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alumina nanofluid using Lattice Boltzmann method
International Communications in Heat and Mass Transfer 40 (2013) 67–77
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Effect of a magnetic field on natural convection in an open cavity subjugated to water/alumina nanofluid using Lattice Boltzmann method☆ GH.R. Kefayati Young Researchers Club, South Tehran Branch, Islamic Azad University, Tehran, Iran
a r t i c l e
i n f o
Available online 9 November 2012 Keywords: Natural convection Open enclosure Lattice Boltzmann method Nanofluid Magnetic field
a b s t r a c t In this paper, the effect of a magnetic field on natural convection in an open enclosure which subjugated to water/alumina nanofluid using Lattice Boltzmann method has been investigated. The cavity is filled with water and nanoparticles of Al2O3 at the presence of a magnetic field. Calculations were performed for Rayleigh numbers (Ra = 10 4–106), volume fractions of nanoparticles (φ = 0,0.02,0.04 and .0.06) and Hartmann number (0 ≤ Ha ≤ 90) with interval 30 while the magnetic field is considered horizontally. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and volume fractions. The magnetic field augments the effect of nanoparticles at Rayleigh number of Ra = 10 6 regularly. Just as the most effect of nanoparticles for Ra = 10 4 is observed at Ha = 30, so the most influence of nanoparticles occurs at Ha = 60 for Ra = 105. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Open cavities are applied in various engineering applications, such as solar thermal receiver, heat convection from extended surfaces in heat exchangers and solar collectors with insulated strips [1]. Numerical studies on open cavities have been performed in simulating various applications. Quere et al. [2] investigated open isothermal square cavities using aspect ratio of unity. Chan and Tien [3] essayed a numerical study on an open square enclosure with isothermal heated side and adiabatic top and bottom walls. Mohamad [4] studied isothermal inclined open cavities with aspect ratios of 0.5 to 2. Polat and Bilgen [5] investigated laminar natural convection in inclined open shallow cavities at inclination range of 0° to 45° with interval 15°. They found that the inclination angle of the heated plate is an important parameter which affecting volumetric flow rate and the heat transfer. Cha and Choi [6] examined an experimental study on an inclined open square enclosure which obtained similar findings with previous results. An appropriate method to accelerate and to shoot plasma into fusion devices or to produce high energy wind tunnels for simulating hypersonic flight is the use of a magnetic field. These types of problems also arise in electronic packages, microelectronic devices during their actions. Although application of a magnetic field does not limit to this applications and it was utilized at multifarious industries such as crystal growth in liquids, cooling of nuclear reactors and so on.
☆ Communicated by W.J. Minkowycz. E-mail addresses: [email protected], [email protected]. 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.10.024
Various investigations on natural and forced convection in the presence of a magnetic field were done by researchers with different numerical methods. For instance, some of them that were published recently can be named. Kahveci and Oztuna [7] investigated MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure. They showed that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. Moreover, they demonstrated the average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic and that flow and heat transfer have little dependence on the Prandtl number. Pirmohammadi and Ghassemi [8] studied the effect of a magnetic field on convection heat transfer inside a tilted square enclosure. They found that for a given inclination angle (θ), as the value of Hartmann number (Ha) increases, the convection heat transfer reduces. Furthermore, they obtained that at Ra = 10 4, value of Nusselt number depends strongly upon the inclination angle for relatively small values of Hartmann number and at Ra = 10 5, the Nusselt number increases up to about φ = 45° and then decrease as θ increases. Sathiyamoorthy and Chamkha [9] have done a numerical study for natural convection flow of electrically conducting liquid gallium in a square cavity where the bottom wall is uniformly heated and the left and right vertical wall are linearly heated while the top wall kept thermally insulated. They exhibited that the magnetic field with inclined angle has effects on the flow and heat transfer rates in the cavity. Sivasankaran et al. [10] investigated mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of a magnetic field numerically. They found that the
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Nomenclature B c ci cp F f f eq g g eq gy Gr Ha M Ma Nu Pr R Ra T x,y v
magnetic field lattice speed discrete particle speeds specific heat at constant pressure external forces density distribution functions equilibrium density distribution functions internal energy distribution functions equilibrium internal energy distribution functions gravity βg H3 ðT −T Þ Grashof number Gr ¼ y v2H C 2 2 Hartmann number Ha2 ¼ B Lμ σ e Lattice number Mach number Nusselt number Prandtl number constant of the gases βg H3 ðT −T Þ Rayleigh number Ra ¼ y vαH C temperature Cartesian coordinates magnitude velocity in y-direction
Greek letters σ electrical conductivity ωi weighted factor indirection i β thermal expansion coefficient τc relaxation time for temperature τv relaxation time for flow ϑ kinematic viscosity Δx lattice spacing Δt time increment α thermal diffusivity φ volume fraction μ dynamic viscosity ψ stream function value
Subscripts avg average C cold H hot f fluid nf nanofluid s solid
heat transfer rate increases with the phase deviation up to θ = π/2 and then it decreases for further increase in the phase deviation. Then, it was obtained that the heat transfer rate increases on increasing the amplitude ratio. Rahman et al. [11] studied the development of a magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure. Their results indicate that various Hartmann, Rayleigh and Reynolds numbers strongly affect the flow phenomenon and temperature field inside the cavity where in the channel these effects are less significant. Oztop et al. [12] studied mixed convection heat transfer characteristics for a lid-driven air flow within a square enclosure having a circular body. The authors found that the circular body has significant effects on flow field and temperature distribution. In addition, Oztop et al. [13] considered Laminar mixed convection flow in the presence of a magnetic field in a top sided lid-driven cavity heated by a corner heater.
They exhibit heat transfer decreases with increasing of Hartmann number. Nasrin and Parvin [14] made a numerical work on hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface. They indicated that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increment of Hartmann number. These investigations showed the magnetic field causes heat transfer to decline; therefore, a heat transfer improvement method is needful. Adding nanoparticles to fluid can be a good solution for this problem. The fluids that are traditionally used for heat transfer applications such as water, mineral oils and ethylene glycol have a rather low thermal conductivity and they can't play as an efficiency heat transfer agent. Nanoparticles are known as an efficient way for the improvement of thermal conductivity of the base fluids. Fluids with nanoparticles suspended in them are called nanofluids. Therefore, nanofluids have an anomalous high thermal conductivity at very low nanoparticle concentrations [15,16]. Many numerical, experimental and theoretical investigations were performed about natural convection flow of nanofluids in different shapes. Khanafer et al. [17] numerically investigated buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. They showed that the Nusselt number for the natural convection of nanofluids is increased by the enhancement of the volume fraction. Putra et al. [18] conducted experimentally about the heat transfer characteristics of nanofluids under natural convection inside a horizontal cylinder that had been heated and cooled from both ends. They reported unlike conduction or forced convection, a systematic and definite deterioration in natural convective heat transfer has been occurred in the presence of nanoparticles where this deterioration depends on particle density and concentration as well as the aspect ratio of the cylinder. Kim et al. [19] analytically researched the convective instability driven by buoyancy and heat transfer characteristics of nanofluids with theoretical models. Jahanshahi et al [20] numerically investigated the effects of the SiO2/water nanofluid on pure fluid as the thermal conductivity of nanofluid was obtained according to the experimental results and theoretical formulations on laminar natural convection heat transfer in a square enclosure. Their comparisons showed that the mean Nusselt number increases with volume fraction for the whole range of Rayleigh numbers in the experimental case but the heat transfer changes marginally in another studied case. Mahmoudi et al. [21] investigated numerical modeling of natural convection in an open enclosure with two vertical thin heat sources subjected to a nanofluid. They demonstrated that the average Nusselt number increases linearly with the increase in the solid volume fraction of nanoparticles. For more than two decades, Lattice Boltzmann method (LBM) has been demonstrated to be a very effective numerical tool for a broad variety of complex fluid flow phenomena that are problematic for conventional methods [22–28]. In comparison with traditional methods of computational fluid dynamic, The LBM algorithms are much easier to be implemented especially in complex geometries and multi component flows than traditional methods [29]. Mohamad et al. [30] studied natural convection in an open enclosure numerically with Lattice Boltzmann method. They investigated the effects of systematic analysis of aspect ratio on the physics of the flow and heat transfer. They have showed that increasing the aspect ratio for a given Rayleigh number decreases the rate of heat transfer up to the conduction limit. Recently, Kefayati et al. [31] studied the effect of SiO2/water nanofluid for heat transfer improvement in tall enclosures by Lattice Boltzmann method. They showed that the average Nusselt number increases with volume fraction for the whole range of Rayleigh numbers and the aspect ratios. They also showed that the effect of nanoparticles on heat transfer augments as the enclosure aspect ratio increases. Moreover, Kefayati et.al [32] investigated Prandtl number effect on natural convection MHD in an open cavity
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which has been filled respectively with liquid gallium, air and water by Lattice Boltzmann method. They exhibited heat transfer declines with the increment of Hartmann number, while this reduction is marginal for Ra = 10 3 by comparison with other Rayleigh numbers. The main aim of the present study is to identify the ability of Lattice Boltzmann method (LBM) for solving nanofluid and magnetic field simultaneously in complicated geometries and boundaries. In fact, it is endeavored to express the best situation for heat transfer with the alterations of multifarious considered parameters. Hence, the AL2O3–water nanofluid on laminar natural convection heat transfer at the presence of a magnetic field in an open cavity by LBM was investigated. The results of LBM are validated with previous numerical investigations and effects of all parameters (Rayleigh number, volume fraction, Hartmann number and aspect ratio) on flow field and temperature distribution are also considered.
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Table 1 Thermophysical properties of water and alumina. Property
Water
Al2O3
μ (kg/ms) cp (j/kg·k) ρ (kg/m3) β (k−1) k (w/m k)s
8.9 × 10−4 4179 997.1 2.1 × 10−4 0.6
– 765 3970 8.5 × 10−6 25
produced by the motion of an electrically conducting fluid is negligible compared to the applied magnetic field. Furthermore, it is assumed that the viscous dissipation and Joule heating are neglected. 2.2. Lattice Boltzmann method
2. Mathematical formulation 2.1. Problem statement The geometry of the present problem is shown in Fig. 1. It displays a two-dimensional enclosure with the height of H. The temperature of the enclosure's left wall is maintained at (TH). An external cold nanofluid enters into the enclosure from the east opening boundaries while the Al2O3–water nanofluid is correlated with the opening boundary at the constant temperature of TC. The top and bottom horizontal walls have been considered to be adiabatic i.e., nonconducting and impermeable to mass transfer. Thermophysical properties of the nanofluid are assumed to be constant (Table 1). The density variation in the nanofluid is approximated by the standard Boussinesq model. The enclosure is filled with a mixture of water and solid Alumina. The nanofluid is assumed to be Newtonian, incompressible and laminar. Moreover, it is considered while the liquid and solid nanoparticles are in thermal equilibrium, flowing at an equal velocity. The uniform magnetic field with constant magnitudes is applied horizontally. It is assumed that the induced magnetic field
For the incompressible problems, Lattice Boltzmann method (LBM) utilizes two distribution functions, f and g, for the flow and temperature fields respectively [33]. For the flow field: f i ðx þ ci Δt; t þ Δt Þ−f i ðx; t Þ ¼ −
1 eq f ðx; t Þ−f i ðx; t Þ þ ΔtF i : τv i
ð1Þ
1 eq g ðx; tÞ−g i ðx; tÞ τc i
ð2Þ
For the temperature field: g i ðx þ ci Δt; t þ ΔtÞ−g i ðx; t Þ ¼ −
where the discrete particle velocity vectors defined ci (Fig. 2a, b), Δt denotes lattice time step which is set to unity. τv, τc are the relaxation time for the flow and temperature fields, respectively. fieq, gieq are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties which are calculated with Eqs. (3) and (4) for flow and temperature fields respectively. Also F is an external force term. " eq
f i ðx; t Þ ¼ ωi ρ 1 þ 3
ci :u 9 ðci :uÞ2 3 u:u þ − 2 c4 2 c2 c2
# ð3Þ
cu eq ′ g i ¼ ω i T 1 þ 3 i2 c
ð4Þ
u and ρ are the macroscopic velocity and density, respectively, c is the . lattice speed and is equals to Δ x t where Δx is lattice space and simΔ ilar to lattice time step is equal to unity, ωi is the weighting factor for flow, and ωi′ is the weighting factor for temperature. D2Q9 model for flow and D2Q4 model for temperature are used in this investigation; therefore, the weighting factors and the discrete particle velocity vectors are different for these two models and they calculate as follows: For D2Q9 8 < 4=9 ωi ¼ 1=9 : 1=36
i¼0 i ¼ 1−4 : i ¼ 5−8
ð5Þ
The discrete velocities, ci, for the D2Q9 (Fig. 2a) are defined as follows: 8 0 i¼0 h . i h . i > < π π c cos ði−1Þ ; sin ði−1Þ i ¼ 1−4 : ci ¼ 2 2 h . . i h . . i > π π π π : cpffiffiffi ; sin ði−5Þ i ¼ 5−8 þ þ 2 cos ði−5Þ 2
Fig. 1. Geometry of the present study.
4
2
4
ð6Þ
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In the simulation the Boussinesq approximation is applied to the buoyancy force term. In this case the external force (F) appearing in Eq. (1) is given by: F i ¼ 3ωi g y βΔT
ð9Þ
where gy is the gravitational vector, ΔT is the temperature difference C that it is equal to (T–Tm) as T m ¼ T H þT and β is the thermal expansion 2 coefficient. Finally, the macroscopic quantities (ρ, u, T) can be calculated by the mentioned variables, with the following formula. Flow density : ρðx; t Þ ¼ ∑ f i ðx; t Þ
ð10Þ
Momentum : ρuðx; t Þ ¼ ∑ f i ðx; t Þci
ð11Þ
Temperature : ρRT ¼ ∑ g i ðx; t Þ:
ð12Þ
i
i
i
2.3. Boundary conditions 2.3.1. Flow Implementation of boundary conditions is very important for the simulation. The unknown distribution functions pointing to the fluid zone at the boundaries nodes must be specified (Fig. 2c). Concerning the no-slip boundary condition, bounce back boundary condition is used on the solid boundaries. The unknown density distribution functions on the east boundary can be determined by the following conditions: f 6;n ¼ f 6;n−1 ; f 7;n ¼ f 7;n−1 ; f 3;n ¼ f 3;n−1
ð13Þ
where n is the lattice on the boundary. 2.3.2. Temperature The north and south of the boundaries are adiabatic, as a consequence; bounce back boundary condition is used on them. Temperatures at the west and east walls are known. In the west wall TH = 1.0.
Fig. 2. (a) The discrete velocity vectors for D2Q9, (b) the discrete velocity vectors for D2Q4 and (c) the domain boundaries.
For D2Q4. The weighting factor for temperature is equal for each main four directions which is ωi′ = 0.25. The discrete velocities, ci, for the D2Q4 (Fig. 2b) are defined as follows: ci ¼
i−1 i−1 π ; sin π c i ¼ 1−4 : cos 2 2
ð7Þ
The kinematic viscosity (ϑ) and the thermal diffusivity (α) are then related to the relaxation times by: 1 2 ϑ ¼ τv − cS Δt 2
1 and α ¼ τc − cS 2 Δt 2
ð8Þ
where cs is the lattice speed of sound in media, it is equals to c =p3ffiffi .
Fig. 3. Grid independent test (A = 1, Pr = 0.71).
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Table 2 Comparison of average Nusselt number at hot wall. Ra
Present (LBM)
LBM [30]
FVM [4]
104 105 106
3.319 7.391 14.404
3.377 7.323 14.380
3.264 7.261 14.076
For the west wall ′ ′ g 1 ¼ T H ω 1 þ ω 3 −g 3 :
ð14Þ
For the east wall if u b 0 then: g 3;n ¼ 0−g 1;n :
ð15aÞ
if u > 0 then: g 3;n ¼ g 3;n−1 Fig. 4. Comparison of the temperature on axial midline between the present results and numerical results by Khanafer et al. [17] and Jahanshahi et al. [20] (pr = 6.2,φ = 0.1,Gr = 104).
Since we are using D2Q4, the unknown internal energy distribution function at the west and the east boundaries can be determined by the following conditions [30]:
ð15bÞ
2.4. Method of solution By fixing Rayleigh number, Prandtl number and Mach number the viscosity and thermal diffusivity are calculated from definition of these. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ma2 M2 Prc2 ϑ¼ Ra
Fig. 5. Comparison of the streamlines and isotherms between (a) numerical results by Mohamad et al. [30] and (b) the present results.
ð16Þ
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Fig. 6. Comparison of the streamlines and isotherms for natural convection at Ra = 105 and Ha = 50 between (a) the present results and (b) numerical results by Sathiyamoorthy and Chamkha [9].
where M is the number of lattices in y-direction (parallel to gravitational acceleration). Rayleigh and Prandtl numbers are defined as βgy M3 ðT H −T C Þ , and Pr ¼ αϑ, ϑα 1 constant ( c ¼ pffiffi3) and Mach
Ra ¼
respectively. Besides, speed of lattice is number was fixed at Ma = 0.1 in the
present study. After defining the whole parameters of Eq. (16), we can get viscosity and subsequently thermal diffusivity. Finally, Eq. (8) is used to calculate the relaxation times for density and temperature distribution functions. 2.5. Lattice Boltzmann method at the presence of a magnetic field The effect of the magnetic field was shown only in the force term where it is added to the buoyancy force term. For natural convection driven flow, the force term is: F n ¼ ρg y βΔT
ð17Þ
where gy is the gravitational vector, ρ is the density, ΔT is the temperature difference between hot and cold boundaries and β is the thermal expansion coefficient. But for the magnetic field at X-direction, the force term is:
The force is added to the collision process as: F i ¼ 3ωi Fci
where F = FB + Fn and the values of ωi and ci were shown before. 2.6. Lattice Boltzmann method for nanofluid The major control parameter of the test case is the Rayleigh number, βg H 3 PrðT −T Þ μc Ra ¼ y ϑ2 H C with Pr ¼ kp : The nanofluids were assumed to be similar to a pure fluid and then nanofluid qualities were gotten and they were applied for the equations. The density varies only in the buoyant force, which has been used only in the body force term by applying the Boussinesq approximation. The pertinent thermophysical properties are given in Table 1. The effective density of a nanofluid is given by [34]: ρnf ¼ ð1−φÞρf þ φρS
ð18Þ
ð20Þ
whereas the heat capacitance of the nanofluid and part of the Boussinesq term are [34]: ρcp
nf
F i ¼ 3ωi Fci :
ð19Þ
¼ ð1−φÞ ρcp þ φ ρcp f
ðρβÞnf ¼ ð1−φÞðρβÞf þ φðρβÞS
S
ð21Þ ð22Þ
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Ra=1E4
Ra=1E5
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Ra=1E6
Ha=0
Ha=30
Ha=60
Ha=90
Fig. 7. Comparison of the isotherms between nanofluids (- - -) (φ = 0.06) and base fluid (—) (φ = 0) at various Hartmann and Rayleigh numbers.
where φ is being the volume fraction of the solid particles, subscripts f, nf and s stand for base fluid, nanofluid and solid, respectively. The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is given by [35]: μ nf ¼
μf
: ð1−φÞ2:5
Nusselt number Nu is one of the most important dimensionless parameters in the description of the convective heat transport. The local Nusselt number and the average value at the hot wall are calculated as: NUy ¼ −
ð23Þ
ð25Þ
H
NUavg ¼ The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model where the nanoparticles are assumed to be the same and have spherical shapes [35]: knf ks þ 2kf þ 2φ kf −kS : ¼ kf kS þ 2kf −φ kf −kS
H ∂T ΔT ∂x
ð24Þ
1 ∫ NUy dy: H0
ð26Þ
Because of the convenience, a normalized average Nusselt number is defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water that is as follows: NUavg ðφÞ ¼
NUavg ðφÞ : NUavg ðφ ¼ 0Þ
ð27Þ
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Ra=1E4
Ra=1E5
Ra=1E6
Ha=0
Ha=30
Ha=60
Ha=90
Fig. 8. Comparison of the streamlines between nanofluids (- - -) (φ = 0.06) and base fluid (—) (φ = 0) at various Hartmann and Rayleigh numbers.
3. Code validation and grid independence The open enclosure was investigated at different Rayleigh numbers of 10 4, 10 5 and 10 6, with four various Hartmann numbers of Ha = 0, 30, 60 and 90. The open boundary was subjugated by nanofluid alumina-water while volume fractions vary from 0 to 0.06 as a magnetic field has been utilized in the open cavity horizontally. The LBM scheme is applied for obtaining the numerical simulations. An extensive mesh testing procedure was conducted to guarantee a grid independent solution. Ten different mesh combinations were
explored for the case of Ra = 10 4 and 10 5. The present code was tested for grid independence by calculating the average Nusselt number on the left wall. It was found that a grid size of 101_101 ensures a grid independent solution. It was confirmed that the grid size (101_101) ensures a grid independent solution as portrayed in Fig. 3. The present numerical method was validated at the three topics of this previous problem. At the first part, the method of the solution for nanofluid by Lattice Boltzmann method was validated by the results of Khanafer et al. [17] and Jahanshahi et al. [20]. Fig. 4 shows a comparison with temperature at the mid section of a cavity
GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77
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Ra=1E4
0.06
0.06
0.06
0.06
0.06
0.06
Ra=1E5
Ra=1E6
Fig. 9. Values of the average Nusselt number (NUavg) and normalized average Nusselt number (NUavg*) at different volume fractions and various Rayleigh and Hartmann numbers.
for Cu-water nanofluid where volume fraction is φ = 0.1. For the second part, the method of the solution for open cavity by Lattice Boltzmann method was compared with the consequences of Mohamad et al. [30] in Fig. 5 for the open cavity. Furthermore, Table 2 shows the comparison of average Nusselt number at the hot wall of the present study with the prediction of LBM [30] and Finite Volume Method (FVM) [4]. For the validation of MHD flow, we utilize it to solve the issue was studied by Sathiyamoorthy and Chamkha [9]. The isotherms and streamlines compared between two methods in Fig. 6. The comparisons demonstrate a good agreement with previous work. It should
be mentioned that the considered matter is MHD natural convection in a cavity with linearly heated west wall which is filled with Liquid Gallium with Pr = 0.025 and Ha = 50. These comparisons show that the present study has a good agreement with previous studies. 4. Results and discussions Fig. 7 shows a comparison between pure fluid (φ = 0) and nanofluid (φ = 0.06) for various Hartmann and Rayleigh numbers in the term of the isotherms. It is obvious that the isotherms near the hot wall as Rayleigh number augments at different Hartmann
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numbers. That's why the gradient of temperature and subsequently heat transfer increases by the enhancement of Rayleigh number. Moreover, at Ha = 0, the increase in Rayleigh number causes to occur an almost isothermal process at down half of the open enclosure (y b 1/2). The effect of the presence of the magnetic field is clear in the counter of the isotherms where the isotherms recede from the hot wall slowly and their gradients on the hot wall declines extremely in which it exposes the decrease in heat transfer in the open cavity. As can be seen, the isotherms of nanofluids incline toward the hot wall more than the isotherm of the pure fluid. In fact, it provokes the heat transfer to ameliorate with the augmentation of the volume fraction in multifarious Hartmann numbers. In addition, It is sensible that nanoparticles influence the isotherm further significantly at Ra = 10 4 toward pure fluid for Ha = 0. The isotherms of nanofluid and fluid overlap on each other perfectly at the open boundary where they get away from each other steadily while they are getting closer to the hot wall. Hence, the effect of nanoparticle at the open boundary partition is negligible and the effect augments gradually with the movement of the fluid within the enclosure. Fig. 8 compares the streamlines between the pure fluid (φ = 0) and nanofluid (φ = 0.06) for various Hartmann and Rayleigh numbers. It exhibits that the streamlines traverse into the open enclosure further as Rayleigh number rises. Furthermore, the maximum value of the stream function declines due to the augmentation of Hartmann number while the rate of the decrease is different for various Rayleigh numbers. For instance, from Ha = 0 to 30 the values of the maximum stream function decrease by 63%, 41% and 29% for Rayleigh numbers of Ra = 10 4, 10 5 and 10 6 respectively. Therefore, the trend exhibits that the effect of the magnetic field on the fluid flow drops with the increase in Rayleigh number. Effect of nanoparticles on the streamlines is clear when the value of the maximum stream function and the streamlines motivation increase for different Hartmann and Rayleigh numbers. Consequently, nanoparticles cause to improve the buoyancy-driven circulations in the open enclosure. Moreover, nanoparticles at different Hartmann and Rayleigh numbers show multifarious performances on the streamlines as they increase the maximum stream function erratically. Fig. 9 illustrates the influence of the nanoparticles volume fraction (φ) on the average Nusselt number (NUavg) and the normalized average Nusselt number (NUavg*) along the heated surface for different Hartmann and Rayleigh numbers. It demonstrates the average Nusselt number is raised steadily and linearly by the augmentation of the nanoparticles at various Hartmann and Rayleigh numbers. It is obvious that the effect of the magnetic field on Nusselt number declines with the enhancement of Rayleigh number. The best way to investigate the amount of the improvement in heat transfer by nanoparticles at different Hartmann numbers is the computation of the normalized average Nusselt number (NUavg*). At Ra = 10 4, the increase in Hartmann number from Ha = 0 to 30 act as an effective parameter to augment heat transfer whereas the other Hartmann numbers (Ha = 60 and 90) plunge it dramatically. At Ra = 10 5, the magnetic field assists the nanoparticles to increase heat transfer more than Ha = 0 but the positive impact vanishes from Ha = 60 to 90. The crucial role of the magnetic field to rise the heat transfer with the addition of nanoparticles is observed evidently at Ra = 10 6 while the normalized average Nusselt number (NUavg*) grows for different volume fractions with the increase in Hartmann number regularly. 5. Conclusions Natural convection in an open enclosure which is filled with a water/AL2O3 nanofluid has been conducted numerically at the presence of a magnetic field by Lattice Boltmann Method (LBM). This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10 4–10 6,
Hartmann number (Ha) Ha = 0, 30, 60 and 90, the volume fraction 0–6% and some conclusions were summarized as follows: a) A proper validation with previous numerical investigations demonstrates that Lattice Boltzmann method is an appropriate method for different applicable problems. b) Heat transfer declines with increase in Hartmann number for various Rayleigh numbers and volume fractions. c) The growth of nanoparticle volume fractions improves heat transfer for various Hartmann and Rayleigh numbers. d) At Ra = 10 4, the enhancement of Hartmann number provokes the influence of nanoparticles to enhance from Ha = 0 to 30 as the effect plummets significantly from Ha = 30 to 90. e) At Ra = 10 5, the weakest improvement in heat transfer with the addition of the nanoparticles happened at Ha = 0 among the investigated Hartmann numbers. f) The increase in the power of the magnetic field ameliorates the effect of nanoparticles on heat transfer at Ra = 10 6 frequently. References [1] I. Sazia, A.A. Mohamad, Suppressing free convection from a flat plate with poor conductor ribs, International Journal of Heat and Mass Transfer 42 (1) (1999) 2041–2051. [2] P. Le Quere, J.A.C. Humphrey, F.S. Sherman, Numerical calculation of thermally driven two-dimensional unsteady laminar flow in cavities of rectangular cross section, Numerical Heat Transfer 4 (1981) 249–283. [3] Y.L. Chan, C.L. Tien, A numerical study of two-dimensional natural convection in square open cavities, Numerical Heat Transfer 8 (1985) 65–80. [4] A.A. Mohamad, Natural convection in open cavities and slots, Numerical Heat Transfer 27 (1995) 705–716. [5] O. Polat, E. Bilgen, Laminar natural convection in inclined open shallow cavities, International Journal of Thermal Science 41 (2002) 360–368. [6] S.S. Cha, K.J. Choi, An interferometric investigation of open enclosure natural convection heat transfer, Experimental Heat Transfer 2 (1989) 27–40. [7] Kamil Kahveci, Semiha Oztuna, MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure, European Journal of Mechanics B/Fluids 28 (2009) 744–752. [8] Mohsen Pirmohammadi, Majid Ghassemi, Effect of magnetic field on convection heat transfer inside a tilted square enclosure, International Communications in Heat and Mass Transfer 36 (2009) 776–780. [9] M. Sathiyamoorthy, A. Chamkha, Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s), International Journal of Thermal Sciences 49 (2010) 1856–1865. [10] S. Sivasankaran, A. Malleswaran, Jinho Lee, Pon Sundar, Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls, International Journal of Heat and Mass Transfer 54 (2011) 512–525. [11] M.M. Rahman, S. Parvin, R. Saidur, N.A. Rahim, Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity, International Communications in Heat and Mass Transfer 38 (2011) 184–193. [12] H.F. Oztop, Z. Zhao, B. Yu, Conduction-combined forced and natural convection in lid-driven enclosures divided by a vertical solid partition, International Communications in Heat and Mass Transfer 36 (2009) 661–668. [13] F. Hakan, Khaled Al-Salem Oztop, Ioan Pop, MHD mixed convection in a lid-driven cavity with corner heater, International Journal of Heat and Mass Transfer 54 (15–16) (2011) 3494–3504. [14] Rehena Nasrin, Salma Parvin, Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface, International Communications in Heat and Mass Transfer 38 (6) (2011) 781–789. [15] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles, Netsu Bussei 7 (4) (1993) 227–233. [16] 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 (2005) 1107–1116. [17] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. [18] N. Putra, W. Roetzel, S.K. Das, Natural convection of nano-fluids, Heat and Mass Transfer 39 (2003) 775–784. [19] J. Kim, Y.T. Kang, C.K. Choi, Analysis of convective instability and heat transfer characteristics of nanofluids, Physics of Fluids 16 (2004) 2395–2401. [20] M. Jahanshahi, S.F. Hosseinizadeh, M. Alipanah, A. Dehghani, G.R. Vakilinejad, Numerical simulation of free convection based on experimental measured conductivity in a square cavity using Water/SiO2 nanofluid, International Communications in Heat and Mass Transfer 37 (2010) 687–694. [21] A.H. Mahmoudi, M. Shahi, A.M. Shahedin, N. Hemati, Numerical modeling of natural convection in an open enclosure with two vertical thin heat sources subjected to a nanofluid, International Communications in Heat and Mass Transfer 38 (2011) 110–118.
GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77 [22] Lin Zheng, Baochang Shi, Zhaoli Guo, Chuguang Zheng, Lattice Boltzmann equation for axisymmetric thermal flows, Computers & Fluids 39 (2010) 945–952. [23] A.A. Mohamad, R. Bennacer, M. El-Ganaoui, Double dispersion, natural convection in an open end cavity simulation via Lattice Boltzmann method, International Journal of Thermal Sciences 49 (2010) 1944–1953. [24] M.A. Mussa, S. Abdullah, C.S. Nor Azwadi, N. Muhamad, Simulation of natural convection heat transfer in an enclosure by the Lattice-Boltzmann method, Computers & Fluids 44 (2011) 162–168. [25] S.K. Choi, S.O. Kim, T.H. Lee, Y.I. Kim, D. Hahn, Computation of turbulent natural convection in a rectangular cavity with the Lattice Boltzmann method, Numerical Heat Transfer Part B 61 (2012) 492–504. [26] S.K. Choi, S.O. Kim, Comparative analysis of thermal models in the Lattice Boltzmann method for the simulation of natural convection in a square cavity, Numerical Heat Transfer Part B 60 (2011) 135–145. [27] H. Bararnia, K. Hooman, D.D. Ganji, Natural convection in a nanofluid filled portion cavity; the Lattice-Boltzmann Method, Numerical Heat Transfer Part A 59 (2011) 487–502. [28] B. Mondal, S.C. Mishra, Simulation of natural convection in the presence of volumetric radiation using the Lattice Boltzmann method, Numerical Heat Transfer Part A 55 (2009) 18–41. [29] H.K. Jeong, H.S. Yoon, M.Y. Ha, M. Tsutahara, An immersed boundary-thermal Lattice Boltzmann method using an equilibrium internal energy density approach
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Effect of a magnetic field on natural convection in an open cavity subjugated to water/alumina nanofluid using Lattice Boltzmann method☆ GH.R. Kefayati Young Researchers Club, South Tehran Branch, Islamic Azad University, Tehran, Iran
a r t i c l e
i n f o
Available online 9 November 2012 Keywords: Natural convection Open enclosure Lattice Boltzmann method Nanofluid Magnetic field
a b s t r a c t In this paper, the effect of a magnetic field on natural convection in an open enclosure which subjugated to water/alumina nanofluid using Lattice Boltzmann method has been investigated. The cavity is filled with water and nanoparticles of Al2O3 at the presence of a magnetic field. Calculations were performed for Rayleigh numbers (Ra = 10 4–106), volume fractions of nanoparticles (φ = 0,0.02,0.04 and .0.06) and Hartmann number (0 ≤ Ha ≤ 90) with interval 30 while the magnetic field is considered horizontally. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and volume fractions. The magnetic field augments the effect of nanoparticles at Rayleigh number of Ra = 10 6 regularly. Just as the most effect of nanoparticles for Ra = 10 4 is observed at Ha = 30, so the most influence of nanoparticles occurs at Ha = 60 for Ra = 105. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Open cavities are applied in various engineering applications, such as solar thermal receiver, heat convection from extended surfaces in heat exchangers and solar collectors with insulated strips [1]. Numerical studies on open cavities have been performed in simulating various applications. Quere et al. [2] investigated open isothermal square cavities using aspect ratio of unity. Chan and Tien [3] essayed a numerical study on an open square enclosure with isothermal heated side and adiabatic top and bottom walls. Mohamad [4] studied isothermal inclined open cavities with aspect ratios of 0.5 to 2. Polat and Bilgen [5] investigated laminar natural convection in inclined open shallow cavities at inclination range of 0° to 45° with interval 15°. They found that the inclination angle of the heated plate is an important parameter which affecting volumetric flow rate and the heat transfer. Cha and Choi [6] examined an experimental study on an inclined open square enclosure which obtained similar findings with previous results. An appropriate method to accelerate and to shoot plasma into fusion devices or to produce high energy wind tunnels for simulating hypersonic flight is the use of a magnetic field. These types of problems also arise in electronic packages, microelectronic devices during their actions. Although application of a magnetic field does not limit to this applications and it was utilized at multifarious industries such as crystal growth in liquids, cooling of nuclear reactors and so on.
☆ Communicated by W.J. Minkowycz. E-mail addresses: [email protected], [email protected]. 0735-1933/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.10.024
Various investigations on natural and forced convection in the presence of a magnetic field were done by researchers with different numerical methods. For instance, some of them that were published recently can be named. Kahveci and Oztuna [7] investigated MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure. They showed that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. Moreover, they demonstrated the average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic and that flow and heat transfer have little dependence on the Prandtl number. Pirmohammadi and Ghassemi [8] studied the effect of a magnetic field on convection heat transfer inside a tilted square enclosure. They found that for a given inclination angle (θ), as the value of Hartmann number (Ha) increases, the convection heat transfer reduces. Furthermore, they obtained that at Ra = 10 4, value of Nusselt number depends strongly upon the inclination angle for relatively small values of Hartmann number and at Ra = 10 5, the Nusselt number increases up to about φ = 45° and then decrease as θ increases. Sathiyamoorthy and Chamkha [9] have done a numerical study for natural convection flow of electrically conducting liquid gallium in a square cavity where the bottom wall is uniformly heated and the left and right vertical wall are linearly heated while the top wall kept thermally insulated. They exhibited that the magnetic field with inclined angle has effects on the flow and heat transfer rates in the cavity. Sivasankaran et al. [10] investigated mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of a magnetic field numerically. They found that the
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Nomenclature B c ci cp F f f eq g g eq gy Gr Ha M Ma Nu Pr R Ra T x,y v
magnetic field lattice speed discrete particle speeds specific heat at constant pressure external forces density distribution functions equilibrium density distribution functions internal energy distribution functions equilibrium internal energy distribution functions gravity βg H3 ðT −T Þ Grashof number Gr ¼ y v2H C 2 2 Hartmann number Ha2 ¼ B Lμ σ e Lattice number Mach number Nusselt number Prandtl number constant of the gases βg H3 ðT −T Þ Rayleigh number Ra ¼ y vαH C temperature Cartesian coordinates magnitude velocity in y-direction
Greek letters σ electrical conductivity ωi weighted factor indirection i β thermal expansion coefficient τc relaxation time for temperature τv relaxation time for flow ϑ kinematic viscosity Δx lattice spacing Δt time increment α thermal diffusivity φ volume fraction μ dynamic viscosity ψ stream function value
Subscripts avg average C cold H hot f fluid nf nanofluid s solid
heat transfer rate increases with the phase deviation up to θ = π/2 and then it decreases for further increase in the phase deviation. Then, it was obtained that the heat transfer rate increases on increasing the amplitude ratio. Rahman et al. [11] studied the development of a magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure. Their results indicate that various Hartmann, Rayleigh and Reynolds numbers strongly affect the flow phenomenon and temperature field inside the cavity where in the channel these effects are less significant. Oztop et al. [12] studied mixed convection heat transfer characteristics for a lid-driven air flow within a square enclosure having a circular body. The authors found that the circular body has significant effects on flow field and temperature distribution. In addition, Oztop et al. [13] considered Laminar mixed convection flow in the presence of a magnetic field in a top sided lid-driven cavity heated by a corner heater.
They exhibit heat transfer decreases with increasing of Hartmann number. Nasrin and Parvin [14] made a numerical work on hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface. They indicated that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increment of Hartmann number. These investigations showed the magnetic field causes heat transfer to decline; therefore, a heat transfer improvement method is needful. Adding nanoparticles to fluid can be a good solution for this problem. The fluids that are traditionally used for heat transfer applications such as water, mineral oils and ethylene glycol have a rather low thermal conductivity and they can't play as an efficiency heat transfer agent. Nanoparticles are known as an efficient way for the improvement of thermal conductivity of the base fluids. Fluids with nanoparticles suspended in them are called nanofluids. Therefore, nanofluids have an anomalous high thermal conductivity at very low nanoparticle concentrations [15,16]. Many numerical, experimental and theoretical investigations were performed about natural convection flow of nanofluids in different shapes. Khanafer et al. [17] numerically investigated buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. They showed that the Nusselt number for the natural convection of nanofluids is increased by the enhancement of the volume fraction. Putra et al. [18] conducted experimentally about the heat transfer characteristics of nanofluids under natural convection inside a horizontal cylinder that had been heated and cooled from both ends. They reported unlike conduction or forced convection, a systematic and definite deterioration in natural convective heat transfer has been occurred in the presence of nanoparticles where this deterioration depends on particle density and concentration as well as the aspect ratio of the cylinder. Kim et al. [19] analytically researched the convective instability driven by buoyancy and heat transfer characteristics of nanofluids with theoretical models. Jahanshahi et al [20] numerically investigated the effects of the SiO2/water nanofluid on pure fluid as the thermal conductivity of nanofluid was obtained according to the experimental results and theoretical formulations on laminar natural convection heat transfer in a square enclosure. Their comparisons showed that the mean Nusselt number increases with volume fraction for the whole range of Rayleigh numbers in the experimental case but the heat transfer changes marginally in another studied case. Mahmoudi et al. [21] investigated numerical modeling of natural convection in an open enclosure with two vertical thin heat sources subjected to a nanofluid. They demonstrated that the average Nusselt number increases linearly with the increase in the solid volume fraction of nanoparticles. For more than two decades, Lattice Boltzmann method (LBM) has been demonstrated to be a very effective numerical tool for a broad variety of complex fluid flow phenomena that are problematic for conventional methods [22–28]. In comparison with traditional methods of computational fluid dynamic, The LBM algorithms are much easier to be implemented especially in complex geometries and multi component flows than traditional methods [29]. Mohamad et al. [30] studied natural convection in an open enclosure numerically with Lattice Boltzmann method. They investigated the effects of systematic analysis of aspect ratio on the physics of the flow and heat transfer. They have showed that increasing the aspect ratio for a given Rayleigh number decreases the rate of heat transfer up to the conduction limit. Recently, Kefayati et al. [31] studied the effect of SiO2/water nanofluid for heat transfer improvement in tall enclosures by Lattice Boltzmann method. They showed that the average Nusselt number increases with volume fraction for the whole range of Rayleigh numbers and the aspect ratios. They also showed that the effect of nanoparticles on heat transfer augments as the enclosure aspect ratio increases. Moreover, Kefayati et.al [32] investigated Prandtl number effect on natural convection MHD in an open cavity
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which has been filled respectively with liquid gallium, air and water by Lattice Boltzmann method. They exhibited heat transfer declines with the increment of Hartmann number, while this reduction is marginal for Ra = 10 3 by comparison with other Rayleigh numbers. The main aim of the present study is to identify the ability of Lattice Boltzmann method (LBM) for solving nanofluid and magnetic field simultaneously in complicated geometries and boundaries. In fact, it is endeavored to express the best situation for heat transfer with the alterations of multifarious considered parameters. Hence, the AL2O3–water nanofluid on laminar natural convection heat transfer at the presence of a magnetic field in an open cavity by LBM was investigated. The results of LBM are validated with previous numerical investigations and effects of all parameters (Rayleigh number, volume fraction, Hartmann number and aspect ratio) on flow field and temperature distribution are also considered.
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Table 1 Thermophysical properties of water and alumina. Property
Water
Al2O3
μ (kg/ms) cp (j/kg·k) ρ (kg/m3) β (k−1) k (w/m k)s
8.9 × 10−4 4179 997.1 2.1 × 10−4 0.6
– 765 3970 8.5 × 10−6 25
produced by the motion of an electrically conducting fluid is negligible compared to the applied magnetic field. Furthermore, it is assumed that the viscous dissipation and Joule heating are neglected. 2.2. Lattice Boltzmann method
2. Mathematical formulation 2.1. Problem statement The geometry of the present problem is shown in Fig. 1. It displays a two-dimensional enclosure with the height of H. The temperature of the enclosure's left wall is maintained at (TH). An external cold nanofluid enters into the enclosure from the east opening boundaries while the Al2O3–water nanofluid is correlated with the opening boundary at the constant temperature of TC. The top and bottom horizontal walls have been considered to be adiabatic i.e., nonconducting and impermeable to mass transfer. Thermophysical properties of the nanofluid are assumed to be constant (Table 1). The density variation in the nanofluid is approximated by the standard Boussinesq model. The enclosure is filled with a mixture of water and solid Alumina. The nanofluid is assumed to be Newtonian, incompressible and laminar. Moreover, it is considered while the liquid and solid nanoparticles are in thermal equilibrium, flowing at an equal velocity. The uniform magnetic field with constant magnitudes is applied horizontally. It is assumed that the induced magnetic field
For the incompressible problems, Lattice Boltzmann method (LBM) utilizes two distribution functions, f and g, for the flow and temperature fields respectively [33]. For the flow field: f i ðx þ ci Δt; t þ Δt Þ−f i ðx; t Þ ¼ −
1 eq f ðx; t Þ−f i ðx; t Þ þ ΔtF i : τv i
ð1Þ
1 eq g ðx; tÞ−g i ðx; tÞ τc i
ð2Þ
For the temperature field: g i ðx þ ci Δt; t þ ΔtÞ−g i ðx; t Þ ¼ −
where the discrete particle velocity vectors defined ci (Fig. 2a, b), Δt denotes lattice time step which is set to unity. τv, τc are the relaxation time for the flow and temperature fields, respectively. fieq, gieq are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties which are calculated with Eqs. (3) and (4) for flow and temperature fields respectively. Also F is an external force term. " eq
f i ðx; t Þ ¼ ωi ρ 1 þ 3
ci :u 9 ðci :uÞ2 3 u:u þ − 2 c4 2 c2 c2
# ð3Þ
cu eq ′ g i ¼ ω i T 1 þ 3 i2 c
ð4Þ
u and ρ are the macroscopic velocity and density, respectively, c is the . lattice speed and is equals to Δ x t where Δx is lattice space and simΔ ilar to lattice time step is equal to unity, ωi is the weighting factor for flow, and ωi′ is the weighting factor for temperature. D2Q9 model for flow and D2Q4 model for temperature are used in this investigation; therefore, the weighting factors and the discrete particle velocity vectors are different for these two models and they calculate as follows: For D2Q9 8 < 4=9 ωi ¼ 1=9 : 1=36
i¼0 i ¼ 1−4 : i ¼ 5−8
ð5Þ
The discrete velocities, ci, for the D2Q9 (Fig. 2a) are defined as follows: 8 0 i¼0 h . i h . i > < π π c cos ði−1Þ ; sin ði−1Þ i ¼ 1−4 : ci ¼ 2 2 h . . i h . . i > π π π π : cpffiffiffi ; sin ði−5Þ i ¼ 5−8 þ þ 2 cos ði−5Þ 2
Fig. 1. Geometry of the present study.
4
2
4
ð6Þ
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In the simulation the Boussinesq approximation is applied to the buoyancy force term. In this case the external force (F) appearing in Eq. (1) is given by: F i ¼ 3ωi g y βΔT
ð9Þ
where gy is the gravitational vector, ΔT is the temperature difference C that it is equal to (T–Tm) as T m ¼ T H þT and β is the thermal expansion 2 coefficient. Finally, the macroscopic quantities (ρ, u, T) can be calculated by the mentioned variables, with the following formula. Flow density : ρðx; t Þ ¼ ∑ f i ðx; t Þ
ð10Þ
Momentum : ρuðx; t Þ ¼ ∑ f i ðx; t Þci
ð11Þ
Temperature : ρRT ¼ ∑ g i ðx; t Þ:
ð12Þ
i
i
i
2.3. Boundary conditions 2.3.1. Flow Implementation of boundary conditions is very important for the simulation. The unknown distribution functions pointing to the fluid zone at the boundaries nodes must be specified (Fig. 2c). Concerning the no-slip boundary condition, bounce back boundary condition is used on the solid boundaries. The unknown density distribution functions on the east boundary can be determined by the following conditions: f 6;n ¼ f 6;n−1 ; f 7;n ¼ f 7;n−1 ; f 3;n ¼ f 3;n−1
ð13Þ
where n is the lattice on the boundary. 2.3.2. Temperature The north and south of the boundaries are adiabatic, as a consequence; bounce back boundary condition is used on them. Temperatures at the west and east walls are known. In the west wall TH = 1.0.
Fig. 2. (a) The discrete velocity vectors for D2Q9, (b) the discrete velocity vectors for D2Q4 and (c) the domain boundaries.
For D2Q4. The weighting factor for temperature is equal for each main four directions which is ωi′ = 0.25. The discrete velocities, ci, for the D2Q4 (Fig. 2b) are defined as follows: ci ¼
i−1 i−1 π ; sin π c i ¼ 1−4 : cos 2 2
ð7Þ
The kinematic viscosity (ϑ) and the thermal diffusivity (α) are then related to the relaxation times by: 1 2 ϑ ¼ τv − cS Δt 2
1 and α ¼ τc − cS 2 Δt 2
ð8Þ
where cs is the lattice speed of sound in media, it is equals to c =p3ffiffi .
Fig. 3. Grid independent test (A = 1, Pr = 0.71).
GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77
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Table 2 Comparison of average Nusselt number at hot wall. Ra
Present (LBM)
LBM [30]
FVM [4]
104 105 106
3.319 7.391 14.404
3.377 7.323 14.380
3.264 7.261 14.076
For the west wall ′ ′ g 1 ¼ T H ω 1 þ ω 3 −g 3 :
ð14Þ
For the east wall if u b 0 then: g 3;n ¼ 0−g 1;n :
ð15aÞ
if u > 0 then: g 3;n ¼ g 3;n−1 Fig. 4. Comparison of the temperature on axial midline between the present results and numerical results by Khanafer et al. [17] and Jahanshahi et al. [20] (pr = 6.2,φ = 0.1,Gr = 104).
Since we are using D2Q4, the unknown internal energy distribution function at the west and the east boundaries can be determined by the following conditions [30]:
ð15bÞ
2.4. Method of solution By fixing Rayleigh number, Prandtl number and Mach number the viscosity and thermal diffusivity are calculated from definition of these. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ma2 M2 Prc2 ϑ¼ Ra
Fig. 5. Comparison of the streamlines and isotherms between (a) numerical results by Mohamad et al. [30] and (b) the present results.
ð16Þ
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GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77
Fig. 6. Comparison of the streamlines and isotherms for natural convection at Ra = 105 and Ha = 50 between (a) the present results and (b) numerical results by Sathiyamoorthy and Chamkha [9].
where M is the number of lattices in y-direction (parallel to gravitational acceleration). Rayleigh and Prandtl numbers are defined as βgy M3 ðT H −T C Þ , and Pr ¼ αϑ, ϑα 1 constant ( c ¼ pffiffi3) and Mach
Ra ¼
respectively. Besides, speed of lattice is number was fixed at Ma = 0.1 in the
present study. After defining the whole parameters of Eq. (16), we can get viscosity and subsequently thermal diffusivity. Finally, Eq. (8) is used to calculate the relaxation times for density and temperature distribution functions. 2.5. Lattice Boltzmann method at the presence of a magnetic field The effect of the magnetic field was shown only in the force term where it is added to the buoyancy force term. For natural convection driven flow, the force term is: F n ¼ ρg y βΔT
ð17Þ
where gy is the gravitational vector, ρ is the density, ΔT is the temperature difference between hot and cold boundaries and β is the thermal expansion coefficient. But for the magnetic field at X-direction, the force term is:
The force is added to the collision process as: F i ¼ 3ωi Fci
where F = FB + Fn and the values of ωi and ci were shown before. 2.6. Lattice Boltzmann method for nanofluid The major control parameter of the test case is the Rayleigh number, βg H 3 PrðT −T Þ μc Ra ¼ y ϑ2 H C with Pr ¼ kp : The nanofluids were assumed to be similar to a pure fluid and then nanofluid qualities were gotten and they were applied for the equations. The density varies only in the buoyant force, which has been used only in the body force term by applying the Boussinesq approximation. The pertinent thermophysical properties are given in Table 1. The effective density of a nanofluid is given by [34]: ρnf ¼ ð1−φÞρf þ φρS
ð18Þ
ð20Þ
whereas the heat capacitance of the nanofluid and part of the Boussinesq term are [34]: ρcp
nf
F i ¼ 3ωi Fci :
ð19Þ
¼ ð1−φÞ ρcp þ φ ρcp f
ðρβÞnf ¼ ð1−φÞðρβÞf þ φðρβÞS
S
ð21Þ ð22Þ
GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77
Ra=1E4
Ra=1E5
73
Ra=1E6
Ha=0
Ha=30
Ha=60
Ha=90
Fig. 7. Comparison of the isotherms between nanofluids (- - -) (φ = 0.06) and base fluid (—) (φ = 0) at various Hartmann and Rayleigh numbers.
where φ is being the volume fraction of the solid particles, subscripts f, nf and s stand for base fluid, nanofluid and solid, respectively. The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is given by [35]: μ nf ¼
μf
: ð1−φÞ2:5
Nusselt number Nu is one of the most important dimensionless parameters in the description of the convective heat transport. The local Nusselt number and the average value at the hot wall are calculated as: NUy ¼ −
ð23Þ
ð25Þ
H
NUavg ¼ The effective thermal conductivity of the nanofluid can be approximated by the Maxwell–Garnetts (MG) model where the nanoparticles are assumed to be the same and have spherical shapes [35]: knf ks þ 2kf þ 2φ kf −kS : ¼ kf kS þ 2kf −φ kf −kS
H ∂T ΔT ∂x
ð24Þ
1 ∫ NUy dy: H0
ð26Þ
Because of the convenience, a normalized average Nusselt number is defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water that is as follows: NUavg ðφÞ ¼
NUavg ðφÞ : NUavg ðφ ¼ 0Þ
ð27Þ
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GH.R. Kefayati / International Communications in Heat and Mass Transfer 40 (2013) 67–77
Ra=1E4
Ra=1E5
Ra=1E6
Ha=0
Ha=30
Ha=60
Ha=90
Fig. 8. Comparison of the streamlines between nanofluids (- - -) (φ = 0.06) and base fluid (—) (φ = 0) at various Hartmann and Rayleigh numbers.
3. Code validation and grid independence The open enclosure was investigated at different Rayleigh numbers of 10 4, 10 5 and 10 6, with four various Hartmann numbers of Ha = 0, 30, 60 and 90. The open boundary was subjugated by nanofluid alumina-water while volume fractions vary from 0 to 0.06 as a magnetic field has been utilized in the open cavity horizontally. The LBM scheme is applied for obtaining the numerical simulations. An extensive mesh testing procedure was conducted to guarantee a grid independent solution. Ten different mesh combinations were
explored for the case of Ra = 10 4 and 10 5. The present code was tested for grid independence by calculating the average Nusselt number on the left wall. It was found that a grid size of 101_101 ensures a grid independent solution. It was confirmed that the grid size (101_101) ensures a grid independent solution as portrayed in Fig. 3. The present numerical method was validated at the three topics of this previous problem. At the first part, the method of the solution for nanofluid by Lattice Boltzmann method was validated by the results of Khanafer et al. [17] and Jahanshahi et al. [20]. Fig. 4 shows a comparison with temperature at the mid section of a cavity
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75
Ra=1E4
0.06
0.06
0.06
0.06
0.06
0.06
Ra=1E5
Ra=1E6
Fig. 9. Values of the average Nusselt number (NUavg) and normalized average Nusselt number (NUavg*) at different volume fractions and various Rayleigh and Hartmann numbers.
for Cu-water nanofluid where volume fraction is φ = 0.1. For the second part, the method of the solution for open cavity by Lattice Boltzmann method was compared with the consequences of Mohamad et al. [30] in Fig. 5 for the open cavity. Furthermore, Table 2 shows the comparison of average Nusselt number at the hot wall of the present study with the prediction of LBM [30] and Finite Volume Method (FVM) [4]. For the validation of MHD flow, we utilize it to solve the issue was studied by Sathiyamoorthy and Chamkha [9]. The isotherms and streamlines compared between two methods in Fig. 6. The comparisons demonstrate a good agreement with previous work. It should
be mentioned that the considered matter is MHD natural convection in a cavity with linearly heated west wall which is filled with Liquid Gallium with Pr = 0.025 and Ha = 50. These comparisons show that the present study has a good agreement with previous studies. 4. Results and discussions Fig. 7 shows a comparison between pure fluid (φ = 0) and nanofluid (φ = 0.06) for various Hartmann and Rayleigh numbers in the term of the isotherms. It is obvious that the isotherms near the hot wall as Rayleigh number augments at different Hartmann
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numbers. That's why the gradient of temperature and subsequently heat transfer increases by the enhancement of Rayleigh number. Moreover, at Ha = 0, the increase in Rayleigh number causes to occur an almost isothermal process at down half of the open enclosure (y b 1/2). The effect of the presence of the magnetic field is clear in the counter of the isotherms where the isotherms recede from the hot wall slowly and their gradients on the hot wall declines extremely in which it exposes the decrease in heat transfer in the open cavity. As can be seen, the isotherms of nanofluids incline toward the hot wall more than the isotherm of the pure fluid. In fact, it provokes the heat transfer to ameliorate with the augmentation of the volume fraction in multifarious Hartmann numbers. In addition, It is sensible that nanoparticles influence the isotherm further significantly at Ra = 10 4 toward pure fluid for Ha = 0. The isotherms of nanofluid and fluid overlap on each other perfectly at the open boundary where they get away from each other steadily while they are getting closer to the hot wall. Hence, the effect of nanoparticle at the open boundary partition is negligible and the effect augments gradually with the movement of the fluid within the enclosure. Fig. 8 compares the streamlines between the pure fluid (φ = 0) and nanofluid (φ = 0.06) for various Hartmann and Rayleigh numbers. It exhibits that the streamlines traverse into the open enclosure further as Rayleigh number rises. Furthermore, the maximum value of the stream function declines due to the augmentation of Hartmann number while the rate of the decrease is different for various Rayleigh numbers. For instance, from Ha = 0 to 30 the values of the maximum stream function decrease by 63%, 41% and 29% for Rayleigh numbers of Ra = 10 4, 10 5 and 10 6 respectively. Therefore, the trend exhibits that the effect of the magnetic field on the fluid flow drops with the increase in Rayleigh number. Effect of nanoparticles on the streamlines is clear when the value of the maximum stream function and the streamlines motivation increase for different Hartmann and Rayleigh numbers. Consequently, nanoparticles cause to improve the buoyancy-driven circulations in the open enclosure. Moreover, nanoparticles at different Hartmann and Rayleigh numbers show multifarious performances on the streamlines as they increase the maximum stream function erratically. Fig. 9 illustrates the influence of the nanoparticles volume fraction (φ) on the average Nusselt number (NUavg) and the normalized average Nusselt number (NUavg*) along the heated surface for different Hartmann and Rayleigh numbers. It demonstrates the average Nusselt number is raised steadily and linearly by the augmentation of the nanoparticles at various Hartmann and Rayleigh numbers. It is obvious that the effect of the magnetic field on Nusselt number declines with the enhancement of Rayleigh number. The best way to investigate the amount of the improvement in heat transfer by nanoparticles at different Hartmann numbers is the computation of the normalized average Nusselt number (NUavg*). At Ra = 10 4, the increase in Hartmann number from Ha = 0 to 30 act as an effective parameter to augment heat transfer whereas the other Hartmann numbers (Ha = 60 and 90) plunge it dramatically. At Ra = 10 5, the magnetic field assists the nanoparticles to increase heat transfer more than Ha = 0 but the positive impact vanishes from Ha = 60 to 90. The crucial role of the magnetic field to rise the heat transfer with the addition of nanoparticles is observed evidently at Ra = 10 6 while the normalized average Nusselt number (NUavg*) grows for different volume fractions with the increase in Hartmann number regularly. 5. Conclusions Natural convection in an open enclosure which is filled with a water/AL2O3 nanofluid has been conducted numerically at the presence of a magnetic field by Lattice Boltmann Method (LBM). This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10 4–10 6,
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