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Numerical study of hydrodynamic and heat transfer of nanofluid flow in microchannels containing micromixer
International Communications in Heat and Mass Transfer 43 (2013) 146–154
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Numerical study of hydrodynamic and heat transfer of nanofluid flow in microchannels containing micromixer☆ S. Baheri Islami a,⁎, B. Dastvareh a, R. Gharraei b a b
Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Mechanical Engineering Department, Azarbaijan Shahid Madani University, Iran
a r t i c l e
i n f o
Available online 13 February 2013 Keywords: Nanofluid Microchannel Micromixer
a b s t r a c t In this study heat transfer and fluid flow of Al2O3/water nanofluid in two dimensional parallel plate microchannel without and with micromixers have been investigated for nanoparticle volume fractions of ϕ =0,ϕ =4% and base fluid Reynolds numbers of Ref =5, 20, 50. One baffle on the bottom wall and another on the top wall work as a micromixer and heat transfer enhancement device. A single-phase finite difference FORTRAN code using Projection method has been written to solve governing equations with constant wall temperature boundary condition. The effect of various parameters such as nanoparticle volume fraction, base fluid Reynolds number, baffle distance, height and order of arrangement have been studied. Results showed that the presence of baffles and also increasing the Re number and nanoparticle volume fraction increase the local and averaged heat transfer and friction coefficients. Also, the effect of nanoparticle volume fraction on heat transfer coefficient is more than the friction coefficient in most of the cases. It was found that the main mechanism of enhancing heat transfer or mixing is the recirculation zones that are created behind the baffles. The size of these zones increases with Reynolds number and baffle height. The fluid pushing toward the wall by the opposed wall baffle and reattaching of separated flow are the locations of local maximum heat transfer and friction coefficients. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Convective heat transfer in a microchannel is a very effective method for the thermal control micro electronic device because of the high surface area to volume ratio of these channels. So, the ability to remove heat from the high rate flux region becomes an important factor in designing microsystems. Gamrat et al. [1] numerically investigated the thermal entrance effect and conduction/convection coupling effects in both three dimensional and two dimensional microchannels. Zhang et al. [2] numerically analyzed the effect of roughness element with different shapes, like triangular, rectangular and semicircular on the thermal and hydrodynamic characteristics. Their results showed improved performance of heat transfer by roughness. Del Guidice et al. [3] numerically investigated the effect of viscous dissipation and temperature dependant viscosity in developing flow of fluids in straight microchannels with different cross sections and found out that these effects cannot be neglected in a wide range of operative conditions. Another approach to enhance the heat transfer in the microchannels may be utilizing nanofluids as working fluids. This can be possible because nanofluids exhibit unusual thermal and fluid properties, which ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (S.B. Islami), [email protected] (B. Dastvareh), [email protected] (R. Gharraei). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.01.002
in conjunction with microchannel systems may provide enhanced heat transfer performances. Yeul Juang et al. [4] experimentally investigated the convective heat transfer and friction factor of Al2O3/water nanofluid with various particle volume fractions in rectangular microchannels. They observed an increase in heat transfer coefficient with increase in volume fraction and Reynolds number. Akbarinia et al. [5] numerically investigated the forced convection slip and none slip nanofluid flow in two dimensional microchannels to study the effect of nanoparticle volume fraction in heat transfer enhancement. They have reported that with stabilizing the nanofluid Reynolds number, major enhancement on the Nusselt number is not due to nanoparticle concentration. Ahmed et al. [6] investigated the heat transfer enhancement and pressure drop of the copper–water nanofluid through isothermally heated corrugated channel. In their study the Reynolds number and nanoparticle volume fraction are ranged from 100 to 1000 and from 0% to 5% respectively. The results demonstrated that the local Nusselt number is higher in the converging section than in the diverging section along the wall. Tahir and Mital [7] numerically investigated the developing laminar forced convection flow of Al2O3/water nanofluid in a circular tube under uniform heat flux. They studied the effect of Reynolds number, volume fraction of nanoparticles and particle diameter by using discrete phase modeling (DPM). A very different application of nanofluids could be in modern medicine, where for example nanodrugs are mixed in microchannels for controlled delivery with bio-MEMS [8]. In such applications (for example, biological processing, lab on the chips, micro-reactors and fuel cells)
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
2. Geometrical configuration
Nomenclature Cp d k NuP Nu
specific heat (J kg −1 K −1) nanoparticle diameter (m) thermal conductivity (W m −1 K −1) peripheral-averaged Nusselt number of microchannel cross section wL Nusselt number ¼ hL ¼ ðT w q−T Þk k f
b
f
S
Po⁎
averaged Nusselt number ¼ 1S ∫0 NuP dx pressure (Pa) Poiseuille number (f Ref) peripheral-averaged of Poiseuille number of microchannel cross section non-dimensional wall shear stress = ρ2τu w 2 Ref
Po q Ref Renf T u v x y
average Poiseuille number = 1S ∫0 PoP dx heat flux fluid Reynolds number = ρuμin L nanofluid Reynolds number = ρnfμuin L nf temperature (K) velocity in the streamwise direction (m s −1) velocity in the normal direction (m s −1) coordinate in the streamwise direction (m) coordinate in the normal direction (m)
Nu p Po PoP
147
S
f
in
Greek letters ϕ nanoparticle volume fraction μ dynamic viscosity (N s m −2) ρ density (kg m −3)
Subscripts B bottom wall f fluid in inlet condition nf nanofluid p solid nanoparticles T top wall w the wall
rapid and complete mixing of fluid is required. Microchannel flows, due to very low flow rate, are characterized by very low Reynolds number. Owing to the predominantly laminar flow, it is difficult to achieve effective mixing fluids. If the mixing is obtained primarily by a diffusion mechanism then fast mixing becomes impossible. Hence microfluidic mixing is a very challenging problem because it requires fast and efficient mixing of low diffusivity fluids [9]. In general, micromixers are classified into two types: active and passive. In order to achieve rapid mixing in passive micromixers, obstacle structures were inserted into microchannels to enhance the advection effect via splitting, stretching, breaking and folding of liquid flows. Afrooz Alam and Kim [10] numerically investigated the mixing of fluids in a microchannel with grooves in its side walls and found out that it has better mixing performance than smooth channel at Re> 10. Chung et al. [11] designed, fabricated and simulated a passive micromixer which contains some baffles with different arrangement. In the present study the effect of baffled micromixer on the flow structure and heat transfer of single phase nanofluid in microchannel has been investigated. The effect of various geometrical and flow parameters such as height and different arrangement of baffles, Reynolds number and nanoparticle volume fraction, etc. have been studied.
The geometrical configuration of the considered problem has been shown in Fig. 1. The microchannel consists of two parallel plates with the distance of L and the length of S. Two baffles with the heights of e1 and e2 are placed inside the channel. The distances of the first and second baffles from the beginning of the channel are sb1 and sb2, respectively. The baffles are assumed adiabatic with zero thickness in the numerical simulation. The steady, laminar flow of nanofluid, enters the constant wall microchannel with hydrodynamically fully developed velocity and uniform temperature. 3. Governing equations and boundary conditions Under the assumptions of ultra-fine particles (b 100 nm) and no slip velocity between the discontinuous phase of the nanoparticles and the continuous liquid and the local thermal equilibrium between them, particle–liquid mixture may be considered as a conventional singlephase pure fluid [11,12]. Also, physical properties of the nanofluid are assumed constant and are evaluated at the reference state corresponding to the fluid inlet temperature. Under the above conditions, the corresponding non-dimensional governing equations are written as follows: ∂U ∂V þ ¼0 ∂X ∂Y
ð1Þ !
∂U ∂U ∂P 1 ρf μ nf þV ¼− þ U ∂X ∂Y ∂X Ref ρnf μ f
∂2 U ∂2 U þ ∂X 2 ∂Y 2
∂V ∂V ∂P 1 ρf μ nf U þV ¼− þ ∂X ∂Y ∂Y Ref ρnf μ f
∂ V ∂ V þ ∂X 2 ∂Y 2
ρC p k ∂θ ∂θ 1 f nf U þV ¼ kf ∂X ∂Y Ref Prf ρC
! ∂2 θ ∂2 θ þ : ∂X 2 ∂Y 2
p
2
2
ð2Þ
! ð3Þ
ð4Þ
nf
The non-dimensional parameters used in the above equations have been defined as follows: x y u v T−T w p ;V ¼ ;θ ¼ ;P ¼ X ¼ ;Y ¼ ;U ¼ L L uin uin T in −T w ρf u2in μ f C pf ρu L Ref ¼ f in ; Prf ¼ : μf kf
ð5Þ
The used boundary conditions to solve the Eqs. (1) to (4) have been given in Fig. 1. 4. Physical properties of the nanofluids The nanofluid in this study is composed of water and 36 nm particles of Al2O3. The physical properties of the nanofluid in the above equations can be obtained as follows. θ = 0 , U=0 , V=0
e2
sb2
U=6Y(1-Y) θ =1
L sb1
Y
e1
X
S=13L Fig. 1. Geometrical configuration and boundary conditions.
∂U =0 ∂X ∂V =0 ∂X ∂θ =0 ∂X
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1.6
According to [14], viscosity ratio ( μnf ) of the Al2O3/water nanofluid f does not change significantly with temperature for particle volume fractions below 4%.
1.4 1.2
2 μ nf ¼ 1 þ 2:5ϕ þ 150ϕ μ f
U
1 0.8
For calculating the thermal conductivity of the nanofluid, an experimental correlation for the thermal conductivity of Al2O3 nanofluids as a function of nanoparticle size and temperature is used [15].
0.6 400*40 300*30 100*10 Analytical
0.4
knf 0:7460 df ¼ 1 þ 64:7ϕ kf dp
0.2 0
0
0.2
0.4
ð8Þ
0.6
0.8
!0:3690 kp 0:7476 0:9955 1:2321 Pr Re kf
ð9Þ
1
Y
The Prandtl number (Pr) and the Reynolds number (Re) are respectively defined as:
Fig. 2. Fully developed non-dimensional velocity distribution in microchannel without baffle, Ref and ϕ = 0.
Pr ¼
25 Akbariniaetal.[5] presentstudy
η ; ρf α f
Re ¼
ρf K B T 3πη2 λf
ð10Þ
where λf is molecular mean free path of water (λf = 0.17 nm), KB is the Boltzmann constant (KB = 1.3807 × 10 −23 J/K) and η is calculated by the following equation:
20
15 B
Nu
η ¼ A:10T−C ;
−5
A ¼ 2:414 10
;
B ¼ 247:8;
C ¼ 140:
ð11Þ
10 5. Numerical method and meshing
5
0
0
1
2
3
In this study, the finite difference method of Projection has been used which has been presented by Chorin [16] and Temam [17], independently. The explicit edition of this method has been given by Fortin et al. [18]. This explicit method is multistage with first order accuracy for time and second order for space. In the first stage the velocity vector, in a middle time step is calculated explicitly, with the omitting of pressure gradient from momentum equations and in the second stage the velocity vector is corrected. It can be shown that the Newman condition for pressure is valid for all boundaries [19]. Since the fluid flow is assumed steady in this work, the above equations have been solved iteratively, until the time derivation term reaches to the order of 10 −4. The staggered marker-and-cell (MAC) mesh [20] has been used. The grid is uniform in both directions but grid size in x direction can differ from that of y direction. The grid numbers have been chosen according to the stability conditions and independency of results from grid numbers.
4
X Fig. 3. Validation of Nu number variation in streamwise direction, Renf = 6.9, ϕ = 5%.
In order to compute the density and specific heat of a classical twophase mixture, Eqs. (6) and (7) are used [13]: ρnf ¼ ð1−ϕÞρf þ ϕρp ¼ ð1−ϕÞ ρC p þ ϕ ρC p : ρC p nf
f
ð6Þ ð7Þ
p
An empirical correlation, based on the experimental data of Nguyen et al. [14], is used for the viscosity of the nanofluid as shown in Eq. (8).
(a)
0
1
2
3
(b)
4
5
6
0
1
2
X
0
1
2
3
X
3
(c)
4
5
6
0
1
2
X
4
5
6
0
1
2
3
X
3
4
5
6
4
5
6
X
4
5
6
0
1
2
3
X
Fig. 4. The effect of Ref on flow pattern, ϕ = 0 (top row) ϕ = 0.04 (bottom row), sb1 = L, sb2 = 3L, e1 = e2 = 0.66L; a) Ref = 5; b) Ref = 20; c) Ref = 50.
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
149
120 15
110 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
100 90 80
10
70
Nu
Po*
Re f =5
Re f =5
60 50 40
5
30 20 10 0
0
0
2
4
6
8
10
12
-10
14
0
2
4
6
X
8
10
12
14
X 160
20 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
140 120
15 100
Po*
Nu
Re f =20 10
Re f =20
80 60 40
5
20 0
0
0
2
4
6
8
10
12
-20
14
0
2
4
6
X
8
10
12
14
X 240
40 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
30
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
180
20
Re f =50
Po*
Nu
120
Re f =50
60 10
0
0
0
2
4
6
8
10
12
14
X
-60
0
2
4
6
8
10
12
14
X
Fig. 5. The effect of Ref on Nu number (left) and Po⁎ number (right), sb1 = L, sb2 = 3L, e1 = e2 = 0.66L (vertical dash-dot lines show the baffle location and curves without symbol show the microchannel without baffles).
6. Results 6.1. Grid independency test and validation All of the results are tested to be independent of grid numbers and one example of this test can be seen in Fig. 2. This figure shows
the fully developed non-dimensional velocity distribution in the wall normal direction for the microchannel without baffles. It should be noted that uniform velocity profile boundary condition has been used in the channel inlet in this case and development of the velocity profile has been simulated. The grid of 30 × 300 agrees well with theoretical results but using finer grid of 40× 400 does not have
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S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
Table 1 The effect of Ref on averaged Nu and Po numbers, sb1 = L, sb2 = 3L, e1 = e2 = 0.66L. Ref = 5
Nu Po Nu (without baffle) Po (without baffle)
Ref = 20
Ref = 50
ϕ= 0
ϕ = 0.04
ϕ=0
ϕ= 0.04
ϕ=0
ϕ = 0.04
4.47 14.47 4.30 12.29
5.72 19.40 5.55 16.46
5.15 16.41 4.67 12.28
6.61 21.34 6.06 16.46
6.49 20.54 5.40 12.29
8.11 25.53 6.87 16.47
considerable effect on the improvement of results. Therefore, the grid number of 40× 400 has been chosen for the above case. Also Fig. 2 shows the excellent agreement between theoretical and present numerical results. In order to validate the heat transfer characteristics of the flow, the local Nu number for Al2O3/water nanofluid flow with Renf = 6.9, uin = 0.0502 m s−1 and ϕ = 5% through microchannels has been calculated and compared with the results of Akbarinia et al. [5]. They have used empirical correlation presented by Maiga et al. [21] for dynamic viscosity of Al2O3/water nanofluid: 2 μ nf ¼ 1 þ 7:3ϕ þ 123ϕ μ f :
ð12Þ
Therefore, the above model has been used in this validation. Fig. 3 shows that the agreement between results is good.
6.2. The effect of baffles The effect of baffles on flow pattern has been shown in Fig. 4 for various Reynolds numbers. There is a recirculation zone downstream of baffles. The size of this zone increases with Reynolds number and decreases with ϕ. For a given Ref, the viscosity of the nanofluid increases with increasing the nanoparticle volume fraction and the size of recirculation zone decreases. As the Reynolds number increases, the vortex behind the bottom baffle forces the flow toward the wall stronger than that of the lower Reynolds numbers and creates a small recirculation zone upstream of the top baffle (Fig. 4c). It can be predicted that the above flow pattern would enhance the heat transfer and mixing rate of the flow. Fig. 5 shows the variation of the local Nusselt number (Nu) and non-dimensional wall shear stress (Po⁎) along the channel with and without baffles for Ref =5, 20, 50 and ϕ = 0%, 4%. The height of the baffles and their location from the beginning of the channel are e1 = e2 = 0.66L and sb1 = L, sb2 = 3L, respectively. The definition of Po⁎ is exactly similar to the Poiseuille number except that the sign of velocity gradient in τw definition has been retained. Therefore the negative values exist in separated zones. The negative values in Fig. 5 show the recirculation zones, and sign changing of Po⁎ indicates the reattachment points. As the Re number increases, the magnitudes of friction coefficient increase. It is clear that the presence of baffles increases the local heat transfer and friction coefficients. This influence is increased with increasing the
360
70
320 60
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
40
240
Po*_bottom
Nu_bottom
50
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
280
30
200 160 120 80
20
40
10
0 -40
0 0
2
4
6
8
10
12
-80
14
0
2
4
6
X
10
12
14
240
30 sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
20
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
180
Po*_Top
25
Nu_Top
8
X
15
120
60
10 0
5 0
0
2
4
6
8
X
10
12
14
-60
0
2
4
6
8
10
12
14
X
Fig. 6. The effect of the distance between baffles on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref = 50, sb1 = L, e1 = e2 = 0.66L (vertical dash-dot lines show the baffles location).
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
sb2=2L
0
1
0
2
1
3
2
4
sb2=3L
5
6
7
8
0
1
2
3
4
X
sb2=5L
sb2=6L
4
5
6
7
8
0
1
2
3
sb2=4L
5
X
3
151
7
8
0
1
2
3
4
5
6
7
8
X
4
X
6
5
6
7
8
X Fig. 7. The effect of the distance between baffles on flow pattern, ϕ = 0.04, Ref = 50, sb1= L, e1 = e2 = 0.66L.
6.3. The effect of the distance between baffles
Table 2 The effect of the distance between baffles on averaged Nu and Po numbers, Ref = 50, sb1 = L, e1 = e2 = 0.66L.
Nu Po
sb2= 2L
sb2 = 3L
sb2 = 4L
sb2= 5L
sb2 = 6L
ϕ= 0
ϕ=0
ϕ=0
ϕ= 0
ϕ=0
ϕ= 0.04
6.86 8.41 21.68 28.84
ϕ= 0.04
6.49 8.11 20.54 25.53
ϕ= 0.04
6.38 8.02 20.50 25.72
ϕ= 0.04
6.37 8.02 20.55 26.29
Fig. 6 shows the variation of local Nu and Po⁎ numbers along the channel with five different locations for the top baffle (sb2 = 2L, 3L, 4L, 5L, 6L). In all of the cases the location of the first baffle is constant, sb1 = L. The height of baffles is e1 = e2 = 0.66L. Fig. 6 illustrates that the value of Nu and Po⁎ numbers in the location of the second baffle and reattachment points first decreases and then is almost constant at both volume fractions. Presence of local minimum or maximum values in Nu and Po⁎ numbers for sb2 = 4L, 5L, 6L reveals that the flow is reattached. It can be concluded that when the distance between baffles is not enough to occur reattachment, increasing the baffle distance decreases the Nu and Po⁎
ϕ= 0.04
6.37 8.02 20.48 26.38
Re number. The Nusselt number has the highest value at the channel entrance, because of the high temperature gradients, and decreases in streamwise direction. By considering the bottom wall in Fig. 5, it can be seen that the Nu number decreases from the entrance to the bottom baffle location at X = 1. Then the recirculation of flow behind the baffle increases the heat transfer. There is a local maximum in Nu and Po⁎ numbers at X = 2 in Fig. 5 (Ref = 5), which is approximately the reattachment point of baffle vortex. The increase of heat transfer and friction after this point is because of the presence of the upper wall baffle. It pushes the flow toward the bottom wall and enhances the heat transfer and friction coefficients. As the Reynolds number increases, the recirculation zone of the bottom baffle becomes larger and the distance between the reattachment point and the top baffle decreases. So, the effect of reattachment point becomes weaker than the presence of the top baffle and the mentioned local maximum disappears in Fig. 5 (Ref = 50). The behavior of flow on the top wall is similar to the bottom wall except that there is a small recirculation zone at Re= 50. The Nu and Po⁎ number variation at X = 2.5–3 is because of the presence of this zone (Fig. 5 (Ref =50)). Also, the use of nanofluids enhances the heat transfer in comparison with the fluid without the nanoparticles. It is clear from Fig. 5 that the effect of nanoparticle volume fraction on friction coefficients is considerable only on the baffle location. The effect of nanoparticle volume fraction on the local Nusselt number becomes lower with the increase of the Reynolds number. Averaged Nu and Po numbers are shown in Table 1. This table shows that the baffles increase the heat transfer and friction coefficients. As mentioned above, it is clear that adding of the nanoparticles and using baffles enhance the averaged heat transfer and friction coefficients.
e1=e2=0.33L
0
1
2
3
4
X
numbers. When reattachment occurs between the baffles, variation of the baffle distance does not change the heat transfer or friction coefficients. The effect of the baffle distance on recirculation zones and reattachment happening is obviously shown in Fig. 7. For sb2 = 2L and 3L on the bottom wall the location of reattachment point and top baffle effect is coincident on each other approximately, so there is not any local minimum or maximum values in Nu and Po⁎ numbers in Fig. 6. But for higher sb2 values the effect of the top wall on the bottom wall becomes considerable. Also, Fig. 7 indicates that as the distance between the baffles is low, the flow after the first baffle does not have enough space to develop and a small recirculation zone in front of the top baffle creates. For sb2 values higher than 4L, this zone disappears. Averaged Nu and Po is shown in Table 2 for various baffle distance values. It is clear from the table values, that the effect of the baffle distance on heat transfer coefficient is more considerable than that of the friction coefficient. As mentioned above, increasing the baffle distance to more than 3L does not have any effect on flow and heat transfer parameters. 6.4. The effect of the baffle height Three pairs of baffles with the heights of e1 = e2 = 0.33L, e1 = e2= 0.5L and e1 = e2 =0.66L have been investigated. The location of baffles is constant (sb1= L, sb2 = 3L) and Ref = 50. The flow pattern
e1=e2=0.5L
5
6
7
0
1
2
3
4
e1=e2=0.66L
5
6
7
0
1
X Fig. 8. The effect of the baffle height on flow pattern ϕ = 0.04, sb1 = L, sb2 = 3L, Ref = 50.
2
3
4
X
5
6
7
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S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
240
35 30
e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
Po*_bottom
Nu_bottom
25
e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
180
20 15
120
60
10 0 5 0
0
2
4
6
8
10
12
-60
14
0
2
4
6
X
10
12
14
240
30
200
e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
25
e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
160
Po*_Top
20
Nu_Top
8
X
15
120 80
10 40 5
0
0
0
2
4
6
8
10
12
14
-40
0
2
X
4
6
8
10
12
14
X
Fig. 9. The effect of the baffle height on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref =50, sb1=L, sb2=3L (vertical dash-dot lines show the baffle location).
in Fig. 8 illustrates that the size of the recirculation zone increases with baffle height. Also, for the baffle heights greater than 0.33L, the inclination of streamlines toward the bottom wall increases and a small recirculation zone starts to create in front of the top baffle. Fig. 9 shows the heat transfer and friction coefficient distribution for various baffle heights on the bottom and top walls. It is obvious that the heat transfer and friction coefficients have been increased with baffle height and volume fraction of the nanoparticles. The location of the small recirculation zone is clear in Nu and Po⁎ distribution of the top wall at X = 2.5–3, for e1= e2 = 0.66L. Also, the figure shows that the rate of increasing Nu and Po⁎ with baffle height, increases. Averaged Nu and Po are shown in Table 3 for various baffle heights. This table shows that the heat transfer improving effect of nanofluids is more evident at high baffle heights. Table 3 The effect of the baffle height on averaged Nu and Po numbers, Ref = 50, sb1 = L, sb2 = 3L.
Nu Po
e1 = e2 = 0.33L
e1 = e2 = 0.5L
e1 = e2= 0.66L
ϕ=0
ϕ = 0.04
ϕ=0
ϕ = 0.04
ϕ=0
ϕ = 0.04
5.40 12.07
6.87 16.26
5.73 14.35
7.28 18.87
6.49 20.54
8.11 25.53
6.5. The effect of non-equal baffle heights Two baffle heights of 0.33L and 0.66L have been chosen in this section and the order of arrangement of these baffles has been studied. In all of the cases sb1 = L, sb2 = 3L and Ref = 50. It is clear from Fig. 10 that for the arrangement of “small baffle on the bottom wall–large baffle on the top wall”, maximum Nu and Po⁎ numbers are higher for the bottom wall and for the reverse arrangement, maximum Nu and Po⁎ numbers are higher for the top wall. It is dependent on the location of the large baffle, because the large baffle pushes the flow toward the opposed wall. This is the location of maximum heat transfer and friction coefficients. Also, Fig. 10 shows that the effect of nanoparticle volume fraction is more considerable at the location of the second baffle than the first one. 7. Conclusions In this study the heat transfer and fluid flow of Al2O3/water nanofluid in two dimensional parallel plate microchannel without and with micromixers have been investigated for nanoparticle volume fractions of ϕ=0,ϕ=4% and base fluid Reynolds numbers of Ref =5, 20, 50. Five baffle distances, three baffle heights and two non-equal baffle heights were chosen and studied. Results show that there is a recirculation zone
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
250
30 e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
25
e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
200 150
20
Po*_bottom
Nu_bottom
153
15
100
10
50
5
0
0
0
2
4
6
8
10
12
-50
14
0
2
4
6
X
8
10
12
14
X
30
240 e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
25
e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
200 160
Po*_Top
Nu_Top
20 15
120 80
10 40 5 0
0
0
2
4
6
8
10
12
14
X
-40
0
2
4
6
8
10
12
14
X
Fig. 10. The effect of non-equal baffle heights on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref =50, sb1=L, sb2=3L (vertical dash-dot lines show the baffle location).
downstream of the baffles. The size of this zone increases with Reynolds number and decreases with ϕ. It is observed that the influence of nanoparticle volume fraction on vortex size is weaker at low Re numbers. It is clear that the presence of baffles increases the local heat transfer and friction coefficients. This influence is increased with increasing the Re number. As the Reynolds number increases, the recirculation zone of the bottom baffle becomes larger and the distance between the reattachment point and the top baffle decreases. When the distance between baffles is not enough to occur reattachment, increasing baffle distance decreases the Nu and Po⁎ numbers. When reattachment occurs between the baffles, variation of baffle distance does not change the heat transfer or friction coefficients. The size of the recirculation zone increases with baffle height. The heat transfer and friction coefficients have been increased with baffle height and volume fraction of the nanoparticles. For the arrangement of “small baffle on the bottom wall–large baffle on the top wall”, maximum Nu and Po⁎ numbers are higher for the bottom wall and for the reverse arrangement, maximum Nu and Po⁎ numbers are higher for the top wall.
References [1] G. Gamrat, M. Favre-Marinet, D. Asendrych, Conduction and entrance effects on laminar liquid flow and heat transfer in rectangular microchannels, International Journal of Heat and Mass Transfer 48 (14) (2005) 2943–2954.
[2] C. Zhang, C.Y. Chen, M. Shi, Effects of roughness elements on laminar flow and heat transfer in microchannels, Chemical Engineering and Processing: Process Intensification 49 (11) (2010) 1188–1192. [3] S. Del Giudice, C. Nonino, S. Savino, Effects of viscous dissipation and temperature dependent viscosity in thermally and simultaneously developing laminar flows in microchannels, International journal of heat and fluid flow 28 (1) (2007) 15–27. [4] J.Y. Jung, H.S. Oh, H.Y. Kwak, Forced convective heat transfer of nanofluids in microchannels, International Journal of Heat and Mass Transfer 52 (1) (2009) 466–472. [5] A. Akbarinia, M. Abdolzadeh, R. Laur, Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes, Applied Thermal Engineering 31 (4) (2010) 556–565. [6] M.A. Ahmed, N.H. Shuaib, M.Z. Yusoff, A.H. Al-Falahi, Numerical investigations of flow and heat transfer enhancement in a corrugated channel using nanofluid, International Communications in Heat and Mass Transfer 38 (10) (2011) 1368–1375. [7] S. Tahir, M. Mital, Numerical investigation of laminar nanofluid developing flow and heat transfer in a circular channel, Applied Thermal Engineering 39 (2012) 8–14. [8] J. Li, Computational analysis of nanofluid flow in microchannels with applications to micro-heat sinks and bio-MEMS. PhD thesis, Graduate Faculty of North Carolina State University, 2008. [9] C.Y. Hsieh, A.S. Yang, Mixing enhancement of a passive micromixer by applying boundary protrusion structures, Advanced Materials Research 74 (2009) 77–80. [10] A. Alam, K.Y. Kim, Analysis of mixing in a curved microchannel with rectangular grooves, Chemical Engineering Journal 181–182 (2012) 708–716. [11] C.K. Chung, C.Y. Wu, T.R. Shih, C.F. Wu, B.H. Wu, Design and simulation of a novel micro-mixer with baffles and side-wall injection into the main channel, 1st IEEE International Conference on Nano/Micro Engineered and Molecular Systems, 2006. [12] Y. Yang, Z.G. Zhang, E.A. Grulke, W.B. Anderson, G. Wu, Heat transfer properties of nanoparticle-in-fluid dispersions (nanofluids) in laminar flow, International Journal of Heat and Mass Transfer 48 (6) (2005) 1107–1116. [13] B.C. Pak, Y.I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Experimental heat transfer: An International Journal 11 (2) (1998) 151–170.
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[14] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare, S. Boucher, H. Angue Mintsa, Temperature and particle-size dependent viscosity data for water-based nanofluids—hysteresis phenomenon, International journal of heat and fluid flow 28 (6) (2007) 1492–1506. [15] 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, Applied Physics Letters 87 (15) (2005) 153107. [16] A.J. Chorin, Numerical solution of the Navier–Stokes equations, Mathematics of Computation 22 (104) (1968) 745–762. [17] R. Temam, Sur l'approximation de la solution des equations de Navier–Stokes par la methode des pas fractionnaires (II), Archive for Rational Mechanics and Analysis 33 (5) (1969) 377–385.
[18] M. Fortin, R. Peyret, R. Temam, Calcul des Ecoulements d'un Fluide Visqueux Incompressibile, Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics Lecture Notes in Physics, 1971, pp. 336–342. [19] R. Peyret, T.D. Taylor, Computational Methods for Fluid Flow, Springer-Verlag, 1983. [20] F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of fluids 8 (1965) 2182. [21] S.E.B. Maiga, C.T. Nguyen, N. Galanis, Gilles Roy, Heat transfer behaviours of nanofluids in a uniformly heated tube, Superlattices and Microstructures 35 (3) (2004) 543–557.
Contents lists available at SciVerse ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Numerical study of hydrodynamic and heat transfer of nanofluid flow in microchannels containing micromixer☆ S. Baheri Islami a,⁎, B. Dastvareh a, R. Gharraei b a b
Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Mechanical Engineering Department, Azarbaijan Shahid Madani University, Iran
a r t i c l e
i n f o
Available online 13 February 2013 Keywords: Nanofluid Microchannel Micromixer
a b s t r a c t In this study heat transfer and fluid flow of Al2O3/water nanofluid in two dimensional parallel plate microchannel without and with micromixers have been investigated for nanoparticle volume fractions of ϕ =0,ϕ =4% and base fluid Reynolds numbers of Ref =5, 20, 50. One baffle on the bottom wall and another on the top wall work as a micromixer and heat transfer enhancement device. A single-phase finite difference FORTRAN code using Projection method has been written to solve governing equations with constant wall temperature boundary condition. The effect of various parameters such as nanoparticle volume fraction, base fluid Reynolds number, baffle distance, height and order of arrangement have been studied. Results showed that the presence of baffles and also increasing the Re number and nanoparticle volume fraction increase the local and averaged heat transfer and friction coefficients. Also, the effect of nanoparticle volume fraction on heat transfer coefficient is more than the friction coefficient in most of the cases. It was found that the main mechanism of enhancing heat transfer or mixing is the recirculation zones that are created behind the baffles. The size of these zones increases with Reynolds number and baffle height. The fluid pushing toward the wall by the opposed wall baffle and reattaching of separated flow are the locations of local maximum heat transfer and friction coefficients. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction Convective heat transfer in a microchannel is a very effective method for the thermal control micro electronic device because of the high surface area to volume ratio of these channels. So, the ability to remove heat from the high rate flux region becomes an important factor in designing microsystems. Gamrat et al. [1] numerically investigated the thermal entrance effect and conduction/convection coupling effects in both three dimensional and two dimensional microchannels. Zhang et al. [2] numerically analyzed the effect of roughness element with different shapes, like triangular, rectangular and semicircular on the thermal and hydrodynamic characteristics. Their results showed improved performance of heat transfer by roughness. Del Guidice et al. [3] numerically investigated the effect of viscous dissipation and temperature dependant viscosity in developing flow of fluids in straight microchannels with different cross sections and found out that these effects cannot be neglected in a wide range of operative conditions. Another approach to enhance the heat transfer in the microchannels may be utilizing nanofluids as working fluids. This can be possible because nanofluids exhibit unusual thermal and fluid properties, which ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (S.B. Islami), [email protected] (B. Dastvareh), [email protected] (R. Gharraei). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.01.002
in conjunction with microchannel systems may provide enhanced heat transfer performances. Yeul Juang et al. [4] experimentally investigated the convective heat transfer and friction factor of Al2O3/water nanofluid with various particle volume fractions in rectangular microchannels. They observed an increase in heat transfer coefficient with increase in volume fraction and Reynolds number. Akbarinia et al. [5] numerically investigated the forced convection slip and none slip nanofluid flow in two dimensional microchannels to study the effect of nanoparticle volume fraction in heat transfer enhancement. They have reported that with stabilizing the nanofluid Reynolds number, major enhancement on the Nusselt number is not due to nanoparticle concentration. Ahmed et al. [6] investigated the heat transfer enhancement and pressure drop of the copper–water nanofluid through isothermally heated corrugated channel. In their study the Reynolds number and nanoparticle volume fraction are ranged from 100 to 1000 and from 0% to 5% respectively. The results demonstrated that the local Nusselt number is higher in the converging section than in the diverging section along the wall. Tahir and Mital [7] numerically investigated the developing laminar forced convection flow of Al2O3/water nanofluid in a circular tube under uniform heat flux. They studied the effect of Reynolds number, volume fraction of nanoparticles and particle diameter by using discrete phase modeling (DPM). A very different application of nanofluids could be in modern medicine, where for example nanodrugs are mixed in microchannels for controlled delivery with bio-MEMS [8]. In such applications (for example, biological processing, lab on the chips, micro-reactors and fuel cells)
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
2. Geometrical configuration
Nomenclature Cp d k NuP Nu
specific heat (J kg −1 K −1) nanoparticle diameter (m) thermal conductivity (W m −1 K −1) peripheral-averaged Nusselt number of microchannel cross section wL Nusselt number ¼ hL ¼ ðT w q−T Þk k f
b
f
S
Po⁎
averaged Nusselt number ¼ 1S ∫0 NuP dx pressure (Pa) Poiseuille number (f Ref) peripheral-averaged of Poiseuille number of microchannel cross section non-dimensional wall shear stress = ρ2τu w 2 Ref
Po q Ref Renf T u v x y
average Poiseuille number = 1S ∫0 PoP dx heat flux fluid Reynolds number = ρuμin L nanofluid Reynolds number = ρnfμuin L nf temperature (K) velocity in the streamwise direction (m s −1) velocity in the normal direction (m s −1) coordinate in the streamwise direction (m) coordinate in the normal direction (m)
Nu p Po PoP
147
S
f
in
Greek letters ϕ nanoparticle volume fraction μ dynamic viscosity (N s m −2) ρ density (kg m −3)
Subscripts B bottom wall f fluid in inlet condition nf nanofluid p solid nanoparticles T top wall w the wall
rapid and complete mixing of fluid is required. Microchannel flows, due to very low flow rate, are characterized by very low Reynolds number. Owing to the predominantly laminar flow, it is difficult to achieve effective mixing fluids. If the mixing is obtained primarily by a diffusion mechanism then fast mixing becomes impossible. Hence microfluidic mixing is a very challenging problem because it requires fast and efficient mixing of low diffusivity fluids [9]. In general, micromixers are classified into two types: active and passive. In order to achieve rapid mixing in passive micromixers, obstacle structures were inserted into microchannels to enhance the advection effect via splitting, stretching, breaking and folding of liquid flows. Afrooz Alam and Kim [10] numerically investigated the mixing of fluids in a microchannel with grooves in its side walls and found out that it has better mixing performance than smooth channel at Re> 10. Chung et al. [11] designed, fabricated and simulated a passive micromixer which contains some baffles with different arrangement. In the present study the effect of baffled micromixer on the flow structure and heat transfer of single phase nanofluid in microchannel has been investigated. The effect of various geometrical and flow parameters such as height and different arrangement of baffles, Reynolds number and nanoparticle volume fraction, etc. have been studied.
The geometrical configuration of the considered problem has been shown in Fig. 1. The microchannel consists of two parallel plates with the distance of L and the length of S. Two baffles with the heights of e1 and e2 are placed inside the channel. The distances of the first and second baffles from the beginning of the channel are sb1 and sb2, respectively. The baffles are assumed adiabatic with zero thickness in the numerical simulation. The steady, laminar flow of nanofluid, enters the constant wall microchannel with hydrodynamically fully developed velocity and uniform temperature. 3. Governing equations and boundary conditions Under the assumptions of ultra-fine particles (b 100 nm) and no slip velocity between the discontinuous phase of the nanoparticles and the continuous liquid and the local thermal equilibrium between them, particle–liquid mixture may be considered as a conventional singlephase pure fluid [11,12]. Also, physical properties of the nanofluid are assumed constant and are evaluated at the reference state corresponding to the fluid inlet temperature. Under the above conditions, the corresponding non-dimensional governing equations are written as follows: ∂U ∂V þ ¼0 ∂X ∂Y
ð1Þ !
∂U ∂U ∂P 1 ρf μ nf þV ¼− þ U ∂X ∂Y ∂X Ref ρnf μ f
∂2 U ∂2 U þ ∂X 2 ∂Y 2
∂V ∂V ∂P 1 ρf μ nf U þV ¼− þ ∂X ∂Y ∂Y Ref ρnf μ f
∂ V ∂ V þ ∂X 2 ∂Y 2
ρC p k ∂θ ∂θ 1 f nf U þV ¼ kf ∂X ∂Y Ref Prf ρC
! ∂2 θ ∂2 θ þ : ∂X 2 ∂Y 2
p
2
2
ð2Þ
! ð3Þ
ð4Þ
nf
The non-dimensional parameters used in the above equations have been defined as follows: x y u v T−T w p ;V ¼ ;θ ¼ ;P ¼ X ¼ ;Y ¼ ;U ¼ L L uin uin T in −T w ρf u2in μ f C pf ρu L Ref ¼ f in ; Prf ¼ : μf kf
ð5Þ
The used boundary conditions to solve the Eqs. (1) to (4) have been given in Fig. 1. 4. Physical properties of the nanofluids The nanofluid in this study is composed of water and 36 nm particles of Al2O3. The physical properties of the nanofluid in the above equations can be obtained as follows. θ = 0 , U=0 , V=0
e2
sb2
U=6Y(1-Y) θ =1
L sb1
Y
e1
X
S=13L Fig. 1. Geometrical configuration and boundary conditions.
∂U =0 ∂X ∂V =0 ∂X ∂θ =0 ∂X
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S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154 μ
1.6
According to [14], viscosity ratio ( μnf ) of the Al2O3/water nanofluid f does not change significantly with temperature for particle volume fractions below 4%.
1.4 1.2
2 μ nf ¼ 1 þ 2:5ϕ þ 150ϕ μ f
U
1 0.8
For calculating the thermal conductivity of the nanofluid, an experimental correlation for the thermal conductivity of Al2O3 nanofluids as a function of nanoparticle size and temperature is used [15].
0.6 400*40 300*30 100*10 Analytical
0.4
knf 0:7460 df ¼ 1 þ 64:7ϕ kf dp
0.2 0
0
0.2
0.4
ð8Þ
0.6
0.8
!0:3690 kp 0:7476 0:9955 1:2321 Pr Re kf
ð9Þ
1
Y
The Prandtl number (Pr) and the Reynolds number (Re) are respectively defined as:
Fig. 2. Fully developed non-dimensional velocity distribution in microchannel without baffle, Ref and ϕ = 0.
Pr ¼
25 Akbariniaetal.[5] presentstudy
η ; ρf α f
Re ¼
ρf K B T 3πη2 λf
ð10Þ
where λf is molecular mean free path of water (λf = 0.17 nm), KB is the Boltzmann constant (KB = 1.3807 × 10 −23 J/K) and η is calculated by the following equation:
20
15 B
Nu
η ¼ A:10T−C ;
−5
A ¼ 2:414 10
;
B ¼ 247:8;
C ¼ 140:
ð11Þ
10 5. Numerical method and meshing
5
0
0
1
2
3
In this study, the finite difference method of Projection has been used which has been presented by Chorin [16] and Temam [17], independently. The explicit edition of this method has been given by Fortin et al. [18]. This explicit method is multistage with first order accuracy for time and second order for space. In the first stage the velocity vector, in a middle time step is calculated explicitly, with the omitting of pressure gradient from momentum equations and in the second stage the velocity vector is corrected. It can be shown that the Newman condition for pressure is valid for all boundaries [19]. Since the fluid flow is assumed steady in this work, the above equations have been solved iteratively, until the time derivation term reaches to the order of 10 −4. The staggered marker-and-cell (MAC) mesh [20] has been used. The grid is uniform in both directions but grid size in x direction can differ from that of y direction. The grid numbers have been chosen according to the stability conditions and independency of results from grid numbers.
4
X Fig. 3. Validation of Nu number variation in streamwise direction, Renf = 6.9, ϕ = 5%.
In order to compute the density and specific heat of a classical twophase mixture, Eqs. (6) and (7) are used [13]: ρnf ¼ ð1−ϕÞρf þ ϕρp ¼ ð1−ϕÞ ρC p þ ϕ ρC p : ρC p nf
f
ð6Þ ð7Þ
p
An empirical correlation, based on the experimental data of Nguyen et al. [14], is used for the viscosity of the nanofluid as shown in Eq. (8).
(a)
0
1
2
3
(b)
4
5
6
0
1
2
X
0
1
2
3
X
3
(c)
4
5
6
0
1
2
X
4
5
6
0
1
2
3
X
3
4
5
6
4
5
6
X
4
5
6
0
1
2
3
X
Fig. 4. The effect of Ref on flow pattern, ϕ = 0 (top row) ϕ = 0.04 (bottom row), sb1 = L, sb2 = 3L, e1 = e2 = 0.66L; a) Ref = 5; b) Ref = 20; c) Ref = 50.
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
149
120 15
110 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
100 90 80
10
70
Nu
Po*
Re f =5
Re f =5
60 50 40
5
30 20 10 0
0
0
2
4
6
8
10
12
-10
14
0
2
4
6
X
8
10
12
14
X 160
20 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
140 120
15 100
Po*
Nu
Re f =20 10
Re f =20
80 60 40
5
20 0
0
0
2
4
6
8
10
12
-20
14
0
2
4
6
X
8
10
12
14
X 240
40 Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
30
Bottom wall with baffle Top wall with baffle phi=0 phi=0.04
180
20
Re f =50
Po*
Nu
120
Re f =50
60 10
0
0
0
2
4
6
8
10
12
14
X
-60
0
2
4
6
8
10
12
14
X
Fig. 5. The effect of Ref on Nu number (left) and Po⁎ number (right), sb1 = L, sb2 = 3L, e1 = e2 = 0.66L (vertical dash-dot lines show the baffle location and curves without symbol show the microchannel without baffles).
6. Results 6.1. Grid independency test and validation All of the results are tested to be independent of grid numbers and one example of this test can be seen in Fig. 2. This figure shows
the fully developed non-dimensional velocity distribution in the wall normal direction for the microchannel without baffles. It should be noted that uniform velocity profile boundary condition has been used in the channel inlet in this case and development of the velocity profile has been simulated. The grid of 30 × 300 agrees well with theoretical results but using finer grid of 40× 400 does not have
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S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
Table 1 The effect of Ref on averaged Nu and Po numbers, sb1 = L, sb2 = 3L, e1 = e2 = 0.66L. Ref = 5
Nu Po Nu (without baffle) Po (without baffle)
Ref = 20
Ref = 50
ϕ= 0
ϕ = 0.04
ϕ=0
ϕ= 0.04
ϕ=0
ϕ = 0.04
4.47 14.47 4.30 12.29
5.72 19.40 5.55 16.46
5.15 16.41 4.67 12.28
6.61 21.34 6.06 16.46
6.49 20.54 5.40 12.29
8.11 25.53 6.87 16.47
considerable effect on the improvement of results. Therefore, the grid number of 40× 400 has been chosen for the above case. Also Fig. 2 shows the excellent agreement between theoretical and present numerical results. In order to validate the heat transfer characteristics of the flow, the local Nu number for Al2O3/water nanofluid flow with Renf = 6.9, uin = 0.0502 m s−1 and ϕ = 5% through microchannels has been calculated and compared with the results of Akbarinia et al. [5]. They have used empirical correlation presented by Maiga et al. [21] for dynamic viscosity of Al2O3/water nanofluid: 2 μ nf ¼ 1 þ 7:3ϕ þ 123ϕ μ f :
ð12Þ
Therefore, the above model has been used in this validation. Fig. 3 shows that the agreement between results is good.
6.2. The effect of baffles The effect of baffles on flow pattern has been shown in Fig. 4 for various Reynolds numbers. There is a recirculation zone downstream of baffles. The size of this zone increases with Reynolds number and decreases with ϕ. For a given Ref, the viscosity of the nanofluid increases with increasing the nanoparticle volume fraction and the size of recirculation zone decreases. As the Reynolds number increases, the vortex behind the bottom baffle forces the flow toward the wall stronger than that of the lower Reynolds numbers and creates a small recirculation zone upstream of the top baffle (Fig. 4c). It can be predicted that the above flow pattern would enhance the heat transfer and mixing rate of the flow. Fig. 5 shows the variation of the local Nusselt number (Nu) and non-dimensional wall shear stress (Po⁎) along the channel with and without baffles for Ref =5, 20, 50 and ϕ = 0%, 4%. The height of the baffles and their location from the beginning of the channel are e1 = e2 = 0.66L and sb1 = L, sb2 = 3L, respectively. The definition of Po⁎ is exactly similar to the Poiseuille number except that the sign of velocity gradient in τw definition has been retained. Therefore the negative values exist in separated zones. The negative values in Fig. 5 show the recirculation zones, and sign changing of Po⁎ indicates the reattachment points. As the Re number increases, the magnitudes of friction coefficient increase. It is clear that the presence of baffles increases the local heat transfer and friction coefficients. This influence is increased with increasing the
360
70
320 60
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
40
240
Po*_bottom
Nu_bottom
50
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
280
30
200 160 120 80
20
40
10
0 -40
0 0
2
4
6
8
10
12
-80
14
0
2
4
6
X
10
12
14
240
30 sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
20
sb2=2L sb2=3L sb2=4L sb2=5L sb2=6L phi=0 phi=0.04
180
Po*_Top
25
Nu_Top
8
X
15
120
60
10 0
5 0
0
2
4
6
8
X
10
12
14
-60
0
2
4
6
8
10
12
14
X
Fig. 6. The effect of the distance between baffles on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref = 50, sb1 = L, e1 = e2 = 0.66L (vertical dash-dot lines show the baffles location).
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
sb2=2L
0
1
0
2
1
3
2
4
sb2=3L
5
6
7
8
0
1
2
3
4
X
sb2=5L
sb2=6L
4
5
6
7
8
0
1
2
3
sb2=4L
5
X
3
151
7
8
0
1
2
3
4
5
6
7
8
X
4
X
6
5
6
7
8
X Fig. 7. The effect of the distance between baffles on flow pattern, ϕ = 0.04, Ref = 50, sb1= L, e1 = e2 = 0.66L.
6.3. The effect of the distance between baffles
Table 2 The effect of the distance between baffles on averaged Nu and Po numbers, Ref = 50, sb1 = L, e1 = e2 = 0.66L.
Nu Po
sb2= 2L
sb2 = 3L
sb2 = 4L
sb2= 5L
sb2 = 6L
ϕ= 0
ϕ=0
ϕ=0
ϕ= 0
ϕ=0
ϕ= 0.04
6.86 8.41 21.68 28.84
ϕ= 0.04
6.49 8.11 20.54 25.53
ϕ= 0.04
6.38 8.02 20.50 25.72
ϕ= 0.04
6.37 8.02 20.55 26.29
Fig. 6 shows the variation of local Nu and Po⁎ numbers along the channel with five different locations for the top baffle (sb2 = 2L, 3L, 4L, 5L, 6L). In all of the cases the location of the first baffle is constant, sb1 = L. The height of baffles is e1 = e2 = 0.66L. Fig. 6 illustrates that the value of Nu and Po⁎ numbers in the location of the second baffle and reattachment points first decreases and then is almost constant at both volume fractions. Presence of local minimum or maximum values in Nu and Po⁎ numbers for sb2 = 4L, 5L, 6L reveals that the flow is reattached. It can be concluded that when the distance between baffles is not enough to occur reattachment, increasing the baffle distance decreases the Nu and Po⁎
ϕ= 0.04
6.37 8.02 20.48 26.38
Re number. The Nusselt number has the highest value at the channel entrance, because of the high temperature gradients, and decreases in streamwise direction. By considering the bottom wall in Fig. 5, it can be seen that the Nu number decreases from the entrance to the bottom baffle location at X = 1. Then the recirculation of flow behind the baffle increases the heat transfer. There is a local maximum in Nu and Po⁎ numbers at X = 2 in Fig. 5 (Ref = 5), which is approximately the reattachment point of baffle vortex. The increase of heat transfer and friction after this point is because of the presence of the upper wall baffle. It pushes the flow toward the bottom wall and enhances the heat transfer and friction coefficients. As the Reynolds number increases, the recirculation zone of the bottom baffle becomes larger and the distance between the reattachment point and the top baffle decreases. So, the effect of reattachment point becomes weaker than the presence of the top baffle and the mentioned local maximum disappears in Fig. 5 (Ref = 50). The behavior of flow on the top wall is similar to the bottom wall except that there is a small recirculation zone at Re= 50. The Nu and Po⁎ number variation at X = 2.5–3 is because of the presence of this zone (Fig. 5 (Ref =50)). Also, the use of nanofluids enhances the heat transfer in comparison with the fluid without the nanoparticles. It is clear from Fig. 5 that the effect of nanoparticle volume fraction on friction coefficients is considerable only on the baffle location. The effect of nanoparticle volume fraction on the local Nusselt number becomes lower with the increase of the Reynolds number. Averaged Nu and Po numbers are shown in Table 1. This table shows that the baffles increase the heat transfer and friction coefficients. As mentioned above, it is clear that adding of the nanoparticles and using baffles enhance the averaged heat transfer and friction coefficients.
e1=e2=0.33L
0
1
2
3
4
X
numbers. When reattachment occurs between the baffles, variation of the baffle distance does not change the heat transfer or friction coefficients. The effect of the baffle distance on recirculation zones and reattachment happening is obviously shown in Fig. 7. For sb2 = 2L and 3L on the bottom wall the location of reattachment point and top baffle effect is coincident on each other approximately, so there is not any local minimum or maximum values in Nu and Po⁎ numbers in Fig. 6. But for higher sb2 values the effect of the top wall on the bottom wall becomes considerable. Also, Fig. 7 indicates that as the distance between the baffles is low, the flow after the first baffle does not have enough space to develop and a small recirculation zone in front of the top baffle creates. For sb2 values higher than 4L, this zone disappears. Averaged Nu and Po is shown in Table 2 for various baffle distance values. It is clear from the table values, that the effect of the baffle distance on heat transfer coefficient is more considerable than that of the friction coefficient. As mentioned above, increasing the baffle distance to more than 3L does not have any effect on flow and heat transfer parameters. 6.4. The effect of the baffle height Three pairs of baffles with the heights of e1 = e2 = 0.33L, e1 = e2= 0.5L and e1 = e2 =0.66L have been investigated. The location of baffles is constant (sb1= L, sb2 = 3L) and Ref = 50. The flow pattern
e1=e2=0.5L
5
6
7
0
1
2
3
4
e1=e2=0.66L
5
6
7
0
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X Fig. 8. The effect of the baffle height on flow pattern ϕ = 0.04, sb1 = L, sb2 = 3L, Ref = 50.
2
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e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
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20 15
120
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e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
25
e1=e2=0.33L e1=e2=0.5L e1=e2=0.66L phi=0 phi=0.04
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Nu_Top
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10 40 5
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0
2
X
4
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8
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Fig. 9. The effect of the baffle height on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref =50, sb1=L, sb2=3L (vertical dash-dot lines show the baffle location).
in Fig. 8 illustrates that the size of the recirculation zone increases with baffle height. Also, for the baffle heights greater than 0.33L, the inclination of streamlines toward the bottom wall increases and a small recirculation zone starts to create in front of the top baffle. Fig. 9 shows the heat transfer and friction coefficient distribution for various baffle heights on the bottom and top walls. It is obvious that the heat transfer and friction coefficients have been increased with baffle height and volume fraction of the nanoparticles. The location of the small recirculation zone is clear in Nu and Po⁎ distribution of the top wall at X = 2.5–3, for e1= e2 = 0.66L. Also, the figure shows that the rate of increasing Nu and Po⁎ with baffle height, increases. Averaged Nu and Po are shown in Table 3 for various baffle heights. This table shows that the heat transfer improving effect of nanofluids is more evident at high baffle heights. Table 3 The effect of the baffle height on averaged Nu and Po numbers, Ref = 50, sb1 = L, sb2 = 3L.
Nu Po
e1 = e2 = 0.33L
e1 = e2 = 0.5L
e1 = e2= 0.66L
ϕ=0
ϕ = 0.04
ϕ=0
ϕ = 0.04
ϕ=0
ϕ = 0.04
5.40 12.07
6.87 16.26
5.73 14.35
7.28 18.87
6.49 20.54
8.11 25.53
6.5. The effect of non-equal baffle heights Two baffle heights of 0.33L and 0.66L have been chosen in this section and the order of arrangement of these baffles has been studied. In all of the cases sb1 = L, sb2 = 3L and Ref = 50. It is clear from Fig. 10 that for the arrangement of “small baffle on the bottom wall–large baffle on the top wall”, maximum Nu and Po⁎ numbers are higher for the bottom wall and for the reverse arrangement, maximum Nu and Po⁎ numbers are higher for the top wall. It is dependent on the location of the large baffle, because the large baffle pushes the flow toward the opposed wall. This is the location of maximum heat transfer and friction coefficients. Also, Fig. 10 shows that the effect of nanoparticle volume fraction is more considerable at the location of the second baffle than the first one. 7. Conclusions In this study the heat transfer and fluid flow of Al2O3/water nanofluid in two dimensional parallel plate microchannel without and with micromixers have been investigated for nanoparticle volume fractions of ϕ=0,ϕ=4% and base fluid Reynolds numbers of Ref =5, 20, 50. Five baffle distances, three baffle heights and two non-equal baffle heights were chosen and studied. Results show that there is a recirculation zone
S.B. Islami et al. / International Communications in Heat and Mass Transfer 43 (2013) 146–154
250
30 e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
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e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
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100
10
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240 e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
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e1=0.33L , e2=0.66L e1=0.66L , e2=0.33L phi=0 phi=0.04
200 160
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Nu_Top
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0
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4
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Fig. 10. The effect of non-equal baffle heights on Nu number (left) and Po⁎ number (right) for the bottom and top walls, Ref =50, sb1=L, sb2=3L (vertical dash-dot lines show the baffle location).
downstream of the baffles. The size of this zone increases with Reynolds number and decreases with ϕ. It is observed that the influence of nanoparticle volume fraction on vortex size is weaker at low Re numbers. It is clear that the presence of baffles increases the local heat transfer and friction coefficients. This influence is increased with increasing the Re number. As the Reynolds number increases, the recirculation zone of the bottom baffle becomes larger and the distance between the reattachment point and the top baffle decreases. When the distance between baffles is not enough to occur reattachment, increasing baffle distance decreases the Nu and Po⁎ numbers. When reattachment occurs between the baffles, variation of baffle distance does not change the heat transfer or friction coefficients. The size of the recirculation zone increases with baffle height. The heat transfer and friction coefficients have been increased with baffle height and volume fraction of the nanoparticles. For the arrangement of “small baffle on the bottom wall–large baffle on the top wall”, maximum Nu and Po⁎ numbers are higher for the bottom wall and for the reverse arrangement, maximum Nu and Po⁎ numbers are higher for the top wall.
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