# Brownian dynamic simulation for the prediction of effective thermal conductivity of nanofluid

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11051-008-9454-4

- Cite this article as:
- Jain, S., Patel, H.E. & Das, S.K. J Nanopart Res (2009) 11: 767. doi:10.1007/s11051-008-9454-4

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## Abstract

Nanofluid is a colloidal solution of nanosized solid particles in liquids. Nanofluids show anomalously high thermal conductivity in comparison to the base fluid, a fact that has drawn the interest of lots of research groups. Thermal conductivity of nanofluids depends on factors such as the nature of base fluid and nanoparticle, particle concentration, temperature of the fluid and size of the particles. Also, the nanofluids show significant change in properties such as viscosity and specific heat in comparison to the base fluid. Hence, a theoretical model becomes important in order to optimize the nanofluid dispersion (with respect to particle size, volume fraction, temperature, etc.) for its performance. As molecular dynamic simulation is computationally expensive, here the technique of Brownian dynamic simulation coupled with the Green Kubo model has been used in order to compute the thermal conductivity of nanofluids. The simulations were performed for different concentration ranging from 0.5 to 3 vol%, particle size ranging from 15 to 150 nm and temperature ranging from 290 to 320 K. The results were compared with the available experimental data, and they were found to be in close agreement. The model also brings to light important physical aspect like the role of Brownian motion in the thermal conductivity enhancement of nanofluids.

### Keywords

NanofluidsSuspensionsThermal conductionBrownian dynamic simulationEffective conductivityNanoparticlesModeling## Introduction

- 1.
They form very stable colloidal system with little settling in static conditions.

- 2.
They show anomalous enhancement in thermal conductivity in comparison to the base fluid.

Various experimental groups have reported enhancement in thermal conductivity with increasing volume fraction (Patel 2007; Xie et al. 2002; Masuda et al. 1993). Also, it has been shown that the thermal conductivity enhancement increases as the particle size decreases (Wang et al. 1999). Das et al. (2003) measured the conductivities of alumina and cupric oxide (in water) for different temperatures in the range of 20–50°C and for different loading conditions. They observed a linear increase in the conductivity with temperature. The viscosity ratio of nanofluid shows quadratic (Wang et al. 1999) to threefold (Pak and Cho 1998) enhancement with varying volume fraction, while the thermal conductivity enhancement is almost linear. Looking at these trends, it is clear that nanofluid dispersions need to be optimized with respect to particle size, volume fraction and temperature for its performance as a heat transfer fluid.

The effort to present theoretical model for nanofluid conductivity started immediately after the invention of nanofluids. Kumar et al. (2004) and Xue (2003) built simple analytical models which can explain the observed effect in nanofluids. However, such simple analytical models run into twofold problems. Firstly, these models are based on certain assumptions of the heat transfer mechanism, and hence, they cannot shed light on complex mechanisms such as particle–particle and fluid–particle interactions. Secondly, they have certain adjustable constants which bring a certain amount of empiricism in the model skirting the physics of the process. Hence, it is important to carry out large-scale simulations based on the fundamental principles to understand the physics of the energy transport in nanofluids. Obviously, conventional CFD simulations in Eulerian–Eulerian or Eulerian–Lagrangian frame which are usually used for slurry transport are inadequate for nanoparticles due to their extremely small sizes, and hence, molecular dynamic simulation, Brownian dynamic simulation or solution of Boltzmann transport equation are necessary.

Bhattacharya et al. (2004) performed the first large-scale microscopic simulation for nanofluids. They assumed that the nanoparticles are much bigger than solvent or base fluid particles. Therefore, the fluid particles can be omitted and their effects are represented by a combination of random and frictional forces. The motion of the solvent particles is computed using the Langevin equation. The results of the Brownian motion coupled with Green Kubo model give the thermal conductivity of the nanofluid. However, in their simulation they did not show the effect of parameters like particle size and temperature on effective thermal conductivity of nanofluids. The empirical inter-particle potential function obtained by curve fitting shows that the magnitude of inter-particle force is negligible when compared to the random force.

The lattice Boltzmann numerical method was developed by Xuan and Yao (2005) in order to investigate the nanoparticle distribution in a stationary nanofluid. The distribution of the suspended nanoparticles was determined by a series of the acting forces and potentials. In the model Brownian force is a dominant factor affecting random displacement and aggregation of the nanoparticles.

Evans et al. (2006) performed molecular dynamic simulations (MDS) of heat flow in a model nanofluid with well-dispersed particles and found that the hydrodynamic effects associated with Brownian motion have only a minor effect on the thermal conductivity of the nanofluid. The thermal conductivity was computed using the Fourier law by applying a source and a sink and computing the temperature profile. However, the number of particles and time steps required were quite large, and the results were not validated with any available experimental result.

Sarkar and Selvam (2007) have used MDS to predict the thermal conductivity of nanofluids. The model was successfully able to predict the expected enhancement in thermal conductivity of the nanofluid, consisting of copper nanoparticle (2 nm) in argon base fluid, with varying concentrations and temperature. But these results were obtained for highly idealized conditions like choice of argon as base fluid and therefore cannot be validated against experimental data that is available in the literature. Also, effect of particle size has not been simulated. Choice of more complex base fluid and greater number of nanoparticles would have resulted in highly computationally expensive simulation.

Thus, it is obvious that there is hardly any computational model which captures all the observed effects on nanofluid conductivity, namely, concentration, particle size and temperature effects.

Molecular dynamic simulation of nanofluids involves considering particles of two different sizes: relatively small fluid molecules and larger nanoparticles. We require longer time steps to capture the evolution of the slow moving nanoparticles, while smaller time steps are required for capturing the motion of fast moving fluid molecules. We cannot use longer time steps as it will result in overlap of fluid particles and thus erroneous results. While smaller time steps would require a very long run in order to allow the complete evolution of slower mode. In Brownian dynamic simulation, the fluid molecules are omitted, and the effect of hydrodynamic interactions mediated by the host solvent is included through a position-dependent inter-particle friction tensor. Based on these observations in the present work, Brownian dynamic simulation has been carried out to bring forward the effects of particle size, concentration and temperature on the anomalous thermal conductivity enhancement of nanofluids.

## Theoretical model

*N*particles whose mass and size are much larger than the host medium particles can be obtained using the 3

*N*coupled Langevin equations. This method can be applied for dilute colloidal solution with large particles of spherical symmetry. The 3

*N*coupled Langevin equation for the system of

*N*Brownian particles as given by Deutch and Oppenheim (1971) is

*i*and

*j*label components (1 ≤

*i, j*≤ 3

*N*) and \( \sum\limits_{j} {\alpha _{{ij}} f_{j} } \) represent the randomly fluctuating force exerted on the particle by the surrounding fluid. The

*f*

_{i}are described by Gaussian distribution with mean and covariance:

_{ij}are related to the hydrodynamic tensor by

*t*, \( D_{{ij}}^{0} \) is the (

*i,j*)th element of the diffusion tensor,

*F*

_{j}is the systematic force experienced by the

*j*th particle,

*f*

_{i}is random variable with Gaussian distribution whose mean and variance are given by Eq. 2, and σ

_{ij}is given by

_{ij}Δ

*t*based on the assumption that \( \tau _{{ii}} \ll \Updelta t \). This assumption holds true for microsized particles as were simulated by Ermak and McCammon (1978), but in case of nanoparticles the above assumption does not hold true. Hence, it is more appropriate to use Eq. 6 for the covariance of displacement for more accurate results. The diffusion tensors are suggested as approximation to the hydrodynamic interaction mediated by the fluid. The two diffusion tensors suggested in the literature are Oseen tensor (Yamakawa 1971) and Rotne–Prager tensor (Rotne and Prager 1969). We chose the Rotne–Prager tensor for the simulation as it is positive definite. Both these tensors have the properiety \( \sum\limits_{{}} {\frac{{\partial D_{{ij}} }}{{\partial r_{j} }} \equiv 0} \) so that this term can be dropped from Eq. 4. The random force term is of order 10

^{−11}N, which is much greater when compared to the inter-particle force term obtained from the empirical correlation function of Bhattacharya et al. (2004)

*A*and

*B*are constants obtained by curve fitting whose values for aluminium oxide–ethylene glycol nanofluid are 2.8 × 10

^{−17}and 1.00 × 10

^{−22}, respectively. Also, MDS by Sarkar and Selvam (2007) where only a single particle was used to model the system predicts thermal conductivity similar to the expected experimental trends. Hence, we assume in our formulation that inter-particle force term is negligible in comparison to the random force term and can be neglected. However, more accurate results could be obtained if a better approximation of inter-particle potential function is used which becomes important when the particles are close to each other. Care has been taken in the code to avoid effect of particle overlaps by finding at the end of each time step the particles which overlap and then finding their new velocities by assuming an elastic collision. The average velocity of a particle during a particular time step is calculated using

Here, Φ_{ij} is the inter-particle potential energy which being much smaller than kinetic energy of the particles has been neglected in our simulation. *h*_{i} is the enthalpy of the nanofluid at given temperature. We can exclude *h*_{i} from the equation as its value is negligible compared to the value of *E*_{i} computed from the simulation in a temperature range we are going to deal with. As the component of random force is much larger than the inter-particle forces, we have neglected the inter-particle potential in comparison to the kinetic energy of the particles.

*k*

_{p}is the thermal conductivity due to the Brownian motion of the nanoparticles,

*k*

_{b}is the Boltzmann’s constant,

*T*is the temperature,

*V*is the volume of the simulation domain,

*n*is the number of time steps used and Δ

*t*is the value of time step used.

*k*

_{eff}is the effective thermal conductivity of the nanofluid,

*k*

_{p}is thermal conductivity due to the Brownian motion of the nanoparticles,

*k*

_{f}is thermal conductivity of the base fluid and Φ is the particle volume fraction.

## Simulation details

Initially, the particles are arranged in FCC lattice which has been used as the starting configuration for simulation of liquids (Allen and Tildesley 1987). In the FCC arrangement, all atoms are at equivalent positions, and the atomic displacements are isotropic. Specular boundary conditions have been applied on all the walls of the domain box.

The choice of time step is critical for any Brownian dynamic simulation as it has to be long enough to ensure the momentum relaxation of the particles, but at the same time the numerical accuracy limits the maximum size of the time step, requiring it to be sufficiently short so that the forces on the particle and the gradient of the diffusion tensor are essentially constant during Δ*t*. The order of time step was taken equal to the time scale of convection \( \tau _{c} \) (Prasher et al. 2006) due to the movement of the particle, which is given as \( \tau _{c} = \frac{{d^{2} }}{\nu } \), where *d* is the diameter of the particle and *v* is the kinematic viscosity of the fluid.

The Brownian dynamic simulation follows *O*(*n*^{3}) time complexity where *n* is number of particles. The simulation was run for 32, 108 and 256 particles, and it was found that the results were almost similar in the latter two cases, although the simulation noise was reduced when the number of particle were increased. Hence, 108 particles were chosen as a compromise between computational time and accuracy in our simulation. The time complexity of BDS is linear with respect to number of time steps for which the simulation is run. The simulation was checked for 100, 1,000 and 10,000 steps, and it was found that as the number of step increases the simulation noise is reduced. Again 1,000 time steps was chosen as a good compromise between accuracy and computational time. Also, as the HCAF decays to zero very fast 1,000 time steps was a good choice for duration of run for the simulation.

## Results and discussion

It can be seen that the present simulation is close to experimental data of Xie et al. (2002) showing a near linear dependence on particle concentration. It can also be seen that at this particle size the enhancement is much higher than that predicted by the traditional Hamilton–Crosser model (Hamilton and Crosser 1962).

## Conclusions

We were successfully able to compute the effective thermal conductivity of nanofluids using Brownian dynamic simulation in which the effect of hydrodynamic interactions mediated by the base fluid is included through a position-dependent inter-particle friction tensor. The simulation based on *N* coupled Langevin equations though very fundamental in its formulation was able to simulate the effects of parameters like particle concentration, particle size and the temperature of the fluid on the effective thermal conductivity of nanofluids. The simulation results matched well with the experimental data available. We can safely conclude that Brownian dynamic simulation is a better alternative to computationally expensive molecular dynamic simulation, and Brownian motion of the particle is the most important phenomena which is responsible for the anomalous enhancement in the thermal conductivity of nanofluids.