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Investigation of nanoparticle aggregation effect on thermal properties of nanofluid by a combined equilibrium and non-equilibrium molecular dynamics simulation
Journal of Molecular Liquids 197 (2014) 14–22
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Investigation of nanoparticle aggregation effect on thermal properties of nanofluid by a combined equilibrium and non-equilibrium molecular dynamics simulation Mina Sedighi, Ali Mohebbi ⁎ Department of Chemical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
a r t i c l e
i n f o
Article history: Received 19 October 2013 Received in revised form 16 March 2014 Accepted 22 April 2014 Available online 5 May 2014 Keywords: Nanofluid Aggregation Thermal conductivity Specific heat Thermal diffusivity Molecular dynamics simulation
a b s t r a c t Many theoretical and experimental studies on heat transfer and flow behavior of nanofluids have been done and the results show that nanofluids significantly increase heat transfer. Nevertheless, there is no accurate understanding from the effect of different mechanisms on nanofluid heat transfer. Computer simulations are a suitable tool for description of physical mechanisms in many processes. In this study, molecular dynamics simulation was used to investigate the effect of nanoparticle aggregation on thermal properties of water-silicon dioxide nanofluid, specifically its thermal conductivity. For calculating nanofluid thermal conductivity a combination of two equilibrium and non-equilibrium molecular dynamics simulations was performed to calculate the specific heat and thermal diffusivity of the nanofluid, respectively. Simulations were performed in NVT ensemble and spherical coordinate. The model was validated by comparison of thermal properties of water base fluid with experimental data in four various temperatures. Results also were compared with theoretical models such as HC model for nanofluid. To investigate the effect of nanoparticle aggregation, two cases of constant and variable volume fractions (i.e. 1.5, 3 and 4.5%) at temperature of 308 K were considered. The results showed that when the aggregation occurs with increasing nanoparticle concentrations, there are an increase in the thermal conductivity and thermal diffusivity of the nanofluid and a decrease in its specific heat. Moreover, when aggregation takes place at constant nanoparticle concentration, the specific heat of nanofluid with suspended nanoparticles did not change with respect to nanofluid with aggregated nanoparticles, but its diffusivity and thermal conductivity increase. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Cooling systems are one of the most important concerns in factories, industries, transportation and each place that deals with heat transfer. Therefore, in new technologies high heat flow process was created in order to enhance heat transfer. There are different methods for heat transfer improvement [1]. One of them is using the nanofluids instead of current heat transfer fluids. Nanofluids have been prepared by dispersing metallic or non-metallic particles at nanoscale size in ordinary heat transfer fluids. Suspended nanoparticles increase heat transfer by means of increasing the values of nanofluid thermophysical properties. Thermal conductivity of nanofluids is the most important parameter to indicate the heat transfer potential. Several mathematical models have been presented to predict the nanofluid thermal conductivity [2]. Maxwell's [3] model is the first model in this context. He had predicted that the thermal conductivity of nanofluids is a function of particle volume fraction and thermal ⁎ Corresponding author. Tel./fax: +98 3412118298. E-mail addresses: [email protected], [email protected] (A. Mohebbi).
http://dx.doi.org/10.1016/j.molliq.2014.04.019 0167-7322/© 2014 Elsevier B.V. All rights reserved.
conductivity of fluid and solid particles. Later Hamilton and Crosser (HC) [4] by development of Maxwell model and adding a shape factor in it have presented a new model. In addition several models such as Jeffrey [5] and Davis [6] models have been created. All of these models have been investigated in macroscale size and do not consider the solid and liquid movements and consequently probable collisions that cause thermal conductivity enhancement, therefore, they obtained underpredict values in comparison with experimental data. According to experimental researches [7–9] some parameters can be effective on thermal conductivity coefficient, such as, particle volume fraction, type of nanoparticles, size of particles, and temperature. Hong et al. [9] prepared Fe nanofluid based on ethylene glycol and indicated that thermal conductivity of Fe nanofluid is increased nonlinearly up to 18% as the volume fraction of particles is increased up to 55%. Daungthongsuk and Wongwises [10] experimentally reported the thermal conductivity and dynamic viscosity of TiO2-water nanofluid. They found that thermal conductivity and viscosity of nanofluids depend on temperature, so that, thermal conductivity of nanofluid increases with increasing temperature and conversely its viscosity decreases. Timofeeva et al. [11] characterized nanofluids of alumina
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Nomenclatures Cp Cv E k KB r t T U α φ ρ
Specific heat at constant pressure Specific heat at constant volume Total energy Thermal conductivity Boltzmann's constant Radius Time Temperature Potential energy Thermal diffusivity Particle volume fraction Density
Subscripts Nf Nanofluid f Fluid p Particle th Theory MD Molecular dynamics
particles in water and ethylene glycol using thermal conductivity, viscosity and dynamic light scattering measurements. Their results showed that the thermal conductivity enhancement is within the range predicted by effective medium theory in which the particles are also agglomerated over time. Kathikeyan et al. [12] synthesized CuO nanoparticles with average diameter of 8 nm by a simple precipitation technique and study the thermal properties of the suspensions. The experimental results showed that the nanoparticle size, polydispersity, cluster size, and the volume fraction of the particles have a significant effect on thermal conductivity. The study also mentioned that nanofluids containing ceramic or metallic nanoparticles showed large enhancement in thermal conductivity that cannot be explained by conventional theories. Except for the parameters that were studied at experimental measurements, investigation of possible mechanisms in molecular scale proves that several other parameters in molecular level are considerable. These factors including Brownian motion of nanoparticles, liquid layering at the liquid/solid interface, nanoparticle clustering and radioactive heat transfer somewhat justify nanofluid's unusual behavior [13]. Nevertheless, generally there is no determinative mechanism in nanofluid studies. Nanoparticle agglomeration is one of the most controversial mechanisms in nanofluid thermal conductivity studies. It has been indicated that when the nanoparticles are suspended in the base fluid, on the effect of Van der Waals forces they are agglomerated over time. This phenomenon has been called nanoparticle aggregation. There are several theories to examine the aggregation effects on the nanofluid's thermal properties and they are described in the following. Xuan et al. [14] applied the theory of Brownian motion and diffusion-limited aggregation model to simulate random motion and the aggregation process of the nanoparticles. They found that morphology of the suspended nanoparticles besides nanoparticle diameter and volume fraction of nanoparticles is one of the several important factors that affect the thermodynamic properties of nanofluid and that formation of aggregates reduces the efficiency of the energy transport enhancement of the suspended nanoparticles. Prasher et al. [15] used aggregation kinetics of nanoscale colloidal solutions combined with physics of thermal transport to capture the effect of aggregation on the thermal conductivity of nanofluids. Their study developed a unified model, which combines the micro convective effects due to Brownian motions with the change in conduction due to aggregation. The results
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showed that colloidal chemistry plays a significant role in deciding the conductivity of colloidal suspensions. Jie et al. [16] proposed a new model for thermal conductivity of nanofluids, which is derived from the fact that nanoparticles and clusters coexist in the fluids. The effects of compactness and perfectness of contact between the particles in clusters on the effective thermal conductivity are analyzed. The model showed that the effective thermal conductivity of nanofluids decreases with the increasing concentration of clusters. Feng et al. [17] proposed a new model for effective thermal conductivity of nanofluids based on nanolayer and nanoparticle aggregation. Their study was derived on a model based on the fact that a nanolayer exists between nanoparticles and fluid and some particles in nanofluids may contact each other to form clusters. A governed equation for effective thermal conductivity was developed by both the agglomerated clusters and nanoparticles suspended in the fluids. Wu et al. [18] verified experimentally and theoretically the significance of the effect of the cluster structure, size distribution, and thermal conductivity of solid particles in water. The aggregation kinetics of SiO2 particles in water base fluid was done by adjusting the pH. Their experiments showed that clustering has no any discernible enhancement in the thermal conductivity even at high volume loading. Investigation of previous researches showed that there are no conclusive theories for the effect of aggregation on thermal properties of nanofluids. Nanoparticle aggregation creates a condition with lower thermal resistance for heat transfer and causes the thermal conductivity enhancement. On the other hand, this phenomenon can decrease the thermal conductivity of nanofluids, in this case, aggregation of the nanoparticles which may cause instability in the suspension, as well as create low efficiency areas in the liquid. To evaluate the nanofluid's behavior and investigation of the effect of various factors on it, two Mont Carlo and molecular dynamics simulations [19] have been created. Molecular dynamics simulation was first introduced to study the interactions of hard spheres by Alder and Wainwright [20] in the late 1950s. Later at 1964 Rahman [21] carried out the first simulation using a realistic potential for liquid argon. This method simulates the system by force, velocity and positions of atoms in the system at each time step and its purpose is calculation of macroscopic state of the system with a microscopic model. Molecular dynamics simulation uses the algebraic method to specify the path of atoms, thus, it has particular computational advantages. Presence of the time variable estimates the required time in each simulation. In Monte Carlo simulation to evaluate the time there is no such estimation. There are two equilibrium molecular dynamics and non-equilibrium molecular dynamics approaches for molecular dynamics simulations that have been used by researchers to study the nanofluid's behavior. Keblinski et al. [13] explored the four possible explanations for increasing thermal conductivity using molecular dynamics simulation: Brownian motion of particles, molecular layering of the fluid at the liquid/solid interface, the nature of heat transport in nanoparticles and the effect of nanoparticle clustering. Eapen et al. [22] performed molecular dynamics simulations of the time-dependent heat current correlation to obtain the systematic, dynamical details at the atomistic level using a model system of Xe base fluid and Pt nanoparticles. Their model indicated that the interatomic interactions between fluid and the nanoparticles can be much stronger than the interaction between fluid particles, as well as nanoparticle interactions, are significant factors to understand the heat transport mechanisms of nanofluids. Galamba et al. [23] calculated the thermal conductivity of molten NaCl and KCl through the Evans–Gillan non-equilibrium molecular dynamics algorithm and Green–Kubo equilibrium molecular dynamics simulations. The EMD simulations performed for a binary ionic mixture and the NEMD simulations assumed a pure system for reasons discussed in their work. They found that the thermal conductivity obtained from NEMD simulations was in very good agreement with that obtained through Green–Kubo EMD simulations for a binary ionic mixture. Galliero and Volz [24] used a non-equilibrium molecular dynamics simulation to
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propose a new algorithm for calculating single particle thermodiffusion. Through their estimation the thermophoretic force has been applied on a solute particle. Their results showed that thermal conductivity decreases with nanoparticle concentration. They also realized that nanoparticle tends to migrate toward the cold area and the single particle thermal diffusion coefficient is independent to the size of the nanoparticle, whereas it increases with the quality of the solvent and is inversely proportional to the viscosity of the fluid. Sarkar and Selvam [25] used an equilibrium molecular dynamics simulation to calculate the thermal conductivity of copper in argon nanofluid. For this nanofluid, they reported the thermal conductivity enhancement up to 20% compared to the base fluid with 1% nanoparticle concentration. Sankar [26] used an equilibrium molecular dynamics modeling for estimation of thermal conductivity enhancement of water due to well dispersed platinum nanoparticle. They compared their results with existing experimental data for copper nanoparticles suspended in water and those predicted by conventional theories. The thermal conductivity of the nanofluid was obtained from the Green Kubo formulation. Kang et al. [27] examined a non-equilibrium molecular dynamics simulation in a copper– argon nanofluid. They introduced two different methods; the physical definition and the curve fitting to calculate the coupling factor between nanoparticles and base fluid. They found that nanoparticle aggregation causes to decrease the coupling factor. Furthermore, they showed that the coupling factors obtained by these two methods are consistent and proportional to the volume fraction of the nanoparticle and inversely proportional to nanoparticle diameter. Jia et al. [28] studied on the characteristics of a nanofluid system composed of argon liquid and copper nanoparticle, such as heat current measurement by its mean value, variance, third moment, and the Shannon entropy. For this purpose, they carried out a molecular dynamics simulation using Green–Kubo method. They exhibited that the thermal conductivity increases as the nanoparticle volume fraction increases. Kang et al. [29] performed a MD simulation using Green–Kubo method to study the effect of nanoparticle aggregation on the thermal conductivity and viscosity in nanofluids. They placed multiple nanoparticles in the simulation box to simulate the aggregation of the nanoparticles. Their results showed that the nanoparticle aggregation induces a significant enhancement of thermal conductivity in nanofluid, while the viscosity increases moderately. Moreover, they indicated that different configurations of the nanoparticle cluster bring about different enhancements of thermal conductivity and increase of viscosity in the nanofluid. Mohebbi [30] predicted the specific heat and thermal conductivity of silicon nitride nanoparticle based on liquid argon by a combined molecular dynamics simulation. He examined the effect of temperature and volume concentration of nanoparticles on thermal conductivity of nanofluid and found that the thermal conductivity of nanofluid increases with the increase of the volume concentration and decrease of the temperature. Heat transfer autocorrelation function or Green–Kubo integral formula [31] and the direct method are the most commonly used for equilibrium and non-equilibrium molecular dynamics simulations respectively. The convergence of Green–Kubo's integral usually requires a lot of time or it may not be possible. Also, this method is not able to perform simulations for the systems with a large number of molecules. Therefore, finding a reliable and simpler method without disadvantages of Green–Kubo's formula, in which it does not need heavy and time consuming programming, seems to be helpful. In this study a new method based on combination of two equilibrium and non-equilibrium molecular dynamics approaches was presented and the effect of nanoparticle aggregation on thermal properties of SiO2-water nanofluid such as its specific heat, thermal diffusivity and thermal conductivity was investigated. Whereas, the aggregation of nanoparticles is dependent on particle volume concentration, calculations were done in two cases of variable and constant volume fractions of nanoparticles. Also, to validate the obtained results, simulation for water in four temperatures was carried out and the results were compared with experimental data [32].
2. Methodology As mentioned before, up to now the main purpose of researchers has been the study of thermal conductivity coefficient (k) that it is obtained directly by current molecular dynamics methods. Using the equilibrium and non-equilibrium molecular dynamics simulations simultaneously, properties of SiO2-water nanofluid including, specific heat and heat diffusivity were calculated besides calculating thermal conductivity. To compute the specific heat and diffusivity (two necessary parameters for the thermal conductivity calculation) EMD and NEMD simulations were applied, respectively that each one is described in the following. 2.1. Thermal conductivity calculation The conduction of heat transfer is created by means of the atom oscillation against each other, thus, in a nanofluid one can estimate that the rate of heat conduction is similar to particle diffusivity in the fluid. According to the relation between conductivity and diffusivity: αnf ¼
knf ρnf Cpnf
ð1Þ
where, knf and Cpnf are the thermal conductivity and specific heat of nanofluid, respectively and ρnf is its density and it is calculated by ρnf = φρnp + (1 − φ)ρf that φ is a volume fraction of nanoparticles. In the present study the thermal conductivity (k) of SiO2-water nanofluid was calculated by rewriting Eq. (1) as: knf ¼ αnf ρnf Cvnf :
ð2Þ
2.2. Specific heat calculation A long EMD simulation in spherical boundary condition and NVT (canonical) ensemble was performed. In canonical ensemble, the specific heat equation has been derived from the definition of constant as below [33]: volume specific heat Cv ¼ ∂E ∂T D Cv ¼
v
E E −hEi2 2
ð3Þ
kB T 2
where, b E N is the average total energy, bE2N is the squared-average total energy, T is temperature (K) and KB is Boltzmann's constant = 0.00198 kcal/mol·K. During the simulation, fluctuations of the energy were recorded as a function of time (t). There are no exact and comprehensive experimental data for thermal properties of SiO2-water nanofluid. Harry O'Hanley et al. [34] offered the following equation to calculate the specific heat of nanofluids:
Cpnf ¼
ρCp ρnf
nf
¼
φ ρCp þ ð1−φÞ ρCp p
φρp þ ð1−φÞρ f
f
:
ð4Þ
This equation provides a theoretical approach for predicting the thermal properties of nanofluids. It has been proved that there is a good agreement between the obtained values from the model of Eq. (4) and limited available experimental data [34]. In this study, the present model was used to compare our results with the calculated values from this model. 2.3. Thermal diffusivity calculation In this step, a NEMD simulation using the results of the last equilibration step in spherical boundary condition was performed and diffusivity was calculated by considering an imaginary shell around the atomic
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Generally, thermal diffusivity, which have been conducted few studies about it, can be obtained from Eq. (9).
Table 1 Bonded parameters for SiO2 and water molecules. Parameters
Si\O\Si
O\Si\O
H2O (TIP3P)
kθ (kcal/mol·rad2) θ0 (degree) kb (kcal/mol·Å2) b0 (Å)
4.66 174.22 885.1 1.61
159.57 110.93 885.1 1.61
55 104.52 545 0.96
αnf ¼
6 R min;ij 12 R −2 min;ij rij rij
εO εH εSi RO min/2 RH min/2 RSi min/2 qO (SiO2) qO (H2O) qH qSi εO 1− 4 εH 1− 4 εSi 1− 4
Cq q
Uelec ¼ ε1−4 ε ir j 0 ij
−0.152 −0.046 −0.3 1.76 0.22 2.1475 −0.834 −0.5 0.417 1.0 −0.15 −0.046 −0.3
* εij (kcal/mol) = sqrt (εi·εj), Rmin,ij (Å) = Rmin,i / 2 + Rmin,j / 2.
sphere. Molecule temperature in the outer layer of the sphere was specified. Temperature of the sphere and the shell around it will be equal at the end and diffusivity was determined by monitoring changes of the system temperature and comparing it to the theoretical expression. In a sphere of radius R the theoretical average temperature as a function of the time in each time step is calculated as below [33]:
TðtÞtheory ¼ Tbath þ 6
∞ . TSim −Tbath X nπ 2 1 exp − αnf t 2 n R π n¼1 2
ð5Þ
where αnf is the thermal diffusivity, and TSim and Tbath are the initial temperature of the system and the temperature of the sphere boundary, respectively. This equation is governed by the heat diffusion equation [33]: ! ∂T r ; t ∂T
2 ! ¼ αnf ∇ T r ; t
ð6Þ
TðR; tÞ ¼ Tbath :
rbR
ρCp Þnf
ð9Þ
3. Simulation procedure In the present study, to predict the thermal properties of SiO2-water nanofluid a combined EMD and NEMD simulation in two steps was performed. Water as the most applicable and the most available heat transfer fluid is a suitable choice for this purpose. The base fluid was considered by the number of 11,225 water atoms including Hydrogen and Oxygen atoms. Sirk et al. [35] indicated that the rigid, three-site (i.e. SPC, SPC/E), and transferable intermolecular potential (i.e. TIP3P) water models are shown to have similar thermal conductivity values at standard conditions, whereas models that include bond stretching and angle bending have higher thermal conductivities. In the present study, TIP3P water model implemented in CHARMM force field was adopted because it is a simple model with higher computational efficiency in comparison with other classical models of water. Also, non-bonded and bonded parameters such as angle, charges and forces are closer to real water molecule. The CHARMM version of TIP3P model places Lennard–Jones parameters on the Hydrogen atoms in addition to Oxygen. The initial arrangement of the atoms was configured based on replication of a unit cell of water molecules in spherical boundary condition and three dimensions [36]. The simulation was performed in canonical ensemble (NVT), which meant the total number of atoms; the system volume and temperature were constant throughout the simulation. To fix the system temperature all over the simulation, Langevin algorithm [37] was employed. Langevin algorithm keeps constant the system temperature by means of the velocity rescaling method. Integration scheme in the present simulation was velocity Verlet algorithm [38]. All of the simulation process was performed by NAMD2 software [39], which is a parallel molecular dynamics program, designed to run efficiently for simulating large molecules. NAMD employs a common potential energy function to determine a force field model that has the following contributions [39]: Utotal ¼ Ubond þ Uangle þ Udihedral þ þUvdw þ Ucoulomb :
with initial and boundary conditions for spherical coordinate: ! T r ; t ¼ TSim ð0Þ
knf
In this equation knf was calculated by HC model, thus, αnf estimated by this method has an underpredict value. We compared our simulation results with the thermal diffusivity that was calculated by Eq. (9).
Table 2 Non-bonded parameters and electrostatic properties for O, H and Si atoms. UL− J ¼ εmin
17
ð7Þ
ð8Þ
ð10Þ
The first three terms describe the stretching, bending and torsional bonded interactions, in which each one is defined by the specific mathematical equation. The last two terms in Eq. (10) describe interactions between non-bonded atom pairs, which correspond to the van der Waal's 6–12 forces and electrostatic interactions, respectively. These interactions are for all of fluid–fluid, solid–fluid and solid–solid atoms.
Fig. 1. Process of constructing a silicon dioxide nanoparticle.
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In this study we used CHARMM22 force field potential function, parameters and file formats [40]. We also considered that all atoms in the system have rigid bonds. Tables 1 and 2 show equations and numerical values of CHARMM22 bonded and non-bonded force field potential functions for water and SiO2 molecules [41,42]. In Table 2, C and ε0 are defined as Coulombs constant and vacuum permittivity respectively. VMD 1.9 software [43] was used to analyze the output results from NAMD software. VMD is a molecular visualization program for displaying, animating, and analyzing molecular systems using 3-D graphics and built-in scripting. 4. Initial configuration and modeling of the nanofluid To prepare the nanofluid, initially a unit cell of SiO2 crystal that contains eight Oxygen and four silicon atoms was built by means of modeling features of VMD software. Parameters of interatomic distances and bond angles have been described by VMD [43]. Then, to generate a crystal membrane in hexagonal geometry oriented perpendicular to the z-axis, the unit cell was replicated in three dimensions with replication numbers of 6, 6 and 12 along the respective pffiffifficrystal axis (i. e. x, y and z). The unit cell vectors were [1.00.0 0.0], [1/2 3=2 0.0] and [0.0 0.0 1.0]. To generate a nanoparticle with the radius of 7.5 nm and the number of 126 atoms, a sphere was cut out of the hexagonal patch. This nanoparticle was placed in the center of water base fluid. Thus, in this case, the concentration of the nanoparticles in the base fluid was 1.5 vol.%. Generation of the nanoparticle is shown in Fig. 1.
Fig. 3. Two (a) and three (b) dispersed nanoparticles in water base fluid for constant volume fractions of 3 and 4.5%.
4.1. Aggregation structure
4.2. Specific heat
Considering the dependency of the nanoparticle aggregation on concentration, the effect of the aggregation on thermal properties of nanofluid in two cases of variable and constant volume fractions was investigated. In the first case, three simulations were performed by increasing the number of aggregated nanoparticles and consequently increasing the concentration of nanoparticles in the base fluid. These simulations were done for one, two and three SiO2 nanoparticles in the base fluid (see Fig. 2) and the volume fractions of nanoparticle in nanofluid were 1.5, 3 and 4.5%, respectively. As one can see in Fig. 2b and c, in two simulations, two and three aggregated nanoparticles were placed in the center of water base fluid. Each nanoparticle contains 126 atoms with a radius of 7.5 nm. In the second case, two simulations were accomplished with two and three suspended nanoparticles in the base fluid for constant particle volume fractions of 3 and 4.5% (see Fig. 3).
To calculate the specific heat of nanofluid a long EMD simulation for 1,500,000 time steps was accomplished. Each time step was considered as 1 fs. The simulation was performed in spherical boundary condition and NVT ensemble. To fix the system temperature at 308 K (i.e. studied temperature) Langevin damping coefficient was considered at 3 ps−1. To improve computational efficiency, cutoff distance was chosen equal to 20 Å, which meant that when the two atoms were further than 20 Å, they had a negligible nonbonded interaction. To equilibrate the system throughout the simulation, initially in first 100 time steps, potential energy minimization at temperature of 0 K was carried out. In all over the simulation, by changing the atomic positions and gradual reduction of the potential energy, equilibration process continued until reaching the system to equilibrium. The relevant data to the total energy versus the time in each time step (i.e. 1 fs)
Fig. 2. 1.5% (a), 3% (b) and 4.5% (c) aggregated SiO2 nanoparticles in water base fluid.
M. Sedighi, A. Mohebbi / Journal of Molecular Liquids 197 (2014) 14–22
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was recorded in NAMD's logfile and specific heat of nanofluid was calculated according to Eq. (3).
The thermal diffusivity was calculated by optimization of an objective function defined below:
4.3. Thermal diffusivity
ο¼
To calculate the thermal diffusivity of nanofluid, a shell with thickness of 4 nm was considered around the spherical system and its temperature was considered by about 30 °C lower than the temperature of the sphere (i.e. 308 K); therefore, the heat was diffused in the system over time. In this step, a short NEMD simulation for 40,000 time steps was performed, because of using the results of the previous equilibration simulation for calculation of specific heat.
n X Tth −TMD 2 Tth i i¼1
ð11Þ
where, Tth and TMD are temperatures obtained from Eq. (5) and MD calculations respectively, and n is the number of time steps. This objective function was defined by investigation of the trend of decreasing system temperature from MD simulation and comparing it with the temperature reduction from theoretical calculations (i.e. Eq. (5)) and fitting the corresponding temperatures to achieve the minimum value of the difference. This was performed by the “goal seek” tool of Microsoft Office Excel 2010. To adjust the temperature of molecules in outer layer of the sphere, we used the feature of temperature coupling of NAMD2 software [33]. 5. Results and discussion 5.1. Validation of simulation In the present study to examine the MD results' accuracy, obtained results were compared with experimental data [32] for water base fluid. To satisfy the density of water, simulation was performed for the number of 11,255 water atoms and the radius of 30 nm in a spherical boundary condition and NVT ensemble. The thermal properties of water in four temperatures at 298, 308, 318 and 333 K were calculated, then the results were compared with corresponding experimental data and the relative errors calculated. Fig. 4a, b and c compares the simulation results with experimental data for specific heat, diffusivity and thermal conductivity of the water base fluid respectively. As one can see from this figure, there is good agreement between the MD results and experimental data and the errors are less than 3.5%. As mentioned previous, equilibration process with an initial minimization was completed in 1,500,000 time steps. Fig. 5 shows decreasing trend of the potential energy in first 1000 time steps and then reaching to equilibration. 5.2. Effect of aggregation on thermal properties of nanofluid by increasing the number of aggregated nanoparticles Three nanofluid simulations with one nanoparticle and then two and three aggregated nanoparticles were performed. Each nanoparticle contains 126 atoms with radius of 7.5 nm. The volume fractions of the nanoparticles in these nanofluids were 1.5, 3 and 4.5%, respectively. 5.2.1. Specific heat and thermal diffusivity To calculate the specific heat and diffusivity of the nanofluid, simulations were performed at constant temperature of 308 K. The density of the nanoparticle was 2369 kg/m3.
Fig. 4. Validation of the MD simulation results for specific heat, diffusivity and thermal conductivity of water base fluid.
Fig. 5. Equilibration process by the decrease of potential energy.
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Fig. 6. Specific heat of nanofluid versus number of aggregated nanoparticles.
Fig. 8. Thermal conductivity of nanofluid versus number of aggregated nanoparticles.
For specific heat the results are shown in bar chart in Fig. 6. As one can see, on increasing the number of aggregated nanoparticles and consequent concentration enhancement, specific heat of SiO2-water nanofluid decreases. The results of present MD model are in good agreement with calculated values from Eq. (4). Here is a point that needs to be considered, the results obtained from Eq. (4) is correct, when the nanoparticles in the base fluid are dispersed but in present simulation, SiO2 nanoparticles in water base fluid are aggregated. The reason of accordance between the calculated values from this model and our simulation is explained in Section 5.3.1. Fig. 7 shows the thermal diffusivity of nanofluid with one nanoparticle, two and three aggregated nanoparticles. This figure shows that by adding one nanoparticle in each step, nanofluid thermal diffusivity increases up to 3% rather than its previous case. This can be because of existence the paths with lower thermal resistance in water base fluid with more aggregated nanoparticles. As one can see from this figure, results are higher than those values predicted using Eq. (9). This is due to knf in Eq. (9). This parameter was calculated from HC model, which under predicts the thermal conductivity of nanofluid.
visible. We found thermal conductivity enhancement up to 9.8% compared to water base fluid by adding one SiO2 nanoparticle to the base fluid. This is corresponding to 1.5% nanoparticle concentration. By adding two and three aggregated nanoparticles (i.e. 3% and 4.5% nanoparticle concentration, respectively) to the base fluid, the thermal conductivity enhancements were 11.3% and 14.9% compared to the base fluid respectively. Fig. 9 shows the thermal conductivity enhancement of the nanofluid versus nanoparticle volume fractions. This figure also illustrates that the conductivity enhancement was steeper at low nanoparticle loading compared to higher loadings as found by Sarkar and Selvam [25] by MD simulation of copper nanoparticles in liquid argon. This result was not predicted by available theoretical models such as HC.
5.2.2. Thermal conductivity As mentioned, thermal conductivity of nanofluid was calculated by multiplying the specific heat and diffusivity of nanofluid in its density at the corresponding concentration and temperature. Fig. 8 shows the results. As one can see the number of nanoparticles can have a significant effect on the values of thermal conductivity of SiO2-water nanofluid, so that, more aggregated nanoparticles in the base fluid cause to increase the thermal conductivity. The results of MD simulation were also compared with HC model and their differences are obviously
Fig. 7. Nanofluid thermal diffusivity enhancement by adding number of aggregated nanoparticles.
5.3. Effect of aggregation on thermal properties of nanofluid at constant nanoparticle loadings To investigate the effect of nanoparticle aggregation on thermal properties of nanofluid at constant nanoparticle loading, two cases were considered. In the first case, two aggregated nanoparticles were located at the water base fluid and the results were compared to the nanofluid with two dispersed nanoparticles at the same concentration (i.e. 3%). Second case is as the first case except instead of two nanoparticles, three nanoparticles with the concentration of 4.5% were used. A 15 nm silica nanoparticle consisting of 126 atoms was considered in all simulations. 5.3.1. Specific heat and thermal diffusivity As before for calculating the specific heat and thermal diffusivity of dispersed SiO2 nanoparticles in water base fluid EMD and NEMD simulations at temperature of 308 K were carried out respectively. Then the results were compared with the results of the nanofluid with aggregated nanoparticles from previous section (i.e. Section 5.2.1). Fig. 10 shows
Fig. 9. Thermal conductivity enhancement versus nanoparticle volume fractions.
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Fig. 12. Comparison of the thermal conductivity of aggregated and dispersed nanoparticles in the nanofluid with HC model and the base fluid. Fig. 10. Comparison of specific heat of SiO2-water nanofluid for two cases of dispersed and aggregated nanoparticles.
the results for specific heat. As one can see from this figure, specific heat of the nanofluid in both cases of suspended and aggregated nanoparticles is almost close. Specific heat depends on system temperature and nanoparticle concentration; hence, when these parameters maintain constant, the specific heat for two cases of nanofluid had no considerable difference. This finding is in accordance with Eq. (4). Fig. 11 compares the thermal diffusivity of SiO2-water nanofluid for the two cases of dispersed and aggregated nanoparticles. As one can see from this figure, the nanofluid diffusivity of suspended nanoparticles for each volume fraction is by about 2% more than that of aggregated case. The smaller and dispersed nanoparticles can move faster; therefore, they create more efficient space for heat transfer. 5.3.2. Thermal conductivity The thermal conductivities of 3% and 4.5% suspended nanoparticles in the nanofluid were calculated and the results compared to the case of aggregated nanoparticles. Fig. 12 shows the results. The thermal conductivity of aggregated SiO2 nanoparticle in water base fluid at the same concentration was less than suspended nanoparticles in the base fluid. This trend is similar to the thermal diffusivity. The suspended nanoparticles have smaller size than aggregated nanoparticles; therefore, they move faster and easier. This causes heat transfer inside the nanofluid to increase. Fig. 12 also compares results of MD simulation with the calculated values from HC model and those experimental data of the base fluid. As one can see from this figure, the predicted thermal conductivity for both cases of the nanofluid is more than HC model. HC model underpredicts the thermal conductivity because it does not consider the movements of atoms and their possible collisions, which can cause
to transfer the heat in nanofluid faster. Contrary to the HC model, MD simulation considers these movements in the system; therefore it is a suitable and reliable tool for studying the heat transfer mechanism in nanofluid. 6. Conclusions and recommendations In this study, a combined EMD and NEMD simulation was used to calculate the specific heat, thermal diffusivity and thermal conductivity for silicon dioxide in water nanofluid system. To validate the MD model, the results were compared with experimental data for water and HC model for nanofluids. Thermal properties of mentioned nanofluid were calculated and found that the nanoparticle aggregation could have an ambivalent effect on specific heat, diffusivity and the thermal conductivity. The results show that when the aggregation takes place by adding one nanoparticle in each step, the specific heat decreases by about 3%, but diffusivity increases by about 3.5%. This increasing trend was also observed for the thermal conductivity. Moreover, at first by adding one nanoparticle to the base fluid, the value of thermal conductivity considerably increased. For the cases of two and three aggregated nanoparticles, this enhancement was less than that of one nanoparticle. Thermal properties of suspended nanoparticles and aggregated nanoparticles at constant nanoparticle concentration in the base fluid were calculated and observed that when the nanoparticles are suspended, specific heat of nanofluid did not change with respect to the aggregated nanoparticles, but its diffusivity and thermal conductivity increase by about 2%. Generally, it can be concluded that aggregation cannot have significant effect on thermal properties of nanofluid. It may be because of this fact that when the nanoparticles become aggregated, their concentration in the nanofluid increases until the aggregates become so large that they separate from each other over time and settling takes place; therefore, improvement of thermal properties of nanofluids via nanoparticle aggregation is temporary. Nevertheless, study about aggregation kinetic on thermal properties of nanofluid is very complex and presenting definite opinion about it, is very difficult. Also, in this study we only used one aggregate and one can investigate by more aggregates. References
Fig. 11. Comparison of thermal diffusivity of SiO2-water nanofluid for two cases of dispersed and aggregated nanoparticles.
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Investigation of nanoparticle aggregation effect on thermal properties of nanofluid by a combined equilibrium and non-equilibrium molecular dynamics simulation Mina Sedighi, Ali Mohebbi ⁎ Department of Chemical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
a r t i c l e
i n f o
Article history: Received 19 October 2013 Received in revised form 16 March 2014 Accepted 22 April 2014 Available online 5 May 2014 Keywords: Nanofluid Aggregation Thermal conductivity Specific heat Thermal diffusivity Molecular dynamics simulation
a b s t r a c t Many theoretical and experimental studies on heat transfer and flow behavior of nanofluids have been done and the results show that nanofluids significantly increase heat transfer. Nevertheless, there is no accurate understanding from the effect of different mechanisms on nanofluid heat transfer. Computer simulations are a suitable tool for description of physical mechanisms in many processes. In this study, molecular dynamics simulation was used to investigate the effect of nanoparticle aggregation on thermal properties of water-silicon dioxide nanofluid, specifically its thermal conductivity. For calculating nanofluid thermal conductivity a combination of two equilibrium and non-equilibrium molecular dynamics simulations was performed to calculate the specific heat and thermal diffusivity of the nanofluid, respectively. Simulations were performed in NVT ensemble and spherical coordinate. The model was validated by comparison of thermal properties of water base fluid with experimental data in four various temperatures. Results also were compared with theoretical models such as HC model for nanofluid. To investigate the effect of nanoparticle aggregation, two cases of constant and variable volume fractions (i.e. 1.5, 3 and 4.5%) at temperature of 308 K were considered. The results showed that when the aggregation occurs with increasing nanoparticle concentrations, there are an increase in the thermal conductivity and thermal diffusivity of the nanofluid and a decrease in its specific heat. Moreover, when aggregation takes place at constant nanoparticle concentration, the specific heat of nanofluid with suspended nanoparticles did not change with respect to nanofluid with aggregated nanoparticles, but its diffusivity and thermal conductivity increase. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Cooling systems are one of the most important concerns in factories, industries, transportation and each place that deals with heat transfer. Therefore, in new technologies high heat flow process was created in order to enhance heat transfer. There are different methods for heat transfer improvement [1]. One of them is using the nanofluids instead of current heat transfer fluids. Nanofluids have been prepared by dispersing metallic or non-metallic particles at nanoscale size in ordinary heat transfer fluids. Suspended nanoparticles increase heat transfer by means of increasing the values of nanofluid thermophysical properties. Thermal conductivity of nanofluids is the most important parameter to indicate the heat transfer potential. Several mathematical models have been presented to predict the nanofluid thermal conductivity [2]. Maxwell's [3] model is the first model in this context. He had predicted that the thermal conductivity of nanofluids is a function of particle volume fraction and thermal ⁎ Corresponding author. Tel./fax: +98 3412118298. E-mail addresses: [email protected], [email protected] (A. Mohebbi).
http://dx.doi.org/10.1016/j.molliq.2014.04.019 0167-7322/© 2014 Elsevier B.V. All rights reserved.
conductivity of fluid and solid particles. Later Hamilton and Crosser (HC) [4] by development of Maxwell model and adding a shape factor in it have presented a new model. In addition several models such as Jeffrey [5] and Davis [6] models have been created. All of these models have been investigated in macroscale size and do not consider the solid and liquid movements and consequently probable collisions that cause thermal conductivity enhancement, therefore, they obtained underpredict values in comparison with experimental data. According to experimental researches [7–9] some parameters can be effective on thermal conductivity coefficient, such as, particle volume fraction, type of nanoparticles, size of particles, and temperature. Hong et al. [9] prepared Fe nanofluid based on ethylene glycol and indicated that thermal conductivity of Fe nanofluid is increased nonlinearly up to 18% as the volume fraction of particles is increased up to 55%. Daungthongsuk and Wongwises [10] experimentally reported the thermal conductivity and dynamic viscosity of TiO2-water nanofluid. They found that thermal conductivity and viscosity of nanofluids depend on temperature, so that, thermal conductivity of nanofluid increases with increasing temperature and conversely its viscosity decreases. Timofeeva et al. [11] characterized nanofluids of alumina
M. Sedighi, A. Mohebbi / Journal of Molecular Liquids 197 (2014) 14–22
Nomenclatures Cp Cv E k KB r t T U α φ ρ
Specific heat at constant pressure Specific heat at constant volume Total energy Thermal conductivity Boltzmann's constant Radius Time Temperature Potential energy Thermal diffusivity Particle volume fraction Density
Subscripts Nf Nanofluid f Fluid p Particle th Theory MD Molecular dynamics
particles in water and ethylene glycol using thermal conductivity, viscosity and dynamic light scattering measurements. Their results showed that the thermal conductivity enhancement is within the range predicted by effective medium theory in which the particles are also agglomerated over time. Kathikeyan et al. [12] synthesized CuO nanoparticles with average diameter of 8 nm by a simple precipitation technique and study the thermal properties of the suspensions. The experimental results showed that the nanoparticle size, polydispersity, cluster size, and the volume fraction of the particles have a significant effect on thermal conductivity. The study also mentioned that nanofluids containing ceramic or metallic nanoparticles showed large enhancement in thermal conductivity that cannot be explained by conventional theories. Except for the parameters that were studied at experimental measurements, investigation of possible mechanisms in molecular scale proves that several other parameters in molecular level are considerable. These factors including Brownian motion of nanoparticles, liquid layering at the liquid/solid interface, nanoparticle clustering and radioactive heat transfer somewhat justify nanofluid's unusual behavior [13]. Nevertheless, generally there is no determinative mechanism in nanofluid studies. Nanoparticle agglomeration is one of the most controversial mechanisms in nanofluid thermal conductivity studies. It has been indicated that when the nanoparticles are suspended in the base fluid, on the effect of Van der Waals forces they are agglomerated over time. This phenomenon has been called nanoparticle aggregation. There are several theories to examine the aggregation effects on the nanofluid's thermal properties and they are described in the following. Xuan et al. [14] applied the theory of Brownian motion and diffusion-limited aggregation model to simulate random motion and the aggregation process of the nanoparticles. They found that morphology of the suspended nanoparticles besides nanoparticle diameter and volume fraction of nanoparticles is one of the several important factors that affect the thermodynamic properties of nanofluid and that formation of aggregates reduces the efficiency of the energy transport enhancement of the suspended nanoparticles. Prasher et al. [15] used aggregation kinetics of nanoscale colloidal solutions combined with physics of thermal transport to capture the effect of aggregation on the thermal conductivity of nanofluids. Their study developed a unified model, which combines the micro convective effects due to Brownian motions with the change in conduction due to aggregation. The results
15
showed that colloidal chemistry plays a significant role in deciding the conductivity of colloidal suspensions. Jie et al. [16] proposed a new model for thermal conductivity of nanofluids, which is derived from the fact that nanoparticles and clusters coexist in the fluids. The effects of compactness and perfectness of contact between the particles in clusters on the effective thermal conductivity are analyzed. The model showed that the effective thermal conductivity of nanofluids decreases with the increasing concentration of clusters. Feng et al. [17] proposed a new model for effective thermal conductivity of nanofluids based on nanolayer and nanoparticle aggregation. Their study was derived on a model based on the fact that a nanolayer exists between nanoparticles and fluid and some particles in nanofluids may contact each other to form clusters. A governed equation for effective thermal conductivity was developed by both the agglomerated clusters and nanoparticles suspended in the fluids. Wu et al. [18] verified experimentally and theoretically the significance of the effect of the cluster structure, size distribution, and thermal conductivity of solid particles in water. The aggregation kinetics of SiO2 particles in water base fluid was done by adjusting the pH. Their experiments showed that clustering has no any discernible enhancement in the thermal conductivity even at high volume loading. Investigation of previous researches showed that there are no conclusive theories for the effect of aggregation on thermal properties of nanofluids. Nanoparticle aggregation creates a condition with lower thermal resistance for heat transfer and causes the thermal conductivity enhancement. On the other hand, this phenomenon can decrease the thermal conductivity of nanofluids, in this case, aggregation of the nanoparticles which may cause instability in the suspension, as well as create low efficiency areas in the liquid. To evaluate the nanofluid's behavior and investigation of the effect of various factors on it, two Mont Carlo and molecular dynamics simulations [19] have been created. Molecular dynamics simulation was first introduced to study the interactions of hard spheres by Alder and Wainwright [20] in the late 1950s. Later at 1964 Rahman [21] carried out the first simulation using a realistic potential for liquid argon. This method simulates the system by force, velocity and positions of atoms in the system at each time step and its purpose is calculation of macroscopic state of the system with a microscopic model. Molecular dynamics simulation uses the algebraic method to specify the path of atoms, thus, it has particular computational advantages. Presence of the time variable estimates the required time in each simulation. In Monte Carlo simulation to evaluate the time there is no such estimation. There are two equilibrium molecular dynamics and non-equilibrium molecular dynamics approaches for molecular dynamics simulations that have been used by researchers to study the nanofluid's behavior. Keblinski et al. [13] explored the four possible explanations for increasing thermal conductivity using molecular dynamics simulation: Brownian motion of particles, molecular layering of the fluid at the liquid/solid interface, the nature of heat transport in nanoparticles and the effect of nanoparticle clustering. Eapen et al. [22] performed molecular dynamics simulations of the time-dependent heat current correlation to obtain the systematic, dynamical details at the atomistic level using a model system of Xe base fluid and Pt nanoparticles. Their model indicated that the interatomic interactions between fluid and the nanoparticles can be much stronger than the interaction between fluid particles, as well as nanoparticle interactions, are significant factors to understand the heat transport mechanisms of nanofluids. Galamba et al. [23] calculated the thermal conductivity of molten NaCl and KCl through the Evans–Gillan non-equilibrium molecular dynamics algorithm and Green–Kubo equilibrium molecular dynamics simulations. The EMD simulations performed for a binary ionic mixture and the NEMD simulations assumed a pure system for reasons discussed in their work. They found that the thermal conductivity obtained from NEMD simulations was in very good agreement with that obtained through Green–Kubo EMD simulations for a binary ionic mixture. Galliero and Volz [24] used a non-equilibrium molecular dynamics simulation to
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propose a new algorithm for calculating single particle thermodiffusion. Through their estimation the thermophoretic force has been applied on a solute particle. Their results showed that thermal conductivity decreases with nanoparticle concentration. They also realized that nanoparticle tends to migrate toward the cold area and the single particle thermal diffusion coefficient is independent to the size of the nanoparticle, whereas it increases with the quality of the solvent and is inversely proportional to the viscosity of the fluid. Sarkar and Selvam [25] used an equilibrium molecular dynamics simulation to calculate the thermal conductivity of copper in argon nanofluid. For this nanofluid, they reported the thermal conductivity enhancement up to 20% compared to the base fluid with 1% nanoparticle concentration. Sankar [26] used an equilibrium molecular dynamics modeling for estimation of thermal conductivity enhancement of water due to well dispersed platinum nanoparticle. They compared their results with existing experimental data for copper nanoparticles suspended in water and those predicted by conventional theories. The thermal conductivity of the nanofluid was obtained from the Green Kubo formulation. Kang et al. [27] examined a non-equilibrium molecular dynamics simulation in a copper– argon nanofluid. They introduced two different methods; the physical definition and the curve fitting to calculate the coupling factor between nanoparticles and base fluid. They found that nanoparticle aggregation causes to decrease the coupling factor. Furthermore, they showed that the coupling factors obtained by these two methods are consistent and proportional to the volume fraction of the nanoparticle and inversely proportional to nanoparticle diameter. Jia et al. [28] studied on the characteristics of a nanofluid system composed of argon liquid and copper nanoparticle, such as heat current measurement by its mean value, variance, third moment, and the Shannon entropy. For this purpose, they carried out a molecular dynamics simulation using Green–Kubo method. They exhibited that the thermal conductivity increases as the nanoparticle volume fraction increases. Kang et al. [29] performed a MD simulation using Green–Kubo method to study the effect of nanoparticle aggregation on the thermal conductivity and viscosity in nanofluids. They placed multiple nanoparticles in the simulation box to simulate the aggregation of the nanoparticles. Their results showed that the nanoparticle aggregation induces a significant enhancement of thermal conductivity in nanofluid, while the viscosity increases moderately. Moreover, they indicated that different configurations of the nanoparticle cluster bring about different enhancements of thermal conductivity and increase of viscosity in the nanofluid. Mohebbi [30] predicted the specific heat and thermal conductivity of silicon nitride nanoparticle based on liquid argon by a combined molecular dynamics simulation. He examined the effect of temperature and volume concentration of nanoparticles on thermal conductivity of nanofluid and found that the thermal conductivity of nanofluid increases with the increase of the volume concentration and decrease of the temperature. Heat transfer autocorrelation function or Green–Kubo integral formula [31] and the direct method are the most commonly used for equilibrium and non-equilibrium molecular dynamics simulations respectively. The convergence of Green–Kubo's integral usually requires a lot of time or it may not be possible. Also, this method is not able to perform simulations for the systems with a large number of molecules. Therefore, finding a reliable and simpler method without disadvantages of Green–Kubo's formula, in which it does not need heavy and time consuming programming, seems to be helpful. In this study a new method based on combination of two equilibrium and non-equilibrium molecular dynamics approaches was presented and the effect of nanoparticle aggregation on thermal properties of SiO2-water nanofluid such as its specific heat, thermal diffusivity and thermal conductivity was investigated. Whereas, the aggregation of nanoparticles is dependent on particle volume concentration, calculations were done in two cases of variable and constant volume fractions of nanoparticles. Also, to validate the obtained results, simulation for water in four temperatures was carried out and the results were compared with experimental data [32].
2. Methodology As mentioned before, up to now the main purpose of researchers has been the study of thermal conductivity coefficient (k) that it is obtained directly by current molecular dynamics methods. Using the equilibrium and non-equilibrium molecular dynamics simulations simultaneously, properties of SiO2-water nanofluid including, specific heat and heat diffusivity were calculated besides calculating thermal conductivity. To compute the specific heat and diffusivity (two necessary parameters for the thermal conductivity calculation) EMD and NEMD simulations were applied, respectively that each one is described in the following. 2.1. Thermal conductivity calculation The conduction of heat transfer is created by means of the atom oscillation against each other, thus, in a nanofluid one can estimate that the rate of heat conduction is similar to particle diffusivity in the fluid. According to the relation between conductivity and diffusivity: αnf ¼
knf ρnf Cpnf
ð1Þ
where, knf and Cpnf are the thermal conductivity and specific heat of nanofluid, respectively and ρnf is its density and it is calculated by ρnf = φρnp + (1 − φ)ρf that φ is a volume fraction of nanoparticles. In the present study the thermal conductivity (k) of SiO2-water nanofluid was calculated by rewriting Eq. (1) as: knf ¼ αnf ρnf Cvnf :
ð2Þ
2.2. Specific heat calculation A long EMD simulation in spherical boundary condition and NVT (canonical) ensemble was performed. In canonical ensemble, the specific heat equation has been derived from the definition of constant as below [33]: volume specific heat Cv ¼ ∂E ∂T D Cv ¼
v
E E −hEi2 2
ð3Þ
kB T 2
where, b E N is the average total energy, bE2N is the squared-average total energy, T is temperature (K) and KB is Boltzmann's constant = 0.00198 kcal/mol·K. During the simulation, fluctuations of the energy were recorded as a function of time (t). There are no exact and comprehensive experimental data for thermal properties of SiO2-water nanofluid. Harry O'Hanley et al. [34] offered the following equation to calculate the specific heat of nanofluids:
Cpnf ¼
ρCp ρnf
nf
¼
φ ρCp þ ð1−φÞ ρCp p
φρp þ ð1−φÞρ f
f
:
ð4Þ
This equation provides a theoretical approach for predicting the thermal properties of nanofluids. It has been proved that there is a good agreement between the obtained values from the model of Eq. (4) and limited available experimental data [34]. In this study, the present model was used to compare our results with the calculated values from this model. 2.3. Thermal diffusivity calculation In this step, a NEMD simulation using the results of the last equilibration step in spherical boundary condition was performed and diffusivity was calculated by considering an imaginary shell around the atomic
M. Sedighi, A. Mohebbi / Journal of Molecular Liquids 197 (2014) 14–22
Generally, thermal diffusivity, which have been conducted few studies about it, can be obtained from Eq. (9).
Table 1 Bonded parameters for SiO2 and water molecules. Parameters
Si\O\Si
O\Si\O
H2O (TIP3P)
kθ (kcal/mol·rad2) θ0 (degree) kb (kcal/mol·Å2) b0 (Å)
4.66 174.22 885.1 1.61
159.57 110.93 885.1 1.61
55 104.52 545 0.96
αnf ¼
6 R min;ij 12 R −2 min;ij rij rij
εO εH εSi RO min/2 RH min/2 RSi min/2 qO (SiO2) qO (H2O) qH qSi εO 1− 4 εH 1− 4 εSi 1− 4
Cq q
Uelec ¼ ε1−4 ε ir j 0 ij
−0.152 −0.046 −0.3 1.76 0.22 2.1475 −0.834 −0.5 0.417 1.0 −0.15 −0.046 −0.3
* εij (kcal/mol) = sqrt (εi·εj), Rmin,ij (Å) = Rmin,i / 2 + Rmin,j / 2.
sphere. Molecule temperature in the outer layer of the sphere was specified. Temperature of the sphere and the shell around it will be equal at the end and diffusivity was determined by monitoring changes of the system temperature and comparing it to the theoretical expression. In a sphere of radius R the theoretical average temperature as a function of the time in each time step is calculated as below [33]:
TðtÞtheory ¼ Tbath þ 6
∞ . TSim −Tbath X nπ 2 1 exp − αnf t 2 n R π n¼1 2
ð5Þ
where αnf is the thermal diffusivity, and TSim and Tbath are the initial temperature of the system and the temperature of the sphere boundary, respectively. This equation is governed by the heat diffusion equation [33]: ! ∂T r ; t ∂T
2 ! ¼ αnf ∇ T r ; t
ð6Þ
TðR; tÞ ¼ Tbath :
rbR
ρCp Þnf
ð9Þ
3. Simulation procedure In the present study, to predict the thermal properties of SiO2-water nanofluid a combined EMD and NEMD simulation in two steps was performed. Water as the most applicable and the most available heat transfer fluid is a suitable choice for this purpose. The base fluid was considered by the number of 11,225 water atoms including Hydrogen and Oxygen atoms. Sirk et al. [35] indicated that the rigid, three-site (i.e. SPC, SPC/E), and transferable intermolecular potential (i.e. TIP3P) water models are shown to have similar thermal conductivity values at standard conditions, whereas models that include bond stretching and angle bending have higher thermal conductivities. In the present study, TIP3P water model implemented in CHARMM force field was adopted because it is a simple model with higher computational efficiency in comparison with other classical models of water. Also, non-bonded and bonded parameters such as angle, charges and forces are closer to real water molecule. The CHARMM version of TIP3P model places Lennard–Jones parameters on the Hydrogen atoms in addition to Oxygen. The initial arrangement of the atoms was configured based on replication of a unit cell of water molecules in spherical boundary condition and three dimensions [36]. The simulation was performed in canonical ensemble (NVT), which meant the total number of atoms; the system volume and temperature were constant throughout the simulation. To fix the system temperature all over the simulation, Langevin algorithm [37] was employed. Langevin algorithm keeps constant the system temperature by means of the velocity rescaling method. Integration scheme in the present simulation was velocity Verlet algorithm [38]. All of the simulation process was performed by NAMD2 software [39], which is a parallel molecular dynamics program, designed to run efficiently for simulating large molecules. NAMD employs a common potential energy function to determine a force field model that has the following contributions [39]: Utotal ¼ Ubond þ Uangle þ Udihedral þ þUvdw þ Ucoulomb :
with initial and boundary conditions for spherical coordinate: ! T r ; t ¼ TSim ð0Þ
knf
In this equation knf was calculated by HC model, thus, αnf estimated by this method has an underpredict value. We compared our simulation results with the thermal diffusivity that was calculated by Eq. (9).
Table 2 Non-bonded parameters and electrostatic properties for O, H and Si atoms. UL− J ¼ εmin
17
ð7Þ
ð8Þ
ð10Þ
The first three terms describe the stretching, bending and torsional bonded interactions, in which each one is defined by the specific mathematical equation. The last two terms in Eq. (10) describe interactions between non-bonded atom pairs, which correspond to the van der Waal's 6–12 forces and electrostatic interactions, respectively. These interactions are for all of fluid–fluid, solid–fluid and solid–solid atoms.
Fig. 1. Process of constructing a silicon dioxide nanoparticle.
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In this study we used CHARMM22 force field potential function, parameters and file formats [40]. We also considered that all atoms in the system have rigid bonds. Tables 1 and 2 show equations and numerical values of CHARMM22 bonded and non-bonded force field potential functions for water and SiO2 molecules [41,42]. In Table 2, C and ε0 are defined as Coulombs constant and vacuum permittivity respectively. VMD 1.9 software [43] was used to analyze the output results from NAMD software. VMD is a molecular visualization program for displaying, animating, and analyzing molecular systems using 3-D graphics and built-in scripting. 4. Initial configuration and modeling of the nanofluid To prepare the nanofluid, initially a unit cell of SiO2 crystal that contains eight Oxygen and four silicon atoms was built by means of modeling features of VMD software. Parameters of interatomic distances and bond angles have been described by VMD [43]. Then, to generate a crystal membrane in hexagonal geometry oriented perpendicular to the z-axis, the unit cell was replicated in three dimensions with replication numbers of 6, 6 and 12 along the respective pffiffifficrystal axis (i. e. x, y and z). The unit cell vectors were [1.00.0 0.0], [1/2 3=2 0.0] and [0.0 0.0 1.0]. To generate a nanoparticle with the radius of 7.5 nm and the number of 126 atoms, a sphere was cut out of the hexagonal patch. This nanoparticle was placed in the center of water base fluid. Thus, in this case, the concentration of the nanoparticles in the base fluid was 1.5 vol.%. Generation of the nanoparticle is shown in Fig. 1.
Fig. 3. Two (a) and three (b) dispersed nanoparticles in water base fluid for constant volume fractions of 3 and 4.5%.
4.1. Aggregation structure
4.2. Specific heat
Considering the dependency of the nanoparticle aggregation on concentration, the effect of the aggregation on thermal properties of nanofluid in two cases of variable and constant volume fractions was investigated. In the first case, three simulations were performed by increasing the number of aggregated nanoparticles and consequently increasing the concentration of nanoparticles in the base fluid. These simulations were done for one, two and three SiO2 nanoparticles in the base fluid (see Fig. 2) and the volume fractions of nanoparticle in nanofluid were 1.5, 3 and 4.5%, respectively. As one can see in Fig. 2b and c, in two simulations, two and three aggregated nanoparticles were placed in the center of water base fluid. Each nanoparticle contains 126 atoms with a radius of 7.5 nm. In the second case, two simulations were accomplished with two and three suspended nanoparticles in the base fluid for constant particle volume fractions of 3 and 4.5% (see Fig. 3).
To calculate the specific heat of nanofluid a long EMD simulation for 1,500,000 time steps was accomplished. Each time step was considered as 1 fs. The simulation was performed in spherical boundary condition and NVT ensemble. To fix the system temperature at 308 K (i.e. studied temperature) Langevin damping coefficient was considered at 3 ps−1. To improve computational efficiency, cutoff distance was chosen equal to 20 Å, which meant that when the two atoms were further than 20 Å, they had a negligible nonbonded interaction. To equilibrate the system throughout the simulation, initially in first 100 time steps, potential energy minimization at temperature of 0 K was carried out. In all over the simulation, by changing the atomic positions and gradual reduction of the potential energy, equilibration process continued until reaching the system to equilibrium. The relevant data to the total energy versus the time in each time step (i.e. 1 fs)
Fig. 2. 1.5% (a), 3% (b) and 4.5% (c) aggregated SiO2 nanoparticles in water base fluid.
M. Sedighi, A. Mohebbi / Journal of Molecular Liquids 197 (2014) 14–22
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was recorded in NAMD's logfile and specific heat of nanofluid was calculated according to Eq. (3).
The thermal diffusivity was calculated by optimization of an objective function defined below:
4.3. Thermal diffusivity
ο¼
To calculate the thermal diffusivity of nanofluid, a shell with thickness of 4 nm was considered around the spherical system and its temperature was considered by about 30 °C lower than the temperature of the sphere (i.e. 308 K); therefore, the heat was diffused in the system over time. In this step, a short NEMD simulation for 40,000 time steps was performed, because of using the results of the previous equilibration simulation for calculation of specific heat.
n X Tth −TMD 2 Tth i i¼1
ð11Þ
where, Tth and TMD are temperatures obtained from Eq. (5) and MD calculations respectively, and n is the number of time steps. This objective function was defined by investigation of the trend of decreasing system temperature from MD simulation and comparing it with the temperature reduction from theoretical calculations (i.e. Eq. (5)) and fitting the corresponding temperatures to achieve the minimum value of the difference. This was performed by the “goal seek” tool of Microsoft Office Excel 2010. To adjust the temperature of molecules in outer layer of the sphere, we used the feature of temperature coupling of NAMD2 software [33]. 5. Results and discussion 5.1. Validation of simulation In the present study to examine the MD results' accuracy, obtained results were compared with experimental data [32] for water base fluid. To satisfy the density of water, simulation was performed for the number of 11,255 water atoms and the radius of 30 nm in a spherical boundary condition and NVT ensemble. The thermal properties of water in four temperatures at 298, 308, 318 and 333 K were calculated, then the results were compared with corresponding experimental data and the relative errors calculated. Fig. 4a, b and c compares the simulation results with experimental data for specific heat, diffusivity and thermal conductivity of the water base fluid respectively. As one can see from this figure, there is good agreement between the MD results and experimental data and the errors are less than 3.5%. As mentioned previous, equilibration process with an initial minimization was completed in 1,500,000 time steps. Fig. 5 shows decreasing trend of the potential energy in first 1000 time steps and then reaching to equilibration. 5.2. Effect of aggregation on thermal properties of nanofluid by increasing the number of aggregated nanoparticles Three nanofluid simulations with one nanoparticle and then two and three aggregated nanoparticles were performed. Each nanoparticle contains 126 atoms with radius of 7.5 nm. The volume fractions of the nanoparticles in these nanofluids were 1.5, 3 and 4.5%, respectively. 5.2.1. Specific heat and thermal diffusivity To calculate the specific heat and diffusivity of the nanofluid, simulations were performed at constant temperature of 308 K. The density of the nanoparticle was 2369 kg/m3.
Fig. 4. Validation of the MD simulation results for specific heat, diffusivity and thermal conductivity of water base fluid.
Fig. 5. Equilibration process by the decrease of potential energy.
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Fig. 6. Specific heat of nanofluid versus number of aggregated nanoparticles.
Fig. 8. Thermal conductivity of nanofluid versus number of aggregated nanoparticles.
For specific heat the results are shown in bar chart in Fig. 6. As one can see, on increasing the number of aggregated nanoparticles and consequent concentration enhancement, specific heat of SiO2-water nanofluid decreases. The results of present MD model are in good agreement with calculated values from Eq. (4). Here is a point that needs to be considered, the results obtained from Eq. (4) is correct, when the nanoparticles in the base fluid are dispersed but in present simulation, SiO2 nanoparticles in water base fluid are aggregated. The reason of accordance between the calculated values from this model and our simulation is explained in Section 5.3.1. Fig. 7 shows the thermal diffusivity of nanofluid with one nanoparticle, two and three aggregated nanoparticles. This figure shows that by adding one nanoparticle in each step, nanofluid thermal diffusivity increases up to 3% rather than its previous case. This can be because of existence the paths with lower thermal resistance in water base fluid with more aggregated nanoparticles. As one can see from this figure, results are higher than those values predicted using Eq. (9). This is due to knf in Eq. (9). This parameter was calculated from HC model, which under predicts the thermal conductivity of nanofluid.
visible. We found thermal conductivity enhancement up to 9.8% compared to water base fluid by adding one SiO2 nanoparticle to the base fluid. This is corresponding to 1.5% nanoparticle concentration. By adding two and three aggregated nanoparticles (i.e. 3% and 4.5% nanoparticle concentration, respectively) to the base fluid, the thermal conductivity enhancements were 11.3% and 14.9% compared to the base fluid respectively. Fig. 9 shows the thermal conductivity enhancement of the nanofluid versus nanoparticle volume fractions. This figure also illustrates that the conductivity enhancement was steeper at low nanoparticle loading compared to higher loadings as found by Sarkar and Selvam [25] by MD simulation of copper nanoparticles in liquid argon. This result was not predicted by available theoretical models such as HC.
5.2.2. Thermal conductivity As mentioned, thermal conductivity of nanofluid was calculated by multiplying the specific heat and diffusivity of nanofluid in its density at the corresponding concentration and temperature. Fig. 8 shows the results. As one can see the number of nanoparticles can have a significant effect on the values of thermal conductivity of SiO2-water nanofluid, so that, more aggregated nanoparticles in the base fluid cause to increase the thermal conductivity. The results of MD simulation were also compared with HC model and their differences are obviously
Fig. 7. Nanofluid thermal diffusivity enhancement by adding number of aggregated nanoparticles.
5.3. Effect of aggregation on thermal properties of nanofluid at constant nanoparticle loadings To investigate the effect of nanoparticle aggregation on thermal properties of nanofluid at constant nanoparticle loading, two cases were considered. In the first case, two aggregated nanoparticles were located at the water base fluid and the results were compared to the nanofluid with two dispersed nanoparticles at the same concentration (i.e. 3%). Second case is as the first case except instead of two nanoparticles, three nanoparticles with the concentration of 4.5% were used. A 15 nm silica nanoparticle consisting of 126 atoms was considered in all simulations. 5.3.1. Specific heat and thermal diffusivity As before for calculating the specific heat and thermal diffusivity of dispersed SiO2 nanoparticles in water base fluid EMD and NEMD simulations at temperature of 308 K were carried out respectively. Then the results were compared with the results of the nanofluid with aggregated nanoparticles from previous section (i.e. Section 5.2.1). Fig. 10 shows
Fig. 9. Thermal conductivity enhancement versus nanoparticle volume fractions.
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Fig. 12. Comparison of the thermal conductivity of aggregated and dispersed nanoparticles in the nanofluid with HC model and the base fluid. Fig. 10. Comparison of specific heat of SiO2-water nanofluid for two cases of dispersed and aggregated nanoparticles.
the results for specific heat. As one can see from this figure, specific heat of the nanofluid in both cases of suspended and aggregated nanoparticles is almost close. Specific heat depends on system temperature and nanoparticle concentration; hence, when these parameters maintain constant, the specific heat for two cases of nanofluid had no considerable difference. This finding is in accordance with Eq. (4). Fig. 11 compares the thermal diffusivity of SiO2-water nanofluid for the two cases of dispersed and aggregated nanoparticles. As one can see from this figure, the nanofluid diffusivity of suspended nanoparticles for each volume fraction is by about 2% more than that of aggregated case. The smaller and dispersed nanoparticles can move faster; therefore, they create more efficient space for heat transfer. 5.3.2. Thermal conductivity The thermal conductivities of 3% and 4.5% suspended nanoparticles in the nanofluid were calculated and the results compared to the case of aggregated nanoparticles. Fig. 12 shows the results. The thermal conductivity of aggregated SiO2 nanoparticle in water base fluid at the same concentration was less than suspended nanoparticles in the base fluid. This trend is similar to the thermal diffusivity. The suspended nanoparticles have smaller size than aggregated nanoparticles; therefore, they move faster and easier. This causes heat transfer inside the nanofluid to increase. Fig. 12 also compares results of MD simulation with the calculated values from HC model and those experimental data of the base fluid. As one can see from this figure, the predicted thermal conductivity for both cases of the nanofluid is more than HC model. HC model underpredicts the thermal conductivity because it does not consider the movements of atoms and their possible collisions, which can cause
to transfer the heat in nanofluid faster. Contrary to the HC model, MD simulation considers these movements in the system; therefore it is a suitable and reliable tool for studying the heat transfer mechanism in nanofluid. 6. Conclusions and recommendations In this study, a combined EMD and NEMD simulation was used to calculate the specific heat, thermal diffusivity and thermal conductivity for silicon dioxide in water nanofluid system. To validate the MD model, the results were compared with experimental data for water and HC model for nanofluids. Thermal properties of mentioned nanofluid were calculated and found that the nanoparticle aggregation could have an ambivalent effect on specific heat, diffusivity and the thermal conductivity. The results show that when the aggregation takes place by adding one nanoparticle in each step, the specific heat decreases by about 3%, but diffusivity increases by about 3.5%. This increasing trend was also observed for the thermal conductivity. Moreover, at first by adding one nanoparticle to the base fluid, the value of thermal conductivity considerably increased. For the cases of two and three aggregated nanoparticles, this enhancement was less than that of one nanoparticle. Thermal properties of suspended nanoparticles and aggregated nanoparticles at constant nanoparticle concentration in the base fluid were calculated and observed that when the nanoparticles are suspended, specific heat of nanofluid did not change with respect to the aggregated nanoparticles, but its diffusivity and thermal conductivity increase by about 2%. Generally, it can be concluded that aggregation cannot have significant effect on thermal properties of nanofluid. It may be because of this fact that when the nanoparticles become aggregated, their concentration in the nanofluid increases until the aggregates become so large that they separate from each other over time and settling takes place; therefore, improvement of thermal properties of nanofluids via nanoparticle aggregation is temporary. Nevertheless, study about aggregation kinetic on thermal properties of nanofluid is very complex and presenting definite opinion about it, is very difficult. Also, in this study we only used one aggregate and one can investigate by more aggregates. References
Fig. 11. Comparison of thermal diffusivity of SiO2-water nanofluid for two cases of dispersed and aggregated nanoparticles.
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