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Natural convection investigation in square cavity filled with nanofluid using dispersion model
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 6 ) 1 e1 3
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Natural convection investigation in square cavity filled with nanofluid using dispersion model Abdelhamid Boualit a,*, Noureddine Zeraibi b, Toufik Chergui a, Mohamed Lebbi a, Lyes Boutina a, Salima Laouar a Unite de Recherche Appliquee en Energies Renouvelables, URAER, Centre de Developpement des Energies Renouvelables, CDER, BP: 88 Gart Taam Z.I, 47133, Ghardaı¨a, Bounoura, Algeria b Universite de Boumerdes, Faculte des hydrocarbures dept, Transport et equipement, 35000 Avenue de l'independance, Boumerdes, Algeria a
article info
abstract
Article history:
A cooling achieved with compact and efficient device is one of the major challenges
Received 3 March 2016
encountered in the promising technique of fuel cell stacks. The safe and reliable use of
Received in revised form
such a system is highly dependent on the efficiency of the assured heat transfer and
29 June 2016
consequently on the quality of the coolant used. To test the possible improvement of the
Accepted 8 July 2016
coolant performances, laminar natural convection in square cavity filled with copper-
Available online xxx
water nanofluid is numerically carried out taking into account the thermal dispersion effect on the heat transfer intensity. The finite element method is used to solve the governing
Keywords:
equations. The hydrodynamic structure of the flow and its thermal behavior are studied for
Nanofluid
a wide range of Rayleigh numbers. The obtained results showed an enhancement of heat
Natural convection
transfer with an increase in nanoparticle volume fraction for all examined Rayleigh
Thermal dispersion
numbers. However, it is found that an increase in nanoparticle diameter enhances heat
Square cavity
transfer only when thermal dispersion is significant. Correlation with 99.94% confidence
Fuel cell
coefficient is proposed to quantify the heat transfer intensity according to the Rayleigh number and particle diameter and concentration. © 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Aiming to improve thermal properties of working fluids, recent works relate that addition of low nanoparticle concentrations leads to a phenomenal improvement in fluid thermal conductivity with very low viscosity increase. For better use, stabilizers or surfactants are generally added to ultrasonically homogenized mixtures to avoid pipe clogging resulting from particle sedimentation and cluster formation.
Considering this significant improvement, such substances seem to be an appropriate solution to overcome the limitation in enhancing performance of engineering devices involving working fluids where heat transfer effectiveness is seriously hampered by the low thermal conductivity of the fluids used. The efficiency and compactness of the resulting devices are valuable assets in the feasibility and reliability of promising new technologies. In hydrogen technology, fuel cell
* Corresponding author. E-mail addresses: [email protected] (A. Boualit), [email protected] (N. Zeraibi), [email protected] (T. Chergui), mohamed. [email protected] (M. Lebbi), [email protected] (L. Boutina), [email protected] (S. Laouar). http://dx.doi.org/10.1016/j.ijhydene.2016.07.132 0360-3199/© 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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stacks are a typical example of this situation. Although they have high energy conversion efficiency and low operating temperature, these systems are particularly penalized by significant heat generation which must be removed efficiently to prevent any risk of overheating [1] or flooding [2,3]. Article [4] gives a clear and detailed review of thermal control processes required to insure safe and efficient use of fuel cells. Since works carried out by Maxwell (1904) [5], several theoretical [6,7] and experimental studies [8,9] have considered the case of liquids and gases containing suspended solid particles. These particles of some micrometers have improved thermal conductivity of the base fluid. On the other hand, several drawbacks in relation with particle size were recorded; such as, increasing pressure loss, sedimentation, pipe erosion and clogging… The last decades have witnessed the advent of new manufacturing processes allowing production of nanoscale particles. As advantages, several inconveniences related to particle size were removed. Fluids containing these fine particles are called nanofluids (Choi [10]). Recent works were focused on the determination of nanofluid effective thermal conductivity and effective viscosity [10e12]. It was reported that thermal conductivity of the base fluid is tremendously increased (more than 20%) with addition of a weak fraction of nanoparticles (1e5%). This increase depends on the particle diameter, particle volume fraction, as well as the thermal properties of the two components (base fluid and nanoparticles). Investigations involving heat transfer in nanofluids remain very difficult owing the interaction between several complex phenomena, such as Brownian motion of the nanoparticles, clusters formation, interaction and collision among particles, liquid layer structure at the liquid/particles interface, etc. The effects of these various phenomena were research topics of several recent works [13,14]. Concerning natural convection heat transfer, this phenomenon occurs widely in many industrial applications such as chemical reactors, heat exchangers, electronic cooling, double pane windows, and solar collectors etc. A comprehensive review was made by Baı¨ri et al. [15] where they underline the scientific and engineering fields where the knowledge of natural convection in enclosures is advantageously applied. The process relevance and its broad spectrum of applications have conducted to extensive studies dealing with mechanisms of heat transfer enhancement and efficiency improvement. Thus, several studies have treated a large variety of enclosure configurations with different shapes and inclinations [16e19], various thermal boundary conditions [20e22], and heat source distributions [23e25]. In the case of natural convection in enclosures filled with nanofluids, few experimental and numerical works were carried out [26e29], where the effort was primarily focused on the quantification of the heat transfer enhancement and the identification of the different mechanisms involved. However, published works have shown contradictory effects of nanoparticles on heat transfer. Khanafer et al. [30] studied Cuewater nanofluids in a two-dimensional rectangular enclosure. They reported that an increase in the nanoparticles volume fraction improves heat transfer at any given Grashof number.
The same study is carried out by Oztop and Abu-Nada [31] and the same remarks were formulated. On the other hand, Abu-Nada et al. [32] demonstrated that the heat transfer enhancement in natural convection depends mainly on the magnitude of Rayleigh number. Using experimental data of CuOewater and Al2O3ewater nanofluids reported by Nguyen et al. [33], they derived a correlation to quantify the effective viscosity and found for AL2O3ewater nanofluid a deterioration of the mean Nusselt number at high Rayleigh numbers by increasing the nanoparticles volume fraction above 5%. However, at low Rayleigh number, the mean Nusselt number is slightly enhanced by increasing the volume fraction of nanoparticles. Concerning CuOewater nanofluid at high Rayleigh numbers, they reported a continuous decrease in mean Nusselt number as the volume fraction of nanoparticles is increased. Experimentally, using Al2O3 and CuO water nanofluids, Putra et al. [34] remarked that the natural convection heat transfer coefficient is lower than that of the clear fluid. Wen and Ding [35] also showed heat transfer deterioration resulting from the addition of nanoscale particles. The present work investigates natural convection heat transfer in partially heated square cavity filled with nanofluid consisting of copper nanoparticles dispersed in water. Simulations are carried out to define the influence exerted by the volume fraction and the diameter of the nanoparticles on the fields of velocity and temperature for various Rayleigh numbers. The main objective of this work is to check whether the thermal dispersion affects the natural convection heat transfer.
Problem formulation The geometrical configuration of the problem considered is represented in Fig. 1. The flow field is a square enclosure with
Fig. 1 e Schematic representation of the examined geometry.
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 6 ) 1 e1 3
an aspect ratio of unity. The horizontal walls of the cavity are assumed to be insulated while the vertical walls are maintained at constant and uniform temperatures. The left vertical wall is kept at a high temperature (Th) and the right wall at a low temperature (Tc). The temperature gradient (DT ¼ TheTc) between the two vertical walls is maintained at a constant value(20 K). The cavity is filled with a nanofluid made up of water and copper nanoparticles (Cu). The physical properties of those components (Table 1) are considered constant and given at the bulk temperature (Tb¼ (ThþTc)/2). In this study, Tb is taken equal to 293 K. The cavity length (L) is used as a control parameter in order to generate natural convection with various intensities since the temperature gradient (DT) and the fluid proprieties are considered invariant. Thus, the length is varied from 1.53 mm to 15.3 mm in order to generate natural convection of Ra ¼ 103 to 106. In this study, the overheat ratio d(d ¼ (ThTc)/Tb) is small enough that the use of the Boussinesq approximation is justified. Assuming the case of an incompressible fluid and a steady two-dimensional flow, the governing equations for laminar natural convection in a square enclosure are given by the following system of partial derivative equations: vu vv þ ¼0 vx vy
(1)
vu vu vp v vu v vu vv ¼ þ mnf ;0 2 þ þ rnf ;0 u þ v vx vy vx vx vx vy vy vx
(2)
vv vv vp v vu vv v vv rnf ;0 u þ v ¼ þ mnf ;0 þ þ 2 vx vy vy vx vy vx vy vy i h þ ∅rp;0 bp;0 þ ð1 ∅Þrf ;0 bf ;0 gðT Tc Þ
rCp
nf ;0
vT vT vT v v ¼ knf ;0 þ kd þ knf ;0 u þv vx vy vx vx vy vT þ kd vy
The physical properties of the nanofluid are given according to those of the base fluid and the nanoparticles. Using the mixture rule, the density and the heat capacitance of the nanofluid are given by: rnf ;0 ¼ ð1 ∅Þrf ;0 þ ∅rp;0
rCp
nf ;0
¼ ð1 ∅Þ rCp f ;0 þ ∅ rCp p;0
(4)
(5) (6)
The effective viscosity measures of Cu-water nanofluid reported by Xuan and Li [36] for a temperature ranging from 20 C to 70 C, match well with the values predicted from Brinkman's model [37]. This model is expressed as follow: mnf ;0 ¼
mf ;0 ð1 ∅Þ2:5
(7)
The nanoparticle volume fraction effect on the dynamic viscosity of the base fluid is represented in Fig. 2. There is shown that the nanoparticle addition increases the fluid viscosity. So, volume fractions of 1%, 10% and 20% give rise to a viscosity increase of 2.5%, 30% and 74% respectively. For successful use, the volume fraction of the nanoparticles must be between 1% and 10% to prevent, on the one hand, the flow deterioration by the drastic increase in the fluid viscosity, and on the other hand, the loss of the fluid homogeneity caused by the formation of nanoparticle clusters. The stagnant thermal conductivity of the nanofluid can be predicted from the following relation suggested by Wasp [38]: knf ;0 ¼ kf ;0
(3)
3
! kp;0 þ 2kf ;0 2∅ kf ;0 kp;0 kp;0 þ 2kf ;0 þ ∅ kf ;0 kp;0
(8)
where kp and kf are respectively the thermal conductivities of the nanoparticles and the base fluid. Based on porous media theory, the following relation predicts the dispersed thermal conductivity kd [39]: kd ¼ C rCp nf ;0 w∅dp
(9)
(x,y) are the cartesian coordinates of the geometry, (u,v) are the velocity components, p is the pressure, T is the temperature. r, m, (rCp), k, b are respectively the density, dynamic viscosity, heat capacitance, thermal conductivity and thermal expansion coefficient. ∅ is the volume fraction of nanoparticles in the base fluid. The indices p, f, nf designate, respectively, nanoparticles, base fluid and nanofluid. The index 0 accompanying a physical property indicates that this property is given at the bulk temperature Tb.
Table 1 e Physical properties of base fluid and Cu nanoparticles at 20 C. Physical properties
Base fluid (water) (at 20 C)
Cu nanoparticles
r (kg/m3) Cp (J/kg/K) k (W/m/K) m (kg/s/m) b (1/K)
997.64 4180.4 0.602 0.001 20.7$105
8954 383.1 386 e 5.1$105
Fig. 2 e Effect of nanoparticle volume fraction on the increase rate of the fluid dynamic viscosity.
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where w ¼ u2 þ v2 , dp is the nanoparticle diameter and C an unknown constant determined experimentally. For Cu-water nanofluids C is taken as equal to 0.4 [39]. According to this relation (Eq. (9)), thermal dispersion is function of flow intensity, volume fraction and physical proprieties of nanoparticles, as well as the proprieties of the base fluid. Introducing the dimensionless variables below to the system of equations (1)e(4) x y u v X ¼ ; Y ¼ ; U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; V ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; P L L gbf ;0 DTL gbf ;0 DTL ¼
p T Tc and q ¼ rf ;0 bf ;0 gDTL Th Tc
leads to the following dimensionless system of equations: vU vV þ ¼0 vX vY U
(10)
rf ;0 vP mnf ;0 1 vU vU v vU v vU vV pffiffiffiffiffiffi þv ¼ þ 2 þ þ vX vY vX vY vY vX rnf ;0 vX mf ;0 Gr vX (11)
U
rf ;0 vP mnf ;0 1 vV vV v vU vV v vV pffiffiffiffiffiffi þV ¼ þ þ þ 2 vX vY vY vY rnf ;0 vY mf ;0 Gr vX vY vX rf ;0 b q þ rnf ;0 r;0 (12)
2 rCp f ;0 knf ;0 vq vq 1 v q v2 q þV ¼ U þ UW pffiffiffiffiffiffiffiffiffiffiffi þ vX vY rCp nf ;0 kf ;0 PrRa vX2 vY 2 rCp f ;0 U vW vq vW vq pffiffiffiffiffiffiffiffiffiffiffi þ þ rCp nf ;0 PrRa vX vX vY vY
Where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rCp nf ;0 dp pffiffiffiffiffiffi kd ∅ Pr Gr; W ¼ U2 þ V2 ; and br UW ¼ ; U¼C kf ;0 rCp f ;0 L ðrbÞp;0 ¼ ð1 ∅ÞðrbÞf ;0 þ ∅ ðrbÞf ;0 The dimensionless numbers are expressed using the physical properties of the base fluid given at the bulk temperature (Tb). The Prandtl number (Pr), the Grash of number (Gr) and the Rayleigh number (Ra) are writing as below: Pr ¼
kf ;0
; Gr ¼
r2f ;0 gbf ;0 DTL3 m2f ;0
; Ra ¼ PrGr
The dimensionless boundary conditions considered in this study are: On the top and bottom walls: U ¼ V ¼ vq/vY ¼ 0, on the left wall: U ¼ V ¼ 0 and q ¼ 1, and on the right wall: U ¼ V ¼ q ¼ 0. To quantify the heat transfer intensity between the hotwall and the fluid, local and mean Nusselt numbers are computed as follow: knf ;0 vq Nu ¼ kf ;0 vX X¼0
NuðYÞdY
(15)
0
Numerical procedure The system of equations (10)e(13) subject to the abovementioned dimensionless boundary conditions is solved using finite element method where the Galerkine formulation is employed for the weighted residues construction. The resolution algorithm is based on a so-called mixed method which allows a simultaneous determination of the velocity and the pressure, unlike projection methods where the two fields are uncoupled. To retain a sufficiently smooth solution for the investigated system, an element free of the LBB (LadyzhenskayaeBabuskaeBrezzi) stability constraint is taken into account. So, a 9-nodes quadrilateral element with the well-paired bi-linear interpolation function for the pressure and the bi-quadratic interpolation function for the velocities (Q9P4) is adopted [40]. As the solver, an iterative method based on the direct substitution scheme (Picard method) is used to solve the resulting system of nonlinear algebraic equations, sometimes with under relaxation to ensure convergence [40]. The convergence criterion ð Fkþ1 Fk < 104 Þ is used for all realized computations, where F represents the variables (U, V, P, q) and k is the iteration number.
Grid testing and code validation (13)
mf ;0 Cp f ;0
Z1 Numean ¼
(14)
To ensure grid independence of the solution, three grid combinations are examined. The grids M1, M2 and M3 containing respectively 20 20, 30 30 and 40 40 elements are explored for Ra ¼ 103, 104, 105, 106 and Pr ¼ 0.71 in the case of pure fluid (∅ ¼ 0). All the grids are non-uniform with more concentration of elements near the walls. Table 2 presents a comparison between the numerical results obtained using the various grids and those reported in literature. This comparison is based on the mean, the maximum and the minimum of the Nusselt number along the hot wall and their corresponding locations. On the other hand, the maximum longitudinal velocity (U) along the line X ¼ 0.5 and the maximum transversal velocity (V) along Y ¼ 0.5 are subject to comparison. The grids M2 and M3 have led to solutions which are in good agreement with the benchmark solution given by De Vahl Davis [41] for all values of Ra, as well as with those given by Fusegi et al. [42]. However, M2 requires less machine memory and less calculation effort. So, the grid M2 is adopted in this study. In the case of natural convection in square enclosures filled with nanofluids, the numerical code is validated by comparing the present results of the mean Nusselt number at the hot wall with those reported by Abu-Nada et al. [26]. This comparison, shown in Fig. 3, indicates clearly that the two sets of results are in good agreement.
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Table 2 e Comparison between the numerical results obtained using the various grids and those reported in literature. M1 Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
1.1178 1.5059 0.6910 0.1367 0.1388
(Y ¼ 0.087) (Y ¼ 1.00) (Y ¼ 0.8226) (X ¼ 0.1774)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
2.2472 3.5332 0.5867 0.1921 0.2317
(Y ¼ 0.1525) (Y ¼ 1.00) (Y ¼ 0.8226) (X ¼ 0.1299)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
4.5499 7.7857 0.7319 0.1301 0.2576
(Y ¼ 0.0870) (Y ¼ 1.00) (Y ¼ 0.8475) (X ¼ 0.0668)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
9.1852 18.3463 (Y ¼ 0.0311) 0.9937 (Y ¼ 1.00) 0.0767 (Y ¼ 0.8475) 0.2557 (X ¼ 0.0311)
M2
M3
Ra ¼ 103 1.1178 1.1180 1.5059 (Y ¼ 0.0899) 1.5078 (Y ¼ 0.0927) 0.6914 (Y ¼ 1.00) 0.6890 (Y ¼ 1.00) 0.1369 (Y ¼ 0.8191) 0.1370 (Y ¼ 0.8127) 0.1387 (X ¼ 0.1809) 0.1387 (X ¼ 0.1740) Ra ¼ 104 2.2452 2.2450 3.5310 (Y ¼ 0.1477) 3.5304 (Y ¼ 0.1421) 0.5859 (Y ¼ 1.00) 0.5849 (Y ¼ 1.00) 0.1920 (Y ¼ 0.8191) 0.1920 (Y ¼ 0.8261) 0.2329 (X ¼ 0.1174) 0.2326 (X ¼ 0.1192) Ra ¼ 105 4.522 4.5227 7.7146 (Y ¼ 0.0769) 7.7217 (Y ¼ 0.0835) 0.7290 (Y ¼ 1.00) 0.7276 (Y ¼ 1.00) 0.1316 (Y ¼ 0.8523) 0.1303 (Y ¼ 0.8518) 0.2590 (X ¼ 0.0653) 0.2576 (X ¼ 0.0659) Ra ¼ 106 9.1141 9.0874 17.6971(Y ¼ 0.0357) 17.5479 (Y ¼ 0.0364) 0.9854 (Y ¼ 1.00) 0.9814 (Y ¼ 1.00) 0.0769 (Y ¼ 0.8523) 0.0769 (Y ¼ 0.8518) 0.2588 (X ¼ 0.0356) 0.2613 (X ¼ 0.0364)
Fig. 3 e Nusselt number evolution versus Ra number and comparison with Abu-Nada et al. [27] work.
Results and discussion In this work, a numerical study is performed for natural convection analysis in square cavity filled with a nanofluid made up water and a volume fraction (∅) of copper nanoparticles. The aim is to analyze the influence exerted by the concentration and the diameter of the nanoparticles on the flow structure and the heat transfer intensity. Thus, simulations are realized for various volume fractions (0 ∅ 0.2), different nanoparticle diameters (10 nm dp 100 nm) and a wide range of Rayleigh numbers. (103 Ra 106).
De Vahl Davis [41]
Fusegi et al. [42]
1.118 1.505 (Y ¼ 0.092) 0.692 (Y ¼ 1.00) 0.136 (Y ¼ 0.813) 0.138 (X ¼ 0.178)
1.105 1.420 0.764 0.132 0.131
(Y ¼ 0.083) (Y ¼ 1.000) (Y ¼ 0.833) (X ¼ 0.200)
2.243 3.528 (Y ¼ 0.143) 0.586 (Y ¼ 1.00) 0.192 (Y ¼ 0.823) 0.234 (X ¼ 0.119)
2.302 3.652 0.611 0.201 0.225
(Y ¼ 0.623) (Y ¼ 1.000) (Y ¼ 0.8 17) (X ¼ 0.1 17)
4.519 7.717 (Y ¼ 0.081) 0.729 (Y ¼ 1.00) 0.153 (Y ¼ 0.855) 0.261 (X ¼ 0.066)
4.646 7.795 0.787 0.147 0.247
(Y ¼ 0.083) (Y ¼ 1.000) (Y ¼ 0.855) (X ¼ 0.065)
8.799 17.925 (Y ¼ 0.0378) 0.989 (Y ¼ 1.00) 0.079 (Y ¼ 0.850) 0.262 (X ¼ 0.038)
9.012 17.670 (Y ¼ 0.0379) 1.257 (Y ¼ 1000) 0.084 (Y ¼ 0.856) 0.259 (X ¼ 0.033)
Fig. 4 presents the streamlines (left) and isotherms (right) for dp ¼ 10 nm and different Ra. Each diagram illustrates the volume fraction effects on the fields of velocity and temperature for a given Ra number (smooth lines for ∅ ¼ 0, long-dash lines for ∅ ¼ 0.1 and dash-dot lines for ∅ ¼ 0.2). According to the Rayleigh number, this figure shows that the increase in the nanoparticles volume fraction causes the displacement of the streamlines towards or far from the cavity center. For Ra ¼ 103, the streamlines are shifted to the cavity center when ∅ increases showing a decrease in the velocity field intensity. In the case of Ra ¼ 104, the streamlines close to the walls express a small shift to the center, while the lines located at the cavity center move in the walls direction. This is due to the velocity decrease near the walls and its increase at the center. For Ra¼105 and 106, all the streamlines move away from the center except those very close to the walls. These findings can be explained by the influence of the nanoparticles volume fraction on the viscous forces and the buoyancy forces interacting during a natural convection flow. Indeed, the nanoparticles addition influences simultaneously the two kinds of force. It increases the viscous forces effect by increasing the fluid viscosity (Eq. (7)) and the buoyancy forces effect by improving the fluid thermal conductivity. The thermal transfer improvement will be analyzed in detail further in this discussion. During natural convection in a closed enclosure, the two kinds of force are present with different intensities. In the flow field, one force can be locally or globally more significant than the other one. For Ra ¼ 103, the viscous forces are dominant throughout the flow field. So, the addition of the nanoparticles increases the fluid viscosity and slows the flow. For Ra ¼ 104, the viscous forces are significant near the walls whereas the buoyancy forces are dominant at the cavity center. In this
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 5 e Plot of nanoparticle volume fraction effect on the vertical profile of the longitudinal velocity at Y ¼ 0.5 for various Rayleigh numbers.
Fig. 6 e Plot of nanoparticle volume fraction effect on the horizontal profile of the transversal velocity at X ¼ 0.5 for various Rayleigh numbers.
Fig. 4 e Streamlines (left) and isotherms (right). dp ¼ 100 nm. ∅ ¼ 0 (smooth lines), ∅ ¼ 0.1 (long-dash lines) and ∅ ¼ 0.2 (dash-dot lines). (a,b) Ra ¼ 103; (c,d) Ra ¼ 104; (e,f) Ra ¼ 105 and (g,h) Ra ¼ 106.
situation, the increase of the nanoparticles volume fraction reduces the velocity field near the walls and inversely at the center. All these findings are well illustrated by the Figs. 5 and 6 which show respectively the vertical profiles of the horizontal velocity at X ¼ 0.5 and the horizontal profiles of the vertical velocity at Y ¼ 0.5 for various nanoparticle volume fractions. Fig. 4 shows also the effect of the volume fraction increase on the heat transfer. For Ra ¼ 103 (Fig. 4-b), the isothermal
lines tend to become perfectly parallel to the vertical walls indicating the trend of the thermal transfer to be purely conductive. When Ra 104 (Fig. 4-d, f and h), the increase of the nanoparticles volume fraction causes widening of the thermal boundary layer located near the vertical walls causing the reduction of the temperature gradient. Fig. 7 illustrates the evolution of the Nusselt number along the hot wall for various values of Ra and ∅. This figure shows that an increase in the nanoparticles volume fraction leads to Nu improvement for all considered Ra numbers. According to Eq. (14), the Nusselt number is the product of the temperature gradient located near the hot wall and the thermal conductivity ratio (thermal conductivity of the nanofluide to that of the pure fluid). The addition of copper nanoparticles decreases the temperature gradient by widening the thermal boundary layer but increases more
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 6 ) 1 e1 3
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Fig. 7 e Effect of volume fraction on the Nusselt number evolution along the hot wall. dp ¼ 100 nm. Ra ¼ 103 (a), Ra ¼ 104 (b), Ra ¼ 105 (c) and Ra ¼ 106 (d).
significantly the thermal conductivity ratio what explains the Nusselt number improvement. Fig. 8 presents the effect of the nanoparticle diameter (dp) on heat transfer intensity (Nu) for various Rayleigh numbers. For Ra ¼ 103 and 104, Nusselt number is insensitive to nanoparticle diameter. However, for Ra ¼ 105 and 106 (Fig. 8-c and
d respectively) the Nusselt number increases with ascending values of dp. This observation is the consequence of the thermal dispersion effect which is detailed in the section consecrated to the mean Nusselt number. The evolution of the mean Nusselt number according to the nanoparticles volume fraction is given in Fig. 9 for various
Fig. 8 e Effect of nanoparticles diameter on the Nusselt number evolution along the hot wall. ∅ ¼ 0.2, Ra ¼ 103(a), Ra ¼ 104(b), Ra ¼ 105 (c) and Ra ¼ 106 (d). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 9 e Effect of volume fraction on the mean Nusselt number of the hot wall for various nanoparticle diameters (dp ¼ 10, 50 and 100 nm), and various Rayleigh numbers (Ra ¼ 103 (a), 104 (b), 105 (c) and 106 (d)).
values of the Rayleigh number and various nanoparticle diameters. Figures a, b, c and d associated, respectively, with the Rayleigh numbers 103,104,105 and 106, show a significant increase of mean Nusselt number which accompanies the increase of nanoparticle volume fraction. The same trend is observed for all Ra numbers studied. However, the effect of nanoparticle diameter on heat transfer intensity (Numean) varies according to Ra and ∅. The Fig. 9 a and b which correspond, respectively, to Ra ¼ 103 and Ra ¼ 104, show an insensitivity of Numean to nanoparticle diameter for all values of nanoparticles volume fraction belonging to the interval [0 , 0.2]. While, for Ra ¼ 105 and Ra ¼ 106 (Fig. 9-c and d respectively), the increase in nanoparticle diameter generates an intensification of heat transfer that is more significant for the high nanoparticle volume fraction and Rayleigh number. This last remark is explained by the contribution of thermal dispersion to Nusselt improvement, which is significant only for high Ra numbers and large nanoparticle diameters (Fig. 10). Furthermore, graphical representation of the effect exerted by the nanoparticle diameter on the mean Nusselt number for different concentrations and high Rayleigh numbers (Fig. 11), shows that the improvement in question is linearly proportional to the nanoparticle diameter. According to the Eq. (9), the thermal dispersion depends on the volume fraction and the diameter of the nanoparticles, but more primarily on the velocity field intensity. Fig. 12 shows the distribution of thermal dispersion along the flow field for ∅ ¼ 0.2, dp ¼ 100 nm and various values of the Rayleigh number (Ra ¼ 103,104,105 and 106). Indeed, this figure shows thermal dispersion intensification accompanying the Ra increase, but the most interesting is the location of its maximum values. For Ra ¼ 103 (Fig. 12-a), the maximum of the thermal dispersion (1.4105) is located coarsely halfway
between the centre and the cavity walls. However, for Ra ¼ 106 (Fig. 12-d), where the dependence of thermal transfer on nanoparticles diameter and thermal dispersion is more intense, the maximum of thermal dispersion (13.52 105) is very close to the vertical walls. Thereby, thermal dispersion affects directly the thermal boundary layer responsible of the heat transfer intensity causing an increase of Numean. To show how thermal dispersion acts on heat transfer intensity, Fig. 13 illustrates the form of the isothermal lines very close to the hot wall for ∅ ¼ 0.2, various values of Rayleigh number (a103, b104,c105 and d106) and different nanoparticle diameters (10 nm smooth lines, 50 nm long-dash lines and 100 nm dash-dot lines). Fig. 13-a and b, representative of Ra ¼ 103 and 104, respectively, show that the configuration of the thermal field is independent of dp, contrary to Ra ¼ 105 and 106 (Fig. 13-c and d respectively) where the isothermal lines are progressively closer with ascending values of dp. Therefore, it is observed that unlike the stagnant thermal conductivity, thermal dispersion reduces the thickness of the thermal boundary layer and generates a more significant temperature gradient, which consequently increases Nu and Numean. This observation is more visible for flows with high Rayleigh numbers (Ra ¼ 105and 106) characterized by an important thermal dispersion near the vertical walls. Moreover, this situation is accentuated by the increase of the nanoparticles diameter, which directly affects the thermal dispersion. The effectiveness of nanoparticle addition on natural convection heat transfer improvement in closed enclosures, is estimated by an increase rate given by
Eð%Þ ¼
Numean;nf Numean;f 100 Numean;f
(16)
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Fig. 10 e Comparison of Numean evolution according to nanofluids volume fraction with and without the thermal dispersion effect. (Ra ¼ 103, 104, 105 and 106, dp ¼ 10 and 100 nm).
Fig. 14 shows the increase rate, according to the nanoparticles volume fraction, for various Rayleigh numbers and different nanoparticle diameters (a10 nm, b50 nm, c100nm). It is shown that E(%) expresses the same trend of increase according to the volume fraction for all considered values of Ra and dp. However, the increase rate obtained for Ra ¼ 103 is much higher than those calculated for 104 Ra 106. This is due, on the one hand, to thermal conduction which dominates flows at low Rayleigh numbers (Ra ¼ 103) and, on the other hand, to the considerable improvement in thermal conductivity resulting from nanoparticles addition.
For 104 Ra 106, the increase rate is enhanced by the increase of the Rayleigh number. However, the effect of the nanoparticles diameter is apparent only for Ra ¼ 105 and 106. In this case, flows characterized by strong thermal dispersion express an enhancement of E(%) with increasing nanoparticle diameter (dp). Using the numerical results of the many simulations carried out in this study, correlation is proposed to predict the mean Nusselt number at the hot wall according to the Rayleigh number (103 Ra 106), the volume fraction (0 ∅ 0.2) and the nanoparticle diameter (10 nm dp 100 nm). This correlation is expressed as follow:
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 15 presents a comparison between the simulated values of Numean (symbols) and those predicted by the proposed correlation (lines). The error bars correspond to an error of 7%. As shown in this figure, the correlated values of Numean are in good agreement with the results of the performed simulations. The confidence coefficient of the above correlation is R2 ¼ 99.94%.
Conclusion
Fig. 11 e Effect of nanoparticle diameter on the mean Nusselt number for various Rayleigh numbers and various nanoparticle volume fractions.
1 þ 0:003Ra ∅ e2:116∅ Numean ¼ 0:1318Ra0:3091 1 þ 33344 1 þ 41:067Ra C dp Ra þ 0:329 ∅ L
(18)
In this study, numerical simulations are carried out in order to analyze the effect of nanoparticles addition on the laminar natural convection in a square enclosure. The hydrodynamic structure of the flow and its thermal behavior are studied for a wide range of Rayleigh numbers and nanoparticle concentrations. The results obtained have shown an enhancement of the mean Nusselt number with an increase of nanoparticle volume fraction for all examined Ra numbers ranging from 103 to 106. Furthermore, the effectiveness of nanoparticle addition on the improvement of natural convection heat transfer is more significant for Ra¼103 due to the dominance of conduction on the heat transfer mechanism. However, in a situation of convection-dominated heat transfer (104Ra106), Numean enhancement is proportional to Ra number.
Fig. 12 e Thermal dispersion field (£10¡5) for dp¼100 nm, ∅ ¼ 0.2 and various Rayleigh numbers (Ra¼103(a), 104 (b),105 (c) and 106 (d)). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 13 e Isotherms configuration near the hot wall for dp¼100 nm, ∅ ¼ 0.2 and various Rayleigh numbers (Ra ¼ 103 (a), 104 (b), 105 (c) and 106 (d)).
Fig. 14 e Evolution of increase rate of the mean Nusselt number according to the volume fraction for various Rayleigh numbers and various diameters (dp ¼ 10 nm (a), 50 nm (b) and 100 nm (c)). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 15 e Comparison between Numean values obtained by numerical simulations and those given by the suggested correlation(17).
For the effect of the nanoparticles size on the mean Nusselt number, it is found that an increase in particle diameter enhances heat transfer only when the thermal dispersion is significant. Thus, Numean is insensitive to nanoparticles size for Ra¼103 and 104, whereas it increases with increases of particle diameter for Ra¼105 and 106.
Nomenclature Cp dp g k L Nu Numean p P Pr Ra T u U v V x X y Y
Specific heat capacity, J/kg/K Nanoparticle diameter, nm Gravitational acceleration, m/s2 Thermal conductivity, W/m/K Cavity length, m Dimensionless local Nusselt number Dimensionless mean Nusselt number Pressure, Pa Dimensionless pressure Prandlt number Rayleigh number Temperature, K Longitudinal velocity, m/s Dimensionless longitudinal velocity Transversal velocity, m/s Dimensionless transversal velocity Longitudinal coordinate, m Dimensionless longitudinal coordinate Transversal coordinate, m Dimensionless transversal coordinate
Greek symbols b Thermal expansion coefficient, 1/K d Overheat ratio
q m r f
Dimensionless temperature Dynamic viscosity, kg/m/s Density, kg/m3 Nanoparticle volume fraction
Subscripts b Bulk c Cold d Dispersion f Fluid h Hot nf Nanofluid p Particle 0 Propriety given at the bulk temperature
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[6] Jeffrey DJ. Conduction through a random suspension of spheres. In: Proc. R. Soc. Lond., Series A; 1973. p. 355e67. [7] Batchelor GK. The effect of brownian motion on the bulk stress in a suspension of spherical particles. J Fluid Mech 1977;83:97e117. [8] Boothroyd RG, Haque H. Fully developed heat transfer to a gaseous suspension of particles flowing turbulently in duct of different size. J Mech Eng Sci 1970;12:191e200. [9] Sohn CW, Chen MM. Micro-convective thermal conductivity in dispersed twoephase mixtures as observed in a low velocity couette flow experiment. J Heat Transf 1981;103:45e51. [10] Choi S. U S. Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer DA, Wang HP, editors. Developments and applications of non-newtonian flowsFEDvol. 231/MD-vol. 66. New York: ASME; 1995. p. 99e105. [11] Pak BC, Cho YI. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp Heat Transf 1998;11:151e70. [12] Xuan Y, Li Q. Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000;21:58e64. [13] Keblinski P, Phillpot SR, Choi SUS, Eastman JA. Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids). Int J Heat Mass Transf 2002;45:855e63. [14] Ohara T, Suzuki D. Intermolecular energy transfer at a solidliquid interface. Microscale Thermophys Eng 2000;4:189e96. [15] Baı¨ri A, Zarco-Pernia E, Garcı´a de Marı´a JM. A review on natural convection in enclosures for engineering applications. The particular case of the parallelogrammic diode cavity. Appl Therm Eng 2014;63:304e22. [16] Basak Tanmay, Roy S, Krishna Babu S, Balakrishnan AR. Finite element analysis of natural convection flow in a isosceles triangular enclosure due to uniform and nonuniform heating at the side walls. Int J Heat Mass Transf 2008;51:4496e505. [17] Roy S, Basak Tanmay, Thirumalesha Ch, Krishna Murali Ch. Finite element simulation on natural convection flow in a triangular enclosure due to uniform and nonuniform bottom heating. ASME J Heat Transf 2008;130. 032501. [18] Mukutmoni D, Yang KT. Thermal convection in small enclosures: an atypical bifurcation sequence. Int J Heat Mass Transf 1995;38:113e26. [19] Kahveci Kamil. Natural convection in a partitioned vertical enclosure heated with a uniform heat flux. ASME J Heat Transf 2009;129:717e26. [20] Sathiyamoorthy M, Tanmay Basak, Roy S, Pop I. Steady natural convection flows in a square cavity with linearly heated side wall(s). Int J Heat Mass Transf 2007;50:766e75. [21] Cheikh Nader Ben, Beya Brahim Ben, Lili Taieb. Aspect ratio effect on natural convection flow in a cavity submitted to a periodical temperature boundary. ASME J Heat Transf 2007;129:1060e8. [22] Shi Xundan, Khodadadi JM. Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall. ASME J Heat Transf 2003;125:624e34. [23] Joshi Mandar V, Gaitonde UN, Mitra Sushanta K. Analytical study of natural convection in a cavity with volumetric heat generation. ASME J Heat Transf 2008;126:176e82. [24] Bae JH, Hyun JM. Time-dependent buoyant convection in an enclosure with discrete heat sources. Int J Therm Sci 2004;43:3e11.
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[25] Da Silva AK, Lorente S, Bejan A. Optimal distribution of discrete sources on a wall with natural convection. Int J Heat Mass Transf 2004;47:203e14. € [26] Abu-Nada Eiyad, Oztop Hakan F. Effects of inclination angle on natural convection in enclosures filled with CueWater nanofluid. Int J Heat Fluid Flow 2009;30:669e78. € Patrice, Yan Wei-Mon, Al[27] Oztop Hakan F, Estelle Salem Khaled, Orfi Jamel, Mahian Omid. A brief review of natural convection in enclosures under localized heating with and without nanofluids. Int Comm Heat Mass Transf 2015;60:37e44. [28] Bouhalleb Mefteh, Abbassi Hassan. . Natural convection of nanofluids in enclosures with low aspect. Int J Hydrogen Energy 2014;39(27):15275e86. [29] Bouhalleb Mefteh, Abbassi Hassan. Natural convection in an inclined rectangular enclosure filled by CuOeH2O nanofluid, with sinusoidal temperature distribution. Int J Hydrogen Energy 2015;40(39):13676e84. [30] Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf 2003;46:3639e53. € [31] Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids. Int J Heat Fluid Flow 2008;29:1326e36. € [32] Abu-Nada E, Masoud Z, Oztop HF, Campo A. Effect of nanofluid variable properties on natural convection in enclosures. Int J Therm Sci 2010;49:479e91. [33] Nguyen CT, Desgranges F, Roy G, Galanis N, Mare T, Boucher S, et al. Temperature and particle-size dependent viscosity data for water-based nanofluids e hysteresis phenomenon. Int J Heat Fluid Flow 2007;28:1492e506. [34] Putra N, Roetzel W, Das SK. Natural convection of nanofluids. Heat Mass Transf 2003;39:775e84. [35] Wen D, Ding Y. Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. Int J Heat Mass Transf 2004;47:5181e8. [36] Xuan Y, Li Q. Experimental research on the viscosity of nanofluids. Chinese: Report of Nanjing University of Science and Technology; 1999. [37] Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys 1952;20:571e81. [38] Wasp FJ. SolideLiquid flow slurry pipeline transportation. Berlin: Trans. Tech. Pub; 1977. [39] Amiri A, Vafai K. Analysis of dispersion effects and nonthermal equilibrium, non-darcian, variable porosity, incompressible flow through porous media. Int J Heat Mass Transf 1994;37:939e54. [40] Chung T. J. Computational fluid dynamics. Cambridge University Press; 2010. [41] De Vahl Davis G. Natural convection of air in a square cavity, a Benchmark numerical solution. Int J Numer Methods Fluids 1983;3:249e64. [42] Fusegi T, Hyun JM, Kuwahara K, Farouk B. A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure. Int J Heat Mass Transf 1991;34:1543e57.
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Natural convection investigation in square cavity filled with nanofluid using dispersion model Abdelhamid Boualit a,*, Noureddine Zeraibi b, Toufik Chergui a, Mohamed Lebbi a, Lyes Boutina a, Salima Laouar a Unite de Recherche Appliquee en Energies Renouvelables, URAER, Centre de Developpement des Energies Renouvelables, CDER, BP: 88 Gart Taam Z.I, 47133, Ghardaı¨a, Bounoura, Algeria b Universite de Boumerdes, Faculte des hydrocarbures dept, Transport et equipement, 35000 Avenue de l'independance, Boumerdes, Algeria a
article info
abstract
Article history:
A cooling achieved with compact and efficient device is one of the major challenges
Received 3 March 2016
encountered in the promising technique of fuel cell stacks. The safe and reliable use of
Received in revised form
such a system is highly dependent on the efficiency of the assured heat transfer and
29 June 2016
consequently on the quality of the coolant used. To test the possible improvement of the
Accepted 8 July 2016
coolant performances, laminar natural convection in square cavity filled with copper-
Available online xxx
water nanofluid is numerically carried out taking into account the thermal dispersion effect on the heat transfer intensity. The finite element method is used to solve the governing
Keywords:
equations. The hydrodynamic structure of the flow and its thermal behavior are studied for
Nanofluid
a wide range of Rayleigh numbers. The obtained results showed an enhancement of heat
Natural convection
transfer with an increase in nanoparticle volume fraction for all examined Rayleigh
Thermal dispersion
numbers. However, it is found that an increase in nanoparticle diameter enhances heat
Square cavity
transfer only when thermal dispersion is significant. Correlation with 99.94% confidence
Fuel cell
coefficient is proposed to quantify the heat transfer intensity according to the Rayleigh number and particle diameter and concentration. © 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Aiming to improve thermal properties of working fluids, recent works relate that addition of low nanoparticle concentrations leads to a phenomenal improvement in fluid thermal conductivity with very low viscosity increase. For better use, stabilizers or surfactants are generally added to ultrasonically homogenized mixtures to avoid pipe clogging resulting from particle sedimentation and cluster formation.
Considering this significant improvement, such substances seem to be an appropriate solution to overcome the limitation in enhancing performance of engineering devices involving working fluids where heat transfer effectiveness is seriously hampered by the low thermal conductivity of the fluids used. The efficiency and compactness of the resulting devices are valuable assets in the feasibility and reliability of promising new technologies. In hydrogen technology, fuel cell
* Corresponding author. E-mail addresses: [email protected] (A. Boualit), [email protected] (N. Zeraibi), [email protected] (T. Chergui), mohamed. [email protected] (M. Lebbi), [email protected] (L. Boutina), [email protected] (S. Laouar). http://dx.doi.org/10.1016/j.ijhydene.2016.07.132 0360-3199/© 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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stacks are a typical example of this situation. Although they have high energy conversion efficiency and low operating temperature, these systems are particularly penalized by significant heat generation which must be removed efficiently to prevent any risk of overheating [1] or flooding [2,3]. Article [4] gives a clear and detailed review of thermal control processes required to insure safe and efficient use of fuel cells. Since works carried out by Maxwell (1904) [5], several theoretical [6,7] and experimental studies [8,9] have considered the case of liquids and gases containing suspended solid particles. These particles of some micrometers have improved thermal conductivity of the base fluid. On the other hand, several drawbacks in relation with particle size were recorded; such as, increasing pressure loss, sedimentation, pipe erosion and clogging… The last decades have witnessed the advent of new manufacturing processes allowing production of nanoscale particles. As advantages, several inconveniences related to particle size were removed. Fluids containing these fine particles are called nanofluids (Choi [10]). Recent works were focused on the determination of nanofluid effective thermal conductivity and effective viscosity [10e12]. It was reported that thermal conductivity of the base fluid is tremendously increased (more than 20%) with addition of a weak fraction of nanoparticles (1e5%). This increase depends on the particle diameter, particle volume fraction, as well as the thermal properties of the two components (base fluid and nanoparticles). Investigations involving heat transfer in nanofluids remain very difficult owing the interaction between several complex phenomena, such as Brownian motion of the nanoparticles, clusters formation, interaction and collision among particles, liquid layer structure at the liquid/particles interface, etc. The effects of these various phenomena were research topics of several recent works [13,14]. Concerning natural convection heat transfer, this phenomenon occurs widely in many industrial applications such as chemical reactors, heat exchangers, electronic cooling, double pane windows, and solar collectors etc. A comprehensive review was made by Baı¨ri et al. [15] where they underline the scientific and engineering fields where the knowledge of natural convection in enclosures is advantageously applied. The process relevance and its broad spectrum of applications have conducted to extensive studies dealing with mechanisms of heat transfer enhancement and efficiency improvement. Thus, several studies have treated a large variety of enclosure configurations with different shapes and inclinations [16e19], various thermal boundary conditions [20e22], and heat source distributions [23e25]. In the case of natural convection in enclosures filled with nanofluids, few experimental and numerical works were carried out [26e29], where the effort was primarily focused on the quantification of the heat transfer enhancement and the identification of the different mechanisms involved. However, published works have shown contradictory effects of nanoparticles on heat transfer. Khanafer et al. [30] studied Cuewater nanofluids in a two-dimensional rectangular enclosure. They reported that an increase in the nanoparticles volume fraction improves heat transfer at any given Grashof number.
The same study is carried out by Oztop and Abu-Nada [31] and the same remarks were formulated. On the other hand, Abu-Nada et al. [32] demonstrated that the heat transfer enhancement in natural convection depends mainly on the magnitude of Rayleigh number. Using experimental data of CuOewater and Al2O3ewater nanofluids reported by Nguyen et al. [33], they derived a correlation to quantify the effective viscosity and found for AL2O3ewater nanofluid a deterioration of the mean Nusselt number at high Rayleigh numbers by increasing the nanoparticles volume fraction above 5%. However, at low Rayleigh number, the mean Nusselt number is slightly enhanced by increasing the volume fraction of nanoparticles. Concerning CuOewater nanofluid at high Rayleigh numbers, they reported a continuous decrease in mean Nusselt number as the volume fraction of nanoparticles is increased. Experimentally, using Al2O3 and CuO water nanofluids, Putra et al. [34] remarked that the natural convection heat transfer coefficient is lower than that of the clear fluid. Wen and Ding [35] also showed heat transfer deterioration resulting from the addition of nanoscale particles. The present work investigates natural convection heat transfer in partially heated square cavity filled with nanofluid consisting of copper nanoparticles dispersed in water. Simulations are carried out to define the influence exerted by the volume fraction and the diameter of the nanoparticles on the fields of velocity and temperature for various Rayleigh numbers. The main objective of this work is to check whether the thermal dispersion affects the natural convection heat transfer.
Problem formulation The geometrical configuration of the problem considered is represented in Fig. 1. The flow field is a square enclosure with
Fig. 1 e Schematic representation of the examined geometry.
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an aspect ratio of unity. The horizontal walls of the cavity are assumed to be insulated while the vertical walls are maintained at constant and uniform temperatures. The left vertical wall is kept at a high temperature (Th) and the right wall at a low temperature (Tc). The temperature gradient (DT ¼ TheTc) between the two vertical walls is maintained at a constant value(20 K). The cavity is filled with a nanofluid made up of water and copper nanoparticles (Cu). The physical properties of those components (Table 1) are considered constant and given at the bulk temperature (Tb¼ (ThþTc)/2). In this study, Tb is taken equal to 293 K. The cavity length (L) is used as a control parameter in order to generate natural convection with various intensities since the temperature gradient (DT) and the fluid proprieties are considered invariant. Thus, the length is varied from 1.53 mm to 15.3 mm in order to generate natural convection of Ra ¼ 103 to 106. In this study, the overheat ratio d(d ¼ (ThTc)/Tb) is small enough that the use of the Boussinesq approximation is justified. Assuming the case of an incompressible fluid and a steady two-dimensional flow, the governing equations for laminar natural convection in a square enclosure are given by the following system of partial derivative equations: vu vv þ ¼0 vx vy
(1)
vu vu vp v vu v vu vv ¼ þ mnf ;0 2 þ þ rnf ;0 u þ v vx vy vx vx vx vy vy vx
(2)
vv vv vp v vu vv v vv rnf ;0 u þ v ¼ þ mnf ;0 þ þ 2 vx vy vy vx vy vx vy vy i h þ ∅rp;0 bp;0 þ ð1 ∅Þrf ;0 bf ;0 gðT Tc Þ
rCp
nf ;0
vT vT vT v v ¼ knf ;0 þ kd þ knf ;0 u þv vx vy vx vx vy vT þ kd vy
The physical properties of the nanofluid are given according to those of the base fluid and the nanoparticles. Using the mixture rule, the density and the heat capacitance of the nanofluid are given by: rnf ;0 ¼ ð1 ∅Þrf ;0 þ ∅rp;0
rCp
nf ;0
¼ ð1 ∅Þ rCp f ;0 þ ∅ rCp p;0
(4)
(5) (6)
The effective viscosity measures of Cu-water nanofluid reported by Xuan and Li [36] for a temperature ranging from 20 C to 70 C, match well with the values predicted from Brinkman's model [37]. This model is expressed as follow: mnf ;0 ¼
mf ;0 ð1 ∅Þ2:5
(7)
The nanoparticle volume fraction effect on the dynamic viscosity of the base fluid is represented in Fig. 2. There is shown that the nanoparticle addition increases the fluid viscosity. So, volume fractions of 1%, 10% and 20% give rise to a viscosity increase of 2.5%, 30% and 74% respectively. For successful use, the volume fraction of the nanoparticles must be between 1% and 10% to prevent, on the one hand, the flow deterioration by the drastic increase in the fluid viscosity, and on the other hand, the loss of the fluid homogeneity caused by the formation of nanoparticle clusters. The stagnant thermal conductivity of the nanofluid can be predicted from the following relation suggested by Wasp [38]: knf ;0 ¼ kf ;0
(3)
3
! kp;0 þ 2kf ;0 2∅ kf ;0 kp;0 kp;0 þ 2kf ;0 þ ∅ kf ;0 kp;0
(8)
where kp and kf are respectively the thermal conductivities of the nanoparticles and the base fluid. Based on porous media theory, the following relation predicts the dispersed thermal conductivity kd [39]: kd ¼ C rCp nf ;0 w∅dp
(9)
(x,y) are the cartesian coordinates of the geometry, (u,v) are the velocity components, p is the pressure, T is the temperature. r, m, (rCp), k, b are respectively the density, dynamic viscosity, heat capacitance, thermal conductivity and thermal expansion coefficient. ∅ is the volume fraction of nanoparticles in the base fluid. The indices p, f, nf designate, respectively, nanoparticles, base fluid and nanofluid. The index 0 accompanying a physical property indicates that this property is given at the bulk temperature Tb.
Table 1 e Physical properties of base fluid and Cu nanoparticles at 20 C. Physical properties
Base fluid (water) (at 20 C)
Cu nanoparticles
r (kg/m3) Cp (J/kg/K) k (W/m/K) m (kg/s/m) b (1/K)
997.64 4180.4 0.602 0.001 20.7$105
8954 383.1 386 e 5.1$105
Fig. 2 e Effect of nanoparticle volume fraction on the increase rate of the fluid dynamic viscosity.
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where w ¼ u2 þ v2 , dp is the nanoparticle diameter and C an unknown constant determined experimentally. For Cu-water nanofluids C is taken as equal to 0.4 [39]. According to this relation (Eq. (9)), thermal dispersion is function of flow intensity, volume fraction and physical proprieties of nanoparticles, as well as the proprieties of the base fluid. Introducing the dimensionless variables below to the system of equations (1)e(4) x y u v X ¼ ; Y ¼ ; U ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; V ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; P L L gbf ;0 DTL gbf ;0 DTL ¼
p T Tc and q ¼ rf ;0 bf ;0 gDTL Th Tc
leads to the following dimensionless system of equations: vU vV þ ¼0 vX vY U
(10)
rf ;0 vP mnf ;0 1 vU vU v vU v vU vV pffiffiffiffiffiffi þv ¼ þ 2 þ þ vX vY vX vY vY vX rnf ;0 vX mf ;0 Gr vX (11)
U
rf ;0 vP mnf ;0 1 vV vV v vU vV v vV pffiffiffiffiffiffi þV ¼ þ þ þ 2 vX vY vY vY rnf ;0 vY mf ;0 Gr vX vY vX rf ;0 b q þ rnf ;0 r;0 (12)
2 rCp f ;0 knf ;0 vq vq 1 v q v2 q þV ¼ U þ UW pffiffiffiffiffiffiffiffiffiffiffi þ vX vY rCp nf ;0 kf ;0 PrRa vX2 vY 2 rCp f ;0 U vW vq vW vq pffiffiffiffiffiffiffiffiffiffiffi þ þ rCp nf ;0 PrRa vX vX vY vY
Where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rCp nf ;0 dp pffiffiffiffiffiffi kd ∅ Pr Gr; W ¼ U2 þ V2 ; and br UW ¼ ; U¼C kf ;0 rCp f ;0 L ðrbÞp;0 ¼ ð1 ∅ÞðrbÞf ;0 þ ∅ ðrbÞf ;0 The dimensionless numbers are expressed using the physical properties of the base fluid given at the bulk temperature (Tb). The Prandtl number (Pr), the Grash of number (Gr) and the Rayleigh number (Ra) are writing as below: Pr ¼
kf ;0
; Gr ¼
r2f ;0 gbf ;0 DTL3 m2f ;0
; Ra ¼ PrGr
The dimensionless boundary conditions considered in this study are: On the top and bottom walls: U ¼ V ¼ vq/vY ¼ 0, on the left wall: U ¼ V ¼ 0 and q ¼ 1, and on the right wall: U ¼ V ¼ q ¼ 0. To quantify the heat transfer intensity between the hotwall and the fluid, local and mean Nusselt numbers are computed as follow: knf ;0 vq Nu ¼ kf ;0 vX X¼0
NuðYÞdY
(15)
0
Numerical procedure The system of equations (10)e(13) subject to the abovementioned dimensionless boundary conditions is solved using finite element method where the Galerkine formulation is employed for the weighted residues construction. The resolution algorithm is based on a so-called mixed method which allows a simultaneous determination of the velocity and the pressure, unlike projection methods where the two fields are uncoupled. To retain a sufficiently smooth solution for the investigated system, an element free of the LBB (LadyzhenskayaeBabuskaeBrezzi) stability constraint is taken into account. So, a 9-nodes quadrilateral element with the well-paired bi-linear interpolation function for the pressure and the bi-quadratic interpolation function for the velocities (Q9P4) is adopted [40]. As the solver, an iterative method based on the direct substitution scheme (Picard method) is used to solve the resulting system of nonlinear algebraic equations, sometimes with under relaxation to ensure convergence [40]. The convergence criterion ð Fkþ1 Fk < 104 Þ is used for all realized computations, where F represents the variables (U, V, P, q) and k is the iteration number.
Grid testing and code validation (13)
mf ;0 Cp f ;0
Z1 Numean ¼
(14)
To ensure grid independence of the solution, three grid combinations are examined. The grids M1, M2 and M3 containing respectively 20 20, 30 30 and 40 40 elements are explored for Ra ¼ 103, 104, 105, 106 and Pr ¼ 0.71 in the case of pure fluid (∅ ¼ 0). All the grids are non-uniform with more concentration of elements near the walls. Table 2 presents a comparison between the numerical results obtained using the various grids and those reported in literature. This comparison is based on the mean, the maximum and the minimum of the Nusselt number along the hot wall and their corresponding locations. On the other hand, the maximum longitudinal velocity (U) along the line X ¼ 0.5 and the maximum transversal velocity (V) along Y ¼ 0.5 are subject to comparison. The grids M2 and M3 have led to solutions which are in good agreement with the benchmark solution given by De Vahl Davis [41] for all values of Ra, as well as with those given by Fusegi et al. [42]. However, M2 requires less machine memory and less calculation effort. So, the grid M2 is adopted in this study. In the case of natural convection in square enclosures filled with nanofluids, the numerical code is validated by comparing the present results of the mean Nusselt number at the hot wall with those reported by Abu-Nada et al. [26]. This comparison, shown in Fig. 3, indicates clearly that the two sets of results are in good agreement.
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Table 2 e Comparison between the numerical results obtained using the various grids and those reported in literature. M1 Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
1.1178 1.5059 0.6910 0.1367 0.1388
(Y ¼ 0.087) (Y ¼ 1.00) (Y ¼ 0.8226) (X ¼ 0.1774)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
2.2472 3.5332 0.5867 0.1921 0.2317
(Y ¼ 0.1525) (Y ¼ 1.00) (Y ¼ 0.8226) (X ¼ 0.1299)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
4.5499 7.7857 0.7319 0.1301 0.2576
(Y ¼ 0.0870) (Y ¼ 1.00) (Y ¼ 0.8475) (X ¼ 0.0668)
Numean Numax (X ¼ 0) Numin (X ¼ 0) Umax (X ¼ 0.5) Vmax (Y ¼ 0.5)
9.1852 18.3463 (Y ¼ 0.0311) 0.9937 (Y ¼ 1.00) 0.0767 (Y ¼ 0.8475) 0.2557 (X ¼ 0.0311)
M2
M3
Ra ¼ 103 1.1178 1.1180 1.5059 (Y ¼ 0.0899) 1.5078 (Y ¼ 0.0927) 0.6914 (Y ¼ 1.00) 0.6890 (Y ¼ 1.00) 0.1369 (Y ¼ 0.8191) 0.1370 (Y ¼ 0.8127) 0.1387 (X ¼ 0.1809) 0.1387 (X ¼ 0.1740) Ra ¼ 104 2.2452 2.2450 3.5310 (Y ¼ 0.1477) 3.5304 (Y ¼ 0.1421) 0.5859 (Y ¼ 1.00) 0.5849 (Y ¼ 1.00) 0.1920 (Y ¼ 0.8191) 0.1920 (Y ¼ 0.8261) 0.2329 (X ¼ 0.1174) 0.2326 (X ¼ 0.1192) Ra ¼ 105 4.522 4.5227 7.7146 (Y ¼ 0.0769) 7.7217 (Y ¼ 0.0835) 0.7290 (Y ¼ 1.00) 0.7276 (Y ¼ 1.00) 0.1316 (Y ¼ 0.8523) 0.1303 (Y ¼ 0.8518) 0.2590 (X ¼ 0.0653) 0.2576 (X ¼ 0.0659) Ra ¼ 106 9.1141 9.0874 17.6971(Y ¼ 0.0357) 17.5479 (Y ¼ 0.0364) 0.9854 (Y ¼ 1.00) 0.9814 (Y ¼ 1.00) 0.0769 (Y ¼ 0.8523) 0.0769 (Y ¼ 0.8518) 0.2588 (X ¼ 0.0356) 0.2613 (X ¼ 0.0364)
Fig. 3 e Nusselt number evolution versus Ra number and comparison with Abu-Nada et al. [27] work.
Results and discussion In this work, a numerical study is performed for natural convection analysis in square cavity filled with a nanofluid made up water and a volume fraction (∅) of copper nanoparticles. The aim is to analyze the influence exerted by the concentration and the diameter of the nanoparticles on the flow structure and the heat transfer intensity. Thus, simulations are realized for various volume fractions (0 ∅ 0.2), different nanoparticle diameters (10 nm dp 100 nm) and a wide range of Rayleigh numbers. (103 Ra 106).
De Vahl Davis [41]
Fusegi et al. [42]
1.118 1.505 (Y ¼ 0.092) 0.692 (Y ¼ 1.00) 0.136 (Y ¼ 0.813) 0.138 (X ¼ 0.178)
1.105 1.420 0.764 0.132 0.131
(Y ¼ 0.083) (Y ¼ 1.000) (Y ¼ 0.833) (X ¼ 0.200)
2.243 3.528 (Y ¼ 0.143) 0.586 (Y ¼ 1.00) 0.192 (Y ¼ 0.823) 0.234 (X ¼ 0.119)
2.302 3.652 0.611 0.201 0.225
(Y ¼ 0.623) (Y ¼ 1.000) (Y ¼ 0.8 17) (X ¼ 0.1 17)
4.519 7.717 (Y ¼ 0.081) 0.729 (Y ¼ 1.00) 0.153 (Y ¼ 0.855) 0.261 (X ¼ 0.066)
4.646 7.795 0.787 0.147 0.247
(Y ¼ 0.083) (Y ¼ 1.000) (Y ¼ 0.855) (X ¼ 0.065)
8.799 17.925 (Y ¼ 0.0378) 0.989 (Y ¼ 1.00) 0.079 (Y ¼ 0.850) 0.262 (X ¼ 0.038)
9.012 17.670 (Y ¼ 0.0379) 1.257 (Y ¼ 1000) 0.084 (Y ¼ 0.856) 0.259 (X ¼ 0.033)
Fig. 4 presents the streamlines (left) and isotherms (right) for dp ¼ 10 nm and different Ra. Each diagram illustrates the volume fraction effects on the fields of velocity and temperature for a given Ra number (smooth lines for ∅ ¼ 0, long-dash lines for ∅ ¼ 0.1 and dash-dot lines for ∅ ¼ 0.2). According to the Rayleigh number, this figure shows that the increase in the nanoparticles volume fraction causes the displacement of the streamlines towards or far from the cavity center. For Ra ¼ 103, the streamlines are shifted to the cavity center when ∅ increases showing a decrease in the velocity field intensity. In the case of Ra ¼ 104, the streamlines close to the walls express a small shift to the center, while the lines located at the cavity center move in the walls direction. This is due to the velocity decrease near the walls and its increase at the center. For Ra¼105 and 106, all the streamlines move away from the center except those very close to the walls. These findings can be explained by the influence of the nanoparticles volume fraction on the viscous forces and the buoyancy forces interacting during a natural convection flow. Indeed, the nanoparticles addition influences simultaneously the two kinds of force. It increases the viscous forces effect by increasing the fluid viscosity (Eq. (7)) and the buoyancy forces effect by improving the fluid thermal conductivity. The thermal transfer improvement will be analyzed in detail further in this discussion. During natural convection in a closed enclosure, the two kinds of force are present with different intensities. In the flow field, one force can be locally or globally more significant than the other one. For Ra ¼ 103, the viscous forces are dominant throughout the flow field. So, the addition of the nanoparticles increases the fluid viscosity and slows the flow. For Ra ¼ 104, the viscous forces are significant near the walls whereas the buoyancy forces are dominant at the cavity center. In this
Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 5 e Plot of nanoparticle volume fraction effect on the vertical profile of the longitudinal velocity at Y ¼ 0.5 for various Rayleigh numbers.
Fig. 6 e Plot of nanoparticle volume fraction effect on the horizontal profile of the transversal velocity at X ¼ 0.5 for various Rayleigh numbers.
Fig. 4 e Streamlines (left) and isotherms (right). dp ¼ 100 nm. ∅ ¼ 0 (smooth lines), ∅ ¼ 0.1 (long-dash lines) and ∅ ¼ 0.2 (dash-dot lines). (a,b) Ra ¼ 103; (c,d) Ra ¼ 104; (e,f) Ra ¼ 105 and (g,h) Ra ¼ 106.
situation, the increase of the nanoparticles volume fraction reduces the velocity field near the walls and inversely at the center. All these findings are well illustrated by the Figs. 5 and 6 which show respectively the vertical profiles of the horizontal velocity at X ¼ 0.5 and the horizontal profiles of the vertical velocity at Y ¼ 0.5 for various nanoparticle volume fractions. Fig. 4 shows also the effect of the volume fraction increase on the heat transfer. For Ra ¼ 103 (Fig. 4-b), the isothermal
lines tend to become perfectly parallel to the vertical walls indicating the trend of the thermal transfer to be purely conductive. When Ra 104 (Fig. 4-d, f and h), the increase of the nanoparticles volume fraction causes widening of the thermal boundary layer located near the vertical walls causing the reduction of the temperature gradient. Fig. 7 illustrates the evolution of the Nusselt number along the hot wall for various values of Ra and ∅. This figure shows that an increase in the nanoparticles volume fraction leads to Nu improvement for all considered Ra numbers. According to Eq. (14), the Nusselt number is the product of the temperature gradient located near the hot wall and the thermal conductivity ratio (thermal conductivity of the nanofluide to that of the pure fluid). The addition of copper nanoparticles decreases the temperature gradient by widening the thermal boundary layer but increases more
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Fig. 7 e Effect of volume fraction on the Nusselt number evolution along the hot wall. dp ¼ 100 nm. Ra ¼ 103 (a), Ra ¼ 104 (b), Ra ¼ 105 (c) and Ra ¼ 106 (d).
significantly the thermal conductivity ratio what explains the Nusselt number improvement. Fig. 8 presents the effect of the nanoparticle diameter (dp) on heat transfer intensity (Nu) for various Rayleigh numbers. For Ra ¼ 103 and 104, Nusselt number is insensitive to nanoparticle diameter. However, for Ra ¼ 105 and 106 (Fig. 8-c and
d respectively) the Nusselt number increases with ascending values of dp. This observation is the consequence of the thermal dispersion effect which is detailed in the section consecrated to the mean Nusselt number. The evolution of the mean Nusselt number according to the nanoparticles volume fraction is given in Fig. 9 for various
Fig. 8 e Effect of nanoparticles diameter on the Nusselt number evolution along the hot wall. ∅ ¼ 0.2, Ra ¼ 103(a), Ra ¼ 104(b), Ra ¼ 105 (c) and Ra ¼ 106 (d). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 9 e Effect of volume fraction on the mean Nusselt number of the hot wall for various nanoparticle diameters (dp ¼ 10, 50 and 100 nm), and various Rayleigh numbers (Ra ¼ 103 (a), 104 (b), 105 (c) and 106 (d)).
values of the Rayleigh number and various nanoparticle diameters. Figures a, b, c and d associated, respectively, with the Rayleigh numbers 103,104,105 and 106, show a significant increase of mean Nusselt number which accompanies the increase of nanoparticle volume fraction. The same trend is observed for all Ra numbers studied. However, the effect of nanoparticle diameter on heat transfer intensity (Numean) varies according to Ra and ∅. The Fig. 9 a and b which correspond, respectively, to Ra ¼ 103 and Ra ¼ 104, show an insensitivity of Numean to nanoparticle diameter for all values of nanoparticles volume fraction belonging to the interval [0 , 0.2]. While, for Ra ¼ 105 and Ra ¼ 106 (Fig. 9-c and d respectively), the increase in nanoparticle diameter generates an intensification of heat transfer that is more significant for the high nanoparticle volume fraction and Rayleigh number. This last remark is explained by the contribution of thermal dispersion to Nusselt improvement, which is significant only for high Ra numbers and large nanoparticle diameters (Fig. 10). Furthermore, graphical representation of the effect exerted by the nanoparticle diameter on the mean Nusselt number for different concentrations and high Rayleigh numbers (Fig. 11), shows that the improvement in question is linearly proportional to the nanoparticle diameter. According to the Eq. (9), the thermal dispersion depends on the volume fraction and the diameter of the nanoparticles, but more primarily on the velocity field intensity. Fig. 12 shows the distribution of thermal dispersion along the flow field for ∅ ¼ 0.2, dp ¼ 100 nm and various values of the Rayleigh number (Ra ¼ 103,104,105 and 106). Indeed, this figure shows thermal dispersion intensification accompanying the Ra increase, but the most interesting is the location of its maximum values. For Ra ¼ 103 (Fig. 12-a), the maximum of the thermal dispersion (1.4105) is located coarsely halfway
between the centre and the cavity walls. However, for Ra ¼ 106 (Fig. 12-d), where the dependence of thermal transfer on nanoparticles diameter and thermal dispersion is more intense, the maximum of thermal dispersion (13.52 105) is very close to the vertical walls. Thereby, thermal dispersion affects directly the thermal boundary layer responsible of the heat transfer intensity causing an increase of Numean. To show how thermal dispersion acts on heat transfer intensity, Fig. 13 illustrates the form of the isothermal lines very close to the hot wall for ∅ ¼ 0.2, various values of Rayleigh number (a103, b104,c105 and d106) and different nanoparticle diameters (10 nm smooth lines, 50 nm long-dash lines and 100 nm dash-dot lines). Fig. 13-a and b, representative of Ra ¼ 103 and 104, respectively, show that the configuration of the thermal field is independent of dp, contrary to Ra ¼ 105 and 106 (Fig. 13-c and d respectively) where the isothermal lines are progressively closer with ascending values of dp. Therefore, it is observed that unlike the stagnant thermal conductivity, thermal dispersion reduces the thickness of the thermal boundary layer and generates a more significant temperature gradient, which consequently increases Nu and Numean. This observation is more visible for flows with high Rayleigh numbers (Ra ¼ 105and 106) characterized by an important thermal dispersion near the vertical walls. Moreover, this situation is accentuated by the increase of the nanoparticles diameter, which directly affects the thermal dispersion. The effectiveness of nanoparticle addition on natural convection heat transfer improvement in closed enclosures, is estimated by an increase rate given by
Eð%Þ ¼
Numean;nf Numean;f 100 Numean;f
(16)
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Fig. 10 e Comparison of Numean evolution according to nanofluids volume fraction with and without the thermal dispersion effect. (Ra ¼ 103, 104, 105 and 106, dp ¼ 10 and 100 nm).
Fig. 14 shows the increase rate, according to the nanoparticles volume fraction, for various Rayleigh numbers and different nanoparticle diameters (a10 nm, b50 nm, c100nm). It is shown that E(%) expresses the same trend of increase according to the volume fraction for all considered values of Ra and dp. However, the increase rate obtained for Ra ¼ 103 is much higher than those calculated for 104 Ra 106. This is due, on the one hand, to thermal conduction which dominates flows at low Rayleigh numbers (Ra ¼ 103) and, on the other hand, to the considerable improvement in thermal conductivity resulting from nanoparticles addition.
For 104 Ra 106, the increase rate is enhanced by the increase of the Rayleigh number. However, the effect of the nanoparticles diameter is apparent only for Ra ¼ 105 and 106. In this case, flows characterized by strong thermal dispersion express an enhancement of E(%) with increasing nanoparticle diameter (dp). Using the numerical results of the many simulations carried out in this study, correlation is proposed to predict the mean Nusselt number at the hot wall according to the Rayleigh number (103 Ra 106), the volume fraction (0 ∅ 0.2) and the nanoparticle diameter (10 nm dp 100 nm). This correlation is expressed as follow:
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Fig. 15 presents a comparison between the simulated values of Numean (symbols) and those predicted by the proposed correlation (lines). The error bars correspond to an error of 7%. As shown in this figure, the correlated values of Numean are in good agreement with the results of the performed simulations. The confidence coefficient of the above correlation is R2 ¼ 99.94%.
Conclusion
Fig. 11 e Effect of nanoparticle diameter on the mean Nusselt number for various Rayleigh numbers and various nanoparticle volume fractions.
1 þ 0:003Ra ∅ e2:116∅ Numean ¼ 0:1318Ra0:3091 1 þ 33344 1 þ 41:067Ra C dp Ra þ 0:329 ∅ L
(18)
In this study, numerical simulations are carried out in order to analyze the effect of nanoparticles addition on the laminar natural convection in a square enclosure. The hydrodynamic structure of the flow and its thermal behavior are studied for a wide range of Rayleigh numbers and nanoparticle concentrations. The results obtained have shown an enhancement of the mean Nusselt number with an increase of nanoparticle volume fraction for all examined Ra numbers ranging from 103 to 106. Furthermore, the effectiveness of nanoparticle addition on the improvement of natural convection heat transfer is more significant for Ra¼103 due to the dominance of conduction on the heat transfer mechanism. However, in a situation of convection-dominated heat transfer (104Ra106), Numean enhancement is proportional to Ra number.
Fig. 12 e Thermal dispersion field (£10¡5) for dp¼100 nm, ∅ ¼ 0.2 and various Rayleigh numbers (Ra¼103(a), 104 (b),105 (c) and 106 (d)). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 13 e Isotherms configuration near the hot wall for dp¼100 nm, ∅ ¼ 0.2 and various Rayleigh numbers (Ra ¼ 103 (a), 104 (b), 105 (c) and 106 (d)).
Fig. 14 e Evolution of increase rate of the mean Nusselt number according to the volume fraction for various Rayleigh numbers and various diameters (dp ¼ 10 nm (a), 50 nm (b) and 100 nm (c)). Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132
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Fig. 15 e Comparison between Numean values obtained by numerical simulations and those given by the suggested correlation(17).
For the effect of the nanoparticles size on the mean Nusselt number, it is found that an increase in particle diameter enhances heat transfer only when the thermal dispersion is significant. Thus, Numean is insensitive to nanoparticles size for Ra¼103 and 104, whereas it increases with increases of particle diameter for Ra¼105 and 106.
Nomenclature Cp dp g k L Nu Numean p P Pr Ra T u U v V x X y Y
Specific heat capacity, J/kg/K Nanoparticle diameter, nm Gravitational acceleration, m/s2 Thermal conductivity, W/m/K Cavity length, m Dimensionless local Nusselt number Dimensionless mean Nusselt number Pressure, Pa Dimensionless pressure Prandlt number Rayleigh number Temperature, K Longitudinal velocity, m/s Dimensionless longitudinal velocity Transversal velocity, m/s Dimensionless transversal velocity Longitudinal coordinate, m Dimensionless longitudinal coordinate Transversal coordinate, m Dimensionless transversal coordinate
Greek symbols b Thermal expansion coefficient, 1/K d Overheat ratio
q m r f
Dimensionless temperature Dynamic viscosity, kg/m/s Density, kg/m3 Nanoparticle volume fraction
Subscripts b Bulk c Cold d Dispersion f Fluid h Hot nf Nanofluid p Particle 0 Propriety given at the bulk temperature
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Please cite this article in press as: Boualit A, et al., Natural convection investigation in square cavity filled with nanofluid using dispersion model, International Journal of Hydrogen Energy (2016), http://dx.doi.org/10.1016/j.ijhydene.2016.07.132