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Streaming potential and heat transfer of nanofluids in parallel plate microchannels
Accepted Manuscript Title: Streaming potential and heat transfer of nanofluids in parallel plate microchannels Author: Guangpu Zhao Yongjun Jian Fengqin Li PII: DOI: Reference:
S0927-7757(16)30194-7 http://dx.doi.org/doi:10.1016/j.colsurfa.2016.03.053 COLSUA 20534
To appear in:
Colloids and Surfaces A: Physicochem. Eng. Aspects
Received date: Revised date: Accepted date:
26-11-2015 17-3-2016 18-3-2016
Please cite this article as: Guangpu Zhao, Yongjun Jian, Fengqin Li, Streaming potential and heat transfer of nanofluids in parallel plate microchannels, Colloids and Surfaces A: Physicochemical and Engineering Aspects http://dx.doi.org/10.1016/j.colsurfa.2016.03.053 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Streaming potential and heat transfer of nanofluids in parallel plate microchannels Guangpu Zhao, Yongjun Jian, Fengqin Li School of Mathematical Science, Inner Mongolia University, Hohhot, Inner Mongolia 010021, PR China
Graphical abstract
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* Corresponding author. Tel. : +86 471 4991251 8313
E-mail address: [email protected] (Y.J. Jian).
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Research Highlights:
Flow and thermal transport characteristics of nanofluid are analytically explored.
The nanofluid is under the effects of pressure gradient and streaming potential.
The nanofluid flow is confined to a parallel plate microchannel.
The impacts of pertinent parameters on the velocity, temperature, Nusselt number and entropy
are discussed.
ABSTRACT:
In the present study, the heat transfer characteristics of thermally developed
nanofluid flow through a parallel plate microchannel are investigated under combined influences of pressure-driven and streaming potential effects. The analytical solution for electrokinetic flow in microchannel is obtained by employing the Debye–Hückel linearization. The classical boundary condition of uniform wall heat flux is considered in the analysis, and the effects of viscous dissipation as well as Joule heating are also taken into account. Furthermore, based upon the velocity field and temperature field, the Nusselt number variations are induced, and the variations of local and total entropy generation of nanofluids are also performed. Concisely, the results show the profiles of streaming potential decrease with the dimensionless EDL thickness, whereas the Nusselt number increases with the dimensionless EDL thickness. An enhanced heat transfer performance with increasing nanoparticle volume fraction can be witnessed. The local entropy generation gradually grows from the centerline toward the wall. Beside, the total entropy generation obviously grows with increasing Br. 2
Keywords: Electrokinetic; Streaming potential; Nanofluid; Heat transfer; Nusselt number; Entropy. Nomenclature Br
the Brinkman number
cp
the heat capacitance of the fluids at the reference pressure (J/kg K)
e
the elementary electric charge (C)
ES
the electric field induced by streaming potential (V/m)
E0
the characteristic electric field (V/m)
f
the ionic friction coefficient for cations and anions
h
the convective heat transfer (W/m2 K)
H
height (m)
IC
the conduction current (A)
IS
the streaming current (A)
k
the thermal conductivity of the nanopaticle (W/mK)
KB
the Boltzmann constant ( J/K)
K
the Debye-Hückel parameter
n0
the bulk volume concentration of positive or negative ions charge (m-3)
n
the number densities of the electrolyte ion (m-3)
Nu
the Nusselt number
Px
the pressure gradient (k Pa)
q
the wall heat flux (W/m2)
r
axis of cylindrical coordinate system
S
the dimensionless Joule heat parameter
3
SG
the volumetric entropy generation rate
STotal
the dimensionless total entropy generation
T
the temperature of the fluids (K)
u
the dimensionless velocity field in axial direction
U
the velocity field (m/s)
Ue
the Helmholtz-Smoluchowski electroosmotic velocity (m/s)
UP
the reference pressure-driven flow velocity (m/s)
z
the valence number of ions
Greek symbols
the fluid permittivity (C/Vm)
γ
the dimensionless number defined the ratio of the nanolayer thickness to the original particle radius
θ
the dimensionless temperature
Θ
the dimensionless parameter
µ
the viscosity of the fluids (kg /m s)
σ
the electrical conductivity of the nanofluid
ρ
density (kg/m3)
ρe
the local volumetric net charge density (C/m3)
ϕ
the volume fraction of the nanoparticles
ψ
the electrical potential (V)
ψ0
the wall zeta potential (V)
Subscripts av
the absolute value
4
eff
effective
f
fluid
m
mean
nf
nanofluid
s
particle
w
wall
J
the Joule heating
H
the heat diffusion
L
local
V
the viscous friction
±
cations and anions
1. Introduction The past decades have witnessed an overwhelming boom in microfluidics and nanofluidics. Due to their inherent advantages such as high transport coefficients, efficient process management, excellent durability, reduction of required sample volumes and processing time and so on, they are widely utilized in various fields of sciences and engineering. The concerned applications cover micropumps, heat exchangers, inkjet printing, chemical separation devices, drug delivery, biomedical diagnostics and biochemical analysis [1-7]. All those devices and instruments involve fluid flow and heat transfer in microchannels. Traditionally, flow manipulation in such systems is
5
usually achieved by using pressure gradient, shear-driven and surface tension [8, 9]. Whereas, flow can also be induced and manipulated precisely by virtue of other interfacial phenomena like electrokinetics which is contributed to the development of the electric double layer (EDL). EDL is formed owing to the interaction of ionized solution with static charge on dielectric surfaces. On application of an external electric field on the EDL, the mobile ions in the diffuse region of this EDL are actuated to move, resulting in a bulk liquid motion via viscous effect. This is usually termed as the electroosmtic flow (EOF). Owing to their diverse application, the study of EOF has received a great of attentions. Theoretical, numerical and experimental investigations of EOF on hydrodynamics and thermal transport phenomena have been well researched in various geometric domains of microchannels [10-24]. Another electrokinetic flow phenomenon is steaming potential effect. The significant character of this electrokinetic phenomenon is not application of externally electric field, but its own induced electric field, known as the streaming potential. The emergence of this potential is attributed to the relative motions of ions of the EDL near the channel wall when a pressure-driven transport is employed. Naturally, the resultant accumulation of ions in the downstream section of the channel builds up a reverse electrical field comparing with the direction of fluid flow. Such streaming potential once again drives the current and gives rise to conduction current, which can balance the streaming potential at steady state, so that the net electrical current becomes zero, consistent with a pure pressure-driven flow condition. An EOF can also be brought due to the combined interactions between the streaming potential and the net mobile charge, and the fluid direction is opposite to that of the imposed pressure-driven flow. Consequently, the overall volumetric flow rate reduces in the channel. This reduction gives the appearance of an increased liquid viscosity, which is referred to as
6
the electroviscous effect [25-35]. Many relvent studies have revealed that the streaming current can be utilized to provide an effective conversion method of transferring kinetic energy of the pressure-driven transport and chemical energy of the EDL into electrical power of the streaming current [36-43]. This novel energy harvesting system in micro and nanochannel by the aid of electrokinetic phenomena refers as electrokinetic energy conversion (EKEC). Although the energy harnessed from a single nanochannel may be really small, a considerable energy can be harnessed by integrating the arrays of nanoporous materials. EKEC in nanofluidic channels showed the maximal conversion efficiency was measured nearly from 3% to 11% [44, 45]. However, in a recent research concerning a ballistic electrostatic generator a very high efficiency of almost 50% was obtained [46]. This energy harvesting system might open a neoteric path to explore new renewable energy in the future. In the industrial application, since the need of higher heat transfer rates is desirable urgently, conventional fluids are not favored any longer and significant improvement in heat transfer performance of working fluids has become an important concern for scientists. A new fluid feathered by notable heat transfer is nanofluid, which is a suspension of solid nanoparticles (normally smaller than 100nm in diameter) in the conventional liquids such as water, oil and ethylene glycol. Many applications involving nanofluids in advanced cooling systems, micro/nanochanical devices have been reported in the publications by Das et al.[47], Wang [48], Ganji [49], Hedayati et al. [50], Saidur et al. [51], ,Shahbeddin et al.[52], Sheikholeslami et al.[53, 54], Mahian et al.[55], Hedayati and Domairry [56], Ganguly and Sarkar [57, 58], Malvandi and Ganji [59], and Turkilmazoglu [60] and so on. From the above review, we can see that most research works related to steaming potential
7
mainly paid attention to the effects of EDL on liquid flow, and the impacts of the streaming potential on heat transfer of pressure-driven flow in microchannels are limited. However, it is imperative to study this flow and convective heat transfer characteristics in microchannels due to their significant engineering applications. Tan and Liu [61] numerically investigated the influences of wall slip and streaming potential on liquid flow and heat transfer in the parallel plate microchannels beyond the Debye-Hückel limitation, which was subjected to the constant heat flux. In addition, the Newtonian fluid was employed in their consideration as the working fluid. The present paper delineates, for the first time, the impacts of the nanofluid on thermally developed flow through a microchannel, by considering the combined effects of applied pressure gradient and streaming potential effects. In this case, the wall potential is assumed to be very small, so that the linear Debye-Hückel limitation can be appropriately utilized. We expect this work to provide a valuable guideline towards thermodynamic idealization of the microfluidic system.
2. Mathematical formulation 2.1. Problem definition Consider the situation where flow of an incompressible viscous nanofluid takes place through a long parallel plate microchannel with channel half height oh H under the effects of imposed pressure-gradient and streaming potential. An illustration of the problem is sketched in Fig. 1. The bottom and upper wall are both made of hydrophobic materials and bear the same constant zeta potential. The flow is assumed to be steady, and thermally fully developed. At the same time the uniform and constant wall heat flux is applied along the channel walls, where qw is considered to be positive when directed into the flow. Since nanoparticles can generate their own EDLs, but in present work, we assume that those EDLs are not to overlap each other. Furthermore, those 8
nanoparticles are supposed to be uniformly distributed and the nanofluid is considered in a thermal equilibrium state. In that case, we can neglect the influences of nanoparticles EDLs [57, 58]
2.2. Electrical potential equation and approximate solution For a symmetric (z : z) electrolyte solution, the electrical potential ψ(Y) and the local volumetric net charge density ρe(Y) are described by the following Poisson-Boltzmann (PB) equations:
2 (Y ) / Y 2 e (Y ) /
(1)
e (Y ) ze(n n )
(2)
where is the fluid permittivity, z is the valence number of ions, e is the elementary electric charge,
n and n are the number densities of the electrolyte cations and anions, respectively, and are given by the Boltzmann distribution, i.e.
n n0 exp[
ez (Y ) ] k BTav
(3)
here n0 is the bulk volume concentration of positive or negative ions which is independent of the surface electrochemistry, kB is the Boltzmann constant, and Tav is the absolute temperature over the entire channel. Assuming the electrical potential is small enough, the Debye–Hückel approximation can be employed for Eq. (3)
exp[
ez (Y ) ez (Y ) ] 1 k BTav k BTav
(4)
by inserting Eq. (2), (3) and (4) into Eq. (1), we get the equations dictating the electrostatic potential distributions as:
d 2 (Y ) / dY 2 k 2 (Y ) with k (2n0 z 2e2 / kBTav )1 / 2 The corresponding boundary conditions are:
9
(5)
(Y ) Y H 0 d (Y ) / dY Y 0 0
(6a, b)
where ψ0 is the wall zeta potential, k is the Debye-Hückel parameter and 1/k denotes the thickness of the EDL. By solving problem (5)(6), then using Eq. (1), the net charge density can be written as [19]:
e (Y ) k 2 0 cosh(kY ) / cosh(kh)
(7)
2.3. Analytical solutions of the velocity field Since the flow has been considered to be fully developed, the effect of vertical component is neglected compared with the axial component. Thus, the velocity of the nanofluid governed by Navier-Stokes equation can be written as:
eff
d 2U dP e ES 0 dY 2 dX
(8)
where µeff is the effective viscosity of the nanofluid, given by Brinkman [62]:
eff
f (1 ) 2.5
(9)
where µf the viscosity of the base fluids , is the volume fraction of the nanoparticles. U is the velocity field in axial direction, P is the press and ES is the stable electric field induced by streaming potential for purely pressure-driven transport. No-slip boundary condition on the wall of microchannel and symmetrical condition about Y axis are
U (Y ) |Y H 0 dU (Y ) / dY |Y 0 0
(10a, b)
Introducing the following dimensionless groups:
y
E Y P H2 U E UP x u U e 0 0 ES s Up f f H E0
where
Px=-dp/dx,
Up
is
a
reference
K kH
pressure-driven 10
(11) flow
velocity,
Ue
is
the
Helmholtz-Smoluchowski electroosmotic velocity, E0 is the characteristic electric field, and K is called electrokinetic width of the EDL denoting the ratio of half-height H of microchannel to Debye length 1/k. After inserting dimensionless parameters (11) into Eq. (8), the dimensionless momentum equation and relevant boundary conditions have the following form: d 2u cosh( Ky ) u r K 2 E S 0 2 dy cosh( K )
u ( y) y 1 0
du dy
y 0
(12)
0
(13a, b)
where
f eff ur U e U p Applying the boundary condition (13), the velocity of the nanofluid can be yielded:
1 cosh( Ky ) 1 u ( y) y 2 ur ES ( ur ES ) 2 cosh( K ) 2
(14)
2.4. Calculation of the streaming potential In order to obtain the streaming potential ES, the net ionic current through the channel section (per unit width) equaling to zero should be utilized, i.e. H
i 2ez (nU nU )dY I S I C 0
(15)
0
where I S is streaming current, I C is conduction current, and U are the ionic velocities expressed as:
U U
ezES f
(16)
where f are the ionic friction coefficient for cations and anions, they are supposed to have equivalent value. Using Eq. (3), (14) and (16) into Eq. (15), we shall finally get the dimensionless streaming potential as:
11
B1 R ur B2
ES
B1 B2 R
K
2
K3
(17)
tanh(K ) 2K
(18a)
tanh(K )
(18b)
2 cosh 2 ( K )
K BTav E 0 f 0U p
(18c)
2.5. Temperature distribution for fully developed flow with uniform wall heat flux Based upon the above obtained velocity field, the energy equation with volumetric joule heating, and energy dissipation can be used to give the temperature distribution in the microchannel. Analytical solution of energy equation is executed with the situation of constant wall heat flux. The governing equation for thermal energy transport by considering axial conduction can be presented as:
( C p )nf U
T 2T 2T dU 2 keff ( 2 2 ) ES2 eff ( ) X X Y dY
(19)
where (ρcp)nf is the effective heat capacitance of the nanofluids at the reference pressure, which is calculated by the equation given by Xuan and Li [63] as:
( C p )nf ( C p ) s (1 )( C p ) f
(20)
where (CP)s and (CP)f are the heat capacities per unit volume of the solid nanoparticles and the base fluid respectively. keff is the effective thermal conductivity of the nanofluids, which is presented by Yu and Chio [64]:
keff
kS 2k f 2(kS k f )(1 )3
(21)
kS 2k f 2(kS k f )(1 )3
12
where kS is the thermal conductivity of the nanopaticle, kf is the thermal conductivity of base fluid. σ is the electrical conductivity of the nanofluid, and γ is a dimensionless number and is defined as the ratio of the nanolayer thickness to the original particle radius. Usually, the value of γ is considered equal to 0.1 [65]. T is the local temperature of the nanofluid, the second term at the right hand side of (19) denotes the volumetric heat generation due to Joule heating effect and the third term is the viscous dissipation. Corresponding boundary conditions in X direction can be expressed as: dT dY
Y 0
0 keff
dT dY
Y H
qw (0r T
yH
Tw )
(22a, b)
where qw =h (Tw-Tm) is the constant wall heat flux, h is the convective heat transfer coefficient.
Furthermore, coupled with uniform wall heat flux condition, for thermally fully developed problem, we have 2T T dTw dTm 0 =const. and X 2 X dX dX
(23)
Under these assumptions, energy equation (19) yields
( C p )nf U (
dTm 2T dU 2 ) keff ES2 eff ( ) 2 dX Y dY
(24)
Considering an over energy balance for an elemental control volume on a length of duct dX along the centerline of the channel, the Eq. (24) can be written as:
H
0
( C p ) nf UdTmdY
H
0
H H 2T dU 2 keff d X d Y E d X d Y eff ( )2 dXdY S 2 0 0 Y dY
(25)
Rearranging (25), the axial bulk temperature gradient in the thermally fully developed situation has
13
the form: dTm qw ES2 H Q dX ( C p ) nf Hum
(26)
where um is axial mean velocity, which can be given by integration of U(Y) across the section of the microchannel
um
1 H
H
0
1
U (Y )dY U P u ( y)dy U P 1
(27a)
0
and Q eff U P2 2 H
(27b) 1 6
1
1 u ( y)dy 0
1
2 ( 0
ur Es K
1 tanh(K ) ( ur E ) 2
du 2 1 u E K sinh(2 K ) 2 22ur ES sinh( K ) ) dy ( r S )2 ( 1) (cosh( K ) ) dy 2 cosh( K ) 2K 3 cosh( K ) K
(27c)
(27d)
Introducing the following dimensionless temperature for fully developed flow:
( y)
T (Y ) Tw qw H / k f
(28)
The dimensionless energy equation and boundary conditions can be described as:
d 2 k f du [(1 S Br 2 )u / 1 S Br ( )2 ] 2 dy keff dy d dy
y 0
0
d dy
y 1
kf keff
(or
y 1
(29)
0)
(30a, b)
where
S
ES2 H qw
Br
f U P2 qw H
Br Br
(31)
14
The parameter S denotes the ratio of Joule heating to heat flux from microchannel wall in physical, usually is termed as the dimensionless Joule heat parameter, and the parameter Br is Brinkman number, which manifests the ratio of heat produced by viscous dissipation to heat transported by molecular conduction. Substituting dimensionless velocity field (14) into non-dimensional energy equation (29), integrating it twice and applying the boundary condition (30), finally, the solution of the temperature field can be obtained as bellow: kf
( y)
keff
{(1 S Br 2 )(
1 1 u E cosh( Ky ) 1 y 4 r 2 S [1 ] ( ur ES )( y 2 1)) / 1 24 24 K cosh( K ) 2 2
2 1 u E K 1 1 2u E Br ( (1 y 4 ) ( r S ) 2 ( 2 (cosh(2 K ) cosh(2 Ky )) ( y 2 1) r S 12 2 cosh( K ) 4 K 2 cosh( K ) (
1 2 1 (sinh( K ) sinh( Ky ) y) 2 (cosh( Ky ) cosh( K ))) S (1 y 2 )} K K 2
(32)
The dimensionless bulk temperature can be defined as: 1
1
0
0
m u( y) ( y)dy / u( y)dy k f (Tm Tw ) / qw H
(33)
The heat transfer rate can be expressed in the form of Nusselt number Nu, which is defined as: Nu
2 Hqw keff (Tw Tm )
(34)
Combing Eq. (33) and Eq. (34), the Nusselt number can be finally derived as: Nu
kf 2 keff m
(35)
Owing to the velocity and temperature distributions having been known from Eq. (19) and Eq. (32) respectively, the Nu number can be calculated from Eq. (35).
15
3. Entropy generation analysis Once the velocity and temperature fields have been reduced, the volumetric entropy generation rate for electrokinetic convection of nanofluids inside the microchannel can be determined. This function characterizes the irreversible behavior of the system. According to Bejan [66], the local entropy generation can be written as follow [67-71]: SG,L SG,H SG,J SG,V
(36)
where SG,L is the volumetric entropy generation rate per unit volume and the terms on the right hand side of equation represent the irreversibility of local volumetric entropy generation rate due to heat diffusion, Joule heating, and viscous friction of the nanofluids, presented respectively by:
SG , H
keff dT 2 ( ) SG , J ES2 2 T dY T
SG ,V
eff dU T
(
dY
)2
(37)
the dimensionless form of the volumetric entropy generation rate can be expressed explicitly as: SG SH S J SV
(38)
where SH
1 d ( )2 2 k f ( ) dy
keff
1 ( )
(39b)
1 du ( )2 ( ) dy
(39c)
S J ES2 S
S V Br
(39a)
Twk f
(39d)
qw H
16
The total entropy generation in dimensionless form is calculated by integrating the local entropy generation over the whole domain, that is 1
Stotal SG dy
(40)
0
4. Results and discussion In our present work, the analytical expressions derived in the preceding section are used to describe the electrokinetic transport in microchannel in the presence of nanofluid (Al2O3-water). We consider the thermophysical properties of nanofluid at reference of temperature (25oC), namely, Tav is 298 K, and postulate the heat capacitance of nanofluids at the reference pressure. kB is 1.38110-23J K-1, ψ0 is -0.025V, ρs =3600 kg m-3, ρf =997.1 kg m-3, µf =8.9110-3 kg m-1 s-1. E0 is set to change from 0 to 10KV m-1. In order to evaluate the value of R, we approximately take f as 10-12 N s m-1 [30, 32]. Applying the Eq. (18c), the dimensionless parameter R can be estimated from -17 to 0. The range of K changes from 5 to 35. Moreover, the order of Brinkman number (Br) is set to be 0~0.1 and the order of Joule heat parameter (S) is 0~10, which are shown to be in accordance with the physically acceptable values established in the studies [57, 58]. In addition, Ue is set to be 10-4 and the parameter ur is taken as 1.0, which indicates the strength of comparable pressure-driven velocity and electroosmotic velocity. The results for the variation of the streaming potential inside a microchannel as a function of the dimensionless EDL thickness and the volume fraction of nanoparticles can be witnessed in Fig. 2(a). We can find that the profiles of dimensionless streaming potential apparently reduce as K rises, and the amplitude of the streaming potential ultimately approaches to the zero. This is attributed to the fact that the augment of K indicates the increase of the ratio of the length scale H to the EDL
17
thickness. Namely, the density of free charged ions decreases in the nanofluid. When a distant region is located, which is far away from the EDL (i.e. approaching the center of the microchannels), the value of streaming potential will attain to the zero, and the nanofluid will display the electroneutral state. Meanwhile, effects of volume fraction of the nanoparticles on streaming potential are presented, and the profiles of streaming potential decrease with the increase in the volume fraction of the nanoparticles. The effect of dimensionless parameter R on streaming potential is displayed in Fig. 2(b). We can see that the small R in magnitude will give rise to the bigger streaming potential, as expected, which is accorded with the Eq. (18). Fig. 3 shows the variations of velocity distribution with the dimensionless EDL thickness and nanoparticle volume fraction. We can observe the velocity distribution of the nanofluid increases with EDL thickness (K) in Fig. 3(a). In fact, the overall velocity profile is a superposition of the pressure driven flow and the counteracting electroosmotic flow induced by streaming potential. But this streaming potential is really small [30], compared with the normal applied electric field. Thus, the pressure driven flow dominates electroosmotic flow, and velocity profile continues to increase in a parabolic manner up to the axis of symmetry of the microchanne. From Fig. 3(b) we can see the velocity of the nanofluid deceases with the increasing of the nanoparticle volume fraction. This mainly is attributed to the fact with enhancement of the nanoparticle volume fraction ( ), the effective viscosity of the nanofluid increases in response to the shear rate, which easily leads to a greater dispersion in the velocity profile. Fig. 4 depicts the variations of temperature distribution with the dimensionless EDL thickness and nanoparticle volume fraction. The result of Fig. 4(a) shows that, the increase in the dimensionless EDL thickness (K) causes an increase in the dimensionless temperature of the fluid 18
inside the microchannel. As K increases, the velocity rises (in spite of tiny augment in Fig. 4(a)), but this augment will strengthen thermal energy in the microchannel, which is transferred by flow from the channel wall to the nanofluid. The ultimate outcome is the local temperature of the nanofluid rises and the wall temperature cuts down. Consequently, the dimensionless temperature increases. The influences of nanoparticle volume fraction on the dimensionless temperature are demonstrated in Fig. 4(b). It is found that dimensionless temperature increases with increase in nanoparticle concentration. One reason is because of the higher effective thermal conductivity of nanoparticles, the thermal diffusion effects in nanofluids can be remarkably enhanced relative to the case of base fluid ( =0). Another explanation of this phenomenon may be that nanoparticle volume fraction changes the velocity profile and so the shear stress at the walls, heat removal ability of nanofluids would alter with the changes of nanoparticle volume fraction. Thus, the increase of nanoparticle volume fraction brings about the gradual augment of local temperature, correspondingly, gives rise to the increasing dimensionless temperature. Fig. 5 depicts the Nusselt number variations with the dimensionless EDL thickness for different values of nanoparticle volume fraction. In such situation, the Nusselt number is found to increase with K. The reason resulting in this behavior can be explained by the Fig. 4. We can observe that the larger K leads to the larger dimensionless temperature of the fluid, and is easy to cause an increase in the bulk mean temperature relative to the channel wall temperature. Based upon the constant wall heat flux condition, defined by qw=h (Tw-Tm), the quantity of (Tw-Tm) will decrease gradually from previous analysis. Thus, the convective heat transfer coefficient h increases naturally. The Nusselt number reflecting convection of heat transfer enhances coherently as K increases. In addition, due to the presence of the nanoparticles, the thermal conductivity of the nanofluid is remarkably higher
19
than the base fluid. Therefore, the heat transfer performance is correspondingly improved, as illustrated in Fig. 5. Fig. 6 shows the variations of the Nusselt number with Brinkman number for different nanoparticle volume fraction when ur =1.0, K=30. In fact, the value of Brinkman number manifests the degree of the viscous dissipation effect, the bigger Brinkman number is achieved, and the more viscous dissipation effects will be engendered. In this situation, the wall temperature Tw will become much bigger than the mean temperature Tm, the magnitude of (Tw-Tm) will become large. Accordingly, the convective heat transfer coefficient h decreases and the Nusselt number reduces with Br for a fixed the EDL thickness. Moreover, an enhancement of the heat transfer performance with different nanoparticle volume fraction is also presented. Fig. 7 depicts the local entropy generation across the channel for different values of the nanoparticle volume fraction. Going through the plot, we can find the local entropy generation expands from the centerline toward the channel walls. A torpid variation of the local entropy generation occurs in the middle region of the microchannel, but a sharp change takes place at channel walls. It reveals that the local entropy generation is strongly affected by the thermal energy near the walls. Furthermore, the maximum of the local entropy generation emerges corresponding to =0% and this trend will cut down with enhancing of the nanoparticle volume fraction. The relvent result also implies that enhancing solid volume fraction can reduce the thermal irreversibility of nanofluids. Fig. 8 presents the total entropy generation across the channel for different values of Brinkman number. We can see that the total entropy generation obviously goes through a sharp increase by enhancing Brinkman number (Br). The result points out that the increase in the dissipation effect
20
can easily lead to higher magnitude of irreversibility of the nanofluid in microchannels. Also, a reduction in thermal irreversibility with increasing nanoparticle volume fraction ( ) can be witnessed in Fig. 8. In view of this cognition, we can properly choose the parametric values of Brinkman number and nanoparticle volume fraction to achieve the minimum total entropy generation on nanofluids in microfluidic system.
5. Conclusions In the present paper, the flow and heat transfer characteristics under the assumption of thermally fully developed nanofluid flow (Al2O3-water) in a microchannel have been investigated. The flow is actuated by combinational effects of the external pressure gradient and the streaming potential. The analytic solutions for velocity and temperature are derived. Furthermore, the variations of Nusselt number and the entropy generation are also discussed. The impacts of pertinent flow parameters such as the electrokinetic parameter (K), nanoparticle volume fraction ( ), Brinkman number (Br) are investigated qualitatively. The results show that the profiles of streaming potential are found to decrease with K, and the amplitude of this potential will ultimately approach to the zero. The velocity of the nanofluid increases with K for a fixed value of . The amplitude of velocity decreases with an increase in the particle concentration. The dimensionless temperature of the nanofluid increases with K and . The Nusselt number is found to increase with K, whereas it decreases with an increase in Br. In addition, an improvement of the heat transfers performance with increasing nanoparticle volume fraction can be witnessed. The local entropy generation grows slowly from the centerline toward the wall and a sharp variation takes place at the walls. Beside, the total entropy generation obviously grows with increasing Br. Both in Fig. 7 and Fig. 8, we can find increasing nanoparticle volume fraction reduces the total entropy generation, namely, drops down 21
the thermal irreversibility in microfluidic systems.
Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos.11472140, 11562014 and 11362012), the Program for Young Talents of Science and Technology in Universities
of
Inner
Mongolia
Autonomous
Region
(No.NJYT-13-A02),
the
Inner
Mongolia Grassland Talent (No. 12000-12102013), and Opening fund of State Key Laboratory of Nonlinear Mechanics.
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qw -
+
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+ -
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+ -
-
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Conduction Current Ic
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Streaming Field Es
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+
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+ -
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｝EDL Streaming Current Is
+ + - + ｝EDL - -
Fig. 1. Geometry of the physical problem, coordinate system, electric double layer and streaming potential.
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Fig. 2. Variation of the dimensionless streaming potential with the dimensionless EDL thickness for different values of volume fraction when ur = 1.0, R=-0.5 (a) and different value of R when ur = 1.0 (b) in the microchannel.
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Fig. 3. Effects of dimensionless EDL thickness at =3% (a) and volume fraction at K=20 (b) on cross-sectional velocity distribution when ur = 1.0, R=-0.5.
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Fig. 4. Effects of dimensionless EDL thickness at
=3% (a) and volume fraction at K=20 (b) on dimensionless
temperature profiles when ur = 1.0, R=-0.5, Br=0.01, S=10.
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Fig. 5. Effects of the volume fraction on the nature of variation of Nusselt number with dimensionless EDL thickness for different nanoparticle volume fraction when ur = 1.0, R=-10, Br=0.01 and S=10.
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Fig. 6. Effects of the volume fraction on the nature of variation of Nusselt number with Brinkman number when ur = 1.0, R=-0.5, S=10 and K=20.
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Fig. 7. Local entropy generation for different volume fraction when ur = 1.0, R=-0.5, Br=0.01, S=10, Θ=1000 and K=20.
36
Fig. 8. Total entropy generation for different volume fraction with Brinkman number when ur =1.0, R=-0.5, S=10, Θ=1000 and K=20.
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S0927-7757(16)30194-7 http://dx.doi.org/doi:10.1016/j.colsurfa.2016.03.053 COLSUA 20534
To appear in:
Colloids and Surfaces A: Physicochem. Eng. Aspects
Received date: Revised date: Accepted date:
26-11-2015 17-3-2016 18-3-2016
Please cite this article as: Guangpu Zhao, Yongjun Jian, Fengqin Li, Streaming potential and heat transfer of nanofluids in parallel plate microchannels, Colloids and Surfaces A: Physicochemical and Engineering Aspects http://dx.doi.org/10.1016/j.colsurfa.2016.03.053 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Streaming potential and heat transfer of nanofluids in parallel plate microchannels Guangpu Zhao, Yongjun Jian, Fengqin Li School of Mathematical Science, Inner Mongolia University, Hohhot, Inner Mongolia 010021, PR China
Graphical abstract
qw -
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Conduction Current Ic
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Streaming Field Es
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* Corresponding author. Tel. : +86 471 4991251 8313
E-mail address: [email protected] (Y.J. Jian).
1
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｝EDL Streaming Current Is
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Research Highlights:
Flow and thermal transport characteristics of nanofluid are analytically explored.
The nanofluid is under the effects of pressure gradient and streaming potential.
The nanofluid flow is confined to a parallel plate microchannel.
The impacts of pertinent parameters on the velocity, temperature, Nusselt number and entropy
are discussed.
ABSTRACT:
In the present study, the heat transfer characteristics of thermally developed
nanofluid flow through a parallel plate microchannel are investigated under combined influences of pressure-driven and streaming potential effects. The analytical solution for electrokinetic flow in microchannel is obtained by employing the Debye–Hückel linearization. The classical boundary condition of uniform wall heat flux is considered in the analysis, and the effects of viscous dissipation as well as Joule heating are also taken into account. Furthermore, based upon the velocity field and temperature field, the Nusselt number variations are induced, and the variations of local and total entropy generation of nanofluids are also performed. Concisely, the results show the profiles of streaming potential decrease with the dimensionless EDL thickness, whereas the Nusselt number increases with the dimensionless EDL thickness. An enhanced heat transfer performance with increasing nanoparticle volume fraction can be witnessed. The local entropy generation gradually grows from the centerline toward the wall. Beside, the total entropy generation obviously grows with increasing Br. 2
Keywords: Electrokinetic; Streaming potential; Nanofluid; Heat transfer; Nusselt number; Entropy. Nomenclature Br
the Brinkman number
cp
the heat capacitance of the fluids at the reference pressure (J/kg K)
e
the elementary electric charge (C)
ES
the electric field induced by streaming potential (V/m)
E0
the characteristic electric field (V/m)
f
the ionic friction coefficient for cations and anions
h
the convective heat transfer (W/m2 K)
H
height (m)
IC
the conduction current (A)
IS
the streaming current (A)
k
the thermal conductivity of the nanopaticle (W/mK)
KB
the Boltzmann constant ( J/K)
K
the Debye-Hückel parameter
n0
the bulk volume concentration of positive or negative ions charge (m-3)
n
the number densities of the electrolyte ion (m-3)
Nu
the Nusselt number
Px
the pressure gradient (k Pa)
q
the wall heat flux (W/m2)
r
axis of cylindrical coordinate system
S
the dimensionless Joule heat parameter
3
SG
the volumetric entropy generation rate
STotal
the dimensionless total entropy generation
T
the temperature of the fluids (K)
u
the dimensionless velocity field in axial direction
U
the velocity field (m/s)
Ue
the Helmholtz-Smoluchowski electroosmotic velocity (m/s)
UP
the reference pressure-driven flow velocity (m/s)
z
the valence number of ions
Greek symbols
the fluid permittivity (C/Vm)
γ
the dimensionless number defined the ratio of the nanolayer thickness to the original particle radius
θ
the dimensionless temperature
Θ
the dimensionless parameter
µ
the viscosity of the fluids (kg /m s)
σ
the electrical conductivity of the nanofluid
ρ
density (kg/m3)
ρe
the local volumetric net charge density (C/m3)
ϕ
the volume fraction of the nanoparticles
ψ
the electrical potential (V)
ψ0
the wall zeta potential (V)
Subscripts av
the absolute value
4
eff
effective
f
fluid
m
mean
nf
nanofluid
s
particle
w
wall
J
the Joule heating
H
the heat diffusion
L
local
V
the viscous friction
±
cations and anions
1. Introduction The past decades have witnessed an overwhelming boom in microfluidics and nanofluidics. Due to their inherent advantages such as high transport coefficients, efficient process management, excellent durability, reduction of required sample volumes and processing time and so on, they are widely utilized in various fields of sciences and engineering. The concerned applications cover micropumps, heat exchangers, inkjet printing, chemical separation devices, drug delivery, biomedical diagnostics and biochemical analysis [1-7]. All those devices and instruments involve fluid flow and heat transfer in microchannels. Traditionally, flow manipulation in such systems is
5
usually achieved by using pressure gradient, shear-driven and surface tension [8, 9]. Whereas, flow can also be induced and manipulated precisely by virtue of other interfacial phenomena like electrokinetics which is contributed to the development of the electric double layer (EDL). EDL is formed owing to the interaction of ionized solution with static charge on dielectric surfaces. On application of an external electric field on the EDL, the mobile ions in the diffuse region of this EDL are actuated to move, resulting in a bulk liquid motion via viscous effect. This is usually termed as the electroosmtic flow (EOF). Owing to their diverse application, the study of EOF has received a great of attentions. Theoretical, numerical and experimental investigations of EOF on hydrodynamics and thermal transport phenomena have been well researched in various geometric domains of microchannels [10-24]. Another electrokinetic flow phenomenon is steaming potential effect. The significant character of this electrokinetic phenomenon is not application of externally electric field, but its own induced electric field, known as the streaming potential. The emergence of this potential is attributed to the relative motions of ions of the EDL near the channel wall when a pressure-driven transport is employed. Naturally, the resultant accumulation of ions in the downstream section of the channel builds up a reverse electrical field comparing with the direction of fluid flow. Such streaming potential once again drives the current and gives rise to conduction current, which can balance the streaming potential at steady state, so that the net electrical current becomes zero, consistent with a pure pressure-driven flow condition. An EOF can also be brought due to the combined interactions between the streaming potential and the net mobile charge, and the fluid direction is opposite to that of the imposed pressure-driven flow. Consequently, the overall volumetric flow rate reduces in the channel. This reduction gives the appearance of an increased liquid viscosity, which is referred to as
6
the electroviscous effect [25-35]. Many relvent studies have revealed that the streaming current can be utilized to provide an effective conversion method of transferring kinetic energy of the pressure-driven transport and chemical energy of the EDL into electrical power of the streaming current [36-43]. This novel energy harvesting system in micro and nanochannel by the aid of electrokinetic phenomena refers as electrokinetic energy conversion (EKEC). Although the energy harnessed from a single nanochannel may be really small, a considerable energy can be harnessed by integrating the arrays of nanoporous materials. EKEC in nanofluidic channels showed the maximal conversion efficiency was measured nearly from 3% to 11% [44, 45]. However, in a recent research concerning a ballistic electrostatic generator a very high efficiency of almost 50% was obtained [46]. This energy harvesting system might open a neoteric path to explore new renewable energy in the future. In the industrial application, since the need of higher heat transfer rates is desirable urgently, conventional fluids are not favored any longer and significant improvement in heat transfer performance of working fluids has become an important concern for scientists. A new fluid feathered by notable heat transfer is nanofluid, which is a suspension of solid nanoparticles (normally smaller than 100nm in diameter) in the conventional liquids such as water, oil and ethylene glycol. Many applications involving nanofluids in advanced cooling systems, micro/nanochanical devices have been reported in the publications by Das et al.[47], Wang [48], Ganji [49], Hedayati et al. [50], Saidur et al. [51], ,Shahbeddin et al.[52], Sheikholeslami et al.[53, 54], Mahian et al.[55], Hedayati and Domairry [56], Ganguly and Sarkar [57, 58], Malvandi and Ganji [59], and Turkilmazoglu [60] and so on. From the above review, we can see that most research works related to steaming potential
7
mainly paid attention to the effects of EDL on liquid flow, and the impacts of the streaming potential on heat transfer of pressure-driven flow in microchannels are limited. However, it is imperative to study this flow and convective heat transfer characteristics in microchannels due to their significant engineering applications. Tan and Liu [61] numerically investigated the influences of wall slip and streaming potential on liquid flow and heat transfer in the parallel plate microchannels beyond the Debye-Hückel limitation, which was subjected to the constant heat flux. In addition, the Newtonian fluid was employed in their consideration as the working fluid. The present paper delineates, for the first time, the impacts of the nanofluid on thermally developed flow through a microchannel, by considering the combined effects of applied pressure gradient and streaming potential effects. In this case, the wall potential is assumed to be very small, so that the linear Debye-Hückel limitation can be appropriately utilized. We expect this work to provide a valuable guideline towards thermodynamic idealization of the microfluidic system.
2. Mathematical formulation 2.1. Problem definition Consider the situation where flow of an incompressible viscous nanofluid takes place through a long parallel plate microchannel with channel half height oh H under the effects of imposed pressure-gradient and streaming potential. An illustration of the problem is sketched in Fig. 1. The bottom and upper wall are both made of hydrophobic materials and bear the same constant zeta potential. The flow is assumed to be steady, and thermally fully developed. At the same time the uniform and constant wall heat flux is applied along the channel walls, where qw is considered to be positive when directed into the flow. Since nanoparticles can generate their own EDLs, but in present work, we assume that those EDLs are not to overlap each other. Furthermore, those 8
nanoparticles are supposed to be uniformly distributed and the nanofluid is considered in a thermal equilibrium state. In that case, we can neglect the influences of nanoparticles EDLs [57, 58]
2.2. Electrical potential equation and approximate solution For a symmetric (z : z) electrolyte solution, the electrical potential ψ(Y) and the local volumetric net charge density ρe(Y) are described by the following Poisson-Boltzmann (PB) equations:
2 (Y ) / Y 2 e (Y ) /
(1)
e (Y ) ze(n n )
(2)
where is the fluid permittivity, z is the valence number of ions, e is the elementary electric charge,
n and n are the number densities of the electrolyte cations and anions, respectively, and are given by the Boltzmann distribution, i.e.
n n0 exp[
ez (Y ) ] k BTav
(3)
here n0 is the bulk volume concentration of positive or negative ions which is independent of the surface electrochemistry, kB is the Boltzmann constant, and Tav is the absolute temperature over the entire channel. Assuming the electrical potential is small enough, the Debye–Hückel approximation can be employed for Eq. (3)
exp[
ez (Y ) ez (Y ) ] 1 k BTav k BTav
(4)
by inserting Eq. (2), (3) and (4) into Eq. (1), we get the equations dictating the electrostatic potential distributions as:
d 2 (Y ) / dY 2 k 2 (Y ) with k (2n0 z 2e2 / kBTav )1 / 2 The corresponding boundary conditions are:
9
(5)
(Y ) Y H 0 d (Y ) / dY Y 0 0
(6a, b)
where ψ0 is the wall zeta potential, k is the Debye-Hückel parameter and 1/k denotes the thickness of the EDL. By solving problem (5)(6), then using Eq. (1), the net charge density can be written as [19]:
e (Y ) k 2 0 cosh(kY ) / cosh(kh)
(7)
2.3. Analytical solutions of the velocity field Since the flow has been considered to be fully developed, the effect of vertical component is neglected compared with the axial component. Thus, the velocity of the nanofluid governed by Navier-Stokes equation can be written as:
eff
d 2U dP e ES 0 dY 2 dX
(8)
where µeff is the effective viscosity of the nanofluid, given by Brinkman [62]:
eff
f (1 ) 2.5
(9)
where µf the viscosity of the base fluids , is the volume fraction of the nanoparticles. U is the velocity field in axial direction, P is the press and ES is the stable electric field induced by streaming potential for purely pressure-driven transport. No-slip boundary condition on the wall of microchannel and symmetrical condition about Y axis are
U (Y ) |Y H 0 dU (Y ) / dY |Y 0 0
(10a, b)
Introducing the following dimensionless groups:
y
E Y P H2 U E UP x u U e 0 0 ES s Up f f H E0
where
Px=-dp/dx,
Up
is
a
reference
K kH
pressure-driven 10
(11) flow
velocity,
Ue
is
the
Helmholtz-Smoluchowski electroosmotic velocity, E0 is the characteristic electric field, and K is called electrokinetic width of the EDL denoting the ratio of half-height H of microchannel to Debye length 1/k. After inserting dimensionless parameters (11) into Eq. (8), the dimensionless momentum equation and relevant boundary conditions have the following form: d 2u cosh( Ky ) u r K 2 E S 0 2 dy cosh( K )
u ( y) y 1 0
du dy
y 0
(12)
0
(13a, b)
where
f eff ur U e U p Applying the boundary condition (13), the velocity of the nanofluid can be yielded:
1 cosh( Ky ) 1 u ( y) y 2 ur ES ( ur ES ) 2 cosh( K ) 2
(14)
2.4. Calculation of the streaming potential In order to obtain the streaming potential ES, the net ionic current through the channel section (per unit width) equaling to zero should be utilized, i.e. H
i 2ez (nU nU )dY I S I C 0
(15)
0
where I S is streaming current, I C is conduction current, and U are the ionic velocities expressed as:
U U
ezES f
(16)
where f are the ionic friction coefficient for cations and anions, they are supposed to have equivalent value. Using Eq. (3), (14) and (16) into Eq. (15), we shall finally get the dimensionless streaming potential as:
11
B1 R ur B2
ES
B1 B2 R
K
2
K3
(17)
tanh(K ) 2K
(18a)
tanh(K )
(18b)
2 cosh 2 ( K )
K BTav E 0 f 0U p
(18c)
2.5. Temperature distribution for fully developed flow with uniform wall heat flux Based upon the above obtained velocity field, the energy equation with volumetric joule heating, and energy dissipation can be used to give the temperature distribution in the microchannel. Analytical solution of energy equation is executed with the situation of constant wall heat flux. The governing equation for thermal energy transport by considering axial conduction can be presented as:
( C p )nf U
T 2T 2T dU 2 keff ( 2 2 ) ES2 eff ( ) X X Y dY
(19)
where (ρcp)nf is the effective heat capacitance of the nanofluids at the reference pressure, which is calculated by the equation given by Xuan and Li [63] as:
( C p )nf ( C p ) s (1 )( C p ) f
(20)
where (CP)s and (CP)f are the heat capacities per unit volume of the solid nanoparticles and the base fluid respectively. keff is the effective thermal conductivity of the nanofluids, which is presented by Yu and Chio [64]:
keff
kS 2k f 2(kS k f )(1 )3
(21)
kS 2k f 2(kS k f )(1 )3
12
where kS is the thermal conductivity of the nanopaticle, kf is the thermal conductivity of base fluid. σ is the electrical conductivity of the nanofluid, and γ is a dimensionless number and is defined as the ratio of the nanolayer thickness to the original particle radius. Usually, the value of γ is considered equal to 0.1 [65]. T is the local temperature of the nanofluid, the second term at the right hand side of (19) denotes the volumetric heat generation due to Joule heating effect and the third term is the viscous dissipation. Corresponding boundary conditions in X direction can be expressed as: dT dY
Y 0
0 keff
dT dY
Y H
qw (0r T
yH
Tw )
(22a, b)
where qw =h (Tw-Tm) is the constant wall heat flux, h is the convective heat transfer coefficient.
Furthermore, coupled with uniform wall heat flux condition, for thermally fully developed problem, we have 2T T dTw dTm 0 =const. and X 2 X dX dX
(23)
Under these assumptions, energy equation (19) yields
( C p )nf U (
dTm 2T dU 2 ) keff ES2 eff ( ) 2 dX Y dY
(24)
Considering an over energy balance for an elemental control volume on a length of duct dX along the centerline of the channel, the Eq. (24) can be written as:
H
0
( C p ) nf UdTmdY
H
0
H H 2T dU 2 keff d X d Y E d X d Y eff ( )2 dXdY S 2 0 0 Y dY
(25)
Rearranging (25), the axial bulk temperature gradient in the thermally fully developed situation has
13
the form: dTm qw ES2 H Q dX ( C p ) nf Hum
(26)
where um is axial mean velocity, which can be given by integration of U(Y) across the section of the microchannel
um
1 H
H
0
1
U (Y )dY U P u ( y)dy U P 1
(27a)
0
and Q eff U P2 2 H
(27b) 1 6
1
1 u ( y)dy 0
1
2 ( 0
ur Es K
1 tanh(K ) ( ur E ) 2
du 2 1 u E K sinh(2 K ) 2 22ur ES sinh( K ) ) dy ( r S )2 ( 1) (cosh( K ) ) dy 2 cosh( K ) 2K 3 cosh( K ) K
(27c)
(27d)
Introducing the following dimensionless temperature for fully developed flow:
( y)
T (Y ) Tw qw H / k f
(28)
The dimensionless energy equation and boundary conditions can be described as:
d 2 k f du [(1 S Br 2 )u / 1 S Br ( )2 ] 2 dy keff dy d dy
y 0
0
d dy
y 1
kf keff
(or
y 1
(29)
0)
(30a, b)
where
S
ES2 H qw
Br
f U P2 qw H
Br Br
(31)
14
The parameter S denotes the ratio of Joule heating to heat flux from microchannel wall in physical, usually is termed as the dimensionless Joule heat parameter, and the parameter Br is Brinkman number, which manifests the ratio of heat produced by viscous dissipation to heat transported by molecular conduction. Substituting dimensionless velocity field (14) into non-dimensional energy equation (29), integrating it twice and applying the boundary condition (30), finally, the solution of the temperature field can be obtained as bellow: kf
( y)
keff
{(1 S Br 2 )(
1 1 u E cosh( Ky ) 1 y 4 r 2 S [1 ] ( ur ES )( y 2 1)) / 1 24 24 K cosh( K ) 2 2
2 1 u E K 1 1 2u E Br ( (1 y 4 ) ( r S ) 2 ( 2 (cosh(2 K ) cosh(2 Ky )) ( y 2 1) r S 12 2 cosh( K ) 4 K 2 cosh( K ) (
1 2 1 (sinh( K ) sinh( Ky ) y) 2 (cosh( Ky ) cosh( K ))) S (1 y 2 )} K K 2
(32)
The dimensionless bulk temperature can be defined as: 1
1
0
0
m u( y) ( y)dy / u( y)dy k f (Tm Tw ) / qw H
(33)
The heat transfer rate can be expressed in the form of Nusselt number Nu, which is defined as: Nu
2 Hqw keff (Tw Tm )
(34)
Combing Eq. (33) and Eq. (34), the Nusselt number can be finally derived as: Nu
kf 2 keff m
(35)
Owing to the velocity and temperature distributions having been known from Eq. (19) and Eq. (32) respectively, the Nu number can be calculated from Eq. (35).
15
3. Entropy generation analysis Once the velocity and temperature fields have been reduced, the volumetric entropy generation rate for electrokinetic convection of nanofluids inside the microchannel can be determined. This function characterizes the irreversible behavior of the system. According to Bejan [66], the local entropy generation can be written as follow [67-71]: SG,L SG,H SG,J SG,V
(36)
where SG,L is the volumetric entropy generation rate per unit volume and the terms on the right hand side of equation represent the irreversibility of local volumetric entropy generation rate due to heat diffusion, Joule heating, and viscous friction of the nanofluids, presented respectively by:
SG , H
keff dT 2 ( ) SG , J ES2 2 T dY T
SG ,V
eff dU T
(
dY
)2
(37)
the dimensionless form of the volumetric entropy generation rate can be expressed explicitly as: SG SH S J SV
(38)
where SH
1 d ( )2 2 k f ( ) dy
keff
1 ( )
(39b)
1 du ( )2 ( ) dy
(39c)
S J ES2 S
S V Br
(39a)
Twk f
(39d)
qw H
16
The total entropy generation in dimensionless form is calculated by integrating the local entropy generation over the whole domain, that is 1
Stotal SG dy
(40)
0
4. Results and discussion In our present work, the analytical expressions derived in the preceding section are used to describe the electrokinetic transport in microchannel in the presence of nanofluid (Al2O3-water). We consider the thermophysical properties of nanofluid at reference of temperature (25oC), namely, Tav is 298 K, and postulate the heat capacitance of nanofluids at the reference pressure. kB is 1.38110-23J K-1, ψ0 is -0.025V, ρs =3600 kg m-3, ρf =997.1 kg m-3, µf =8.9110-3 kg m-1 s-1. E0 is set to change from 0 to 10KV m-1. In order to evaluate the value of R, we approximately take f as 10-12 N s m-1 [30, 32]. Applying the Eq. (18c), the dimensionless parameter R can be estimated from -17 to 0. The range of K changes from 5 to 35. Moreover, the order of Brinkman number (Br) is set to be 0~0.1 and the order of Joule heat parameter (S) is 0~10, which are shown to be in accordance with the physically acceptable values established in the studies [57, 58]. In addition, Ue is set to be 10-4 and the parameter ur is taken as 1.0, which indicates the strength of comparable pressure-driven velocity and electroosmotic velocity. The results for the variation of the streaming potential inside a microchannel as a function of the dimensionless EDL thickness and the volume fraction of nanoparticles can be witnessed in Fig. 2(a). We can find that the profiles of dimensionless streaming potential apparently reduce as K rises, and the amplitude of the streaming potential ultimately approaches to the zero. This is attributed to the fact that the augment of K indicates the increase of the ratio of the length scale H to the EDL
17
thickness. Namely, the density of free charged ions decreases in the nanofluid. When a distant region is located, which is far away from the EDL (i.e. approaching the center of the microchannels), the value of streaming potential will attain to the zero, and the nanofluid will display the electroneutral state. Meanwhile, effects of volume fraction of the nanoparticles on streaming potential are presented, and the profiles of streaming potential decrease with the increase in the volume fraction of the nanoparticles. The effect of dimensionless parameter R on streaming potential is displayed in Fig. 2(b). We can see that the small R in magnitude will give rise to the bigger streaming potential, as expected, which is accorded with the Eq. (18). Fig. 3 shows the variations of velocity distribution with the dimensionless EDL thickness and nanoparticle volume fraction. We can observe the velocity distribution of the nanofluid increases with EDL thickness (K) in Fig. 3(a). In fact, the overall velocity profile is a superposition of the pressure driven flow and the counteracting electroosmotic flow induced by streaming potential. But this streaming potential is really small [30], compared with the normal applied electric field. Thus, the pressure driven flow dominates electroosmotic flow, and velocity profile continues to increase in a parabolic manner up to the axis of symmetry of the microchanne. From Fig. 3(b) we can see the velocity of the nanofluid deceases with the increasing of the nanoparticle volume fraction. This mainly is attributed to the fact with enhancement of the nanoparticle volume fraction ( ), the effective viscosity of the nanofluid increases in response to the shear rate, which easily leads to a greater dispersion in the velocity profile. Fig. 4 depicts the variations of temperature distribution with the dimensionless EDL thickness and nanoparticle volume fraction. The result of Fig. 4(a) shows that, the increase in the dimensionless EDL thickness (K) causes an increase in the dimensionless temperature of the fluid 18
inside the microchannel. As K increases, the velocity rises (in spite of tiny augment in Fig. 4(a)), but this augment will strengthen thermal energy in the microchannel, which is transferred by flow from the channel wall to the nanofluid. The ultimate outcome is the local temperature of the nanofluid rises and the wall temperature cuts down. Consequently, the dimensionless temperature increases. The influences of nanoparticle volume fraction on the dimensionless temperature are demonstrated in Fig. 4(b). It is found that dimensionless temperature increases with increase in nanoparticle concentration. One reason is because of the higher effective thermal conductivity of nanoparticles, the thermal diffusion effects in nanofluids can be remarkably enhanced relative to the case of base fluid ( =0). Another explanation of this phenomenon may be that nanoparticle volume fraction changes the velocity profile and so the shear stress at the walls, heat removal ability of nanofluids would alter with the changes of nanoparticle volume fraction. Thus, the increase of nanoparticle volume fraction brings about the gradual augment of local temperature, correspondingly, gives rise to the increasing dimensionless temperature. Fig. 5 depicts the Nusselt number variations with the dimensionless EDL thickness for different values of nanoparticle volume fraction. In such situation, the Nusselt number is found to increase with K. The reason resulting in this behavior can be explained by the Fig. 4. We can observe that the larger K leads to the larger dimensionless temperature of the fluid, and is easy to cause an increase in the bulk mean temperature relative to the channel wall temperature. Based upon the constant wall heat flux condition, defined by qw=h (Tw-Tm), the quantity of (Tw-Tm) will decrease gradually from previous analysis. Thus, the convective heat transfer coefficient h increases naturally. The Nusselt number reflecting convection of heat transfer enhances coherently as K increases. In addition, due to the presence of the nanoparticles, the thermal conductivity of the nanofluid is remarkably higher
19
than the base fluid. Therefore, the heat transfer performance is correspondingly improved, as illustrated in Fig. 5. Fig. 6 shows the variations of the Nusselt number with Brinkman number for different nanoparticle volume fraction when ur =1.0, K=30. In fact, the value of Brinkman number manifests the degree of the viscous dissipation effect, the bigger Brinkman number is achieved, and the more viscous dissipation effects will be engendered. In this situation, the wall temperature Tw will become much bigger than the mean temperature Tm, the magnitude of (Tw-Tm) will become large. Accordingly, the convective heat transfer coefficient h decreases and the Nusselt number reduces with Br for a fixed the EDL thickness. Moreover, an enhancement of the heat transfer performance with different nanoparticle volume fraction is also presented. Fig. 7 depicts the local entropy generation across the channel for different values of the nanoparticle volume fraction. Going through the plot, we can find the local entropy generation expands from the centerline toward the channel walls. A torpid variation of the local entropy generation occurs in the middle region of the microchannel, but a sharp change takes place at channel walls. It reveals that the local entropy generation is strongly affected by the thermal energy near the walls. Furthermore, the maximum of the local entropy generation emerges corresponding to =0% and this trend will cut down with enhancing of the nanoparticle volume fraction. The relvent result also implies that enhancing solid volume fraction can reduce the thermal irreversibility of nanofluids. Fig. 8 presents the total entropy generation across the channel for different values of Brinkman number. We can see that the total entropy generation obviously goes through a sharp increase by enhancing Brinkman number (Br). The result points out that the increase in the dissipation effect
20
can easily lead to higher magnitude of irreversibility of the nanofluid in microchannels. Also, a reduction in thermal irreversibility with increasing nanoparticle volume fraction ( ) can be witnessed in Fig. 8. In view of this cognition, we can properly choose the parametric values of Brinkman number and nanoparticle volume fraction to achieve the minimum total entropy generation on nanofluids in microfluidic system.
5. Conclusions In the present paper, the flow and heat transfer characteristics under the assumption of thermally fully developed nanofluid flow (Al2O3-water) in a microchannel have been investigated. The flow is actuated by combinational effects of the external pressure gradient and the streaming potential. The analytic solutions for velocity and temperature are derived. Furthermore, the variations of Nusselt number and the entropy generation are also discussed. The impacts of pertinent flow parameters such as the electrokinetic parameter (K), nanoparticle volume fraction ( ), Brinkman number (Br) are investigated qualitatively. The results show that the profiles of streaming potential are found to decrease with K, and the amplitude of this potential will ultimately approach to the zero. The velocity of the nanofluid increases with K for a fixed value of . The amplitude of velocity decreases with an increase in the particle concentration. The dimensionless temperature of the nanofluid increases with K and . The Nusselt number is found to increase with K, whereas it decreases with an increase in Br. In addition, an improvement of the heat transfers performance with increasing nanoparticle volume fraction can be witnessed. The local entropy generation grows slowly from the centerline toward the wall and a sharp variation takes place at the walls. Beside, the total entropy generation obviously grows with increasing Br. Both in Fig. 7 and Fig. 8, we can find increasing nanoparticle volume fraction reduces the total entropy generation, namely, drops down 21
the thermal irreversibility in microfluidic systems.
Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos.11472140, 11562014 and 11362012), the Program for Young Talents of Science and Technology in Universities
of
Inner
Mongolia
Autonomous
Region
(No.NJYT-13-A02),
the
Inner
Mongolia Grassland Talent (No. 12000-12102013), and Opening fund of State Key Laboratory of Nonlinear Mechanics.
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qw -
+
-
+ -
-
-
+ -
-
- + + -
++
Conduction Current Ic
-
-
Streaming Field Es
- + +
-
Y
+
-
+
+
X -
+ + + - - -
-
-
+ -
-
-
+
+ -
-
+ + + - - qw
-
- + + -
+
-
+ -
-
-
+ -
｝EDL Streaming Current Is
+ + - + ｝EDL - -
Fig. 1. Geometry of the physical problem, coordinate system, electric double layer and streaming potential.
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Fig. 2. Variation of the dimensionless streaming potential with the dimensionless EDL thickness for different values of volume fraction when ur = 1.0, R=-0.5 (a) and different value of R when ur = 1.0 (b) in the microchannel.
31
Fig. 3. Effects of dimensionless EDL thickness at =3% (a) and volume fraction at K=20 (b) on cross-sectional velocity distribution when ur = 1.0, R=-0.5.
32
Fig. 4. Effects of dimensionless EDL thickness at
=3% (a) and volume fraction at K=20 (b) on dimensionless
temperature profiles when ur = 1.0, R=-0.5, Br=0.01, S=10.
33
Fig. 5. Effects of the volume fraction on the nature of variation of Nusselt number with dimensionless EDL thickness for different nanoparticle volume fraction when ur = 1.0, R=-10, Br=0.01 and S=10.
34
Fig. 6. Effects of the volume fraction on the nature of variation of Nusselt number with Brinkman number when ur = 1.0, R=-0.5, S=10 and K=20.
35
Fig. 7. Local entropy generation for different volume fraction when ur = 1.0, R=-0.5, Br=0.01, S=10, Θ=1000 and K=20.
36
Fig. 8. Total entropy generation for different volume fraction with Brinkman number when ur =1.0, R=-0.5, S=10, Θ=1000 and K=20.
37