Journal of Nanoparticle Research

, 11:767

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

Authors

  • Shashi Jain
    • Department of Mechanical EngineeringIndian Institute of Technology Madras
  • Hrishikesh E. Patel
    • Department of Mechanical EngineeringIndian Institute of Technology Madras
    • Department of Mechanical EngineeringIndian Institute of Technology Madras
    • Department of Mechanical EngineeringMassachusetts Institute of Technology (MIT)
Research Paper

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

Nanoparticles form very stable colloidal systems known as nanofluids because of their extreme sizes. They have two interesting properties which make them an exciting possibility for the next generation heat transfer fluid:
  1. 1.

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

     
  2. 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

Dynamic behaviour of N particles whose mass and size are much larger than the host medium particles can be obtained using the 3N coupled Langevin equations. This method can be applied for dilute colloidal solution with large particles of spherical symmetry. The 3N coupled Langevin equation for the system of N Brownian particles as given by Deutch and Oppenheim (1971) is
$$ m_{i} \mathop v\limits^{ \bullet } _{i} = - \sum\limits_{j} {\varsigma _{{ij}} v_{j} + {\mathbf{F}}_{i} + \sum\limits_{j} {\alpha _{{ij}} f_{i} } } $$
(1)
The indices i and j label components (1 ≤ i, j ≤ 3N) and \( \sum\limits_{j} {\alpha _{{ij}} f_{j} } \) represent the randomly fluctuating force exerted on the particle by the surrounding fluid. The fi are described by Gaussian distribution with mean and covariance:
$$ \begin{gathered} \left\langle {f_{i} } \right\rangle = 0 \hfill \\ \left\langle {f_{i} (t)f_{j} (t')} \right\rangle = 2\delta _{{ij}} \delta (t - t') \hfill \\ \end{gathered} $$
(2)
The coefficients αij are related to the hydrodynamic tensor by
$$ \varsigma _{{ij}} = \frac{1}{{kT}}\sum\limits_{l} {\alpha _{{il}} \alpha _{{jl}} } $$
(3)
We can obtain the particle position Langevin equations from the particle momentum Langevin equations by making use of the assumption of rapid momentum relaxation, as suggested by Ermak and McCammon (1978).
$$ \Updelta r_{i} = \sum\limits_{j} {\frac{{\partial D_{{ij}}^{0} }}{{\partial r_{j} }}t + \sum\limits_{j} {\frac{{D_{{ij}}^{0} F_{j}^{0} }}{{kT}}t + \sum\limits_{j} {\int\limits_{0}^{t} {{\text{d}}s\left[ {1 - \exp \left( {\frac{{ - (t - s)}}{{\tau _{{ii}}^{0} }}} \right)} \right]} } } } \sigma _{{ij}}^{0} f_{j} (s) $$
(4)
where the superscript “0” refers to the value of the variable at the beginning of the time step Δt, \( D_{{ij}}^{0} \) is the (i,j)th element of the diffusion tensor, Fj is the systematic force experienced by the jth particle, fi is random variable with Gaussian distribution whose mean and variance are given by Eq. 2, and σij is given by
$$ D_{{ij}} = \sum\limits_{l} {\sigma _{{il}} \sigma _{{jl}} } $$
(5)
The covariance of the displacement is
$$ \begin{gathered} \left\langle {\Updelta r_{i} \Updelta r_{j} } \right\rangle = 2\sum\limits_{l} {\int\limits_{0}^{t} {{\text{d}}s\left[ {1 - \exp \left( {\frac{{ - (t - s)}}{{\tau _{{ii}}^{0} }}} \right)} \right]} \times \left[ {1 - \exp \left( {\frac{{ - (t - s)}}{{\tau _{{jj}}^{0} }}} \right)} \right]\sigma _{{il}}^{0} \sigma _{{jl}}^{0} } \hfill \\ = 2D_{{ij}}^{0} \int\limits_{0}^{t} {{\text{d}}s\left[ {1 - \exp \left( {\frac{{ - (t - s)}}{{\tau _{{ii}}^{0} }}} \right)} \right]} ^{2} \hfill \\ = 2D_{{ij}}^{0} \left( {t - \tau _{{ii}}^{0} \left( {\frac{3}{2} + \frac{1}{2}\exp \left( {\frac{{ - 2t}}{{\tau _{{ii}}^{0} }}} \right) - 2\exp \left( {\frac{{ - t}}{{\tau _{{ii}}^{0} }}} \right)} \right)} \right) \hfill \\ \end{gathered} $$
(6)
where \( \tau _{{ii}}^{0} = \frac{{D_{{ij}}^{0} m_{j} }}{{kT}} \). In the formulation by Bhattacharya et al. (2004), the covariance of displacement was taken as 2DijΔt 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)
$$ \Upphi _{{ij}} = A\,{ \exp }\left( { - B\left( {\frac{{|r_{{ji}} | - d}}{d}} \right)} \right), $$
where 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
$$ v_{i} = \frac{{\Updelta r_{i} }}{{\Updelta t}} $$
(7)
The total excess energy of the Brownian particle is
$$ E_{i} = \frac{1}{2}\sum\limits_{j} {\Upphi _{{ij}} + \frac{1}{2}m_{i} v_{i}^{2} - h_{i} } $$
(8)

Here, Φij is the inter-particle potential energy which being much smaller than kinetic energy of the particles has been neglected in our simulation. hi is the enthalpy of the nanofluid at given temperature. We can exclude hi from the equation as its value is negligible compared to the value of Ei 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.

The Fluctuation Dissipation theorem relates a macroscopic transport coefficient with the time integral of a particular microscopic autocorrelation function measured in equilibrium state. Specifically, thermal conductivity is computed using the microscopic heat flux operator (McQuarrie 1976) and is given by
$$ {\mathbf{Q}}(t) = \frac{{\text{d}}}{{{\text{d}}t}}\sum\limits_{i} {{\mathbf{r}}_{i} E_{i} } $$
(9)
which can be written as given by Lee et al. (1991)
$$ {\mathbf{Q}}(t) = \sum\limits_{i} {{\mathbf{v}}_{i} E_{i} } + \frac{1}{2}\sum\limits_{{i,l,i \ne l}} {\left( {{\mathbf{F}}_{{il}} \cdot {\mathbf{v}}_{i} } \right){\mathbf{r}}_{{il}} } $$
(10)
Green Kubo equations give the relation between the microscopic heat flux operator and thermal conductivity (McQuarrie 1976)
$$ k_{{\text{p}}} (T) = \frac{1}{{3k_{{\text{b}}} T^{2} V}}\int\limits_{0}^{\infty } {{\text{d}}t\left\langle {{\mathbf{Q}}(0) \cdot {\mathbf{Q}}(t)} \right\rangle } , $$
(11)
The above integral can be computed as
$$ k_{{\text{p}}} = \frac{1}{{k_{{\text{b}}} T^{2} V}}\sum\limits_{{j = 0}}^{n} {\left( {\frac{1}{{3(n - j)}}\sum\limits_{{i = 0}}^{{n - j}} {{\mathbf{Q}}(i\Updelta t) \cdot {\mathbf{Q}}\left[ {\left( {i + j} \right)\Updelta t} \right]} } \right)} \Updelta t $$
(12)
where kp is the thermal conductivity due to the Brownian motion of the nanoparticles, kb 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 Δt is the value of time step used.
The thermal conductivity computed above is just the part of the conductivity due to the Brownian motion of the nanoparticle. To find the effective thermal conductivity of the nanofluid, we also need to include the thermal conductivity of the base fluid. We assume that thermal conduction due to the motion of nanoparticles and the base fluid molecules happens in parallel and therefore use the simple correlation to compute the effective thermal conductivity.
$$ k_{{{\text{eff}}}} = \Upphi k_{{\text{p}}} + \left( {1 - \Upphi } \right)k_{{\text{f}}} $$
(13)
where keff is the effective thermal conductivity of the nanofluid, kp is thermal conductivity due to the Brownian motion of the nanoparticles, kf 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 Δt. 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(n3) 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

The thermal conductivity due to the Brownian motion of the particle is computed by integrating the heat flux operator. Figure 1 shows that the time autocorrelation decays to zero quite fast, and thereafter it oscillates about zero which could be due to the inherent noise in the simulation. The curves are normalized against their zero time value, and time is normalized against \( \tau _{c} \). We used the first dip method (Li et al. 1998) to evaluate the thermal conductivity.
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig1_HTML.gif
Fig. 1

Normalized heat flux operator versus time

Figure 2 shows effective thermal conductivity obtained after integration of the heat flux operator. It can be seen in the plot that effective thermal conductivity reaches its equilibrium value in the first few time steps and thereafter oscillates about this value due to the inherent noise in the simulation. We took this mean value as the effective thermal conductivity of the nanofluid.
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig2_HTML.gif
Fig. 2

Effective thermal conductivity obtained from the heat flux operator for alumina–ethylene glycol against the number of time steps

The simulations were performed for alumina–ethylene glycol based nanofluid, and the results were compared with the available experimental values. Our simulations showed similar results as obtained by Bhattacharya et al. (2004) with respect to particle concentration for particles of size 60 nm at 300 K as can be seen in Fig. 3.
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig3_HTML.gif
Fig. 3

Model predictions for the effect of particle volume fraction on the effective thermal conductivity of nanofluid

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).

Figure 4 shows the comparison of variation of effective thermal conductivity for 150 nm particles with changing volume fraction as obtained by our simulation with the experimental results (Patel 2007) and with Hamilton–Crosser model. The simulation results are again very close to the experimental values, and it can be noticed that the value of effective thermal conductivity gets closer to the value predicted by Hamilton–Crosser model for such large particle size.
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig4_HTML.gif
Fig. 4

Model predictions for the effect of particle volume fraction on the effective thermal conductivity of nanofluid

Figure 5 shows the temperature dependence of thermal conductivity as obtained by our model and its comparison with the available experimental values (Patel 2007). The simulation was performed for 11 nm particles with 1% volume fraction and 150 nm particles with 0.5% volume fraction. In both the cases, the simulation results are in close agreement with the experimental values, showing the temperature effect as observed by transient hot wire measurement with temperature control using a thermostatic bath (Patel 2007). Temperature effect is one of the important effects, which usual static models of simulation fail to predict. In the present formulation, the Brownian dynamics which take particles’ random movement into consideration bring this effect quite accurately.
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig5_HTML.gif
Fig. 5

Model predictions for the effect of temperature on the effective thermal conductivity of nanofluid

Size effect is the most important effect in nanofluids, which traditional models fail to predict. The model was used to successfully predict the variation of effective thermal conductivity with particle size as can be seen in Fig. 6. The simulation was run for particles of size 11, 45 and 150 nm for 1% volume fraction and at a temperature of 300 K. Even for such varied range of particle sizes, the results obtained were in agreement with the experimental results of Patel (2007).
https://static-content.springer.com/image/art%3A10.1007%2Fs11051-008-9454-4/MediaObjects/11051_2008_9454_Fig6_HTML.gif
Fig. 6

Model predictions for the effect of particle size on the effective thermal conductivity of nanofluid

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.

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© Springer Science+Business Media B.V. 2008