Numerical study of instability of nanofluids: the coagulation effect and sedimentation effect
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This study is a numerical study on the coagulation as well as the sedimentation effect of nanofluids using the Brownian dynamics method. Three cases are simulated, focusing on the effects of the sizes, volume fraction, and ζ potentials of nano-particles on the formation of coagulation and sedimentation of nanofluids. The rms fluctuation of the particle number concentration, as well as the flatness factor of it, is employed to study the formation and variation of the coagulation process. The results indicate a superposition of coagulation and sedimentation effect of small nano-particles. Moreover, it is stable of nanofluids with the volume fraction of particles below the limit of "resolution" of the fluids. In addition, the effect of ζ potentials is against the formation of coagulation and positive to the stability of nanofluids.
KeywordsSedimentation Number Concentration Hydrodynamic Interaction Gravity Effect Critical Heat Flux
The nanofluid is characterized by the fluid with nanometer-sized solid particles dispersed in solution , which can increase the heat transfer coefficient [2, 3, 4, 5, 6], enhance the critical heat flux in boiling heat transfer [7, 8, 9], reduce the wall friction force , improve the optical characteristics , etc. Nano-sized particles are utilized because of its better stability than the suspension of micro-sized particles. For a badly stable suspension, sedimentation or coagulation (agglomeration) may occur. It compromises the above-mentioned advantages of the nano-suspension.
As is well known , the occurrences of coagulation and sedimentation are the two main factors for the instability of nanofluid. The phenomenon of coagulation is characterized by the formation of particle clusters, i.e., particles are in contact with each other and the cohesion takes place. Then, the clusters grow up. Many researchers investigated the coagulation effect of particles by the Brownian dynamics simulation, focusing on the formation of gelation , coagulation rates , particle network , etc. For example, Hütter  identified the characteristic coagulation time scales in colloidal suspensions, and measured their dependencies on the solid content and potential interaction parameters. He also deduced different cluster-cluster bonding mechanisms in the presence of an energy barrier, etc. Besides, the sedimentation always occurs after a big particle cluster is established, i.e., the particles within the cluster sediment flow downward because of the increased effect of the gravity of the cluster over the buoyancy force of it, and reduced the effect of Brownian motion to the big cluster. Many researches were devoted to the sedimentation [16, 17, 18] using the Brownian dynamics simulation too. For example, Soppe and Jannsen  studied the sediment formation of colloidal particle by a process of irreversible single-particle accretion. They used the algorithm of Ermak and McCammon, incorporating the inter-particle forces and hydrodynamic interaction on the two-particle level, and analyzed the effect of two-particle hydrodynamic interactions on the sediment structure, etc. They found that the process of sediment formation by colloidal particle is the result of a delicate balance of sediment field strength, DLVO interactions, and hydrodynamic interactions.
However, there is an important issue about which few researches have been concerned: the interaction between the coagulation and sedimentation for the instability of nanofluids. For example, the processing of coagulation causes the particle clusters to grow up, and then the large clusters are more prone to sedimentation than that of small clusters because of the intensive gravity effect. In other words, the coagulation effect is able to augment the sedimentation effect. Thus, this study is intended to carry out some research on this issue, exploring the complex interaction as well as the close relation between the coagulation and the sedimentation phenomena.
where η is the viscosity, a is the particle radius, δ ij is the Kronecker delta, is the vector from the center of particle i to the center of particle j, and is the unit tensor.
where A, d, ε, κ, ζ, ρf, ρ, and g are the Hamaker constant, the particle diameter, the electric permittivity of the fluid, the inverse of the double-layer thickness, the zeta potential of the suspension, the density of the fluid, the density of the particle, and the gravity acceleration, respectively.
It is noted in Equation (4) that the results for r ij - d = 0 is meaningless when the contact between the two particles occur, and they will adhere to each other or rebound back. Thus, we treat the condition with as the situation when the two particles are separated, so that Equation (4) works. Otherwise, it results in coalescence between the two particles. Once the coalescence between colliding particles takes place, the clusters start growing up.
Three cases with different diameters of particles, volume fractions, and zeta potentials
Case 1: under different diameters d
(a) d0 = 10 nm, N p = 1200, ψ = 0.153, ζ = 0.0 eV
(b) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0.0 eV
(c) d2 = 50 nm, N p = 1200, ψ = 0.153, ζ = 0.0 eV
Case 2: under different volume fractions ψ
(a) d1 = 25 nm, N p = 400, ψ = 0.051, ζ = 0.0 eV
(b) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0.0 eV
(c) d1 = 25 nm, N p = 2100, ψ = 0.268, ζ = 0.0 eV
(d) d1 = 25 nm, N p = 4200, ψ = 0.537, ζ = 0.0 eV
Case 3: under different zeta potentials ζ
(a) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0 eV
(b) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0.01 eV
(c) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0.025 eV
(d) d1 = 25 nm, N p = 1200, ψ = 0.153, ζ = 0.05 eV
Parameters used in this simulation
Simulation domains (L x , L y , L z )
(4d, 4d, 256d)
Mesh sizes (δ x , δ y , δ z )
(4d, 4d, 4d)
Temperature T (°C)
Hamaker constant A (J)
2 × 10-20
Diameters of particle (d0, d1, d2), (nm)
(10, 25, 50)
Density of fluid ρf (kg/m3)
0.993 × 103
Density of particle (ρp) (kg/m3)
6.4 × 103
Viscosity of fluid η (Pa s)
1.0 × 10-3
Simulation time step Δt (ns)
In this simulation, the boundary conditions in the x and y directions (Table 2) in the horizontal plane are both periodic, whereas the top and bottom walls of the simulation domain in the z-direction are treated as adhesive walls to which the particles adhere immediately once they come into contact with them. It is reasonable to conclude thus, since the agglomerated particle clusters always adhere to the bottom walls or the top interfaces.
Initially, a random distribution is given to the particles. As time advances, the possible movements of particles are computed through solution of the governing equations.
Case 1: effect of particle sizes
It is necessary to mention that Figure 1i, j, k, l does not indicate the stability of the nanofluids. Alternatively, it indicates a relatively stable characteristic compared to Figure 1a, b, c, d. After the evolution over a long time, possible coagulation or sedimentation may also occur.
The rms of concentration means the fluctuation of the number concentration of particles, and it is closely related to the formation of particle clusters due to coagulation. The flatness factor means the intensity of fluctuation of the number concentration, thereby indicating the intensity of coagulation. Thus, these two functions are helpful in enabling the quantification of particle coagulation.
Case 2: effects of volume fractions
In this section, the effect of volume fraction, i.e., the concentration of particles, is studied. As aforementioned, the smaller particles are more prone to coagulate than the larger particles, under the same condition of the volume fractions. However, the process of coagulation is also closely related to the number of particles contained in it.
With the increased number of particles, it is seen that the R1 and R4 are increased too (n p = 1200 and 2100, respectively, Figure 4). However, when the particle number is extremely large, all the spaces are almost stuffed with particles, leading to a homogeneous distribution and a low fluctuation in the number concentration (n p = 4200, in Figure 4).
Case 3: effects of ζ potentials
The previous sections showed the results with ζ = 0 eV. As seen from Equation (5), no repulsive effect has been considered between the particles since fe = 0. Thus, this section will focus on the effect of the repulsive effect by varying the ζ potentials.
A complicated superposition of the coagulation and sedimentation effects for small particle is observed. The mechanisms of sedimentation for the larger and the smaller particles are different. The former is caused mainly by the great gravity effect of any individual particle, whereas the latter is mainly due to the coagulation process, and the superposition of coagulation causes the sedimentation of the whole agglomeration of particles.
There exists a superior limit of the fluid for particle content. When the volume fraction is below the limit, it is hard for the coagulation to occur. In contrast, the coagulation will certainly take place when the concentration of nanoparticles is beyond the capacity of "resolution" of the fluids.
The effect of ζ potentials is beneficial for the stability of nanofluid, since it resists the formation of coagulation. In other words, increase in the value of ζ potentials is helpful to make the nanofluid more stable.
This study is supported by the National High Technology Research and Development of China 863 Program (2007AA05Z254), for which the authors are grateful.
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