Lattice Boltzmann simulation of alumina-water nanofluid in a square cavity
© He et al; licensee Springer. 2011
Received: 30 October 2010
Accepted: 28 February 2011
Published: 28 February 2011
A lattice Boltzmann model is developed by coupling the density (D2Q9) and the temperature distribution functions with 9-speed to simulate the convection heat transfer utilizing Al2O3-water nanofluids in a square cavity. This model is validated by comparing numerical simulation and experimental results over a wide range of Rayleigh numbers. Numerical results show a satisfactory agreement between them. The effects of Rayleigh number and nanoparticle volume fraction on natural convection heat transfer of nanofluid are investigated in this study. Numerical results indicate that the flow and heat transfer characteristics of Al2O3-water nanofluid in the square cavity are more sensitive to viscosity than to thermal conductivity.
List of symbols
c Reference lattice velocity
c s Lattice sound velocity
c p Specific heat capacity (J/kg K)
e α Lattice velocity vector
f α Density distribution function
F α External force in direction of lattice velocity
g Gravitational acceleration (m/s2)
G Effective external force
k Thermal conductivity coefficient (Wm/K)
L Dimensionless characteristic length of the square cavity
Ma Mach number
Pr Prandtl number
r Position vector
Ra Rayleigh number
t Time (s)
T α Temperature distribution function
T Dimensionless temperature
T0 Dimensionless average temperature (T0 = (TH + TC)/2)
TH Dimensionless hot temperature
TC Dimensionless cold temperature
u Dimensionless macrovelocity
uc Dimensionless characteristic velocity of natural convection
w α Weight coefficient
x, y Dimensionless coordinates
β Thermal expansion coefficient (K-1)
ρ Density (kg/m3)
ν Kinematic viscosity coefficient (m2/s)
χ Thermal diffusion coefficient (m2/s)
μ Kinematic viscosity (Ns/m2)
φ Nanoparticle volume fraction
δ x Lattice step
δ t Time step t
τ f Dimensionless collision-relaxation time for the flow field
τ T Dimensionless collision-relaxation time for the temperature field
ΔT Dimensionless temperature difference (ΔT = TH - TC)
Error1 Maximal relative error of velocities between two adjacent time layers
Error2 Maximal relative error of temperatures between two adjacent time layers
α Lattice velocity direction
The most common fluids such as water, oil, and ethylene-glycol mixture have a primary limitation in enhancing the performance of conventional heat transfer due to low thermal conductivities. Nanofluids, using nanoscale particles dispersed in a base fluid, are proposed to overcome this drawback. Nanotechnology has been widely studied in recent years. Wang and Fan  reviewed the nanofluid research in the last 10 years. Choi and Eastman  are the first author to have proposed the term nanofluids to refer to the fluids with suspended nanoparticles. Yang and Liu  prepared a kind of functionalized nanofluid with a method of surface functionalization of silica nanoparticles, and this nanofluid with functionalized nanoparticles have merits including long-term stability and good dispersing. Pinilla et al.  used a plasma-gas-condensation-type cluster deposition apparatus to produce nanometer size-selected Cu clusters in a size range of 1-5 nm. With this method, it is possible to produce nanoparticles with a strict control on size by controlling the experimental conditions. Using the covalent interaction between the fatty acid-binding domains of BSA molecule with stearic acid-capped nanoparticles, Bora and Deb  proposed a novel bioconjugate of stearic acid-capped maghemite nanoparticle with BSA molecule, which will give a huge boost to the development of non-toxic iron oxide nanoparticles using BSA as a biocompatible passivating agent. Wang et al.  showed the method of synthesizing stimuli-responsive magnetic nanoparticles and analyzed the influence of glutathione concentration on its cleavage efficiency. Huang and Wang  produced ε-Fe3N-magnetic fluid by chemical reaction of iron carbonyl and ammonia gas. Guo et al.  investigated the thermal transport properties of the homogeneous and stable magnetic nanofluids containing γ-Fe2O3 nanoparticles.
Many experiments and common numerical simulation methods have been carried out to investigate the nanofluids. Teng et al.  examined the influence of weight fraction, temperature, and particle size on the thermal conductivity ratio of alumina-water nanofluids. Nada et al.  investigated the heat transfer enhancement in a horizontal annuli of nanofluid containing various volume fractions of Cu, Ag, Al2O3, and TiO2 nanoparticles. Jou and Tzeng  studied the natural convection heat transfer enhancements of nanofluid containing various volume fractions, Grashof numbers, and aspect ratios in a two-dimensional enclosure. Heris et al.  investigated experimentally the laminar flow-forced convection heat transfer of Al2O3-water nanofluid inside a circular tube with a constant wall temperature. Ghasemi and Aminossadati  showed the numerical study on natural convection heat transfer of CuO-water nanofluid in an inclined enclosure. Hwang et al.  theoretically investigated the natural convection thermal characteristics of Al2O3-water nanofluid in a rectangular cavity heated from below. Tiwari and Das  numerically investigated the behavior of Cu-water nanofluids inside a two-sided lid-driven differentially heated square cavity and analyzed the convective recirculation and flow processes induced by the nanofluid. Putra et al.  investigated the natural convection heat transfer characteristics of CuO-water nanofluids inside a horizontal cylinder heated and cooled from both of ends, respectively. Bianco et al.  showed the developing laminar forced convection flow of a water-Al2O3 nanofluid in a circular tube with a constant and uniform heat flux at the wall. Polidori et al.  investigated the flow and heat transfer of Al2O3-water nanofluids under a laminar-free convection condition. It has been found that two factors, thermal conductivity and viscosity, play a key role on the heat transfer behavior. Oztop and Nada  investigated the heat transfer and fluid flow characteristic of different types of nanoparticles in a partially heated enclosure. Ho et al.  carried out an experimental study to show the natural convection heat transfer of Al2O3-water nanofluids in square enclosures of different sizes.
The lattice Boltzmann method applied to investigate the nanofluid flow and heat transfer characteristic has been studied in recent years. Hao and Cheng  simulated water invasion in an initially gas-filled gas diffusion layer using lattice Boltzmann method to investigate the effect of wettability on water transport dynamics in gas diffusion layer. Xuan and Yao  developed a lattice Boltzmann model to simulate flow and energy transport processes inside the nanofluids. Xuan et al.  also proposed another lattice Boltzmann model by considering the external and internal forces acting on the suspended nanoparticles as well as mechanical and thermal interactions among the nanoparticles and fluid particles. Arcidiacono and Mantzaras  developed a lattice Boltzmann model for simulating finite-rate catalytic surface chemistry. Barrios et al.  analyzed natural convective flows in two dimensions using the lattice Boltzmann equation method. Peng et al.  proposed a simplified thermal energy distribution model whose numerical results have a good agreement with the original thermal energy distribution model. He et al.  proposed a novel lattice Boltzmann thermal model to study thermo-hydrodynamics in incompressible limit by introducing an internal energy density distribution function to simulate the temperature field.
In this study, a lattice Boltzmann model is developed by coupling the density (D2Q9) and the temperature distribution functions with 9-speed to simulate the convection heat transfer utilizing nanofluids in a square cavity.
Lattice Boltzmann method
where τf is the dimensionless collision-relaxation time for the flow field; eα is the lattice velocity vector; the subscript α represents the lattice velocity direction; f α ( r ,t) is the population of the nanofluid with velocity eα (along the direction α) at lattice r and time t; is the local equilibrium distribution function; δ t is the time step t; F α is the external force term in the direction of lattice velocity; G= -β(Tnf-T0)g is the effective external force, where g is the gravity acceleration; β is the thermal expansion coefficient; T is the temperature of nanofluid; and T0 is the mean value of the high and low temperatures of the walls.
where c = δ x / δ t is the reference lattice velocity, δ x is the lattice step, and the order numbers α = 1, ..., 4 and α = 5, ..., 8, respectively, represent the rectangular directions and the diagonal directions of a lattice.
where is the lattice sound velocity, and w alpha is the weight coefficient.
where τT is the dimensionless collision-relaxation time for the temperature field.
where ΔT is the temperature difference between the high temperature wall and the low temperature wall, and L is the characteristic length of the square cavity.
where is the characteristic velocity of natural convection. For natural convection, the Boussinesq approximation is applied; to ensure that the code works in near incompressible regime, the characteristic velocity must be small compared with the fluid speed of sound. In this study, the characteristic velocity is selected as 0.1 times of speed of the sound.
Lattice Boltzmann model for nanofluid
Thermo-physical properties of water and Al2O3 
Fluid phase (water)
cp (J/kg K)
where ρnf is the density of nanofluid, φ is the volume fraction of Al2O3 nanoparticles, ρbf is the density of water, and ρp is the density of Al2O3 nanoparticles.
where Cpnf is the heat capacity of nanofluid, Cpf is the heat capacity of water, and Cpp is the heat capacity of Al2O3 nanoparticles.
where μnf is the viscosity of nanofluid, and μf is the viscosity of water.
where knf is the thermal conductivity of nanofluid, and kf is the thermal conductivity of water.
Results and discussion
where ε is a small number, for example, for Ra = 8 × 104, ε1 = 10-7, and ε2 = 10-7; for Ra = 8 × 105, ε1 = 10-8, and ε2 = 10-8.
In the lattice Boltzmann method, the time step t = 1.0, the lattice step δ = 1.0, the total computational time of the numerical simulation is 100 s, and the data of equilibrium state is chosen in the simulation.
Comparison of the mean Nusselt number with different grids
192 × 192
256 × 256
300 × 300
Figure 6 illustrates the relation between the average Nusselt number and the volume fraction of nanoparticles at two different Rayleigh numbers. It is observed that the average Nusselt number decreases with the increase of the volume fraction of nanoparticles for Ra = 8 × 104 and Ra = 8 × 105. In addition, it can be seen that the average Nusselt number decreases less at a low Rayleigh number. For the case of Ra = 8 × 104 and Ra = 8 × 105, it is indicated that the high values of φ cause the fluid to become more viscous which causes reduced convection effect accordingly resulting in a decreasing average Nusselt number, and the flow and heat transfer characteristics of nanofluids are more sensitive to the viscosity than to the thermal conductivity at a high Ra.
A lattice Boltzmann model for single phase fluids is developed by coupling the density and temperature distribution functions. A satisfactory agreement between the numerical results and experimental results is observed.
In addition, the heat transfer and flow characteristics of Al2O3-water nanofluid in a square cavity are investigated using the lattice Boltzmann model. It is found that the heat transfer characteristics transform from conduction to convection as the Rayleigh number increases, the average Nusselt number is reduced with increasing volume fraction of nanoparticles, especially at a high Rayleigh number. The flow and heat transfer characteristics of Al2O3-water nanofluid in a square cavity are demonstrated to be more sensitive to viscosity than to thermal conductivity.
This study is financially supported by Natural Science Foundation of China through Grant No. 51076036, the Program for New Century Excellent Talents in University NCET-08-0159, the Scientific and Technological foundation for distinguished returned overseas Chinese scholars, and the Key Laboratory Opening Funding (HIT.KLOF.2009039).
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