Numerical simulation of natural convection in a square enclosure filled with nanofluid using the two-phase Lattice Boltzmann method
© Qi et al.; licensee Springer. 2013
Received: 23 October 2012
Accepted: 22 January 2013
Published: 4 February 2013
Considering interaction forces (gravity and buoyancy force, drag force, interaction potential force, and Brownian force) between nanoparticles and a base fluid, a two-phase Lattice Boltzmann model for natural convection of nanofluid is developed in this work. It is applied to investigate the natural convection in a square enclosure (the left wall is kept at a high constant temperature (TH), and the top wall is kept at a low constant temperature (TC)) filled with Al2O3/H2O nanofluid. This model is validated by comparing numerical results with published results, and a satisfactory agreement is shown between them. The effects of different nanoparticle fractions and Rayleigh numbers on natural convection heat transfer of nanofluid are investigated. It is found that the average Nusselt number of the enclosure increases with increasing nanoparticle volume fraction and increases more rapidly at a high Rayleigh number. Also, the effects of forces on nanoparticle volume fraction distribution in the square enclosure are studied in this paper. It is found that the driving force of the temperature difference has the biggest effect on nanoparticle volume fraction distribution. In addition, the effects of interaction forces on flow and heat transfer are investigated. It is found that Brownian force, interaction potential force, and gravity-buoyancy force have positive effects on the enhancement of natural convective heat transfer, while drag force has a negative effect.
Compared with common fluids such as water, nanofluid, using nanoscale particles dispersed in a base fluid, has an effect of enhancing the performance of natural convection heat transfer due to its high heat conductivity coefficient. Many researchers investigated nanoparticles and nanofluid in recent years. Wang et al.  synthesized stimuli-responsive magnetic nanoparticles and investigated the effect of nanoparticle fraction on its cleavage efficiency. Bora and Deb  developed a novel bioconjugate of stearic acid-capped maghemite nanoparticle (γ-Fe2O3) with bovine serum albumin. Guo et al.  produced magnetic nanofluids containing γ-Fe2O3 nanoparticles using a two-step method, measured their thermal conductivities and viscosity, and tested their convective heat transfer coefficients. Pinilla et al.  investigated the growth of Cu nanoparticles in a plasma-enhanced sputtering gas aggregation-type growth region. Yang and Liu  produced a kind of stable nanofluid by surface functionalization of silica nanoparticles. Zhu et al.  developed a wet chemical method to produce stable CuO nanofluids. Nadeem and Lee  investigated the steady boundary layer flow of nanofluid over an exponential stretching surface. Wang and Fan  reviewed the nanofluid research in the last 10 years.
Natural convection is applied in many fields, and extensive researches have been performed. Oztop et al.  and Ho et al.  respectively investigated natural convection in partially heated rectangular enclosures and discussed the effects of viscosity and thermal conductivity of nanofluid on laminar natural convection heat transfer in a square enclosure by a finite-volume method. Saleh et al.  investigated heat transfer enhancement utilizing nanofluids in a trapezoidal enclosure by a finite difference approach. Ghasemi et al. , Santra et al. , and Aminossadati et al.  numerically simulated natural convection in a triangular enclosure and studied the behavior of natural convection heat transfer in a differentially heated square cavity, described a study on natural convection of a heat source embedded in the bottom wall of an enclosure, and used the SIMPLE algorithm to solve the governing equation. Kargar et al.  used computational fluid dynamics and an artificial neural network to investigate the cooling performance of two electronic components in an enclosure. Abu-Nada et al.  investigated the effect of variable properties on natural convection in enclosures filled with nanofluid, and the governing equations are solved by an efficient finite-volume method. Hwang et al.  investigated the thermal characteristics of natural convection in a rectangular cavity heated from below by Jang and Choi's model .
The Lattice Boltzmann method is a new way to investigate natural convection. Compared with the above traditional methods, the Lattice Boltzmann method has many merits including that boundary conditions can be conveniently dealt with, the transform between macroscopic and microscopic equations is easily achieved, the details of the fluid can be presented, and so on. In addition, nanofluid as the media can enhance heat transfer due to factors such as nanofluids having higher thermal conductivity and the nanoparticles in the fluid disturbing the laminar flow. Therefore, many researchers undertook investigations on the natural convection of nanofluids by the Lattice Boltzmann method. Barrios et al.  developed a Lattice Boltzmann model and applied it to investigate the natural convection of an enclosure with a partially heated left wall. Peng et al.  presented a simple a Lattice Boltzmann model without considering thermal diffusion, and this model is easily applied because it does not contain a gradient term. He et al.  proposed a new Lattice Boltzmann model which introduced an internal energy distribution function to simulate the temperature field, and the result has a good agreement with the benchmark solution. Nemati et al.  simulated the natural convection of a lid-driven flow filled with Cu-water, CuO-water, and Al2O3-water nanofluids and discussed the effects of nanoparticle volume fraction and Reynolds number on the heat transfer. Wang et al.  presented a Lattice Boltzmann algorithm to simulate the heat transfer of a fluid-solid fluid, and the result has a satisfactory agreement with the published data. Dixit et al.  applied the Lattice Boltzmann method to investigate the natural convection of a square cavity at high Rayleigh numbers. Peng et al.  developed a 3D incompressible thermal Lattice Boltzmann model for natural convection in a cubic cavity. The above Lattice Boltzmann methods are all single-phase models, and the nanofluid was seen as a single-phase fluid without considering the interaction forces between nanoparticles and water. In addition, the effects of these interaction forces on heat transfer were disregarded.
There are few two-phase lattice Boltzmann models that consider the interaction forces between nanoparticles and a base fluid for natural convection in an enclosure. Xuan et al.  proposed a two-phase Lattice Boltzmann model to investigate sudden-start Couette flow and convection in parallel plate channels without researching the effect of forces on volume fraction distribution of nanoparticles. Because these forces were not investigated before our work, the effects of forces between water and nanoparticles on the fluid flow patterns were unknown. In addition, as we know, the nanoparticles in the fluid easily gather together and deposit, especially at high volume fraction. Hence, the nanoparticle distribution in the fluid flow is important for nanofluid application, which is another objective in our paper. However, the single-phase model cannot be used to investigate nanoparticle distribution. Furthermore, natural convection of a square enclosure (left wall kept at a high constant temperature (TH), and top wall kept at a low constant temperature (TC)) filled with nanofluid is not investigated in the published literatures. In this paper, a two-phase Lattice Boltzmann model is proposed and applied to investigate the natural convection of a square enclosure (left wall kept at a high constant temperature (TH), and top wall kept at a low constant temperature (TC)) filled with Al2O3-water nanofluid and the inhomogeneous distribution of nanoparticles in the square enclosure.
Lattice Boltzmann method
where is the dimensionless collision-relaxation time for the flow field, e α is the lattice velocity vector, the subscript α represents the lattice velocity direction, is the distribution function of the nanofluid with velocity e α (along the direction α) at lattice position r and time t, is the local equilibrium distribution function, δ t is the time step, δ x is the lattice step, the order numbers α = 1,…,4 and α = 5,…,8, respectively represent the rectangular directions and the diagonal directions of the lattice, is the external force term in the direction of the lattice velocity without interparticle interaction, G = - β(T nf - T0)g is the effective external force, where g is the gravity acceleration, β is the thermal expansion coefficient, T nf is the temperature of the nanofluid, and T0 is the mean value of the high and low temperature of the walls.
where represents the total interparticle interaction forces, and B α is one of the weight coefficients. is a coefficient. Because the total interparticle interaction forces cannot be optionally added in the lattice Boltzmann equation, we introduce an unknown coefficient in the total interparticle interaction forces. In order to enable the lattice Boltzmann equation including the total interparticle interaction forces to recover to the Navier-Stokes equation, based on the mass and momentum conservation, we used multi-scale technique to deduce the unknown coefficient which is equal to . Due to the very long derivation process, we directly gave the final result in the paper.
where is the lattice's sound velocity, and w α is the weight coefficient.
where τ T is the dimensionless collision-relaxation time for the temperature field.
where Fp represents the total forces acting on the nanoparticles, Fw represents the total forces acting on the base fluid, and L x L y represents the total number of lattices.
where Φ αβ is the energy exchange between nanoparticles and base fluid, , and h αβ is the convective heat transfer coefficient of the nanofluid.
From Equations 18 and 19, the collision-relaxation time for the flow field and the temperature field can be calculated. For water phase, the τ f collision-relaxation times are respectively 0.51433 and 0.501433 at Ra = 103 and Ra = 105, and the collision-relaxation time τ T is 0.5. For nanoparticle phase, the τ f collision-relaxation times are respectively 0.50096 and 0.500096 at Ra = 103 and Ra = 105, and the collision-relaxation time τ T is 0.500025.
Interaction forces between base fluid and nanoparticles
As noted before, a nanofluid is, in reality, a kind of two-phase fluid. There are interaction forces between liquid and nanoparticles which affect the behavior of the nanofluid. The external forces include gravity and buoyancy forces FH, and the interparticle interaction forces include drag force (Stokes force) FD, interaction potential FA, and Brownian force FB. We introduce them as follows.
where a is the radius of a nanoparticle, and Δρ' is the mass density difference between the suspended nanoparticle and the base fluid.
where μ is the viscosity of the fluid, and ∆u is the velocity difference between the nanoparticle and the base fluid.
where A is the Hamaker constant, and L cc is the center-to-center distance between particles.
where n i is the number of the particles within the adjacent lattice i, n i = ρ σ V/m σ , m σ is the mass of a single nanoparticle, and V is the volume of a single lattice.
where G i is a Gaussian random number with zero mean and unit variance, which is obtained from a program written by us, and C = 2γk B T = 2 × (6πηa)k B T, γ is the surface tension, k B is the Boltzmann constant, T is the absolute temperature, and η is the dynamic viscosity.
where n is the number of the particles in the given lattice, and V is the lattice volume.
Results and discussion
Thermo-physical properties of water and Al 2 O 3 
Fluid phase (H2O)
cp (J/kg k)
where ε is a small number, for example, for Ra = 1 × 103, ε1 = 10-6, and ε2 = 10-6. About 2 weeks is needed to achieve the equilibrium state for the low Rayleigh number (Ra = 1 × 103), and about 1 month for the high Rayleigh number (Ra = 1 × 105).
In order to perform a grid independence test and validate the Lattice Boltzmann model proposed in this work, we used another square enclosure, because there are exact solutions for this square enclosure. The left wall is kept at a high constant temperature (TH), and the right wall is kept at a low constant temperature (TC). The boundary conditions of the other walls (top wall and bottom wall) are all adiabatic, and the other conditions are the same as those in Figure 1.
Comparison of the mean Nusselt numbers with different grids ( Ra = 1 × 10 5 , Pr = 0.7)
128 × 128
192 × 192
256 × 256
320 × 320
Comparison of average Nusselt numbers with other published data ( Pr = 0.7)
Comparison of different forces ( Ra = 10 5 , φ = 0.03)
It is also found that almost all the isolines behave with oscillations in Figures 6, 7, 8, 9, but smooth isolines are given in Figures 3 and 5. Due to the ruleless Brownian movement of nanoparticles, it is difficult for nanofluid to achieve a complete equilibrium state, which is the difference compared with other common two-phase fluids. In order to expediently judge the equilibrium state and save time, we choose the temperature equilibrium states of water phase and nanoparticle phase as the whole nanofluid equilibrium state in the computation. When the water-phase and nanoparticle-phase temperatures all achieve equilibrium state, the whole nanofluid (temperature distribution, velocity vectors, density distribution, and nanoparticle volume fraction distribution) is considered as being in an equilibrium state. Hence, the temperature isolines in Figures 3 and 5 look smooth due to a complete equilibrium state, and the density distribution in Figures 6 and 7 and nanoparticle volume fraction distribution in Figures 8 and 9 behave with oscillations due to an approximate equilibrium state. Although the interparticle interaction forces have little effect on heat transfer, they play an important role on the nanoparticle distribution.
A 2D two phase Lattice Boltzmann model has been developed for nanofluids and the simulation results of this two-phase Lattice Boltzmann model are in good agreement with published experimental results. This model is applied to investigate the natural convection of a square enclosure filled with Al2O3 nanofluid. The effects of different nanoparticle fractions and Rayleigh numbers on natural convection heat transfer of nanofluid are investigated. In addition the effects of forces on the nanoparticles volume fraction distribution and the heat transfer are also investigated.
It is found that the Nusselt number distribution along the heated surface firstly increases, and then decreases with Y at both low and high Rayleigh numbers. Average Nusselt numbers of the whole square enclosure both increase with nanoparticles volume fraction at a low and a high Rayleigh number. In addition, the enhancement of the average Nusselt numbers is much more pronounced at a high Rayleigh number than at a low Rayleigh number.
It is found that the temperature difference driving force is the biggest force and has the greatest effect on nanoparticle volume fraction distribution. For a low Rayleigh number, the nanoparticle volume fraction is low in the lower right corner and high in the top right corner and lower left corner. For a high Rayleigh number, the nanoparticle volume fraction is low at the bottom and high at the top.
Apart from the temperature difference driving force, Brownian force, interaction potential force, and gravity-buoyancy force contribute to the enhanced natural convective heat transfer, while the drag force contributes to the attenuation of heat transfer.
a radius of nanoparticle (m)
A Hamaker constant
B a weight coefficient
c reference lattice velocity
cs lattice sound velocity
cp specific heat capacity (J/kg K)
FS dimensionless temperature difference driving forces
FB dimensionless Brownian force
FH dimensionless gravity and buoyancy force
FD dimensionless drag force
FA dimensionless interaction potential force
g dimensionless gravitational acceleration
G dimensionless effective external force
G i Gaussian random number
h a β convective heat transfer coefficient (W/(m2 K))
H dimensionless characteristic length of the square cavity
k thermal conductivity coefficient (W/m/K)
k B Boltzmann constant
Lcc center-to-center distance between particles (m)
Ma Mach number
m σ mass of a single nanoparticle (kg)
n i number of the particles within the adjacent lattice i
Nu Nusselt number
Pr Prandtl number
r position vector
Ra Rayleigh number
T dimensionless temperature
T0 dimensionless average temperature (T0 = (TH + TC)/2)
TH dimensionless hot temperature
TC dimensionless cold temperature
u σ dimensionless macro-velocity
uc dimensionless characteristic velocity of natural convection
V A dimensionless interaction potential
V volume of a single lattice (m3)
wα weight coefficient
x, y dimensionless coordinates
β σ thermal expansion coefficient (K-1)
ρ σ density (kg/m3)
v kinematic viscosity (m2/s)
η dynamic viscosity (Pa s)
χ thermal diffusion coefficient (m2/s)
γ surface tension (N/m)
φ nanoparticle volume fraction
δ x lattice step
δ t time step
σ components (σ = 1, 2, water and nanoparticles)
τ f dimensionless collision-relaxation time for the flow field
τ T dimensionless collision-relaxation time for the temperature field
∆T dimensionless temperature difference (∆T = T H – T C )
Δρ’ dimensionless mass density difference between nanoparticles and base fluid
∆u dimensionless velocity difference between nanoparticles and base fluid
Φ αβ dimensionless energy exchange between nanoparticles and base fluid
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
w base fluid
This work is financially supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (grant no. 51121004).
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