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Technical Note

Thermally fully developed electroosmotic flow through a rectangular microchannel

Jie Su a, Yongjun Jian a,, Long Chang a,b

a School of Mathematical Science, Inner Mongolia University, Hohhot, Inner Mongolia 010021, Chinab School of Mathematics and Statistics, Inner Mongolia Finance and Economics College, Hohhot, Inner Mongolia 010051, China

a r t i c l e i n f o

Article history:

Received 11 January 2012Received in revised form 14 May 2012

Accepted 18 May 2012

Available online 23 June 2012

Keywords:

Rectangular microchannel

Electroosmotic flow

Fully developed

Joule heating

a b s t r a c t

This study investigates the velocity and temperature distributions of the thermally fully developed

electroosmotic flow through a rectangular microchannel. Based on linearized PoissonBoltzmann (PB)equation, NavierStokes equation and thermally fully developed energy equation, analytical solutions

of normalized velocity, temperature and Nusselt number are derived. They greatly depend on the ratio

Kof characteristic scale of the rectangular microchannel to Debye length, width to height ratio a andJoule heating to heat flux ratioS. By numerical computation, we found that for prescribed electrokinetic

width K, increased S yields greater temperature. For small K, the variations of the temperatureh are larger

than those of large K. The dependence of temperature on Sis more significant for a small K, while at a

larger Kthe temperature profiles are almost identical. In addition, we illustrate the Nusselt number Nu

variations withS,a andK. 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Microfluidics is a very important research area due to numerouspotential applications in separation and analysis. Most substances

acquire surface electric charges when in contact with an aqueous

medium. The rearrangement of the charges on the solid surface

and in the liquid results in the formation of the electrical double

layer (EDL) [1]. If an electric field is applied tangentially along a

charged surface, the electric field will exert a body force on the ions

in the diffuse layer, resulting in electroosmotic flow (EOF).

Many experimental observations have shown that flow phe-

nomena and heat transfer in microscale are quite different from

those in macroscale. Many hydrodynamics of a fully-developed

and transient EOF have been analyzed in various micro-capillaries

geometric domains[27]. Recently, Investigations have been con-

ducted to better understand the heat transfer characteristics in

microchannel for applications in electronic cooling. The effect of

viscous dissipation in fully developed electroosmotic heat transfer

has been analyzed by Maynes and Webb [8]. Closed form expres-

sions in a parallel plate or circular microchannel for Nusselt num-

ber were obtained for electroosmotic heat transfer with or without

Joule heating effects[911].

Understanding the heat transfer characteristics of EOF through

a rectangular microchannel is very important. However, there

appear to be no studies reported in the literature treating convec-

tive heat transfer for EOF through a rectangular microchannel. The

purpose of this study is to present analytical expressions for ther-

mally and hydrodynamically fully developed EOF in a rectangular

microchannel with constant wall heat flux.

2. Mathematical modeling

2.1. EDL potential distribution

A straight two dimensional rectangular microchannel of width

2W, height 2H and length L is shown in Fig. 1. The fluid flow is

acted upon by an axial steady electric field of strength E. Due to

the symmetry of the problem, the solution domain can be reduced

to a quarter cross section of the channel. For a symmetric binary

electrolyte solution, the electrical potential w of the EDL isdescribed by the PB equation

@2w

@x2@2w

@y22n0zme

e sinh

zmew

kbT0 1

which is subject to the following boundary conditions

wjxW f; wjyH f; dw

dx

x0

0; dwdy

y0

0 2

wheree is the dielectric constant of the electrolyte liquid,n0is theion density of bulk liquid,zmis the valence, e is the electron charge,

kbis the Boltzmann constant, T0is the absolute temperature andfis

the wall zeta potential. We assume the electrical potential is small

enough, so that the DebyeHckel linearization approximation can

be used. Introducing the following dimensionless groups:

0017-9310/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.056

Corresponding author.

E-mail address:[email protected](Y. Jian).

International Journal of Heat and Mass Transfer 55 (2012) 62856290

Contents lists available atSciVerse ScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.056mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.056http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.056mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.056

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x xH

; y yH

; w zvewkbT0

; f zvefkbT0

;

j 2

n0

z

2

me

2

ekbT0 1=2

; K jH 3

Wherej is DebyeHckel parameter, Kis called the electrokineticwidth. Thus the normalized Eq. (1) and boundary conditions Eq.

(2)can now be expressed as

@2w

@x2 @

2w

@y2 K2w 4

@w0; y@x

0; @wx;0@y

0 5a

wa; y f; wx;1 f 5bwherea = W/His the ratio of the width to height of the channel sec-tion,fis the dimensionless wall zeta potential. The solution to the

linearized PB Eq. (4) subjected to the boundary conditions Eq. (5)

can be obtained as[12]

w 4fX1m1

cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 b2m

q x

Em

cosbmy

4fX1n1

cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 c2n

q y

Fn

coscnx 6

where bm2m 1p

2 ; cn

2n 1p2a

; m; n 1;2;3; . . .

7a

Em2m 1p cosh


q a

1m1

;

Fn2n 1p cosh


q 1n1

7b

The detail derivation can be found in Ref. [13]. The ionic net charge

density gives

qe ej2wkbT0zve

8

Once the distributions of charge density are known, the Cauchymomentum can be solved analytically.

2.2. The analytical solutions of the Cauchy momentum equation

We consider the only axial velocity component u(x,y) along

positivezdirection in the rectangular microchannel. For a hydro-

dynamic fully developed pure EOF, the Cauchy momentum equa-

tion can be simplified to

l

@2u

@x2@2u

@y2 ! qeE 9

where qeis the net volume charge density, andl is the fluid viscos-ity. Eq. (9) is subjected to the following no-slip and symmetric

boundary conditions

uW;y 0; ux;H 0 10a

@u0;y@x

0; @ux;0@y

0 10b

Using the following dimensionless steady EOF velocity

U uUeo

; Ueo ekbET0lzve 11

The normalized form of Eqs. (9)and (10) are

@2U

@x2 @

2U

@y2 K2 w 12

Ua; y 0; Ux;1 0; @U0; y@x

0; @Ux;0@y

0 13

Similar to our recent work [13], the solution of Eq. (12) can be

written as

Ux; y X1j1

cosffiffiffiffikj

p x

Yjy 14

where

Yjy

A cosh ffiffiffiffikjp y ffiffiffiffi

kjp

coshffiffiffiffikj

p X1

n1

QnjJnj

Fn X1

m1

PmjImj

Em( )

AX1n1

Qnj cosh


q y

cosh ffiffiffiffikjp y

FnK2 c2n kj

8>>>:

X1m1

Pmj coshffiffiffiffikj

p y

cosbmy Emkj b2m

) 15a

A 8aK2f; kj j 1=2pa

2; j 1;2;3; . . . 15b

Imjffiffiffiffikj

pkj

b2m

cosh

ffiffiffiffikjp

cosbmh i 15c

Jnjffiffiffiffikj

pK2 c2n kj

cosh


q cosh

ffiffiffiffikj

p 15d

Qnj

a=2 n j1

2

1

cnffiffiffikj

p sincn

ffiffiffiffikj

p a

1cn ffiffiffikjp sincn ffiffiffiffikjp a

nj

8>>>>>>>>>:

15e

Pmj 1K2 b2m kjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 b2m

q sinh


q a cos

ffiffiffiffikj

p a

ffiffiffiffikjp cosh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 b2mq a sin ffiffiffiffikjp a 15f

Fig. 1. Sketch of thermally and hydrodynamically fully developed EOF through a

rectangular microchannel.

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2.3. Temperature distribution

Based on the above velocity field, the energy equation can be

written as

qcpu@T

@z kr2T rE2 lU 16

whereq is the fluid density,cp is the specific heat at constant pres-sure, Tis the temperature,kis thermal conductivity,ris the electri-cal conductivity, and U is the viscous dissipation function. In Eq.

(16), the second term in right hand side represents a volumetric

heat generation due to Joule heating and the third term represents

a local volumetric heating due to viscous dissipation [9]. For dis-

tilled water, Gleeson [14]found the ratio of Joule heating to viscous

heating is very large. Hence, for the present analysis, we neglect the

viscous heating and consider only the Joule heating effects.

Assuming the EOF within the rectangular microchannel is ther-

mally fully developed with constant physical properties, we have

@

@z

Twz Tx;y;zTwz Tmz

0 17

whereTw(z) andTm(z) are the local wall and the bulk temperatures,

respectively. Under imposed constant heat flux boundary condi-

tions on the wall, we have

@Tx;y;z@z

dTmzdz

dTwzdz

const: and @2Tx;y;z

@z2

0 18Under these situations, energy equation(16)reduces to

qcpudTmdz

k @2T

@x2 @

2T

@y2

! rE2 19

The relevant boundary conditions for the energy equation are as

follows

@T0;y

@x 0; q00s k@T

W;y

@x 20a

@Tx;0@y

0; q00s k@Tx;H

@y or TjyH Twz 20b

whereq00s is constant wall heat flux. It should be noted that for the

thermal fully developed flow in microchannels, usually the wall

axial thermal conduction cannot be neglected, especially at very

low Reynolds number. Thus the temperature on the wall is thefunc-

tion ofz.

Furthermore, from an overall energy balance for an elemental

control volume on a length of duct dz, it follows that

4q00s H Wdz 4rE2HWdz q~uHWcpdTmdz

dz 21

where~u is axial mean velocity, and its dimensionless form can be

written as

u ~uUeo

R10

Ra0Udxdy

HW 22

Rearranging Eq. (21), the axial bulk temperature gradient in the

thermally fully developed situation is

dTmdz

q00s H W rE2HW

mcp23

Introducing the following dimensionless temperature for fully

developed flow

hx;

y T

Tw

q00sH=k 24

The normalized energy equation may be written as

@2h

@x2 @

2h

@y2 a1U S 25

where

S rE2H

q00s

; a1 H WuW

Su

26

The parameterSdenotes the ratio of Joule heating to wall heat flux

from rectangular wall. Corresponding normalized temperature

boundary conditions are

@h0; y@x

0; @ha; y@x

1 27a

@hx;0@y

0; @hx;1@y

1 or hjy1 0 27b

Similar to our recent work[13], the solution of normalized temper-

ature of Eq.(25)is

hx; y X1

j1

cos ffiffiffiffikjp xYjy 28

where theYjy is expressed as

Yjy C

1j Pj C2jffiffiffiffikj

p cosh ffiffiffiffikjp cosh

ffiffiffiffikj

p y C

1j y sinh ffiffiffiffikjp y

2ffiffiffiffikj

p Bcosh

ffiffiffiffikj

p y 1

kj Aa1ffiffiffiffi

kjp X1

n1

Qnj

FnK2 c2n kj

ffiffiffiffikj

pK2 c2n kj

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 c2n

q y cosh

ffiffiffiffikj

p y

y sinhffiffiffiffikj

p y

2

! Aa1

ffiffiffiffikjpX1m1

Pmj

Emb2m kj

y sinhffiffiffiffikj

p y

2

ffiffiffiffikj

pb2

m kjcosh

ffiffiffiffikj

p y cosbmy

!29

where Pj 12sinh

ffiffiffiffikj

p ; Sj 1ffiffiffiffi

kjp cosh ffiffiffiffikjp 1 30a

Rnjffiffiffiffikj

pK2 c2n kj

coshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 c2n

q cosh

ffiffiffiffikj

p

12sinh

ffiffiffiffikj

p

30b

Tmj 12sinh

ffiffiffiffikj

p

ffiffiffiffikj

pb2m kj

coshffiffiffiffikj

p cosbm 30c

C1j

a1Affiffiffiffikj

p cosh ffiffiffiffikjp

X1n1

QnjJnjFn

X1m1

PmjImjEm

" # 30d

C2j a1A

X1n1

QnjRnj

FnK2 c2n kjX1m1

PmjTmj

Emb2m kj

! BSj 30e

Finally, the solution ofhx; y can be obtained by inserting Eq.(29)

into Eq.(28).

The dimensionless bulk temperature is given by

hmR10

Ra0Uhdxdy

R1

0

Ra0Udxdy

31

The heat transfer rate can be expressed in terms of Nusselt numberas

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Nu hDhk

q00sDhkTw Tm

1hm

32a

where q 00s hTw Tm; hmTm Twq00sH=k

32b

whereh is heat transfer coefficient, Dh is so called hydrodynamic

diameter and Dh= Hfor a square microchannel. The Nusselt number

obtained in the present study and the one given in previous work

[15]are in quite agreement.

3. Numerical results and discussion

Two dimensionless parameters have been defined above for the

general flow behavior of thermally fully developed electroosmoticflow through a rectangular microchannel. They are the ratio of the

characteristic scale of the rectangular microchannel to Debyelength K= Hj, and the ratio of Joule heating to wall heat fluxS rE2H=q00s . The dependence of normalized temperature distribu-tions and Nusselt number on Kand Swill be investigated over their

relevant ranges in this section. In all computational results, the

normalized wall zeta potential f 1 in order to use the linearized

PB equation.

Firstly, the normalized velocity distributions across the rectan-

gular microchannel should be computed. For brevity, we ignored

these similar figures which can be found in Ref. [13]. Based on

velocity distributions, temperature distributions can be calculated.

Fig. 2[(a), (c) and (e) forK= 5; (b), (d) and (f) forK = 50] illus-

trates the three dimensional temperature distributions for differ-

ent ratioS(5, 0 and 5). Both positive and negative values of the

wall heat flux are considered, where the negative value of the Scorresponds to fluid cooling scenario, and the positive value of

Fig. 2. Dimensionless temperature distribution for different ratio S(a= 1). (a)K= 5,S= 5, (b)K= 50,S= 5, (c)K= 5,S= 0, (d)K= 50,S= 0, (e)K= 5,S= 5, (f)K= 50, S= 5.


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the Srelates to fluid heating case. From Fig. 2 we can see that a po-sitive value ofS(surface heating) will result in a greater tempera-

ture variation across the microchannel, while the opposite trend is

true forS(surface cooling) comparing with the case ofS= 0.

Moreover, from the form of the surface heat flux q0s= koT(x,H)/

oy= h(Tw Tm), which requires that the surface temperature gradi-

ent should be kept constant. Importantly, with an increasing Joule

heating parameter S, Tw also increases along with Tm in order to

maintain the constant heat flux condition.

For small electrokinetic widthK[seeFig. 2(a), (c) and (e)], the

variations of the amplitude of temperatureh are larger than those

of large electrokinetic width K[see Fig. 2(b), (d) and (f)]. The reason

is that when the thickness of the EDL approaches to half height of

the microchannel, the effects of Joule heating are prominently felt

almost over the entire channel section, unlike for the case withinfinitesimally thin EDL. These figures clearly demonstrate the

appreciable change in the local temperature across almost the en-

tire channel cross section with increasing magnitude of the Sdue

to appreciable penetration of the EDL into the bulk fluid for small

K. Moreover, the dependence of temperature on Sis more signifi-

cant for a small K, while for larger Kthe temperature profiles are

almost identical.

Fig. 3 shows fully developed Nusselt number Nuvariations with

S, a and K. It can be seen from Fig. 3(a) that Nu monotonicallydecreases with increases Swhatever the electrokinetic width Kis

larger or small. IncreasedKyields the increase of the Nu over the

rectangular microchannel cross section for a prescribed S. This

behavior may be explained byFig. 3(b). Increasing values ofSlead

to higher bulk temperatures with negative sign. This can be ex-plained from Eq. (32) that the larger S produces to higher mean

temperature with negative sign, which leads to lower values ofNusselt number. In addition,Fig. 3(c) and (d) shows Nu variations

with electrokinetic width Kfor different ratio a of the width toheight of the channel section for different S(2,2). It can be seen

thatNu decreases with the increase of width to height ratio a nomatter whetherSis positive or negative.

4. Conclusions

This study investigates the velocity and temperature distribu-

tions of the thermally fully developed electroosmotic flow through

a rectangular microchannel. Based on linear PB equation, Navier

Stokes equation and thermally fully developed energy equation,

analytical solutions of normalized velocity, temperature and Nus-selt number are derived using the method of variable separation.

They greatly depend on the electrokinetic width K, width to height

ratioa and Joule heating to heat flux ratio S.By numerical computations, we found that for prescribed K, in-

creasedSyields greater variation of the temperature over the rect-

angular microchannel cross section. The dependence of

temperature on Sis more significant for a smallK, while at a larger

K the temperature profiles are almost identical. In addition, we

illustrate theNu number variations with Joule heating to heat flux

ratioS, width to height ratio a and electrokinetic widthK. TheNunumber monotonically decreases with increases S no matter the

electrokinetic widthKis larger or small. IncreasedKyields the in-

crease of theNu number over the rectangular microchannel cross

section for a given S. Moreover,Nu decreases with the increase ofwidth to height ratioa no matter whetherSis positive or negative.

Fig. 3. The variations ofNu withS, Kand a.a = 1 for (a) and (b), (c)S= 2, (d) S= 2.

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Acknowledgments

The work was supported by Ph.D. Programs Foundation of Min-

istry of Education of China (No. 20111501120001), Opening fund of

State Key Laboratory of Nonlinear Mechanics, the National Natural

Science Foundation of China (No. 11062005), the Natural Science

Foundation of Inner Mongolia (Grant No: 2010BS0107), the

research start up fund for excellent talents at Inner Mongolia Uni-versity (Grant No. Z20080211) and the support of Natural Science

Key Fund of Inner Mongolia (Grant No: 2009ZD01).

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