features are analyzed considering the Joule heating effects and assuming the H1 thermal boundary conditions. 2.1. Electrical potential distribution The combination of externally applied potential U and EDL potential w will constitute the electrostatic potential field u in the channel. The former is only dependent upon the axial direction so that

uðr; h; zÞ ¼ UðzÞ þ wðr; hÞ

ð1Þ

The relationship between the electrostatic potential and the net electrical charge density qe is given by the Poisson equation:

r2 u ¼ 

qe

ð2Þ



where e is the permittivity constant of the solution. In general, the connection between the electrostatic potential and the electric charge density should be described by the Nernst-Planck equations. However, at the hydrodynamically developed conditions, the spatial distribution of the electric charge density is expressed by the Boltzmann equation, despite the fact that it assumes the thermodynamic equilibrium [41]. This may be attributed to the orthogonality of the velocity vector and the ion concentration gradient at the fully developed conditions. Utilizing the Boltzmann distribution, the electric charge density for a solution containing N ionic species becomes [42]: N X zi ni0 e

qe ¼ e



k

ezi w B T av



ð3Þ

i¼1

where e represents the proton charge, zi and ni0 are the valence number and concentration of the ith species at neutral conditions, respectively, kB is the Boltzmann constant, and Tav is the average absolute temperature over the channel cross section. Substituting the charge density expression into the Poisson equation, and considering a constant voltage gradient in the z-direction, Eq. (2) reduces to the Poisson–Boltzmann equation N @ 2 w 1 @w 1 @ 2 w eX þ 2 þ ¼ zn e 2 2 @r r @r r @h  i¼1 i i0



ezi w B T av

k



ð4Þ

Eq. (4) is nonlinear and cannot be solved analytically; nevertheless, for small potentials, it can be linearized by replacing the exponential terms with their Taylor series and discarding all the terms of the order w2 and higher. This approximation, first introduced by Debye and Hückel [3], has been shown to be valid for zeta potentials of up to 50 mV [39]. Making use of the electroneutrality conditions in the bulk where w = 0, the linearized form of the Poisson– Boltzmann equation in dimensionless form becomes

r2 w ¼

@ 2 w 1 @w 1 @ 2 w þ   þ 2 ¼ K 2 w 2 r @r r @h2 @r

ð5Þ

where w ¼ w=f, r ¼ r=R, and K ¼ R=kD is the dimensionless DebyeP 1=2 N 2 2 as the Debye Hückel parameter with kD ¼ i¼1 ni0 e zi =kB T av length, a measure of the extent of EDL. The boundary conditions for the dimensionless electrical potential equation are as follows

w ¼ 1 at r  ¼ 1 þ e sinðMhÞ

ð6Þ

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Fig. 1. Schematic of the rough micropipe including the coordinate system and EDL. 



w – 1 at r ¼ 0

ð7Þ

We seek an approximate solution in the form of a Poincare type asymptotic expansion for the electrical potential as

w ðr  ; hÞ ¼ w0 ðr  ; hÞ þ ew1 ðr ; hÞ þ e2 w2 ðr  ; hÞ þ   

ð8Þ

Substituting the above expansion into Eq. (5) and equating the terms having the same powers of e, the governing equation of the electrical potential can be split into the following equations

r2 w0 ¼ K 2 w0 ;

r2 w1 ¼ K 2 w1 ;

r2 w2 ¼ K 2 w2

ð9Þ

where those corresponding to the higher powers of e have been ignored. To derive the associated boundary conditions, w⁄ in Eq. (6) is expanded in a Taylor series to obtain

w ð1 þ e sinðMhÞ; hÞ ¼ w ð1; hÞ þ e sinðMhÞ þ

e2 sin2 ðMhÞ @ 2 w

ð1; hÞ þ . . . ¼ 1

@r 2

2

@w ð1; hÞ @r ð10Þ

2.2. Velocity distribution The equation governing the exchange of momentum within a fluid in the absence of pressure effects is written as

q

Du ¼rsþF Dt

ð19Þ

where q stands for the fluid density, s is the stress tensor, and u and F are the velocity and body force vectors, respectively. Here, the only body force is the Coulomb force due to the action of the electric field E ¼ ru on the ions and is given by qe E. At the fully developed conditions, the effects of the transverse velocity components can be ignored. Considering this fact and the continuity equation, that is r  u ¼ 0, the velocity vector adopts the form u ¼ ½0; 0; uz ðr; hÞ. Thus, the term on the left hand side of Eq. (19) obviates for a steady flow. Accordingly, the momentum equation in the axial direction reads

l

@ 2 uz 1 @uz 1 @ 2 uz þ þ r @r r2 @h2 @r2

!

¼ qe Ez

ð20Þ

Substitution of Eq. (8) into Eq. (10) provides   @w  w0 ð1;hÞ þ e w1 ð1; hÞ þ sinðMhÞ 0 ð1; hÞ @r " # 2  @w sin ðMhÞ @ 2 w0 þ e2 w2 ð1;hÞ þ sinðMhÞ 1 ð1;hÞ þ ð1;hÞ þ  ¼ 1 ð11Þ 2 @r @r 2

where l is the dynamic viscosity and Ez ¼ dU=dz denotes the externally applied electric field. Using the linearized form of Eq. (3), the momentum equation can be written in the following dimensionless form

Equating the terms having the same powers of e, the boundary conditions associated to w0 , w1 , and w2 are derived as

r2 u ¼

w0 ð1; hÞ

¼1

w1 ð1; hÞ þ

ð12Þ

@w sinðMhÞ 0 @r

w2 ð1; hÞ þ sinðMhÞ

ð1; hÞ ¼ 0

ð13Þ

@w1 sin ðMhÞ @ 2 w0 ð1; hÞ þ ð1; hÞ ¼ 0  2 @r @r 2 2

ð14Þ

Solving Eq. (9) subject to the boundary conditions (12)–(14) yields

w0 ðr  ; hÞ ¼

I0 ðKr  Þ I0 ðKÞ

w1 ðr  ; hÞ ¼  w2 ðr  ; hÞ ¼ b

ð15Þ

KI1 ðKÞ IM ðKr  Þ sinðMhÞ I0 ðKÞIM ðKÞ 

ð16Þ

b¼

ð21Þ

where u ¼ uz =uHS in which uHS ¼  2 fEz =l denotes the maximum possible electroosmotic velocity known as the HelmholtzSmoluchowski velocity. Besides the finiteness of the velocity at the origin, the dimensionless momentum equation is constrained by the no-slip boundary condition at the wall, that is

u ¼ 0 at r ¼ 1 þ e sinðMhÞ

ð22Þ

Even though it is possible to directly solve the momentum equation in a manner done for the electrical potential equation, it is easier to substitute the term K 2 w from Eq. (5) to reach the following form of the dimensionless momentum equation

r2 uþ ¼ 0

ð23Þ

where



I0 ðKr Þ I2M ðKr Þ b cosð2MhÞ I0 ðKÞ I2M ðKÞ

ð17Þ

wherein 2

@ 2 u 1 @u 1 @ 2 u þ   þ 2 ¼ K 2 w 2 r @r r @h2 @r

2

K I1 ðKÞ K ½IM1 ðKÞ þ IMþ1 ðKÞ  ½I0 ðKÞ þ I2 ðKÞ 4I0 ðKÞIM ðKÞ 8I0 ðKÞ

ð18Þ

uþ ¼ u þ w

ð24Þ

With the consideration of Eqs. (6) and (22), it is obvious that u+ should have the following value at the wall

uþ ¼ 1 at r  ¼ 1 þ e sinðMhÞ

ð25Þ

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We readily deduce that a solution of the form uþ ðr  ; hÞ ¼ 1 satisfies both the governing equation and the associated boundary conditions. Accordingly the velocity field becomes

u ¼ 1  w

ð26Þ

The dimensionless average velocity over the channel cross section is evaluated by

R

uav ¼

u dA A



ð27Þ

wherein A⁄ is the dimensionless cross-sectional area of the channel and is calculated as

A ¼ M

Z

2p M

1þe sinðMhÞ

0

0

Thus

uav ¼ 1 

Z

M

  e2 r  dr  dh ¼ p 1 þ 2

R 2Mp R 1þe sinðMhÞ 0

0



p

w r  dr  dh

2 1þe

ð28Þ

ð30Þ

where cp and k are the specific heat at constant pressure and the thermal conductivity, respectively. Moreover, s ¼ E2z =r represents the rate of volumetric heat generation due to Joule heating with r being the electrical resistivity of the liquid defined as [43]

r0

ð31Þ

coshw

where r0 is the electrical resistivity of the neutral liquid. Since the zeta potential is considered to be low in the present study, the hyperbolic term in the denominator of the above equation tends to one. Therefore, the Joule heating term may be approximated by the constant s ¼ E2z =r0 [15]. Considering a fluid of constant thermal conductivity along with a steady fully developed flow field for which DT=Dt ¼ uz ð@T=@zÞ, the energy equation simplifies to

qcp uz

@T E2 ¼ k r2 T þ z @z r0

S¼

T  T w ðzÞ qav R k

ð32Þ

ð37Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 dr w P ¼ r 2 dh w þ dh perimeter 0 Z 2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 2 1 þ 2e sinðMhÞ þ e2 sin ðMhÞ þ M 2 e2 cos2 ðMhÞdh ð38Þ ¼M Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 r2 w dh þ dr w ¼ M

Z



2p M

0

p

ð64 þ 16M2 e2  34M4 e4 Þ

32

The dimensionless hydraulic diameter, therefore, becomes

Dh ffi

128 þ 64e2

64 þ 16M2 e2  3M4 e4

ð40Þ

The constant wall heat flux boundary condition for the energy equation has been already utilized in the derivation of the expression of dTb/dz and using it one more time does not offer anything new; nevertheless, we still have a supplementary condition at the wall as a result of defining the dimensionless temperature based on Tw(z). The supplementary condition reads

T  ¼ 0 at r ¼ 1 þ e sinðMhÞ

ð41Þ

Substituting the velocity distribution in Eq. (36) from Eq. (26) followed by the application of Eq. (5), we come up with a modified energy equation of the following form

r2 T þ ¼

  @ 2 T þ 1 @T þ 1 @2Tþ 1 4 þ þ ¼ þ S S uav Dh @r 2 r @r  r 2 @h2

where

Tþ ¼ T þ

w uav K 2



4 þS Dh

ð42Þ



Tþ ¼

1 uav K 2



4 þS Dh



at r  ¼ 1 þ e sinðMhÞ

ð34Þ

where Tb is the bulk mean temperature. The energy balance on an element of the fluid with the length of dz gives rise to the following expression for dTb/dz

dT b qav P=A þ E2z =r0 ¼ dz qcp uav

ð39Þ

ð43Þ

Combining Eqs. (6) and (41), it is easy to verify that

ð33Þ

Here, Tw is the wall temperature which is constant at a given cross section of the channel, and qav is the average inward wall R heat flux over the channel perimeter P, that is qav ¼ qdP=P. Further, for thermally fully developed flow with constant wall heat flux, one may write

@T dT w dT b ¼ ¼ @z dz dz

ð36Þ

In addition, Dh ¼ 4A =P  denotes the dimensionless channel hydraulic diameter. This parameter cannot be evaluated analytically because of P⁄ which is, according to Fig. 2, given by

We define a dimensionless temperature of the form

T  ðr  ; hÞ ¼

 4  þS S Dh

E2z R qav r0

P ffi

The conservation of energy including the effects of Joule heating reads

r¼



Nevertheless, for small values of e, one can expand the integrand in Eq. (38) to get an approximate expression of the form

2

DT ¼ r  ðkrTÞ þ s Dt

@ 2 T  1 @T  1 @2T  u þ   þ 2 ¼  2 2 r @r r @h @r uav

wherein S, called the dimensionless Joule heating parameter, is given by

ð29Þ

2.3. Temperature distribution and Nusselt number

qcp

r2 T  ¼

ð35Þ

Since @T=dz is a constant, the axial conduction term in the energy equation vanishes. So, with the help of Eqs. (34) and (35), the energy equation in dimensionless form may be written as

Fig. 2. Schematic showing the arc length in polar coordinate.

ð44Þ

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H. Keramati et al. / International Journal of Heat and Mass Transfer 92 (2016) 244–251

K=5

K = 10 1

1

0.8

0.8

0.6

u*

u*

0.6

0.4

0.4

0.2

0.2

θ

0

0

r

K = 15

K = 20 1 0.8 0.8

u*

u*

0.6 0.4

0.4 0.2 0 0

Fig. 3. Dimensionless velocity distribution for M = 10 and e = 0.02 at different K.

In a similar way as for the electrical potential, T+ is replaced with a Poincare type asymptotic expansion leading to the following þ þ governing equations for T þ 0 , T 1 , and T 2

  1 4 r2 T þ0 ¼   þ S  S; uav Dh

r2 T þ1 ¼ 0;

r2 T þ2 ¼ 0

ð45Þ

T þ1 ð1; hÞ

1 uav K 2



4 þS Dh



ð46Þ

@T þ þ sinðMhÞ 0 ð1; hÞ ¼ 0 @r

ð47Þ

T 1 ðr  ; hÞ ¼

1

T 2 ðr  ; hÞ ¼

K = 10 M = 10 0.8

0.6

u* 0

ε=0 ε = 0.05, θ = π/2M ε = 0.05, θ = 3π/2M

0

0.2

0.4

0.6

2

ð48Þ

  1 4 S 2  1Þ   þ   S ðr 4 Dh uav uav     1 4 I0 ðKr Þ  2 1  þS I0 ðKÞ K uav Dh

  1 4 I1 ðKÞ þ S IM ðKr  Þ sinðMhÞ uav Dh KI0 ðKÞIM ðKÞ   1 4 S þ  S rM sinðMhÞ  2 Dh uav uav    2M  1 4 S þ  S 1  r 2M cosð2MhÞ 8 Dh uav uav   b 4 I0 ðKr  Þ   þS 2 I0 ðKÞ uav K Dh   b 4 I2M ðKr  Þ þS þ cosð2MhÞ 2 D  I2M ðKÞ uav K h

ð49Þ

ð50Þ

ð51Þ

After obtaining the temperature distribution, the quantities of physical interest, including the bulk temperature of the fluid and the heat transfer rate can be obtained. The heat transfer rate can be expressed in terms of the Nusselt number which can be written as

0.4

0.2

@T þ1 sin ðMhÞ @ 2 T þ0 ð1; hÞ þ ð1; hÞ ¼ 0  2 @r @r 2

þ þ After constructing the solutions of T þ 0 , T 1 , and T 2 , the resultant expressions for T 0 , T 1 , and T 2 turn out to be

T 0 ðr  ; hÞ ¼

with the relevant boundary conditions expressed as

T þ0 ð1; hÞ ¼

T þ2 ð1; hÞ þ sinðMhÞ

Nu ¼

0.8

* *

r /rw Fig. 4. Dimensionless velocity profile at different angular positions.

1

hDh qav Dh D ¼ ¼  h k kðT w  T b Þ Tb

ð52Þ

with the dimensionless bulk temperature given as R 2p R 1þe sinðMhÞ     Z 2p Z 1þe sinðMhÞ M M 0M 0 u T r dr dh M ¼  T b ¼ u T  r dr dh 2 R 2Mp R 1þe sinðMhÞ e  u    p 1 þ 0 0 v a u r dr dh M 0 0 2 ð53Þ

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H. Keramati et al. / International Journal of Heat and Mass Transfer 92 (2016) 244–251

K = 10

K=5

0.1

T*

0.4

T*

0.2

0

0 -0.1

-0.2

K = 20

K = 15 0.1

0.1

T*

T*

0 -0.1

0

-0.2 -0.3

-0.1

Fig. 5. Dimensionless temperature distribution for M = 10, e = 0.02, S = 15 at different K.

It was found in the previous section that four dimensionless factors including M, e, K, and S govern the electroosmotic flow and heat transfer in rough microtubes. Here, their interactive influences on the properties of interest are examined. Although we are primarily interested in the heat transfer physics, nevertheless, the presentation of the results is started by a review of the hydrodynamic features because of their inevitable effects on the temperature field. The dimensionless velocity distribution for M = 10 and e = 0.02 at different values of the dimensionless Debye-Hückel parameter is shown in Fig. 3. It can be observed that increasing the dimensionless Debye-Hückel parameter leads to higher dimensionless velocities in the wall vicinity. This may be attributed to the greater concentration of the electroosmotic body force neighboring the wall for thinner EDLs. Note that the minimum value of K is cautiously set to 5 to avoid the double layer overlap [44] and its maximum is considered to be 20 since, as may be inferred from the figure, more increase of K does not significantly alter the velocity pattern. The dimensionless velocity profile for M = 10 and K = 10 at different angular positions is illustrated in Fig. 4. It can be inferred that the concave region of the channel ð0 < h < p=MÞ, represented by h ¼ p=2M in the figure, attains lower velocities than that of the smooth channel case, whereas the opposite is true for the convex region of the channel ðp=M < h < 2p=MÞ which is represented by

0.5

K=5 M = 10 S = -15

0.4

0.3

T

3. Results and discussion

h ¼ 3p=2M. This result can be interpreted simply by the fact that the fluid particles are more influenced by the surface affects in the convex region. It is to be noted that the velocity profiles merge together in the core region of the channel, and coincide with that of the smooth channel case as a consequence of the depletion of the wall corrugation effects. Fig. 5 depicts the dimensionless temperature distribution at different values of K assuming a negative heat flux at the microduct surface. It can be observed that by increasing K, the sign of the dimensionless temperature changes from positive to negative. For low values of K, the wall temperature is higher than that of the fluid particles with the exception of those traveling near the wall. This temperature field is formed because of considerable

*

Due to the complexity of the integrands in Eqs. (29) and (53), only the integration with respect to r⁄ is taken analytically and the integral with respect to h is then found numerically by means of the Simpson method. It is worth noting that the present results for e = 0 are in agreement with the available literature data. For example, the smooth tube velocity profile derived here is the well-known expression for electroosmotic flow in circular microducts [14] and the predicted temperature field and Nusselt number are similar to those of Yavari et al. [19] under the same conditions.

0.2

ε = 0.04, θ = 3π/2M ε=0

0.1

ε = 0.04, θ = π/2M 0 0

0.2

0.4

0.6

0.8

1

r*/r*w Fig. 6. Radial distribution of dimensionless temperature at different angular positions.

H. Keramati et al. / International Journal of Heat and Mass Transfer 92 (2016) 244–251

Joule heating created along with low values of the velocity at the wall vicinity. As stated before, by increasing K the velocity grows especially near the channel surface, thereby rigorously enhancing the energy transferred by the flow therein, resulting in lower wall temperatures. This trend goes on up to the point that the wall temperature becomes lower than that of the fluid particles for a high value of the dimensionless Debye-Hückel parameter such as the case with K = 20 for which h is negative everywhere. In order to acquire more insight into the thermal physics, the radial distribution of dimensionless temperature at different angular positions is shown in Fig. 6 considering M = 10, K = 5, and S = 15. The trends are in accordance with the velocity variations given in Fig. 4: a flatter velocity profile is accompanied by a more uniform temperature distribution. Again, away from the wall, the temperature profiles of the concave and convex regions collapse to one, due to the decadence of the roughness effects. Another subtle point is that the dimensionless temperature in the core region

7.4

K = 20, 10, 5 M=8 S = -2

7.2

7

6.8

Nu

250

6.6

6.4

6.2

6 0

0.02

0.04

0.06

ε Fig. 9. Nusselt number versus e at different K.

7

S = -2, -1, 1, 2 K = 10 M=8

Nu

6.5

6

5.5

5 0

0.01

0.02

0.03

0.04

0.05

0.06

ε Fig. 7. Nusselt number versus e at different Joule heating rates.

1

M = 6, 8, 10 0.98

K = 10 S=-2

Nu/Nu0

0.96

0.94

0.92

0.9

of the rough tube is slightly higher than that of the smooth channel. Now, we turn our attention to the heat transfer rate which is studied through the Nusselt number. Fig. 7 depicts the Nusselt number values as a function of the relative roughness at different Joule heating rates while keeping K = 10 and M = 8. It is observed that the Nusselt number decreases as the relative roughness increases. The main reason is that increasing the relative roughness leads to a decrease in dimensionless hydraulic diameter. It also can be observed that an increase in the dimensionless Joule heating parameter leads to smaller Nusselt number values. This is mainly because of the fact that higher Joule heating rates lead to higher temperature differences between the wall and bulk flow due to the accumulation of energy at the channel surface. The dependency of the ratio of Nusselt number to the associated value for the smooth microtube Nu0 on e at different corrugation numbers is shown in Fig. 8, revealing that the Nusselt number decreases by more than 13% for M = 10 and e = 0.06. This figure also confirms that the decreasing trend of the Nusselt number with the relative roughness, shown in Fig. 7, is not dependent upon M. An increase in the corrugation number is accompanied by a decrease in Nusselt number. This reduction is more pronounced at higher values of the relative roughness. The same trend was observed for the average velocity, in agreement with the findings of Messinger and Squires [28] and Yang and Liu [25], indicating the suppression of electroosmotic flow by the surface roughness. It can therefore be concluded that the corrugated roughness is unfavorable for both the heat transfer and mass flow rate of electroosmotic flow. In the last illustration, that is Fig. 9, the effect of the dimensionless Debye-Hückel parameter on the Nusselt number is presented. It is found that the Nusselt number is an increasing function of K. This is expected since a higher K is accompanied by more uniform velocity fields, thereby flattening the temperature profile. The ultimate effect of enhancing K is to reduce the difference between the wall and bulk temperatures, that is to raise the Nusselt number for a given wall heat flux.

0.88

0

0.01

0.02

0.03

0.04

0.05

0.06

ε Fig. 8. Nusselt number ratio versus e at different corrugation numbers; Nu0 is the Nusselt number of the smooth microtube which, here is 6.932.

4. Conclusions Both hydrodynamically and thermally fully developed electroosmotic flow in a circular microchannel of rough surface was theoretically investigated in this work. A surface of corrugated

H. Keramati et al. / International Journal of Heat and Mass Transfer 92 (2016) 244–251

shape was considered to account for the roughness effects and the thermal boundary condition was assumed to be the constant wall heat flux of the first kind, H1. The governing equations were established based on the Debye-Hückel linearization and then solved analytically by means of the straightforward perturbation method. A comprehensive parametric study revealed that an increase in the Joule heating rate leads to a decrease in the Nusselt number, whereas the opposite is true for the dimensionless Debye-Hückel parameter. Moreover, it was found that the Nusselt number is a decreasing function of both the corrugation number and the relative roughness. The same trend was observed for the mean velocity; accordingly, one may conclude that roughness has unfavorable influences on both the heat transfer and mass flow rates of electroosmotic flow.

[20]

[21]

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[25]

Acknowledgement

[26]

The second author sincerely thanks Iran’s National Elites Foundation (INEF) for their supports during the course of this work.

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