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Numerical investigation of forced convection heat transfer through microchannels with non-Newtonian nanofluids
International Journal of Thermal Sciences 75 (2014) 76e86
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Numerical investigation of forced convection heat transfer through microchannels with non-Newtonian nanofluids Ali Esmaeilnejad a, *, Habib Aminfar b, Mahdieh Shafiee Neistanak c a
School of Mechanical Engineering, Iran University of Science and Technology (IUST), P.O. Box: 16846-13114, Narmak, Tehran, Iran Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran c Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Canada b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 October 2012 Received in revised form 21 July 2013 Accepted 21 July 2013 Available online
In this paper, convection heat transfer and laminar flow of nanofluids with non-Newtonian base fluid in a rectangular microchannel have been investigated numerically using two-phase mixture model. This research investigates the advantages of using nanoparticles in non-Newtonian fluids with particles size equal to 30 nm. The factor that makes nanoparticles feasible is the significant increase in rate of heat transfer within the fluids that are common in today’s industry. The power law model is used both Newtonian and non-Newtonian fluids. The flow behavior and rate of heat transfer performance of microchannel heat sink have been taken into account by looking into the effects of Al2O3 nanoparticles concentrations, Peclet number and flow behavior index. Our results demonstrate significant enhancement of heat transfer of non-Newtonian fluids using nanoparticles particularly in the entrance region. By increasing the volume fraction, higher heat transfer enhancement can be observed. The thermal resistance with Peclet number of 700 and 4% volume fraction reduces approximately 27.2% with shear thinning non-Newtonian base fluid and pressure drop will increase approximately 50.7%. Further analysis on particles type effect is also implemented with Al2O3 and CuO nanoparticles. Ó 2013 Elsevier Masson SAS. All rights reserved.
Keywords: Non-Newtonian nanofluid Two-phase mixture model Laminar forced convection Heat transfer enhancement Microchannel
1. Introduction The liquid coolants are essential for heat transfer in many engineering equipments such as electronic devices, heat exchangers and vehicles. In recent years, many attempts has been made to minimize the scales and improve the performance of instruments. Convective heat transfer in microelectromechanical systems (MEMS) has been confirmed to be an effective method for the thermal control of these devices such as microflow sensors, microchannels heat sink (MCHS), biomedical and biochemical systems [1e3]. The advantages of MCHS consist of ability to produce very high heat transfer coefficient, compactness, high surfaceto-volume ratios of microscale devices and small coolant requirements [4e6]. So far, the most common coolants in the MCHS studies have been done for Newtonian fluids such as water, ethylene glycol and so on. Heat transfer processing of non-Newtonian fluids is encountered in various industrial sectors including chemicals, petrochemicals, * Corresponding author. Tel.: þ98 912 590 6431, þ98 21 77 240 197; fax: þ98 21 77 240 488. E-mail addresses: [email protected], [email protected] (A. Esmaeilnejad). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.07.020
polymers, and pharmaceuticals [7]. High heat transfer capacity and low pumping power of some non-Newtonian fluids make them attractive as a coolant for various applications such as microchannel heat exchangers. Non-Newtonian fluids exhibit a non-linear relation between shear stress and shear rate. The simplest model of nonNewtonian fluids is the Ostwaldede Waele power law model used for intermediate ranges of the shear rate and various fluids [8e10]. Fluid flow and heat transfer problems involving non-Newtonian fluids have been reviewed by Metzner [11], Skelland [12], Cho and Hartnett [13] and among many others. Hartnett and Kostic [14] have collected the results about laminar and turbulent fluid flow and heat transfer of non-Newtonian fluids through rectangular channels with different aspect ratios. Schechter [15] obtained velocity profiles using a variational principle, for laminar flow through rectangular ducts of various aspect ratios. Wheeler and Wissler [16] applied an overrelaxation procedure to obtain more accurate velocity distributions and friction factors for aspect ratios of 0.5, 0.75 and 1. One approach to augment the convective heat transfer coefficient in the microchannel may be utilizing nanofluids as common fluids. Nanofluid is a suspension of solid nanoparticles (with diameter of 1e100 nm) in conventional liquids like water and oil. Depending on shape, size, and thermal properties of the solid nanoparticles, the thermal conductivity can be increased by about
A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
40% with low concentration (1e5% by volume) of solid nanoparticles in the mixture [17e19]. Extensive theoretical and experimental studies have been done to explain behavior of effective thermal conductivity of nanofluids. Some experimental studies [20,21] show that the measured thermal conductivity of nanofluids is much larger than the classical theoretical predictions [22]. Other experimental investigations [23,24] revealed that the thermal conductivity has not shown any anomalous enhancement and for lower volume fractions, the results agree well with the classical equations [22,25]. Many attempts have been made to formulate efficient theoretical models for the prediction of the effective thermal conductivity [26e28]. Heris et al. [29,30] studied the effects of alumina and copper oxide nanofluids on laminar heat transfer in a circular tube with considering water as a base fluid. They reported that heat transfer coefficient enhances for both nanofluids with increasing nanoparticles concentrations as well as Peclet number, and observed higher enhancement in alumina nanofluid than copper oxide. Numerical investigations on nanofluids are carried out in general using two approaches; single-phase or two-phase approach. Single-phase approach assumes that the continuum assumption is still valid for fluids with suspended nano size particles and fluid phase and nanoparticles are in thermal and hydrodynamic equilibrium. The other approach considers a two-phase model, which encloses a better description of the fluid and the solid phases. Since the solid particles sizes are classified as nanoparticles, they can easily fluidize and be approximately considered to behave as a fluid. However, because the effective properties of nanofluids are not known precisely, the numerical predictions of this approach depends on the effective physical properties. Xuan and Li [31] studied the single-phase flow and heat transfer performance of nanofluids under turbulent flow in tubes. Their experimental results showed that the convective heat transfer coefficient and the Nusselt number of nanofluids increase with the Reynolds number and the volume fraction of nanoparticles under turbulent flow. They have observed about 39% increase in Nusselt number when volume fraction was increased from 0 to 2% under same Reynolds number. The two-phase approach seems to be a better model for describing of the nanofluid heat transfer. In other words, the slip velocity between the fluid and nanoparticles might not be zero [19] due to several factors such as friction between the fluid and solid particles, Brownian forces, gravity, Brownian diffusion, sedimentation and dispersion. Behzadmehr et al. [32] studied the turbulent forced convection of a nanofluid in a circular tube by using a two-phase approach. They implemented the two-phase mixture model for the first time to study nanofluid. They examined the axial evolution of the flow field and fully developed velocity profiles at different Reynolds numbers. Their comparison with the experimental results showed that the mixture model is more precise than the single-phase model. Bianco et al. [33] numerically worked on developing laminar forced convection flow of a watereAl2O3 nanofluid in a circular tube. They found that the maximum difference in the average heat transfer coefficient between single-phase and twophase models results was approximately 11%. They concluded that heat transfer enhancement increases with the particle volume concentration as well as shear stress values. In recent years, the rapid development of engineering technologies has contributed significantly to the convective heat transfer enhancement of nanofluids with Newtonian base fluids. However, the effect of nanoparticles on the convective heat transfer in the non-Newtonian base fluids has not been investigated yet. The objective of this study is to numerically investigate the convective heat transfer coefficient of nanofluid with non-Newtonian base
77
fluid in the rectangular microchannel in laminar flow regime. Initially, heat is supplied to the silicon substrate through the heating area, then is removed by flowing non-Newtonian nanofluid through a microchannel. Three dimensional steady state flow is considered for the rectangular microchannel heated uniformly from the bottom. Throughout this research, the base fluid is a nonNewtonian Ostwaldede Waele power law model with spherical nanoparticles with the diameter of 30 nm. Two-phase mixture model approach is employed to evaluate the laminar forced convection flow by considering the temperature variable thermophysical properties. The numerical simulation results are compared with other works presented in literature for flow of non-Newtonian base fluid in microchannel. This paper is structured in the following manner. Section 2 outlining the model formulation, numerical method and simulation conditions, Section 3 discussing the numerical results, and finally, in Section 4, a summary of the main conclusions are presented. 2. Mathematical modeling 2.1. Governing equations Laminar forced convection of a nanofluid consisting of nonNewtonian base fluid with nanoparticles has been considered. In this numerical investigation, the gravitation force is considered as an important parameter. Because of symmetry and for simplicity and reduction of calculation time, only one microchannel is selected and analyzed as computational domain. It is clear that the heat sink exhibits geometrical symmetries. Due to the heat sink module chosen in this study, planes of symmetry can also be identified for the heat transfer part of the problem. With the above simplifications, only one microchannel is modeled as shown in Fig. 1b. The geometrical configuration in this simulation work is depicted schematically in Fig. 1. Due to the heat sink module chosen in this study, symmetrical planes are used for the heat transfer part of the problem. The nanofluid flows through the one rectangular microchannel with imposing of constant heat flux to the silicon substrate at the bottom part of the solid microchannel. The top surface is assumed to be adiabatic. A complete description of thermal behavior of microchannel include three dimensional conduction analysis in solid parts of microchannel and three dimensional solutions of conservation equations of non-Newtonian nanofluid under steady state conditions. For temperature profiles in solid microchannel, it is necessary to solve the conduction equation within solid microchannel with neglecting the effect of dissipation and pressure work. Heat sink performance is commonly measured by its thermal resistance that is defined as below [34].
Rth ¼
Rth ¼ ¼
Tw;max Tf ;in qw
DTmax
Tw;max Tf;in qw Tw;max Tf;out þ Tf;out Tf;in qw
¼
¼ Rcon þ Rcap
qw
¼
1 hAfin
þ
1 _ p mC (1)
where Tw,max, Tf,in, Tf,out, qw, Rcon and Rcap are the maximum temperature of surface (wall), the inlet fluid temperature, the outlet fluid temperature, the heat flux at the heating area, convective and capacitive resistance, respectively.
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V$
n X
4k rk ! v k! v k ¼ V$ðrm ! v m! v m Þ þ V$
k¼1
n X
4k rk ! v dr;k ! v dr;k
k¼1
(6) In terms of mixture variable (n ¼ 2 for two-phase) the momentum equation takes the following form:
! V$ðrm ! v m! v m Þ ¼ Vpm þ V$ðsk Þ þ V$sDm þ rm g
(7)
where sm and sDm represent the average viscous stress and stress diffusion due to the phase slip, respectively. The tree stress tensors are defined as:
sm ¼
n X
4k sk
(8)
k¼1
sDm ¼
n X
4k rk vdr;k vdr;k
(9)
k¼1
where vdr,k is the diffusion velocity, i.e., the velocity of phase k relative to the center of the mixture mass
vdr;k ¼ vk vm
(10)
In non-Newtonian fluid, the so-called apparent viscosity is defined in the same manner as for Newtonian fluids, although it is no longer constant, but depends on flow conditions such as shear rate magnitude. For the simplest power law fluid model, the (apparent) viscosity is expressed as:
s ¼ f ðg_ ; r; nÞ g_
h¼
Fig. 1. (a) Schematic of the microchannel, (b) dimensions of the microchannel.
The two-phase mixture model is also employed to analyze the thermal and fluid dynamic behavior of the considered nonNewtonian nanofluids. The following formulation represents the mathematical description of the mixture model governing equations. The continuity equation for mixture model is as follow:
V$ðrm $! v mÞ ¼ 0
(2)
The mixture density and the mixture velocity for n phase are defined as: n X
rm ¼
4k rk
n 1 X
rm
4k rk ! vk
V$
4k rk ! v k! vk ¼
k¼1
þ
k¼1 n X
4k Vpk þ V$
n X
(12)
Here, r and n are flow consistency index and flow behavior index (or power law index) respectively; both are constant for a given fluid. In special case: r ¼ m ¼ constant and n ¼ 1, corresponds to a Newtonian fluid with h ¼ m ¼ r. The diffusion stress term, V$sDm, in Eq. (7) compared to the onephase equation is representing the momentum diffusion due to the relative motions. The energy equation for the mixture is defined as:
i
h
4rp Cp;p ! v p þ ð1 4Þrf Cp;f ! v f T ¼ V$ keff VT þ Se
where Se, T and keff represent thermal resources, temperature and effective thermal conductivity of nanofluid, respectively. The equation of volume fraction of phases is defined as below:
4k ðsk þ sTk Þ
k¼1
4k rk ! g
(4)
k¼1
n X
s ¼ r g_ n1 g_
V$
where vm represents the velocity of mass center. The momentum equation for mixture model (n phase) is as follow: n X
where h is the apparent viscosity of fluid and the relationship between the shear stress and shear rate for non-Newtonian base fluid according to the Ostwaldede Waele model is:
(3)
k¼1
vm ¼
(11)
V$ 4rp ! v m ¼ V$ 4rp ! v dr;p
k¼1
Using the definitions (2), (3) and (4), the first term of Eq. (5) can be written as:
(14)
The slip velocity (or relative velocity) is defined as the velocity of a secondary phase (p) relative to the velocity of the primary phase (f):
! v mf ¼ ! vp! vf (5)
(13)
(15)
The drift velocity is related to the relative velocity
4rp ! ! ! vf vp v dr;p ¼ ! v pf rm
(16)
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correct velocities at outlet that are used for correcting the pressure fields. - At the lateral walls (vertical symmetry plane), symmetry condition is considered as below:
The relative velocity is determined from equation proposed by Manninen et al. [35] and equation by Schiller and Naumann [36] is used to calculate the drag function fdrag. The effective density and effective thermal diffusivity of the nanofluid at reference temperature is defined as:
rm ¼ 4rp þ ð1 4Þrf
rCp
¼ 4 rCp p þ ð1 4Þ rCp f
m
79
(17)
wc þ ww vTw / kw ¼ 0 z ¼ 2 vz
(23)
(18)
wc þ ww vTw / kw ¼ 0 z ¼ 2 vz
(24)
Thermal diffusivity of nanofluid:
aeff
k eff ¼ 4 rCp p þ 1 4 rCp f
(19)
The effective thermal conductivity of fluid has been determined by the model proposed by Hamilton and Crosser (1962).
" keff ¼
kp þ ðj 1Þkf ðj 1Þ4 kf kp kp þ ðj 1Þkf þ 4 kf kp
# kf
(20)
where j is a shape factor and equal to 3 for spherical nanoparticles. In this equation, the thermal conductivity of fluid (non-Newtonian base fluid) and particles are considered to vary with temperature. As shown by Zhang et al. [24], this model is in good agreement with the experimental results at the low value of the nanoparticles volume fraction (4 < 5%).
xþ ¼
x Dh
yþ ¼
Pr ¼
Cp;m $r vm n1 $ Dh keff
y Dh
rm $vm $Dh w f $Re ¼ 1 rs$v 2 $ h m 2
zþ ¼
pþ ¼ 1 rp$v2 m 2
z Dh
r $C
Tþ ¼
$v $D
TTin q$Dh keff
Re ¼
m h Pe ¼ Re$Pr ¼ m p;m f $Re ¼ 14 keff D ðdpÞ$ h ¼ 1dxr$v2 4 $rm $vhm $Dh Dh ¼ ð4Ach =pch Þ 2
The set of non-linear elliptical governing equations has been solved using the following boundary conditions: - At the microchannel inlet (x ¼ 0):
vz ¼ 0
T ¼ Tin
(21)
where Tin refers to the reference (inlet) condition. - At fluidewall interface:
kw
(25) (26)
In addition, no slip boundary condition is used on the microchannel walls. A fully developed condition is used at outlet boundary. In this study, the flow regime is laminar, incompressible and steady. At the exit plane, (x ¼ L) the pressure is assumed atmospheric pressure. 2.3. Numerical method The governing equations can be converted to non-dimensional form, using the following dimensionless parameters:
ð2nÞ rm $vm $Dnh
r
(27)
m
2.2. Boundary conditions
vy ¼ 0
vTw ¼ qw vy vTw ¼ 0 y ¼ H/ kw vy y ¼ 0/ kw
dpþ $Re dxþ
In this study, the fluid properties of non-Newtonian nanofluid such as thermal conductivity, density and thermal capacity are varied with temperature.
vx ¼ v0
- At the bottom of the microchannel, constant heat flux condition is imposed and upper wall is treated as an adiabatic wall.
vTw ðx; y; zÞ vT ðx; y; zÞ ¼ keff nf vs vs
where Re, Pr, Pe, r and n represent Reynolds number, Prandtl number, Peclet number, flow consistency index and flow behavior index for nanofluid with non-Newtonian base fluid, respectively. In addition, Dh is hydraulic diameter of the microchannel and it is defined as Dh ¼ (4Ach/pch), where pch and Ach is the perimeter of the microchannel cross-section and finned area used in heat transfer, respectively. This set of coupled non-linear differential equations is discretized with the finite difference method based on the control volume approach. The SIMPLE1 algorithm was introduced for the velocityepressure coupling and convection terms were estimated using the QUICK method. The discretization grid is considered to be non-uniform. In addition, structured non-uniform grids have been used to discretize the computational domain and are finer in the vicinity of the heated wall where more accurate solutions is necessary. It is important to note that most of the parameters and the results obtained from this study are non-dimensional.
(22)
where s refers to the normal direction to the surface. - At the microchannel outlet (x ¼ L), the diffusion flux in the direction normal to the exit plane is assumed to be zero for all variables. An overall mass balance correction is utilized to
2.3.1. Grid independency study In order to evaluate the effect of the number of the mesh points on the accuracy of the results, several different grid distributions
1
Semi-Implicit Method for Pressure Linked Equations.
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have been tested to ensure the calculated results are grid independent. Fig. 2 shows the local Nusselt number and nondimensional local pressure drop in fully developed region at different grids, where local Nusselt number is calculated according to the following definition:
NuðxÞ ¼
hðxÞ$Dh keff
(28)
and h(x) is defined as:
qw hðxÞ ¼ Tw ðxÞ Tb ðxÞ
Table 1 Case studies specification.
Lt (cm) Wfin (mm) Wch (mm) Wt (cm) H (mm) hch (mm) Qa (cm3/s) q (W/cm2) R (cm2 C/W)
(29) a
Tuckerman’s experiment [1] Present calculation
Case 1
Case 2
Case 3
1.4 44 56 2 533 320 4.7 181 0.110 0.114
1.4 45 55 2 430 287 6.5 277 0.113 0.115
1.4 50 50 2 458 302 8.6 790 0.090 0.087
Total flow rate through the heat sink assembly.
From the previous equation the hav is calculated as follows:
hav
1 ¼ Lch
ZLch hðxÞdx
(30)
0
and the average Nusselt number is defined as:
Nuav ¼
hav $Dh keff
(31)
As results show, a minimum number of 244,800 non-uniform grid points were necessary before a satisfactory and accurate result could be obtained. Such grid has 170, 45 and 32 nodes, respectively in x, y and z directions with highly packed grid points in the vicinity of the microchannel wall, especially at the entrance region. Hence, further increase in the number of grids does not affect the results.
2.3.2. Validation of the present code In order to demonstrate the validity and precision of the model as well as numerical procedure, comparisons with the available experimental and numerical simulation have also been done. There is a limitation for comparison of basic results because of some differences such as geometry, fluid and numerical method between present study and other related studies. Table 1 is outlining the calculated thermal resistances with the experimental results of Tuckerman [1] in microchannel, that the geometry of this numerical study is similar to their works. As it is shown, the predicted thermal resistance at x ¼ 1.33 is in good agreement with the experimental results. Table 2 is clarifying of the error percentage of thermal resistance between the obtained results and the experimental results. The maximum and minimum errors in comparison with Tuckerman’s experiment are 3.64% and 1.77% in case 1, respectively. This error is considerably low, respect to the amount of parameters. This minimal difference could be caused by using numerical method in this investigation and errors in Tuckerman’s experiments. Therefore, it is safe to say that the predicted thermal resistance is in good agreement with the experimental results. In addition to the thermal resistance, variation of friction factor and Reynolds number (f$Re) is compared with the corresponding results [15,37,38]. The results have been summarized in Table 3 for fully developed region of non-Newtonian fluid flowing through microchannel under laminar conditions. In addition, the differences between the obtained results and the experimental and numerical results of f$Re are noticeably low. The maximum and minimum errors are 2.22% and 0.16% in case of Schechter’s results [15] for n ¼ 0.7 and n ¼ 1, respectively, in comparison with present results, as presented in Table 4. The concordance between the results is good. Therefore, the numerical code is proven to be reliable and can predict forced convection flow of nanofluids with non-Newtonian base fluid through a microchannel.
3. Results and discussion Numerical simulations of flow and heat transfer for a range of Pe and particles volume fractions have been studied. Nanofluid is referred as the mixture of solid spherical alumina nanoparticles of 30 nm diameter and non-Newtonian base fluid with various flow behavior index. In this study, a microchannel heat sink with exerted constant heat flux to the silicon substrate is considered. Results Table 2 Error percentage of thermal resistance between the obtained results in comparison with Tuckerman’s experiment. Tuckerman’s experiment [1] Fig. 2. Grid independence tests.
Case 1
Case 2
Case 3
3.64%
1.77%
3.33%
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81
Table 3 Variation of f$Re for different values of n for a microchannel (aspect ratio ¼ 6.43) in fully developed region. Case studies specification
Gao and Hartnett [37] Schechter [15] Tiu et al. [38] Present calculation
f$Re n ¼ 0.7
n¼1
n ¼ 1.3
e 9.956 10.105 10.177
17.952 18.152 18.039 18.182
32.087 32.377 32.495 32.303
were carried out employing the two-phase mixture model and temperature-dependent properties. In order to show the presence impact of nanoparticles in non-Newtonian base fluid flowing through microchannel, several aspects of these fluids are examined. Fig. 3 shows the dimensionless axial velocity profiles at x ¼ 1.26 cm for Pe ¼ 680 and qw ¼ 790 W/cm2with different volume fractions (Al2O3 nanoparticles) and different flow behavior index. It can be observed that for two-phase mixture model, while properties are dependent on temperature, the dimensionless velocity profiles are nearly independent on concentration value. In fact, dimensional axial velocity increases with nanoparticles volume fractions. This arises from the fact that the physical properties of nanofluid changes with the volume fraction. Therefore, different mean velocity is needed for different nanoparticle volume fraction to keep the Peclets number constant. However, the dimensionless velocity profile shows no differences. This shows that the dimensionless velocity profile remains constant and nanoparticle volume fraction does not have significant effect on the velocity profile. In addition, increasing the value of n (by moving from a shear thinning fluid to a shear thickening fluid) in non-Newtonian base fluids, the velocity gradient of the microchannel wall decreases and consequently the maximum dimensionless axial velocity increases. Fig. 4 shows the pressure drop of non-Newtonian nanofluids using Al2O3 nanoparticles. Nanoparticles do not affect the pressure drop of non-Newtonian nanofluids in low Reynolds number and there is no noticeable increase in comparison to non-Newtonian base fluid. Besides, as the Reynolds number is increased, the pressure drop increases further, and this is more evident at higher values of Reynolds number. In other words, using nanoparticles causes higher pressure drop in high Reynolds number in comparison to lower Reynolds number. However, nanofluid with 4% solid volume fraction causes high-pressure drop especially in high Reynolds number. Examination of results shows that at a fixed Reynolds number of 730 for shear thinning fluids with n ¼ 0.7 and 4 ¼ 4%, the pressure drop could increase as much as 53%, while for shear thickening fluids, it could increase to 50.5% at a value of n ¼ 1.3. As seen in Fig. 4aec, it is observed that by increasing the value of n at fixed Reynolds number, the pressure drop increases, but this augmentation of pressure drop in low Reynolds number is higher than in high Reynolds number. In other words, with increasing Reynolds number, pressure drop receive low effect from variation of power law index. Variation of power law index plays somewhat small in the pressure drop as the Reynolds number increases. Fig. 5 shows the local convective heat transfer coefficient as a function of the non-dimensional microchannel length for three Table 4 Error percentage of f$Re between the obtained results and the experimental and numerical results.
Gao and Hartnett [37] Schechter [15] Tiu et al. [38]
n ¼ 0.7
n¼1
n ¼ 1.3
e 2.22% 0.71%
1.28% 0.16% 0.79%
0.67% 0.23% 0.59%
Fig. 3. Profiles of axial velocity at x ¼ 1.26 cm, for Pe ¼ 680, qw ¼ 790 W/cm2 and several concentration (Al2O3 nanoparticles) and power law index values: (a) 4 ¼ 0, (b) 4 ¼ 1%, 4%.
volume concentrations of Al2O3 nanoparticles at fixed Peclet number of 400. Fig. 5aec is for the quantities of flow behavior index of non-Newtonian base fluids equal to 0.7, 1 and 1.3, respectively. It can be seen that the heat transfer coefficient increases with nanoparticle concentration at a given non-dimensional axial position and Peclet number. Given the Peclet number and nanoparticle concentration, the heat transfer coefficient decreases with axial distance as expected. For the nanofluid with 1% by volume nanoparticles and a shear thinning non-Newtonian base fluid (n ¼ 0.7) the enhancement decreases from 8.11% at position x/Dh ¼ 5 to 3.28% at x/Dh ¼ 173 with Pe ¼ 400 in comparison with the case of nonNewtonian base fluid only. This difference is noticed to be higher for high volume concentration of nanoparticles. For example, for the same Peclet number and flow behavior index, the augmentation of local heat transfer coefficients with 4% volume fraction of Al2O3 nanoparticles are, respectively, 23.39% and 15.57% at position x/Dh ¼ 5 and x/Dh ¼ 173, respectively, in comparison with the case of non-Newtonian base fluid only. As it can be seen from Figs. 5 and 6, the enhancement in the local heat transfer coefficient of nanofluids becomes appreciable at higher Peclet number. Also, the entrance regime effect becomes more pronounced at the high Peclet number of 1385, as shown in Fig. 6. One important parameter in the investigating of performance of heat sinks is their thermal resistance. Thermal resistance, the main parameter of microchannel heat sink performance, is determined
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Fig. 5. Convective heat transfer coefficient as a function of the non-dimensional microchannel length for Pe ¼ 400 with two-phase mixture model (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
Fig. 4. Pressure drop of non-Newtonian nanofluids for several concentration (Al2O3 nanoparticles) and power law index values: (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
by using the maximum temperature difference of cooling fluid and the solid surface, using Eq. (1). The microchannel with lower thermal resistance shows more performance i.e. for fixed inlet fluid temperature and thermal power, the fluid has more potential to remove heat from bottom heat sink when the maximum temperature of heat sink is lower. According to Eq. (1), this leads to lower thermal resistance. The variation of the thermal resistance versus the Peclet number for various Al2O3 nanoparticles volume concentrations are shown in Fig. 7. It is observed that the thermal resistance of microchannel heat sink decreases with increasing the nanoparticle volume fractions in the non-Newtonian base fluid. This is more pronounced at lower Peclet number. In addition, with increasing Peclet number, thermal resistance decreases gradually. It is due to this fact that with increasing of Peclet number, convective heat transfer coefficient increases. Therefore, more heat transfer can be removed from microchannel heat sink and consequently the temperature of fluid increases and thermal resistance decreases. As it is seen in the figure, when the value of n increases, the thermal resistance of the microchannel heat sink rises as well. Therefore, for a proper selection of a
microchannel heat exchanger about using non-Newtonian nanofluid, it is recommended to search for a shear thinning nonNewtonian base fluid coolant at a high Peclet number. Because of wide discrepancies between the given values of thermophysical properties of nanoparticle, these properties of CuO nanoparticles used for present numerical study are measured in chemical engineering laboratory that these values are very close to values stated in literature [40,41]. In addition, physical properties of Al2O3 such as thermal conductivity and specific heat nanosized particles are used from Ref. [40]. Table 5 presents the summary of thermophysical properties of Al2O3 and CuO nanoparticles used for comparing the particle types. The base fluid for the results is an aqueous solution containing different percentages of Carboxy Methyl Cellulose (CMC) which was made by adding CMC powder to distilled water and thoroughly mixed. Nanofluids were considered by dispersing dry nanoparticles into the base liquid. In reality, nanoparticles can be dispersed in CMC solution (carboxy methylcellulose in water) using ultrasonic mixing and mechanical mixer. In this two-phase numerical study, different concentrations of CMC non-Newtonian base fluids with nanoparticles were investigated under forced convection heat transfer conditions and properties of non-Newtonian base fluid was used from Ref. [42] as shown in Table 6.
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83
Fig. 6. Convective heat transfer coefficient as a function of the non-dimensional microchannel length for Pe ¼ 1385with two-phase mixture model (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
Average heat transfer coefficient and pressure drop are reported in Table 7. Comparison is carried out for 1% and 4% volume concentrations of nanoparticles for Peclet number of 1385 and flow behavior index of 0.7. The nanofluid is a mixture of non-Newtonian base fluid with solid spherical particles of 30 nm diameter of both Al2O3 and CuO. The results clearly show the significant enhancement of average heat transfer coefficient for both nanofluids with various kinds of nanoparticles concentrations. The scattering of Al2O3 nanoparticles in non-Newtonian base fluid results in higher average heat transfer coefficient in comparison with CuO nanoparticles. However, the pressure drop using CuO nanoparticles is less than Al2O3 nanoparticles in the same non-Newtonian base fluid. The temperature profiles along the bottom wall of microchannel for qw ¼ 790 W/cm2 and fixed Peclet numbers with variable properties of non-Newtonian nanofluids using Al2O3 nanoparticles is shown in Fig. 8. The reduction of wall temperature, with respect to the non-Newtonian base fluid, increases with the x coordinate. At the exit section, the reduction of the wall temperature of microchannel for n ¼ 0.7 and Pe ¼ 400 are about 0.16% and 0.6%, in comparison with non-Newtonian base fluid, for variable properties in the case of 4 ¼ 1% and 4 ¼ 4%, respectively. Based on the results, using nanoparticles with higher Peclet number results in lower wall temperature. For example, for non-Newtonian nanofluid (n ¼ 0.7) at Pe ¼ 1385 the reduction of temperature are 0.07% and
Fig. 7. Thermal resistance of microchannel in terms of Peclet number for nonNewtonian nanofluids (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
0.25%, in comparison with non-Newtonian base fluid, at the exit of microchannel for 4 ¼ 1% and 4 ¼ 4%, respectively. The results highlights the significant advantage of using nanoparticles for lowering wall temperature is particularly great at the exit of microchannel. For the nanofluid with 4% by volume nanoparticles with Pe ¼ 400 and a shear thinning non-Newtonian base fluid (n ¼ 0.7), the temperature decreases from 8.11% at position x ¼ 0.7 cm to 3.28% at x ¼ 1.4 cm in comparison with the nonNewtonian base fluid. These results have indicated that the beneficial effect of nanoparticles considerably improves the thermal properties of the mixture. It is important to note that additional effects such as gravity, drag on the particles, diffusion and Brownian forces play important roles [39]. Fig. 9 shows heat transfer coefficient enhancement for n ¼ 0.7 and qw ¼ 790 W/cm2 for constant and variable properties using two-phase mixture model with Al2O3 nanoparticles. In constant properties state, the properties of non-Newtonian nanofluid are calculated in constant fluid mean temperature. The variation of fluid temperature along the microchannel is more noticeable in low Reynolds number and high heat flux.
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A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
Table 5 Properties of the studied particles in non-Newtonian base fluids [40]. Property
Table 7 Average heat transfer coefficients and pressure drop with two kinds of nanoparticles for Pe ¼ 1385, n ¼ 0.7 and 4 ¼ 1%, 4%.
Value Al2O3
rp [kg/m3]
3820 880 46
Cp,p [J/kg K] kp [W/m K]
CuO 6320 670 27
4 ¼ 1%
Property
2
hav (W/m K)
DP (pa)
4 ¼ 4%
Al2O3
CuO
Al2O3
CuO
57,334 211,344
56,827 207,493
63,850 273,471
61,973 246,967
Below equation supports the dependency of Re number on Tout Tin:
rnf Vm Cp;nf Ach ðTout Tin Þ ¼ qw $Abottom
(32)
As it is known, Dh ¼ (4Ach/pch) is hydraulic diameter of the microchannel where pch is the perimeter of the microchannel cross-section and thus:
Ach ¼
Dh pch 4
(33)
By submitting Eq. (33) in Eq. (32):
rnf Vm Cp;nf
Dh pch ðTout Tin Þ ¼ qw $Abottom 4
rnf Vm Dh Cp;nf
(34)
pch mnf ðTout Tin Þ ¼ qw $Abottom 4 mnf
(35)
rnf Vm Dh p Cp;nf ch mnf ðTout Tin Þ ¼ qw $Abottom mnf 4
(36)
where Re ¼ ðrnf Vm Dh =mnf Þ is Reynolds number. By submitting in the above equation:
p ReCp;nf ch mnf ðTout Tin Þ ¼ qw $Abottom 4 Tout Tin ¼
(37)
4qw $Abottom Re mnf Cp;nf pch
(38)
Therefore, according to Eq. (38), the variation of fluid temperature along the microchannel is more noticeable in low Reynolds number and high heat flux. Assumption of constant properties of nanofluid along the microchannel can result in error. According to Eq. (38), the temperature difference of nanofluid through microchannel is noticeable in low Reynolds number. In addition, increasing of temperature difference can cause more significant increase in thermal conductivity. Therefore, the difference of heat transfer coefficient between the constant and variable properties is becoming small when the Reynolds number is increased. Unfortunately, neglecting the effect of variable properties of nanofluid
Fig. 8. Effects of the Peclet number and volume concentration of Al2O3 nanoparticles on the bottom wall temperature of microchannel for qw ¼ 790 W/cm2 and n ¼ 0.7: (a) Pe ¼ 400, (b) Pe ¼ 1385.
along the microchannel leads to a major misconception about the effect of parameters. Fig. 10 shows heat transfer coefficient enhancement of nanofluid versus Peclet number at different concentrations of Al2O3 nanoparticles. It is clear that the constant Peclet number heat transfer coefficient increases with nanoparticle concentration. Increased
Table 6 Experimental rheological properties for CMC solutions, expressed as parameters of the power law model [42]. Property
Value r
n
CMC (1%) CMC (0.8%) CMC (0.5%)
0.532 0.651 0.729
1.052 0.308 0.192
Fig. 9. Enhancement of heat transfer coefficient in comparison with non-Newtonian base fluid as a function of Reynolds number for n ¼ 1 and qw ¼ 790 W/cm2 with constant and variable properties using two-phase mixture model) with Al2O3 nanoparticles).
A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
85
Nomenclature
Fig. 10. Heat transfer coefficient as a function of Peclet number for n ¼ 0.7 and qw ¼ 790 W/cm2 using two-phase mixture model (with Al2O3 nanoparticles).
thermal conductivity probably is not the only mechanism responsible for heat transfer enhancement, particularly at high nanoparticles concentration and high Peclet number. It seems that nanoparticles presence may increase the temperature gradient at the wall and hence augment heat transfer of nanofluid. Particles migration, clustering process due to non-uniform shear rate across the microchannel cross-section along with particle interactions and dispersion effects are also probable causes for increased heat transfer of nanofluids. Based on this figure, the heat transfer coefficient for nanofluid is greater than heat transfer coefficient for non-Newtonian base fluid and the heat transfer enhancement is higher for higher concentrations of particles. These results are also in agreement with Bianco et al. [33]. In other words, heat transfer enhancement increases with the particle volume concentration. 4. Conclusion In this paper, the hydrodynamic and thermal behaviors of nonNewtonian nanofluids flowing inside microchannel were numerically investigated in stationary conditions and for laminar forced flow. Two-phase mixture model has been employed to investigate numerically the effect of nanoparticles volume fraction on the hydrodynamic and thermal parameters of non-Newtonian base fluid with various flow behavior index. It is shown that, for a given Peclet number, despite changing the mean inlet velocity, the dimensionless velocity profile does not vary with nanoparticle volume fraction. In addition, in lower Reynolds number using nanoparticles do not affect the pressure drop of non-Newtonian nanofluids and there is no significant increase in comparison to non-Newtonian base fluid. However, as the Reynolds number is increased, the pressure drop consequently increased significantly. Results clearly indicate the beneficial effects of nanoparticles due to the presence of such particles, the thermal properties of the resulting mixture have considerably improved. Additional effects such as gravity, drag on the particles, diffusion and Brownian forces also play an important role. In general, using nanofluid with higher nanoparticle volume fraction improves the convective heat transfer coefficient, thermal resistance of microchannel and wall temperatures. Results were also compared with two kinds of nanoparticles. Using Al2O3 nanoparticles in nonNewtonian base fluid shows more increment in average heat transfer coefficient in comparison with CuO nanoparticles. However, the pressure drop in using CuO nanoparticles is less than Al2O3 nanoparticles in the same non-Newtonian base fluid. In addition, the effect of variable and constant properties along the microchannel is further investigated throughout this study.
Afin Cp Dh f g H h h k L _ m Nu n p Pe Pr pch qw Rth Re r Se T t ! v ! v dr ! v pf w
total finned area used in heat transfer (m2) specific heat (J kg1 K1) hydraulic microchannel diameter Dh ¼ (4Ach/pch) friction factor gravitational acceleration (m s2) microchannel height (see Fig. 1) convective heat transfer coefficient (W m2 K1) average heat transfer coefficient along the channel length thermal conductivity (W m1 K1) microchannel length 1 mass flow rate (kg s ) Nusselt number qw $Dh = keff $ T w Tb flow behavior index (power law index) pressure Peclet number (rm$Cp,m$vm$Dh/keff) Prandtl number (Cp,m$r$(vm/Dh)n1/keff) perimeter of the microchannel cross-section wall heat flux (W cm2) thermal resistance (cm2 C W1) ð2nÞ Reynolds number ðrm $vm $Dnh =rÞ flow consistency index thermal resource temperature wall thickness (see Fig. 1) velocity (m s1) diffusion (drift) velocity vector (m s1) slip velocity vector (m s1) wall thickness (see Fig. 1)
Greek symbols 4 particle volume concentration a thermal diffusivity (m2s1) h apparent viscosity (N s m2) r density (kg m3) s stress tensor (N m2) g_ shear rate (s1) Subscripts av average value b bulk value bottom bottom of microchannel ch microchannel eff effective f refers to base fluid i,in refers to the reference (inlet) condition k number of phase m refers to mixture out refers to outlet condition p particle th thermal w value at microchannel wall x x direction y y direction z z direction þ non-dimensionalized parameter References [1] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electron Device Lett. (1981) 126e129. EDL-2. [2] T. Kishimoto, T. Ohsaki, VLSI packaging technique using liquid-cooled channels, IEEE Trans. Compon. Hybr. (1986) 328e335. CHMT-9. [3] B.X. Wang, X.F. Peng, Experimental investigation on forced convection heat transfer through microchannels, Int. J. Heat Mass Transfer 17 (1994) 73e82.
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[4] H.Y. Wu, P. Cheng, An experimental study of convective heat transfer in silicon microchannels with different surface conditions, Int. J. Heat Mass Transfer 46 (2003) 2547e2556. [5] W. Qu, I. Mudawar, Experimental and numerical study of pressure drop and heat transfer in a single-phase microchannel heat sink, Int. J. Heat Mass Transfer 45 (2002) 2549e2565. [6] C. Perret, J. Boussey, C. Schaeffer, M. Coyaud, Analytic modeling, optimization, and realization of cooling devices in silicon technology, IEEE Trans. Compon. Packag. Tech. 23 (2000) 665e672. [7] R.P. Chhabra, J.F. Richardson, Non-Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications, VCH Publishers, New York, 1999. [8] C.W. Macosko, Rheology: Principles, Measurements and Applications, VCH Publishers, Munich, 1994. [9] R.M. Manglik, P. Fang, Thermal processing of viscous non-Newtonian fluids in annular ducts: effects of power-law rheology, duct eccentricity and thermal boundary conditions, Int. J. Heat Mass Transfer 45 (4) (2002) 803e814. [10] S. Shin, Y.I. Cho, Laminar heat transfer in a rectangular duct with a nonNewtonian fluid with temperature dependent viscosity, Int. J. Heat Mass Transfer 37 (Suppl. 1) (1996) 55e79. [11] A.B. Metzner, Heat Transfer in Non-Newtonian Fluids, Advances in Heat Transfer, Academic Press, New York, 1965. [12] A.H.P. Skelland, Non-Newtonian Flow and Heat Transfer, John Wiley & Sons, 1967. [13] Y.I. Cho, J.P. Hartnett, Handbook of Heat Transfer Applications, McGraw-Hill, New York, 1985. [14] J.P. Hartnett, M. Kostic, Heat Transfer to Newtonian and Non-Newtonian Fluids in Rectangular Ducts, in: Advances in Heat Transfer, Academic Press, New York, 1989. [15] R.S. Schechter, On the steady flow of a non-Newtonian fluid in cylinders, AIChE J. (1961) 445e454. [16] J.A. Wheeler, E.H. Wissler, The friction factor-Reynolds number relation for the steady flow of pseudoplastic fluids through rectangular ducts, AIChE J. 11 (1966) 207e219. [17] S. Lee, S.U.S. Choi, S. Li, J.A. Eastman, Measuring thermal conductivity of fluids containing oxide nanoparticles, Trans. ASME J. Heat Transfer 121 (1999) 280e 289. [18] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718e720. [19] Y.M. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow 21 (2000) 58e64. [20] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalous thermal conductivity enhancement in nanotube suspension, Appl. Phys. Lett. 79 (2001) 2252e2254. [21] D.H. Yoo, K.S. Hong, H.S. Yang, Study of thermal conductivity of nanofluids for the application of heat transfer fluids, Thermochim. Acta 455 (2007) 66. [22] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two component systems, IEC Fundam. 1 (3) (1962) 187e191.
[23] S.A. Putman, D.G. Cahill, P.V. Braun, Z. Ge, R.G. Shimmin, Thermal conductivity of nanoparticle suspensions, J. Appl. Phys. 99 (2006) 084308. [24] X. Zhang, H. Gu, M. Fujii, Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles, Exp. Thermal Fluid Sci. 31 (2007) 593e599. [25] E. Yamada, T. Ota, Effective thermal conductivity of dispersed materials, Wärme-und Stoffübetragung 13 (1980) 27e37. [26] Q.Z. Xue, Model for effective thermal conductivity of nanofluids, Phys. Lett. A 307 (2003) 313e317. [27] Y.M. Xuan, Q. Li, W. Hu, Aggregation structure and thermal conducting of nanofluids, AIChE J. 49 (2004) 1038e1043. [28] C.H. Chon, K.D. Kihm, S.P. Lee, S.U.S. Choi, Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement, Appl. Phys. Lett. 87 (2005) 153107. [29] S.Z. Heris, S. Gh. Etemad, M.N. Esfahany, Experimental investigation of oxide nanofluids laminar flow convective heat transfer, Int. Commun. Heat Mass Transfer 33 (2006) 529e535. [30] S.Z. Heris, M.N. Esfahany, S. Gh. Etemad, Experimental investigation of convective heat transfer of Al2O3/water nanofluid in circular tube, Int. J. Heat Fluid Flow 28 (2007) 203e210. [31] Y. Xuan, Q. Li, Investigation on convective heat transfer and flow features of nanofluids, J. Heat Transfer 125 (2003) 151e155. [32] A. Behzadmehr, M. Saffar-Avval, N. Galanis, Prediction of turbulent forced convection of a nanofluid in a tube with uniform heat flux using a two-phase approach, Int. J. Heat Fluid Flow 28 (2007) 211e219. [33] V. Bianco, F. Chiacchio, O. Manca, S. Nardini, Numerical investigation of nanofluids forced convection in circular tubes, Int. J. Appl. Thermal Eng. 29 (2009) 3632e3642. [34] K.C. Toh, X.Y. Chen, J.C. Chai, Numerical computation of fluid flow and heat transfer in microchannels, Int. J. Heat Mass Transfer 45 (2002) 5133e5141. [35] M. Manninen, V. Taivassalo, S. Kallio, On the Mixture Model for Multiphase Flow, vol. 288, Technical Research Center of Finland, VTT Publications, 1996, p. 3, 9e18. [36] L. Schiller, A. Naumann, A drag coefficient correlation, Z. Ver. Deutsch. Ing. 77 (1935) 318e320. [37] S.X. Gao, J.P. Hartnett, Non-Newtonian fluid laminar flow and forced convection heat transfer in rectangular ducts, Int. Commun. Heat Mass Transfer 19 (1992) 673e686. [38] C. Tiu, W. Kozicki, T.Q. Phung, Geometric parameters for some flow channels, Can. J. Chem. Eng. Sci. 46 (1968) 389e393. [39] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids Science and Technology, John Wiley & Sons, Hoboken, NJ, 2008. [40] S.M. Fotukian, M. Nasr Esfahany, Experimental investigation of turbulent convective heat transfer of dilute g- Al2O3/water nanofluid inside a circular tube, Int. J. Heat Fluid Flow 31 (2010) 606e612. [41] Elif Büyük Ögüt, Natural convection of water-based nanofluids in an inclined enclosure with a heat Source, Int. J. Thermal Sci. 48 (2009) 2063e2073. [42] S. Soltani, S. Gh. Etemad, J. Thibault, Pool boiling heat transfer of nonNewtonian nanofluids, Int. Commun. Heat Mass Transfer 37 (2010) 29e33.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Numerical investigation of forced convection heat transfer through microchannels with non-Newtonian nanofluids Ali Esmaeilnejad a, *, Habib Aminfar b, Mahdieh Shafiee Neistanak c a
School of Mechanical Engineering, Iran University of Science and Technology (IUST), P.O. Box: 16846-13114, Narmak, Tehran, Iran Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran c Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Canada b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 October 2012 Received in revised form 21 July 2013 Accepted 21 July 2013 Available online
In this paper, convection heat transfer and laminar flow of nanofluids with non-Newtonian base fluid in a rectangular microchannel have been investigated numerically using two-phase mixture model. This research investigates the advantages of using nanoparticles in non-Newtonian fluids with particles size equal to 30 nm. The factor that makes nanoparticles feasible is the significant increase in rate of heat transfer within the fluids that are common in today’s industry. The power law model is used both Newtonian and non-Newtonian fluids. The flow behavior and rate of heat transfer performance of microchannel heat sink have been taken into account by looking into the effects of Al2O3 nanoparticles concentrations, Peclet number and flow behavior index. Our results demonstrate significant enhancement of heat transfer of non-Newtonian fluids using nanoparticles particularly in the entrance region. By increasing the volume fraction, higher heat transfer enhancement can be observed. The thermal resistance with Peclet number of 700 and 4% volume fraction reduces approximately 27.2% with shear thinning non-Newtonian base fluid and pressure drop will increase approximately 50.7%. Further analysis on particles type effect is also implemented with Al2O3 and CuO nanoparticles. Ó 2013 Elsevier Masson SAS. All rights reserved.
Keywords: Non-Newtonian nanofluid Two-phase mixture model Laminar forced convection Heat transfer enhancement Microchannel
1. Introduction The liquid coolants are essential for heat transfer in many engineering equipments such as electronic devices, heat exchangers and vehicles. In recent years, many attempts has been made to minimize the scales and improve the performance of instruments. Convective heat transfer in microelectromechanical systems (MEMS) has been confirmed to be an effective method for the thermal control of these devices such as microflow sensors, microchannels heat sink (MCHS), biomedical and biochemical systems [1e3]. The advantages of MCHS consist of ability to produce very high heat transfer coefficient, compactness, high surfaceto-volume ratios of microscale devices and small coolant requirements [4e6]. So far, the most common coolants in the MCHS studies have been done for Newtonian fluids such as water, ethylene glycol and so on. Heat transfer processing of non-Newtonian fluids is encountered in various industrial sectors including chemicals, petrochemicals, * Corresponding author. Tel.: þ98 912 590 6431, þ98 21 77 240 197; fax: þ98 21 77 240 488. E-mail addresses: [email protected], [email protected] (A. Esmaeilnejad). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.07.020
polymers, and pharmaceuticals [7]. High heat transfer capacity and low pumping power of some non-Newtonian fluids make them attractive as a coolant for various applications such as microchannel heat exchangers. Non-Newtonian fluids exhibit a non-linear relation between shear stress and shear rate. The simplest model of nonNewtonian fluids is the Ostwaldede Waele power law model used for intermediate ranges of the shear rate and various fluids [8e10]. Fluid flow and heat transfer problems involving non-Newtonian fluids have been reviewed by Metzner [11], Skelland [12], Cho and Hartnett [13] and among many others. Hartnett and Kostic [14] have collected the results about laminar and turbulent fluid flow and heat transfer of non-Newtonian fluids through rectangular channels with different aspect ratios. Schechter [15] obtained velocity profiles using a variational principle, for laminar flow through rectangular ducts of various aspect ratios. Wheeler and Wissler [16] applied an overrelaxation procedure to obtain more accurate velocity distributions and friction factors for aspect ratios of 0.5, 0.75 and 1. One approach to augment the convective heat transfer coefficient in the microchannel may be utilizing nanofluids as common fluids. Nanofluid is a suspension of solid nanoparticles (with diameter of 1e100 nm) in conventional liquids like water and oil. Depending on shape, size, and thermal properties of the solid nanoparticles, the thermal conductivity can be increased by about
A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
40% with low concentration (1e5% by volume) of solid nanoparticles in the mixture [17e19]. Extensive theoretical and experimental studies have been done to explain behavior of effective thermal conductivity of nanofluids. Some experimental studies [20,21] show that the measured thermal conductivity of nanofluids is much larger than the classical theoretical predictions [22]. Other experimental investigations [23,24] revealed that the thermal conductivity has not shown any anomalous enhancement and for lower volume fractions, the results agree well with the classical equations [22,25]. Many attempts have been made to formulate efficient theoretical models for the prediction of the effective thermal conductivity [26e28]. Heris et al. [29,30] studied the effects of alumina and copper oxide nanofluids on laminar heat transfer in a circular tube with considering water as a base fluid. They reported that heat transfer coefficient enhances for both nanofluids with increasing nanoparticles concentrations as well as Peclet number, and observed higher enhancement in alumina nanofluid than copper oxide. Numerical investigations on nanofluids are carried out in general using two approaches; single-phase or two-phase approach. Single-phase approach assumes that the continuum assumption is still valid for fluids with suspended nano size particles and fluid phase and nanoparticles are in thermal and hydrodynamic equilibrium. The other approach considers a two-phase model, which encloses a better description of the fluid and the solid phases. Since the solid particles sizes are classified as nanoparticles, they can easily fluidize and be approximately considered to behave as a fluid. However, because the effective properties of nanofluids are not known precisely, the numerical predictions of this approach depends on the effective physical properties. Xuan and Li [31] studied the single-phase flow and heat transfer performance of nanofluids under turbulent flow in tubes. Their experimental results showed that the convective heat transfer coefficient and the Nusselt number of nanofluids increase with the Reynolds number and the volume fraction of nanoparticles under turbulent flow. They have observed about 39% increase in Nusselt number when volume fraction was increased from 0 to 2% under same Reynolds number. The two-phase approach seems to be a better model for describing of the nanofluid heat transfer. In other words, the slip velocity between the fluid and nanoparticles might not be zero [19] due to several factors such as friction between the fluid and solid particles, Brownian forces, gravity, Brownian diffusion, sedimentation and dispersion. Behzadmehr et al. [32] studied the turbulent forced convection of a nanofluid in a circular tube by using a two-phase approach. They implemented the two-phase mixture model for the first time to study nanofluid. They examined the axial evolution of the flow field and fully developed velocity profiles at different Reynolds numbers. Their comparison with the experimental results showed that the mixture model is more precise than the single-phase model. Bianco et al. [33] numerically worked on developing laminar forced convection flow of a watereAl2O3 nanofluid in a circular tube. They found that the maximum difference in the average heat transfer coefficient between single-phase and twophase models results was approximately 11%. They concluded that heat transfer enhancement increases with the particle volume concentration as well as shear stress values. In recent years, the rapid development of engineering technologies has contributed significantly to the convective heat transfer enhancement of nanofluids with Newtonian base fluids. However, the effect of nanoparticles on the convective heat transfer in the non-Newtonian base fluids has not been investigated yet. The objective of this study is to numerically investigate the convective heat transfer coefficient of nanofluid with non-Newtonian base
77
fluid in the rectangular microchannel in laminar flow regime. Initially, heat is supplied to the silicon substrate through the heating area, then is removed by flowing non-Newtonian nanofluid through a microchannel. Three dimensional steady state flow is considered for the rectangular microchannel heated uniformly from the bottom. Throughout this research, the base fluid is a nonNewtonian Ostwaldede Waele power law model with spherical nanoparticles with the diameter of 30 nm. Two-phase mixture model approach is employed to evaluate the laminar forced convection flow by considering the temperature variable thermophysical properties. The numerical simulation results are compared with other works presented in literature for flow of non-Newtonian base fluid in microchannel. This paper is structured in the following manner. Section 2 outlining the model formulation, numerical method and simulation conditions, Section 3 discussing the numerical results, and finally, in Section 4, a summary of the main conclusions are presented. 2. Mathematical modeling 2.1. Governing equations Laminar forced convection of a nanofluid consisting of nonNewtonian base fluid with nanoparticles has been considered. In this numerical investigation, the gravitation force is considered as an important parameter. Because of symmetry and for simplicity and reduction of calculation time, only one microchannel is selected and analyzed as computational domain. It is clear that the heat sink exhibits geometrical symmetries. Due to the heat sink module chosen in this study, planes of symmetry can also be identified for the heat transfer part of the problem. With the above simplifications, only one microchannel is modeled as shown in Fig. 1b. The geometrical configuration in this simulation work is depicted schematically in Fig. 1. Due to the heat sink module chosen in this study, symmetrical planes are used for the heat transfer part of the problem. The nanofluid flows through the one rectangular microchannel with imposing of constant heat flux to the silicon substrate at the bottom part of the solid microchannel. The top surface is assumed to be adiabatic. A complete description of thermal behavior of microchannel include three dimensional conduction analysis in solid parts of microchannel and three dimensional solutions of conservation equations of non-Newtonian nanofluid under steady state conditions. For temperature profiles in solid microchannel, it is necessary to solve the conduction equation within solid microchannel with neglecting the effect of dissipation and pressure work. Heat sink performance is commonly measured by its thermal resistance that is defined as below [34].
Rth ¼
Rth ¼ ¼
Tw;max Tf ;in qw
DTmax
Tw;max Tf;in qw Tw;max Tf;out þ Tf;out Tf;in qw
¼
¼ Rcon þ Rcap
qw
¼
1 hAfin
þ
1 _ p mC (1)
where Tw,max, Tf,in, Tf,out, qw, Rcon and Rcap are the maximum temperature of surface (wall), the inlet fluid temperature, the outlet fluid temperature, the heat flux at the heating area, convective and capacitive resistance, respectively.
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A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
V$
n X
4k rk ! v k! v k ¼ V$ðrm ! v m! v m Þ þ V$
k¼1
n X
4k rk ! v dr;k ! v dr;k
k¼1
(6) In terms of mixture variable (n ¼ 2 for two-phase) the momentum equation takes the following form:
! V$ðrm ! v m! v m Þ ¼ Vpm þ V$ðsk Þ þ V$sDm þ rm g
(7)
where sm and sDm represent the average viscous stress and stress diffusion due to the phase slip, respectively. The tree stress tensors are defined as:
sm ¼
n X
4k sk
(8)
k¼1
sDm ¼
n X
4k rk vdr;k vdr;k
(9)
k¼1
where vdr,k is the diffusion velocity, i.e., the velocity of phase k relative to the center of the mixture mass
vdr;k ¼ vk vm
(10)
In non-Newtonian fluid, the so-called apparent viscosity is defined in the same manner as for Newtonian fluids, although it is no longer constant, but depends on flow conditions such as shear rate magnitude. For the simplest power law fluid model, the (apparent) viscosity is expressed as:
s ¼ f ðg_ ; r; nÞ g_
h¼
Fig. 1. (a) Schematic of the microchannel, (b) dimensions of the microchannel.
The two-phase mixture model is also employed to analyze the thermal and fluid dynamic behavior of the considered nonNewtonian nanofluids. The following formulation represents the mathematical description of the mixture model governing equations. The continuity equation for mixture model is as follow:
V$ðrm $! v mÞ ¼ 0
(2)
The mixture density and the mixture velocity for n phase are defined as: n X
rm ¼
4k rk
n 1 X
rm
4k rk ! vk
V$
4k rk ! v k! vk ¼
k¼1
þ
k¼1 n X
4k Vpk þ V$
n X
(12)
Here, r and n are flow consistency index and flow behavior index (or power law index) respectively; both are constant for a given fluid. In special case: r ¼ m ¼ constant and n ¼ 1, corresponds to a Newtonian fluid with h ¼ m ¼ r. The diffusion stress term, V$sDm, in Eq. (7) compared to the onephase equation is representing the momentum diffusion due to the relative motions. The energy equation for the mixture is defined as:
i
h
4rp Cp;p ! v p þ ð1 4Þrf Cp;f ! v f T ¼ V$ keff VT þ Se
where Se, T and keff represent thermal resources, temperature and effective thermal conductivity of nanofluid, respectively. The equation of volume fraction of phases is defined as below:
4k ðsk þ sTk Þ
k¼1
4k rk ! g
(4)
k¼1
n X
s ¼ r g_ n1 g_
V$
where vm represents the velocity of mass center. The momentum equation for mixture model (n phase) is as follow: n X
where h is the apparent viscosity of fluid and the relationship between the shear stress and shear rate for non-Newtonian base fluid according to the Ostwaldede Waele model is:
(3)
k¼1
vm ¼
(11)
V$ 4rp ! v m ¼ V$ 4rp ! v dr;p
k¼1
Using the definitions (2), (3) and (4), the first term of Eq. (5) can be written as:
(14)
The slip velocity (or relative velocity) is defined as the velocity of a secondary phase (p) relative to the velocity of the primary phase (f):
! v mf ¼ ! vp! vf (5)
(13)
(15)
The drift velocity is related to the relative velocity
4rp ! ! ! vf vp v dr;p ¼ ! v pf rm
(16)
A. Esmaeilnejad et al. / International Journal of Thermal Sciences 75 (2014) 76e86
correct velocities at outlet that are used for correcting the pressure fields. - At the lateral walls (vertical symmetry plane), symmetry condition is considered as below:
The relative velocity is determined from equation proposed by Manninen et al. [35] and equation by Schiller and Naumann [36] is used to calculate the drag function fdrag. The effective density and effective thermal diffusivity of the nanofluid at reference temperature is defined as:
rm ¼ 4rp þ ð1 4Þrf
rCp
¼ 4 rCp p þ ð1 4Þ rCp f
m
79
(17)
wc þ ww vTw / kw ¼ 0 z ¼ 2 vz
(23)
(18)
wc þ ww vTw / kw ¼ 0 z ¼ 2 vz
(24)
Thermal diffusivity of nanofluid:
aeff
k eff ¼ 4 rCp p þ 1 4 rCp f
(19)
The effective thermal conductivity of fluid has been determined by the model proposed by Hamilton and Crosser (1962).
" keff ¼
kp þ ðj 1Þkf ðj 1Þ4 kf kp kp þ ðj 1Þkf þ 4 kf kp
# kf
(20)
where j is a shape factor and equal to 3 for spherical nanoparticles. In this equation, the thermal conductivity of fluid (non-Newtonian base fluid) and particles are considered to vary with temperature. As shown by Zhang et al. [24], this model is in good agreement with the experimental results at the low value of the nanoparticles volume fraction (4 < 5%).
xþ ¼
x Dh
yþ ¼
Pr ¼
Cp;m $r vm n1 $ Dh keff
y Dh
rm $vm $Dh w f $Re ¼ 1 rs$v 2 $ h m 2
zþ ¼
pþ ¼ 1 rp$v2 m 2
z Dh
r $C
Tþ ¼
$v $D
TTin q$Dh keff
Re ¼
m h Pe ¼ Re$Pr ¼ m p;m f $Re ¼ 14 keff D ðdpÞ$ h ¼ 1dxr$v2 4 $rm $vhm $Dh Dh ¼ ð4Ach =pch Þ 2
The set of non-linear elliptical governing equations has been solved using the following boundary conditions: - At the microchannel inlet (x ¼ 0):
vz ¼ 0
T ¼ Tin
(21)
where Tin refers to the reference (inlet) condition. - At fluidewall interface:
kw
(25) (26)
In addition, no slip boundary condition is used on the microchannel walls. A fully developed condition is used at outlet boundary. In this study, the flow regime is laminar, incompressible and steady. At the exit plane, (x ¼ L) the pressure is assumed atmospheric pressure. 2.3. Numerical method The governing equations can be converted to non-dimensional form, using the following dimensionless parameters:
ð2nÞ rm $vm $Dnh
r
(27)
m
2.2. Boundary conditions
vy ¼ 0
vTw ¼ qw vy vTw ¼ 0 y ¼ H/ kw vy y ¼ 0/ kw
dpþ $Re dxþ
In this study, the fluid properties of non-Newtonian nanofluid such as thermal conductivity, density and thermal capacity are varied with temperature.
vx ¼ v0
- At the bottom of the microchannel, constant heat flux condition is imposed and upper wall is treated as an adiabatic wall.
vTw ðx; y; zÞ vT ðx; y; zÞ ¼ keff nf vs vs
where Re, Pr, Pe, r and n represent Reynolds number, Prandtl number, Peclet number, flow consistency index and flow behavior index for nanofluid with non-Newtonian base fluid, respectively. In addition, Dh is hydraulic diameter of the microchannel and it is defined as Dh ¼ (4Ach/pch), where pch and Ach is the perimeter of the microchannel cross-section and finned area used in heat transfer, respectively. This set of coupled non-linear differential equations is discretized with the finite difference method based on the control volume approach. The SIMPLE1 algorithm was introduced for the velocityepressure coupling and convection terms were estimated using the QUICK method. The discretization grid is considered to be non-uniform. In addition, structured non-uniform grids have been used to discretize the computational domain and are finer in the vicinity of the heated wall where more accurate solutions is necessary. It is important to note that most of the parameters and the results obtained from this study are non-dimensional.
(22)
where s refers to the normal direction to the surface. - At the microchannel outlet (x ¼ L), the diffusion flux in the direction normal to the exit plane is assumed to be zero for all variables. An overall mass balance correction is utilized to
2.3.1. Grid independency study In order to evaluate the effect of the number of the mesh points on the accuracy of the results, several different grid distributions
1
Semi-Implicit Method for Pressure Linked Equations.
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have been tested to ensure the calculated results are grid independent. Fig. 2 shows the local Nusselt number and nondimensional local pressure drop in fully developed region at different grids, where local Nusselt number is calculated according to the following definition:
NuðxÞ ¼
hðxÞ$Dh keff
(28)
and h(x) is defined as:
qw hðxÞ ¼ Tw ðxÞ Tb ðxÞ
Table 1 Case studies specification.
Lt (cm) Wfin (mm) Wch (mm) Wt (cm) H (mm) hch (mm) Qa (cm3/s) q (W/cm2) R (cm2 C/W)
(29) a
Tuckerman’s experiment [1] Present calculation
Case 1
Case 2
Case 3
1.4 44 56 2 533 320 4.7 181 0.110 0.114
1.4 45 55 2 430 287 6.5 277 0.113 0.115
1.4 50 50 2 458 302 8.6 790 0.090 0.087
Total flow rate through the heat sink assembly.
From the previous equation the hav is calculated as follows:
hav
1 ¼ Lch
ZLch hðxÞdx
(30)
0
and the average Nusselt number is defined as:
Nuav ¼
hav $Dh keff
(31)
As results show, a minimum number of 244,800 non-uniform grid points were necessary before a satisfactory and accurate result could be obtained. Such grid has 170, 45 and 32 nodes, respectively in x, y and z directions with highly packed grid points in the vicinity of the microchannel wall, especially at the entrance region. Hence, further increase in the number of grids does not affect the results.
2.3.2. Validation of the present code In order to demonstrate the validity and precision of the model as well as numerical procedure, comparisons with the available experimental and numerical simulation have also been done. There is a limitation for comparison of basic results because of some differences such as geometry, fluid and numerical method between present study and other related studies. Table 1 is outlining the calculated thermal resistances with the experimental results of Tuckerman [1] in microchannel, that the geometry of this numerical study is similar to their works. As it is shown, the predicted thermal resistance at x ¼ 1.33 is in good agreement with the experimental results. Table 2 is clarifying of the error percentage of thermal resistance between the obtained results and the experimental results. The maximum and minimum errors in comparison with Tuckerman’s experiment are 3.64% and 1.77% in case 1, respectively. This error is considerably low, respect to the amount of parameters. This minimal difference could be caused by using numerical method in this investigation and errors in Tuckerman’s experiments. Therefore, it is safe to say that the predicted thermal resistance is in good agreement with the experimental results. In addition to the thermal resistance, variation of friction factor and Reynolds number (f$Re) is compared with the corresponding results [15,37,38]. The results have been summarized in Table 3 for fully developed region of non-Newtonian fluid flowing through microchannel under laminar conditions. In addition, the differences between the obtained results and the experimental and numerical results of f$Re are noticeably low. The maximum and minimum errors are 2.22% and 0.16% in case of Schechter’s results [15] for n ¼ 0.7 and n ¼ 1, respectively, in comparison with present results, as presented in Table 4. The concordance between the results is good. Therefore, the numerical code is proven to be reliable and can predict forced convection flow of nanofluids with non-Newtonian base fluid through a microchannel.
3. Results and discussion Numerical simulations of flow and heat transfer for a range of Pe and particles volume fractions have been studied. Nanofluid is referred as the mixture of solid spherical alumina nanoparticles of 30 nm diameter and non-Newtonian base fluid with various flow behavior index. In this study, a microchannel heat sink with exerted constant heat flux to the silicon substrate is considered. Results Table 2 Error percentage of thermal resistance between the obtained results in comparison with Tuckerman’s experiment. Tuckerman’s experiment [1] Fig. 2. Grid independence tests.
Case 1
Case 2
Case 3
3.64%
1.77%
3.33%
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81
Table 3 Variation of f$Re for different values of n for a microchannel (aspect ratio ¼ 6.43) in fully developed region. Case studies specification
Gao and Hartnett [37] Schechter [15] Tiu et al. [38] Present calculation
f$Re n ¼ 0.7
n¼1
n ¼ 1.3
e 9.956 10.105 10.177
17.952 18.152 18.039 18.182
32.087 32.377 32.495 32.303
were carried out employing the two-phase mixture model and temperature-dependent properties. In order to show the presence impact of nanoparticles in non-Newtonian base fluid flowing through microchannel, several aspects of these fluids are examined. Fig. 3 shows the dimensionless axial velocity profiles at x ¼ 1.26 cm for Pe ¼ 680 and qw ¼ 790 W/cm2with different volume fractions (Al2O3 nanoparticles) and different flow behavior index. It can be observed that for two-phase mixture model, while properties are dependent on temperature, the dimensionless velocity profiles are nearly independent on concentration value. In fact, dimensional axial velocity increases with nanoparticles volume fractions. This arises from the fact that the physical properties of nanofluid changes with the volume fraction. Therefore, different mean velocity is needed for different nanoparticle volume fraction to keep the Peclets number constant. However, the dimensionless velocity profile shows no differences. This shows that the dimensionless velocity profile remains constant and nanoparticle volume fraction does not have significant effect on the velocity profile. In addition, increasing the value of n (by moving from a shear thinning fluid to a shear thickening fluid) in non-Newtonian base fluids, the velocity gradient of the microchannel wall decreases and consequently the maximum dimensionless axial velocity increases. Fig. 4 shows the pressure drop of non-Newtonian nanofluids using Al2O3 nanoparticles. Nanoparticles do not affect the pressure drop of non-Newtonian nanofluids in low Reynolds number and there is no noticeable increase in comparison to non-Newtonian base fluid. Besides, as the Reynolds number is increased, the pressure drop increases further, and this is more evident at higher values of Reynolds number. In other words, using nanoparticles causes higher pressure drop in high Reynolds number in comparison to lower Reynolds number. However, nanofluid with 4% solid volume fraction causes high-pressure drop especially in high Reynolds number. Examination of results shows that at a fixed Reynolds number of 730 for shear thinning fluids with n ¼ 0.7 and 4 ¼ 4%, the pressure drop could increase as much as 53%, while for shear thickening fluids, it could increase to 50.5% at a value of n ¼ 1.3. As seen in Fig. 4aec, it is observed that by increasing the value of n at fixed Reynolds number, the pressure drop increases, but this augmentation of pressure drop in low Reynolds number is higher than in high Reynolds number. In other words, with increasing Reynolds number, pressure drop receive low effect from variation of power law index. Variation of power law index plays somewhat small in the pressure drop as the Reynolds number increases. Fig. 5 shows the local convective heat transfer coefficient as a function of the non-dimensional microchannel length for three Table 4 Error percentage of f$Re between the obtained results and the experimental and numerical results.
Gao and Hartnett [37] Schechter [15] Tiu et al. [38]
n ¼ 0.7
n¼1
n ¼ 1.3
e 2.22% 0.71%
1.28% 0.16% 0.79%
0.67% 0.23% 0.59%
Fig. 3. Profiles of axial velocity at x ¼ 1.26 cm, for Pe ¼ 680, qw ¼ 790 W/cm2 and several concentration (Al2O3 nanoparticles) and power law index values: (a) 4 ¼ 0, (b) 4 ¼ 1%, 4%.
volume concentrations of Al2O3 nanoparticles at fixed Peclet number of 400. Fig. 5aec is for the quantities of flow behavior index of non-Newtonian base fluids equal to 0.7, 1 and 1.3, respectively. It can be seen that the heat transfer coefficient increases with nanoparticle concentration at a given non-dimensional axial position and Peclet number. Given the Peclet number and nanoparticle concentration, the heat transfer coefficient decreases with axial distance as expected. For the nanofluid with 1% by volume nanoparticles and a shear thinning non-Newtonian base fluid (n ¼ 0.7) the enhancement decreases from 8.11% at position x/Dh ¼ 5 to 3.28% at x/Dh ¼ 173 with Pe ¼ 400 in comparison with the case of nonNewtonian base fluid only. This difference is noticed to be higher for high volume concentration of nanoparticles. For example, for the same Peclet number and flow behavior index, the augmentation of local heat transfer coefficients with 4% volume fraction of Al2O3 nanoparticles are, respectively, 23.39% and 15.57% at position x/Dh ¼ 5 and x/Dh ¼ 173, respectively, in comparison with the case of non-Newtonian base fluid only. As it can be seen from Figs. 5 and 6, the enhancement in the local heat transfer coefficient of nanofluids becomes appreciable at higher Peclet number. Also, the entrance regime effect becomes more pronounced at the high Peclet number of 1385, as shown in Fig. 6. One important parameter in the investigating of performance of heat sinks is their thermal resistance. Thermal resistance, the main parameter of microchannel heat sink performance, is determined
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Fig. 5. Convective heat transfer coefficient as a function of the non-dimensional microchannel length for Pe ¼ 400 with two-phase mixture model (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
Fig. 4. Pressure drop of non-Newtonian nanofluids for several concentration (Al2O3 nanoparticles) and power law index values: (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
by using the maximum temperature difference of cooling fluid and the solid surface, using Eq. (1). The microchannel with lower thermal resistance shows more performance i.e. for fixed inlet fluid temperature and thermal power, the fluid has more potential to remove heat from bottom heat sink when the maximum temperature of heat sink is lower. According to Eq. (1), this leads to lower thermal resistance. The variation of the thermal resistance versus the Peclet number for various Al2O3 nanoparticles volume concentrations are shown in Fig. 7. It is observed that the thermal resistance of microchannel heat sink decreases with increasing the nanoparticle volume fractions in the non-Newtonian base fluid. This is more pronounced at lower Peclet number. In addition, with increasing Peclet number, thermal resistance decreases gradually. It is due to this fact that with increasing of Peclet number, convective heat transfer coefficient increases. Therefore, more heat transfer can be removed from microchannel heat sink and consequently the temperature of fluid increases and thermal resistance decreases. As it is seen in the figure, when the value of n increases, the thermal resistance of the microchannel heat sink rises as well. Therefore, for a proper selection of a
microchannel heat exchanger about using non-Newtonian nanofluid, it is recommended to search for a shear thinning nonNewtonian base fluid coolant at a high Peclet number. Because of wide discrepancies between the given values of thermophysical properties of nanoparticle, these properties of CuO nanoparticles used for present numerical study are measured in chemical engineering laboratory that these values are very close to values stated in literature [40,41]. In addition, physical properties of Al2O3 such as thermal conductivity and specific heat nanosized particles are used from Ref. [40]. Table 5 presents the summary of thermophysical properties of Al2O3 and CuO nanoparticles used for comparing the particle types. The base fluid for the results is an aqueous solution containing different percentages of Carboxy Methyl Cellulose (CMC) which was made by adding CMC powder to distilled water and thoroughly mixed. Nanofluids were considered by dispersing dry nanoparticles into the base liquid. In reality, nanoparticles can be dispersed in CMC solution (carboxy methylcellulose in water) using ultrasonic mixing and mechanical mixer. In this two-phase numerical study, different concentrations of CMC non-Newtonian base fluids with nanoparticles were investigated under forced convection heat transfer conditions and properties of non-Newtonian base fluid was used from Ref. [42] as shown in Table 6.
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83
Fig. 6. Convective heat transfer coefficient as a function of the non-dimensional microchannel length for Pe ¼ 1385with two-phase mixture model (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
Average heat transfer coefficient and pressure drop are reported in Table 7. Comparison is carried out for 1% and 4% volume concentrations of nanoparticles for Peclet number of 1385 and flow behavior index of 0.7. The nanofluid is a mixture of non-Newtonian base fluid with solid spherical particles of 30 nm diameter of both Al2O3 and CuO. The results clearly show the significant enhancement of average heat transfer coefficient for both nanofluids with various kinds of nanoparticles concentrations. The scattering of Al2O3 nanoparticles in non-Newtonian base fluid results in higher average heat transfer coefficient in comparison with CuO nanoparticles. However, the pressure drop using CuO nanoparticles is less than Al2O3 nanoparticles in the same non-Newtonian base fluid. The temperature profiles along the bottom wall of microchannel for qw ¼ 790 W/cm2 and fixed Peclet numbers with variable properties of non-Newtonian nanofluids using Al2O3 nanoparticles is shown in Fig. 8. The reduction of wall temperature, with respect to the non-Newtonian base fluid, increases with the x coordinate. At the exit section, the reduction of the wall temperature of microchannel for n ¼ 0.7 and Pe ¼ 400 are about 0.16% and 0.6%, in comparison with non-Newtonian base fluid, for variable properties in the case of 4 ¼ 1% and 4 ¼ 4%, respectively. Based on the results, using nanoparticles with higher Peclet number results in lower wall temperature. For example, for non-Newtonian nanofluid (n ¼ 0.7) at Pe ¼ 1385 the reduction of temperature are 0.07% and
Fig. 7. Thermal resistance of microchannel in terms of Peclet number for nonNewtonian nanofluids (with Al2O3 nanoparticles): (a) n ¼ 0.7, (b) n ¼ 1 and (c) n ¼ 1.3.
0.25%, in comparison with non-Newtonian base fluid, at the exit of microchannel for 4 ¼ 1% and 4 ¼ 4%, respectively. The results highlights the significant advantage of using nanoparticles for lowering wall temperature is particularly great at the exit of microchannel. For the nanofluid with 4% by volume nanoparticles with Pe ¼ 400 and a shear thinning non-Newtonian base fluid (n ¼ 0.7), the temperature decreases from 8.11% at position x ¼ 0.7 cm to 3.28% at x ¼ 1.4 cm in comparison with the nonNewtonian base fluid. These results have indicated that the beneficial effect of nanoparticles considerably improves the thermal properties of the mixture. It is important to note that additional effects such as gravity, drag on the particles, diffusion and Brownian forces play important roles [39]. Fig. 9 shows heat transfer coefficient enhancement for n ¼ 0.7 and qw ¼ 790 W/cm2 for constant and variable properties using two-phase mixture model with Al2O3 nanoparticles. In constant properties state, the properties of non-Newtonian nanofluid are calculated in constant fluid mean temperature. The variation of fluid temperature along the microchannel is more noticeable in low Reynolds number and high heat flux.
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Table 5 Properties of the studied particles in non-Newtonian base fluids [40]. Property
Table 7 Average heat transfer coefficients and pressure drop with two kinds of nanoparticles for Pe ¼ 1385, n ¼ 0.7 and 4 ¼ 1%, 4%.
Value Al2O3
rp [kg/m3]
3820 880 46
Cp,p [J/kg K] kp [W/m K]
CuO 6320 670 27
4 ¼ 1%
Property
2
hav (W/m K)
DP (pa)
4 ¼ 4%
Al2O3
CuO
Al2O3
CuO
57,334 211,344
56,827 207,493
63,850 273,471
61,973 246,967
Below equation supports the dependency of Re number on Tout Tin:
rnf Vm Cp;nf Ach ðTout Tin Þ ¼ qw $Abottom
(32)
As it is known, Dh ¼ (4Ach/pch) is hydraulic diameter of the microchannel where pch is the perimeter of the microchannel cross-section and thus:
Ach ¼
Dh pch 4
(33)
By submitting Eq. (33) in Eq. (32):
rnf Vm Cp;nf
Dh pch ðTout Tin Þ ¼ qw $Abottom 4
rnf Vm Dh Cp;nf
(34)
pch mnf ðTout Tin Þ ¼ qw $Abottom 4 mnf
(35)
rnf Vm Dh p Cp;nf ch mnf ðTout Tin Þ ¼ qw $Abottom mnf 4
(36)
where Re ¼ ðrnf Vm Dh =mnf Þ is Reynolds number. By submitting in the above equation:
p ReCp;nf ch mnf ðTout Tin Þ ¼ qw $Abottom 4 Tout Tin ¼
(37)
4qw $Abottom Re mnf Cp;nf pch
(38)
Therefore, according to Eq. (38), the variation of fluid temperature along the microchannel is more noticeable in low Reynolds number and high heat flux. Assumption of constant properties of nanofluid along the microchannel can result in error. According to Eq. (38), the temperature difference of nanofluid through microchannel is noticeable in low Reynolds number. In addition, increasing of temperature difference can cause more significant increase in thermal conductivity. Therefore, the difference of heat transfer coefficient between the constant and variable properties is becoming small when the Reynolds number is increased. Unfortunately, neglecting the effect of variable properties of nanofluid
Fig. 8. Effects of the Peclet number and volume concentration of Al2O3 nanoparticles on the bottom wall temperature of microchannel for qw ¼ 790 W/cm2 and n ¼ 0.7: (a) Pe ¼ 400, (b) Pe ¼ 1385.
along the microchannel leads to a major misconception about the effect of parameters. Fig. 10 shows heat transfer coefficient enhancement of nanofluid versus Peclet number at different concentrations of Al2O3 nanoparticles. It is clear that the constant Peclet number heat transfer coefficient increases with nanoparticle concentration. Increased
Table 6 Experimental rheological properties for CMC solutions, expressed as parameters of the power law model [42]. Property
Value r
n
CMC (1%) CMC (0.8%) CMC (0.5%)
0.532 0.651 0.729
1.052 0.308 0.192
Fig. 9. Enhancement of heat transfer coefficient in comparison with non-Newtonian base fluid as a function of Reynolds number for n ¼ 1 and qw ¼ 790 W/cm2 with constant and variable properties using two-phase mixture model) with Al2O3 nanoparticles).
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85
Nomenclature
Fig. 10. Heat transfer coefficient as a function of Peclet number for n ¼ 0.7 and qw ¼ 790 W/cm2 using two-phase mixture model (with Al2O3 nanoparticles).
thermal conductivity probably is not the only mechanism responsible for heat transfer enhancement, particularly at high nanoparticles concentration and high Peclet number. It seems that nanoparticles presence may increase the temperature gradient at the wall and hence augment heat transfer of nanofluid. Particles migration, clustering process due to non-uniform shear rate across the microchannel cross-section along with particle interactions and dispersion effects are also probable causes for increased heat transfer of nanofluids. Based on this figure, the heat transfer coefficient for nanofluid is greater than heat transfer coefficient for non-Newtonian base fluid and the heat transfer enhancement is higher for higher concentrations of particles. These results are also in agreement with Bianco et al. [33]. In other words, heat transfer enhancement increases with the particle volume concentration. 4. Conclusion In this paper, the hydrodynamic and thermal behaviors of nonNewtonian nanofluids flowing inside microchannel were numerically investigated in stationary conditions and for laminar forced flow. Two-phase mixture model has been employed to investigate numerically the effect of nanoparticles volume fraction on the hydrodynamic and thermal parameters of non-Newtonian base fluid with various flow behavior index. It is shown that, for a given Peclet number, despite changing the mean inlet velocity, the dimensionless velocity profile does not vary with nanoparticle volume fraction. In addition, in lower Reynolds number using nanoparticles do not affect the pressure drop of non-Newtonian nanofluids and there is no significant increase in comparison to non-Newtonian base fluid. However, as the Reynolds number is increased, the pressure drop consequently increased significantly. Results clearly indicate the beneficial effects of nanoparticles due to the presence of such particles, the thermal properties of the resulting mixture have considerably improved. Additional effects such as gravity, drag on the particles, diffusion and Brownian forces also play an important role. In general, using nanofluid with higher nanoparticle volume fraction improves the convective heat transfer coefficient, thermal resistance of microchannel and wall temperatures. Results were also compared with two kinds of nanoparticles. Using Al2O3 nanoparticles in nonNewtonian base fluid shows more increment in average heat transfer coefficient in comparison with CuO nanoparticles. However, the pressure drop in using CuO nanoparticles is less than Al2O3 nanoparticles in the same non-Newtonian base fluid. In addition, the effect of variable and constant properties along the microchannel is further investigated throughout this study.
Afin Cp Dh f g H h h k L _ m Nu n p Pe Pr pch qw Rth Re r Se T t ! v ! v dr ! v pf w
total finned area used in heat transfer (m2) specific heat (J kg1 K1) hydraulic microchannel diameter Dh ¼ (4Ach/pch) friction factor gravitational acceleration (m s2) microchannel height (see Fig. 1) convective heat transfer coefficient (W m2 K1) average heat transfer coefficient along the channel length thermal conductivity (W m1 K1) microchannel length 1 mass flow rate (kg s ) Nusselt number qw $Dh = keff $ T w Tb flow behavior index (power law index) pressure Peclet number (rm$Cp,m$vm$Dh/keff) Prandtl number (Cp,m$r$(vm/Dh)n1/keff) perimeter of the microchannel cross-section wall heat flux (W cm2) thermal resistance (cm2 C W1) ð2nÞ Reynolds number ðrm $vm $Dnh =rÞ flow consistency index thermal resource temperature wall thickness (see Fig. 1) velocity (m s1) diffusion (drift) velocity vector (m s1) slip velocity vector (m s1) wall thickness (see Fig. 1)
Greek symbols 4 particle volume concentration a thermal diffusivity (m2s1) h apparent viscosity (N s m2) r density (kg m3) s stress tensor (N m2) g_ shear rate (s1) Subscripts av average value b bulk value bottom bottom of microchannel ch microchannel eff effective f refers to base fluid i,in refers to the reference (inlet) condition k number of phase m refers to mixture out refers to outlet condition p particle th thermal w value at microchannel wall x x direction y y direction z z direction þ non-dimensionalized parameter References [1] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electron Device Lett. (1981) 126e129. EDL-2. [2] T. Kishimoto, T. Ohsaki, VLSI packaging technique using liquid-cooled channels, IEEE Trans. Compon. Hybr. (1986) 328e335. CHMT-9. [3] B.X. Wang, X.F. Peng, Experimental investigation on forced convection heat transfer through microchannels, Int. J. Heat Mass Transfer 17 (1994) 73e82.
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