Numerical simulation of aggregation effect on nanofluids thermal conductivity using the lattice Boltzmann method

International Communications in Heat and Mass Transfer 110 (2020) 104408

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Numerical simulation of aggregation effect on nanofluids thermal conductivity using the lattice Boltzmann method

T

Hamed Tahmooressia,b, Alibakhsh Kasaeianb, Ali Tarokhc, , Roya Rezaeib, Mina Hoorfara ⁎

a

School of Engineering, University of British Columbia, Kelowna, Canada Faculty of New Science and Technologies, University of Tehran, Tehran, Iran c Department of Mechanical Engineering, Lakehead University, Thunder Bay, ON, Canada b

ARTICLE INFO

ABSTRACT

Keywords: Nanofluid Thermal conductivity Lattice Boltzmann method Nanoparticle aggregation

The standard enhancement in nanofluids thermal conductivity due to the addition of nanoparticles is well understood. Despite this, the reason behind observed anomalous increases is still controversial. Limitations in nanoscale experimental observations would make it even harder to approach into this topic. To address this issue, researchers have proposed many different macroscopic (continuum-based)/microscopic (molecular scale) numerical schemes as an alternative for experimental investigations. However, the overall thermal effect of suspended nano-scale particles cannot be observed in neither macroscopic nor microscopic scale due to collective interrelated behaviors such as nanoparticles aggregation. In this paper, a mesoscopic approach, Lattice Boltzmann method (LBM), aims to consider microscopic phenomena in a broader context (mesoscopic scale), been implemented to investigate the nanoparticles aggregation as a probable working mechanism behind the anomalous increase in nanofluids thermal conductivity. The stochastic and dynamic nature of nanoparticles aggregation is captured through generation of fractal random microstructures. The effects of size, shape and distribution regime of aggregates are studied and optimum values are calculated. The results indicate that the aggregation can anomalously enhance nanofluids effective thermal conductivity (ETC). The LBM results are found to be in great agreement with the available numerical/experimental data in the literature.

1. Introduction Nanofluid, as a modern engineering material, has shown to provide promising solutions in heat transfer applications due to its significant thermal properties. In comparison to other multi-component thermal materials (such as composites) nanofluids have shown a much higher thermal energy conduction characteristic [1], due to, undoubtedly, the fluidity of the base material (referred to as the matrix). However, the experimental limitations in the nano-scale investigations have hindered the understanding of the primary mechanism behind this characteristic. This shortcoming has led to the proposition of different microscopic theories such as Browning motion, liquid layering, nanoparticles clustering, and ballistic phonon transport [2–5]. In this situation, numerical simulation can be implemented as a powerful tool to observe the working function of these mechanisms explicitly. Among all these theories, nanoparticles aggregation appears to be more plausible in explaining the reason behind this phenomena base on the numerous studies conducted [2,6–13]. Therefore, conduction heat transfer in nanofluids must be simulated by employing a numerical method which

⁎

Corresponding author. E-mail address: [email protected] (A. Tarokh).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104408

0735-1933/ © 2019 Elsevier Ltd. All rights reserved.

considers not only the microscopic nature of heat conduction in nanofluids but also the complicated stochastic geometrical structures in the computational domain. For such a numerical approach two main steps must be taken into account: generating the microstructure (modeling of the aggregation process), and choosing the proper numerical solver (simulating the aggregation effect on nanofluids thermal conductivity). Nanoparticles may go under different processes in an aquatic environment such as homo/hetero-aggregation, dissolution and chemical transformation [14]. In essence, aggregation is a process during which nanoparticles may stick together and form random structures (called aggregate) as they migrate through the base fluid. This phenomenon happens due to the interparticle forces and their interactions [15]. The first strategy to introduce the effect of aggregates in the thermal conductivity calculations is through a size parameter in theoretical models [16,17]. This size parameter (equivalent radii) can be both experimentally and theoretically extracted. For instance, researchers [18] have proposed a relationship between different parameters: gyration radius, hydrodynamic radius and smallest sphere enclosing radius of an aggregate. Knowing two of these parameters through

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Nomenclature

v dp v m E c e R ωi g e c A Φp Φint Φi Nint Nc df dl R

Parameter Definition φ Advection-diffusion equation dependent variable t Time Γ Diffusion coefficient u Velocity h Enthalpy x Length variable k Thermal conductivity coefficient ρ Density T Temperature Cp Pressure constant specific heat n Unit normal vector a Interface of material a b Interface of material b q Heat flux f Fluid index s Solid index τg Relaxation factor f Distribution function

Particles velocity Particles diameter Volume Mass Total energy Random thermal motion Macroscopic internal Gas constant Weight coefficient Energy distribution function Lattice unit vector Lattice pseudo sound speed Cross section Solid volume fraction Aggregation level Local filling factor Number of particles in a cluster Number of particles in a cluster’s backbone Fractal dimension Chemical Dimension Radius of the inspection

wide variety of combined methods of MD simulations [29–31]. A combined method of EMD (equilibrium molecular dynamics) and NEMD (non-equilibrium molecular dynamics) have been used by Mohebbi et al. [32]. Their simulation results revealed that the thermal conductivity computed in two separated temperatures was increased as the result of nanoparticles aggregation. In MD simulations, selecting a proper inter-atomic potential function can affect the outcome results. This function defines the position of every single particle with respect to time by considering their interactions. Kang et al. [33] studied this issue and improved the accuracy of MD simulation by implementing two potentials, the Lennard-Jones (L-J) and the Embedded Atom Method (EAM). In another study, Wang et al. studied the aggregation morphology of nanoparticles using the EMD method for copper‑argon nanofluids [34]. They modeled two types of nanoparticles aggregations (loose and compact) and showed that the effective thermal conductivity increases with decrease in aggregates fractal dimension, a parameter that mathematically defines the complex geometry of aggregates. Although MD is one of the best candidates to simulate the transport phenomena in nano-scale, the collective behavior of large body of aggregates cannot be investigated due to the significant computational cost of the MD method for simulation of the behavior of several particles. The Lattice Boltzmann method (LBM) is another microscopic method that can handle complex geometries and requires a reasonable computational time (compared to MD) for simulation of transport phenomena in a medium. Considering the advantages of LBM for microscale simulations, there is a great opportunity to utilize this method to investigate nanofluids thermal conductivity which is a complex function of the nanoparticles size, aspect ratio, distribution regime, and material. LBM is capable of importing the effect of all these parameters into simulations while keeping the simulation cost comparable to conventional CFD methods (this can be observed in a few simulations conducted for other two-component systems, such as porous media, cement paste or aerogels [35–37]). For instance, Fang et al. [38] have generated microstructures for cement pastes using the open-cell and diffusion-limited cluster-cluster aggregation (DLCA) methods. Then, the LBM is implemented to solve the conduction-radiation equation to acquire the effective thermal conductivity (ETC) coefficient. The main difference between these materials and the nanofluids is the ability of nanoparticles to migrate in the base fluid which may lead to a higher heat transfer rate. Therefore, the application of LBM in nanofluids thermal energy conduction simulations would not be different from that

experimental investigations, the other one can essentially be derived, leading to the equivalent radii. Then, this parameter can be implemented in a mean-field theory model to import the effect of aggregation phenomenon. In another example, Wei et al. [19] derived a theoretical Hamilton-Crosser based model to import the aggregation shape effect into the effective thermal conductivity (ETC) calculation. They used the shape factor in the Hamilton-Crosser model to represent the aggregates configurations. But, since these are theoretical models, aggregates shape cannot be fully investigated in the first strategy (the theoretical approach) as it is imported within a sole number or some representative parameters. Therefore, the second strategy, numerical microstructure generation (NMG), sounds to be more promising for studying the aggregations shape effect on nanofluids ETC. Comprehensive NMG methods include those that implement the fractal theory. In the Fractal-based NMG methods, the results are more consonant with the natural behavior of the suspended nanoparticles. The Diffusion-Limited Cluster Aggregation (DLCA) and ReactionLimited Cluster Aggregation (RLCA) are among the mostly used fractalbased algorithms to model aggregation process [20–23]. Aside from fractal representation of aggregates, researchers [24] have implemented Population Balance Equation (PBE) method to capture the dynamic behavior of nanoparticles in terms of their aggregation, nucleation and breakage. However, a Maxwell-based model is implemented for the ETC calculation through the Monte Carlo method. It should be noted that there are also some NMG reports in which aggregates are not modeled based on their natural behavior [25,26]. Lotfizadeh et al. [25] investigated the effect of simple regular aggregates on nanofluids ETC using Monte Carlo simulations. The aggregation growth trend and its shape evolution (aggregation middle steps) are not consonant with the real behavior of nanoparticles in nanofluids. Therefore, the exact effect of aggregation processes on the nanofluids thermal conductivity cannot be observed. Since the sizes of the nanoparticles are in the Nano range, continuum assumption for the domain does not conform to the physics of the problem, and utilizing the numerical approaches is not feasible to predict the transport phenomena in the nanofluids. In this case, the Discrete Simulation of Fluid Dynamic (DSDF) approaches such as Monte Carlo, molecular dynamics (MD), and lattice Boltzmann schemes must be employed to simulate heat transfer in the nanofluids [22,27,28]. Molecular Dynamics (MD) simulation is one of the well-known numerical methods that has been implemented by many researchers to investigate the effective thermal conductivity of nanofluids. There is a 2

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of other materials if the migration of nanoparticles is carefully considered. The LBM solves the governing equations in a mesoscopic scale and has been used in simulation of convection heat transfer in nanofluids by different research groups [39–41]. However, the application of this method in investigation of conduction heat transfer is very limited. Pelevic et al. [42] utilized LBM in simulation of conduction heat transfer in nanofluids. In this study, nanoparticles are dispersed into the computational domain based on the Poisson distribution function, and the effect of their size on ETC is calculated for different nanoparticle sizes and distributions. The effect of nanoparticles aggregation, however, is not considered. Also, they have not provided any evidence related to the selected distribution function of the nanoparticles. In this paper, a lattice Boltzmann method is implemented to simulate conduction of thermal energy in suspended aggregations and the base fluid. Since the effect of the aggregations' shape is of a great importance in thermal energy conduction, an algorithm is developed to model the aggregation process and to capture their dynamic behavior with respect to time. As a result, LBM provides a platform to have the heat conduction problem converged properly for complex geometries including a collection of complicated random aggregates. Finally, the configurations of aggregations are fully defined using the fractal theory to demonstrate the correlation between aggregations size, shape, distribution regime and the heat conduction enhancements in nanofluids.

t

xi

Ka

(

ui) =

xi

xi

( h) +

xi

( hu i ) =

xi

k

T xi

k

T n

qint =

T xi

(3)

T =0 xi

(4)

(5)

int . a

= Kb

T n

int . b

(6)

2ka kb Tb (ka + kb ) xb

Ta xa

(7)

Here the harmonic mean of the thermal conductivities (kb,ka) and the piecewise linear temperature profile have been employed to evaluate the heat flux at the interface to ensure the overall energy balance of the computed results. Since this is a steady-state conduction heat transfer in solid, the heat flux at the interface is dependent only on thermal conductivity of each phase, as it was declared by Eq. (4). As mentioned earlier, effective thermal conductivity of nanofluids can be modeled through a steady-state solid-fluid conjugate heat transfer problem. Therefore, Eq. (3) is implemented in the following form (Eq. (8)):

xi

( Tu i ) =

k xi Cp

T xi

(8)

In contrast to Eq. (4), it can be seen that Cp remains in the equation (Eq. (8)). Following the same procedure, Eq. (9) can be derived for solid-liquid conjugate heat transfer [46]:

qint =

Ts

Tf

(ks C pf + kf C ps ) xs

2k s k f

xf

(9)

In the above equation Cpsand Cpf are the specific heat capacity for solid and fluid, respectively. Since steady-state heat conduction is sought, Eq. (9) cannot be accepted due to that fact that it wrongly shows the dependence of the heat flux to Cp in the solid phase (in contradiction to Eq. (4)). This error can be avoided by replacing these two specific heat capacity terms with that of the fluid. As a result, the actual solid specific heat capacity does not affect the computed temperature field. However, as it is shown by Eq. (4), the heat flux at the steady-state condition is only a function of thermal conductivity, rather than heat capacity (Cp). This is physically accepted by the macroscopic conservation equation of energy. Since LBM tries to resemble this equation, it is expected that implementing Cp in LBM mathematical machinery would not affect calculated temperature field, if the system has been allowed to reach the steady condition.

(1)

where Γ is the diffusion coefficient, t is time, and u is velocity. Considering φ = h (h represents enthalpy) and ∇φ = k ∇ T, (where k and T are thermal conductivity coefficient and temperature, respectively), Eq. (1) can be written in the following form:

t

k

where the subscript “int” corresponds to the interfaces, n represents the unit normal vector, and a and b represent each phase. The thermal conductivity variation can be due to non-homogeneity of the material, as in a composite material. To satisfy heat flux continuity, our main objective is to obtain a valid representation for the heat flux qint at the interface:

The Bhatnagar–Gross–Krook (BGK) lattice Boltzmann method fundamentally attempts to retrieve conservation equations of mass, momentum and energy [43]. Therefore, this method cannot violate the basic principles of heat transfer physics achieved by the macroscopic conservation equations. As an open topic, many researchers are presenting innovative methods to capture deficiencies of LBM when it comes about unsteady conjugate heat transfer [44–50]. In this paper, however, the interest is around effective properties (thermal conductivity coefficient) rather than time-dependent properties (in case of reacting fluids or combustion). Thus, there is no need to investigate unsteady conditions. The advection-diffusion equation with the dependent variable φ in the general form without the source term is [46,51,52].

xi

xi

Tint . a = Tint . b

2.1. Governing equations

)+

( Cp Tui ) =

Eq. (4) shows that steady-state heat conduction in solids is a function of thermal conductivity (k), rather than the specific heat (Cp). This rule should remain unchanged in conjugate heat transfer for any combination of materials (solid-solid, solid-fluid). In solid-solid conjugate heat transfer, temperature and heat flux continuities must be satisfied when there is no thermal resistance at the interfaces:

In heat transfer simulations of nanofluids, thermal energy transfers through domains with different thermal properties such as thermal conductivity. In the following, a review on the governing equations of conjugate heat transfer and related issues are provided. Then, these problems are addressed through implementation of LBM. In addition, LBM does not face the problem of two-component boundary tracking for conjugate heat transfer, which is one of the main issues in conventional methods.

(

xi

Effective properties of two-component systems (e.g. thermal conductivity of nanofluids) should be modeled through a steady-state solidfluid conjugate heat transfer problem. For steady-state heat conduction in solid, Eq. (3) can be simplified as:

2. Numerical simulations

t

( Cp T ) +

(2)

Then, substituting h = CpT (Cp is the specific heat capacity) in Eq. (2) yields: 3

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2.2. Thermal lattice Boltzmann method LBM can be used to simulate heat transfer phenomenon by proposing a different approach: as it is a bottom-to-top approach, one can relate the molecules' velocity to their thermal energy content. In fact, temperature can be retrieved through definitions found in the kinetic theory [51,53]. LBM is founded on a parameter called distribution function (f). There are different explanations for this term, however a simple approach to define this parameter in an intuitive way is through dimensional analysis ([f] = kg× 13 × 1 3 ) [51]. In fact, it shows density of a

( ms )

m

medium ρ(x.t) consisting of particles traveling with the three-dimensional velocity v (molecules velocity) at time t. Thus, this function can be defined as:

(x . t ) =

f (x . v . t ) d 3v

(10)

By knowing the distribution function (f), other macroscopic parameters such as the momentum density, total energy density and internal energy can be defined. At the thermodynamic equilibrium state, the distribution function is denoted by feq and calculated by the MaxwellBoltzmann distribution function, as shown below:

f eq (x . v. t ) =

3 2

2 RT

exp

(v

u )2 2RT

Fig. 1. Computational domain and boundary conditions. Two walls are adiabatic, and the other opposite walls are at fix-temperature TC (300 °C) and TH (350 °C).

s

v

f=

1

(f

(12)

keff =

In the above equation, the time that it takes for the distribution function (f) to have the equilibrium state is relaxation factor (τ) that links material thermal properties to LBM. In this paper, a D2Q9 standard single relaxation-time lattice Boltzmann method is used [53]. In fluid flow application of LBM, the distribution function (f) stands for mass ensembles, while in Thermal LBM (TLBM) ‘g’ shows the energy distribution function. In pure thermal conduction problems, fluid macroscopic velocity is zero (u = 0) that leads to a simplified form of Eqs. (13) and (14), the diffusion equation with the temperature (T) as the diffusion variable:

T eqi =

iT

T=

T eq i =

1

(gi

gi eq)

(15)

gi neq = gi

where ei is the lattice velocity vector for i = [2,8] in a D2Q9 LBM model. The relaxation factor (τg) can be introduced using either physical parameters or lattice parameters (lattice units). However, in both τg is dimensionless. The relaxation factor defines material thermal properties. Having said that, each node in the computation domain should be assigned to a specific relaxation factor regarding its property constants. The solver distinguishes each node property, then decides which of the following relations should be used: f

=

kf 3 + 0.5 2 ( Cp)f c 2 t

dA

(18)

Fig. 1 shows the computational domain and its boundary conditions (BCs). To investigate the effective thermal conductivity, the temperature-constant boundary conditions (Dirichlet BCs) are assigned to the right and left walls, and the zero-flux boundary conditions (Neumann BCs) are assigned to the top and bottom walls. In this figure, g3, g6, g7 are known and g1, g5, g8 are unknown distribution functions on the left isothermal wall (higher temperature wall with TH = 350 ° C). For the right isothermal wall (lower temperature wall with TC = 300 ° C), known and unknown distribution functions are reversed. These unknown distribution functions in TLBM are defined using the flux conservation principle (see the following equation) for a bounce-back boundary condition scheme [53,55].

(14)

gi (x . t ) =

qdA T

2.3. Boundary conditions

where ωi is a weighting factor. The above equation is obtained from the Chapman–Enskog Expansion [53]. Thus, the local temperature can be calculated using the discretized Boltzmann equation below:

gi (x + ei t . t + t )

L

where q is the heat flux, dA is the area, L is the distance, and ∆T is the temperature difference.

(13)

gi

(17)

where c is the pseudo sound speed (c = As it was declared before, pure conduction in the solid phase should be independent of (ρCp)s in steady-state condition which is violated in Eqs. (16) and (17). Therefore, the term ρCp in the solid phase is assumed to be identical to that of the fluid phase [46]. After enough time steps, the solution reaches steady state, and the effective thermal conductivity (keff) can be estimated using the following equation [42,54]:

(11)

f eq )

3 ks + 0.5 2 ( Cp)s c 2 t x ). t

where R is gas constant, T is temperature, u is macroscopic velocity, and ρ is macroscopic density. The LBM algorithm calculates the evolution of the distribution function (f) at every node and defines its deviation from the equilibrium distribution function (feq). This can happen through the Boltzmann equation that tracks the distribution functions in a phase space consisting of time, location and velocity.

f +v f+F t

=

gi eq

(19)

For the insulated walls, the Neumann boundary condition is used:

T =0 x

(20)

Also, the initial condition of the whole domain is set to be at TC. All the simulations in this paper are conducted in 2D. To evaluate the effect of the third dimension, we should study the geometry and the boundary conditions to see if there are any parameters that may affect. The first parameter to evaluate the probable effect is the nanoparticles aggregates. Nanoparticles aggregates are fully defined through their fractal dimensions, chemical dimension, etc. It is believed that as far as the fractal theory-based parameters of aggregates stay the same,

(16)

and 4

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H. Tahmooressi, et al.

all cases, the effective thermal conductivity decreased with increase in particles size. They claimed that this enhancement is because of larger specific surface area of small size nanoparticles which can lead to the augmentation of NLL effect. This conclusion however, is in contrast to the previous findings that state NLL accrues when nanoparticles diameter is below 10 nm [56,62]. Liu et al. developed an analytical model to investigate the effect of aggregation and interfacial thermal resistance on nanofluid thermal conductivity for different nanoparticles including Alumina [56]. They showed that larger nanoparticles work better for enhancing the effective thermal conductivity. The reason was stated that the larger nanoparticles decrease the total interfacial surface geometrically. In another study, a molecular dynamics simulation by Xue et al. indicated that NLL cannot be happen at solid-liquid interfaces [63], while many other studies indicated that Nano liquid layering exists [60,64,65]. Several MD simulations showed that surface wettability is the main parameter to distinguished whether interfacial thermal resistance is decreasing the thermal transport efficacy or formation of a nanometer liquid layer around particles enhances thermal transport property of nanofluids [63,66–68]. Hydrophobic interfaces can have 2–3 times greater ITR in comparison to hydrophilic ones [69]. Surface geometry is another interfering parameter. In an investigation, Gao et al. measured the thermal conductivity of graphene nanoplatelet (GNP) ethylene glycol (EG) base nanofluids [70]. They proposed that the lower thermal conductivity enhancement at subzero temperatures may be due to the larger interfacial thermal resistance between GNP and EG caused by high viscosity of EG. In a recent study, Milanese et al. investigated metal and metal oxide nanoparticles/water for the effect of nanolayer both experimentally and using MD simulations [71]. In contrast to others, their results showed that formation of nanolayer is not a function of nanoparticles diameter. Also, they showed that a nanolayer forms around Cu nanoparticles while no specific layering happens for CuO nanoparticles. This indicate the fact that Nano liquid layering can have a crucial role in some nanofluids but no effect in some other types. Atomic structure of nanoparticles/base fluid is another influencing parameter that defines the dominant phenomenon at the solid-liquid interface. For instance, monoatomic molecules are not much affected by the interfacial phenomena while the thermal transport efficacy in complex liquids such as polymers are shown to be more sensitive to the thickness of interfacial layer around nanoparticles [72,73]. The debate on the liquid nanolayer does not limit to its existence. There are conflicting hypothesizes about the thickness of this layer (0.5 to 2 nm) [74,75] as well as its thermal conductivity value (1.6–3 times of base fluid's) [76,77].

disregarding the third dimension would not miss any extra information in explaining the geometry of aggregates. Therefore, their effect on thermal energy conduction would be the same. Horizontal walls of the domain are adiabatic (zero heat flux) and the other two walls are at constant temperature. Therefore, they can be simplified to 2D [54]. Second, the accuracy of TLBM should be checked to see if it can add any unexpected error into the results. Another parameter is the complex geometry of interfaces (curved boundaries). In an extensive study conducted by Demuth et al., it was demonstrated that this parameter does not affect the accuracy of results [54]. More important than that, they showed that D2Q9 (as implemented here) can produce the most accurate results as well as D3Q15 in two-component thermal energy conduction (D2Q9: 2D LBM, D3Q15: 3D LBM). Therefore, two-dimensional LBM can be used interchangeably by three-dimensional LBM if the characteristics of geometry of the problem can be defined in two dimensions (which is the case for fractal objects). 2.4. Solid-liquid interaction There are two main interfacial phenomenon that may enhance or hinder thermal energy conduction in nanofluids; interfacial thermal resistance (ITR) [56,57] and Nano liquid layering (NLL) [58,59]. NLL is expected to enhance the effective thermal conductivity of nanofluids as it forms a crystal-like layer of base fluid molecules around nanoparticles. On the other hand, ITR works as an obstacle for thermal waves (phonons) at the solid-fluid interface that reduces the nanofluids effective thermal conductivity. To explain the complexity of this issue, a review on literature is provided here. There are diverse theoretical and experimental contradictory reports on the existence of each of these mechanisms. In a recent study, Guo and Zhao investigated the probability of formation of the interfacial layer around 2,4 and 6 nm nanoparticles in a CueAr nanofluid using the large-scale atomic/molecular massively parallel simulator molecular dynamics package [60]. They concluded that the interfacial nanolayer (NLL) can be formed and it is thicker for larger nanoparticles. However, the effective thermal conductivity is said to be increased when nanoparticles dimeter decreases due to larger solid surface area. On the contrary, in [2] authors showed that long sonication time can break nanoaggregates into small pieces and reduce the thermal conductivity of nanofluid. They inferred that generation of more solid surface area, and consequently, higher interfacial thermal resistance (ITR) oversees this occurrence. Alawi et al. measured the thermal conductivity of four types of distilled water based nanofluids (Al2O3, CuO, SiO2 and ZnO) with nanoparticles diameter of 20–100 nm [61]. In

Fig. 2. (a) The parallel mode and the theoretical solution. Thermal energy is conducted from the wall with the higher temperature (TH) to the lower one (TC). (b) The series mode and the theoretical solution. 5

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H. Tahmooressi, et al.

It can be inferred from these inconsistent reports that not all nanofluid systems can benefit from Nano liquid layering or suffer from interfacial thermal resistance. In LBM, Importing the effect of NLL and ITR is through adding a source term to Boltzmann equation. However, the exact value of thermal conductivity for the Nano liquid layer and interfacial thermal resistance as well as the thickness of NLL are yet to be defined by researchers. Therefore, it would lead to more uncertainty about the results. In this case, one should consider the overall effect of these mechanisms on aggregation and see how they interact if they are supposed to be considered simultaneously. If ITR exits, aggregation (solid-solid contact) would diminish ITR resulting in TC enhancement and, if NLL exists, aggregation would enhance TC more than the amount that NLL can do. Considering the mentioned overall uncertainties in literature about these two mechanisms and the ultimate target of this paper which is investigating the pure effect of aggregation through a mesoscopic approach, these nanoscale solid-liquid interactions are not considered.

(loose aggregation) and reaction limited aggregation (compact aggregation) [34,103]. Generating clusters using these models is time consuming and computationally expensive. Also, it is not possible to easily tune the generated clusters with preferable chemical or fractal dimensions. To overcome these issues, a growth model proposed by [104], in which the fractal dimension of a cluster can be determined in advance, has been used and further modified, resulting in a new growth model with certain nuance changes in definitions. Therefore, instead of using fractal dimension, the number of particles in a cluster Nint is used, and, for the case of chemical dimension, the number of particles that constitutes the cluster's backbone Nc (i.e., the main heat-carrying structure isolated from dangling ends or dead ends [98,105]) is utilized to define aggregates. In the growth model developed here, not only the fractal dimension (number of cluster's particles) can be pre-determined, but also the cluster chemical dimension (number of backbone's particles) is defined in advance. 3.1. Growth model

2.5. Benchmarks for TLBM

The aggregation growth model is based on the calculation of local filling factor Φi which is a criterion to select the next site to be occupied. The concept is to calculate the ratio between the volume of particles within an inspection sphere to the volume of the inspection sphere. It is defined as:

To validate the TLBM model, LBM results are compared with those obtained using a Finite Difference Method (FDM) and a theoretical solution for a standard test problem, i.e., a two-component heat conduction in parallel and series modes (at the end of this paper, the nanofluid system is modeled and the simulations results are validated with experimental data found in literature [6,78]). Fig. 2 (a) and (b) show the parallel and series modes, respectively, and their theoretical solution used for calculation of the effective thermal conductivity (keff). The two-component system resembles a nanofluid system with solid and fluid components. Thermal conductivity of the fluid phase is selected as of water, kf=0.613 (W/mk), i.e., the most common base fluid in nanofluids [79–87]. The thermal conductivity coefficients of the solid phase are assumed as ks1= 40[W/mk] and ks2=11.7 [W/mk] for Al2O3 [26,88–93] and TiO2 [58,81,85,94–97], respectively (used as nanoparticles). Tables 1 and 2 list the effective thermal conductivity values calculated using LBM, FDM and theoretical solution for the parallel and series modes for Al2O3/water and TiO2/water combinations, respectively. Two numerical methods (FDM and LBM) are in great agreement with the theoretical results. The largest deviation of the LBM results from the theoretical solution ( LBM Value Theoretical Value ) is < 0.03.

i

=

Nint 4 Rgrain3/3 (21)

4 R cluster 3/3

where R is radius of the inspection sphere. The algorithm calculates the local filling factor for each occupied site, and then selects a site with the smallest Φi to locate the next particle in the cluster. A complete explanation can be found in [104]. Although the simulations are performed in a 2D space, a 3D representation of the constructed clusters are first provided to illustrate better the aggregation algorithm's outcome. Fig. 3 (a) and (b) show two single aggregates ((a) with Nint = 30 and (b) Nint = 20). In each case, three different configurations (with different Nc are shown. The larger the number of Nc the more likely hood of obtaining a chain-like cluster (with a long arm). Fig. 3 (c) shows multiple, two, and single aggregates generated. This shows different possible aggregate regimes with the same nanoparticles volume fraction (Φp). The generated microstructures in these figures are relatively small aggregates in comparison to those found in aerosol or sol gel [106]. The reason is that in nanofluids the aim is to disperse nanoparticles as much as possible using methods like ultrasonication [85]. This would reduce the chance of formation of long aggregates that can be observed in aerosol where nanoparticles spontaneously stick together and form large aggregates (referred to as coagulation of agglomerates) [106].

Theoretical Value

3. Numerical microstructure generation (NMG) The first step to numerically investigate the effect of nanoparticles aggregation on nanofluids thermal conductivity is to understand and characterize the stochastic patterns of the complex nanoscale aggregations. Fractal geometry, as a novel mathematical tool, has been implemented by a few researchers to quantitatively explain aggregation of the suspended nanoparticles and then generate microstructures [21]. Fractal objects are categorized into deterministic and random fractals [98]. Fractal dimension (df) and chemical dimension (dl) are two main parameters defining precisely the characteristics of deterministic fractal objects. However, calculation of these parameters for random fractal objects is more rigorous. Fractal dimension defines the way an aggregate's mass scales with length; chemical dimension provides information about the shape of the fractals [99]. For instance, when particles highly concentrate in an accumulated fashion in a cluster, the chemical dimension decreases. In contrast, when particles are aggregated in a chain-like manner with long arms, chemical dimension tends to grow [6,16], [100,101]. To generate nanoparticles aggregates, growth algorithms need to be developed based on fractal parameters. For the case of nanofluids, researchers believe that the behavior of nanoparticles aggregation can be simulated through a diffusion limited cluster aggregation (DLCA) model [21,102]. Wang et al. modeled both diffusion limited aggregation

4. Aggregation scenario The aggregation process can be modeled under different regimes, evolution from a fully dispersed to a fully aggregated system. There are three main strategies devised to model an aggregation process in this paper: (1) changes of the rate by which the particles aggregate into Table 1 Comparison of the effective thermal conductivity values calculated using LBM, FDM and theoretical solution for parallel and series Al2O3/water system. Al2O3/water keff [W/mK] Deviations from Theoretical Solution Solid Volume Fraction

6

Theoretical Solution FDM LBM FDM LBM

Parallel

Series

6.9169 6.9210 6.9177 5 × 10−4 1 × 10−4 0.15

0.728 0.729 0.738 0.001 0.013

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modeled through different cluster's mass for which the effective thermal conductivity is calculated by the TLBM solver. Figs. 4 to 6 depicts the process involving the generation of constant-mass aggregations (from Φint = 0 to Φint = 1) with Nint = 6, Nc = 5; Nint = 11, Nc = 7; and Nint = 18, Nc = 10, repectivley. The difference between these figures is the number of particles in each newborn aggregate (6, 11, and 18 in Figs. 4, 5, 6, respectively). The Nint and Nc quantities chosen in these microstructures correspond to the fractal and chemical dimensions found for nanofluid aggregations in literature (df = 1.8 and dl = 1.4) [57]. Thus, the effective thermal conductivity calculated based on these microstructures (see Section 5) represents the real behavior of nanofluids.

Table 2 Comparison of the effective thermal conductivity values calculated using LBM, FDM and theoretical solution for parallel and series TiO2/water system. TiO2/water keff [W/mK] Deviations from Theoretical Solution Solid Volume Fraction

Theoretical Solution FDM LBM FDM LBM

Parallel

Series

2.387 2.393 2.4094 0.002 0.009 0.15

0.7226 0.7290 0.7429 0.008 0.028

constant-mass clusters, (2) changes of the clusters' mass in a fully aggregated system, and (3) changes of the clusters' shape of a fully aggregated system (chemical dimension value growth). In all the generated microstructures, the solid volume fraction is 0.1 (Φp= 0.1) for 10 nm (dp = 10 nm) Al2O3 nanoparticles.

4.2. Changes of clusters' shape in a fully aggregated system The clusters' shape can be characterized through alteration of its chemical dimension (dl). This parameter defines distribution of nanoparticles in a cluster's backbone and dead-ends. However, the number of backbone particles (Nc) is preferred over chemical dimension (dl) to tune clusters' shape (as discussed earlier). To our knowledge, this is the first study on nanofluids thermal conductivity in which the generated aggregations with the alteration of their shape are reported pictorially. Fig. 7 illustrates generated nanoparticles aggregations with Nint = 18 and Nc = 10, 11, 13, 15, 18. Fig. 7 (a) shows a typical aggregate (Nint = 18, Nc = 10), while (e) is an aggregate with Nint = Nc = 18 in

4.1. Changes of rate of particle aggregation into constant-mass cluster We have modeled nanoparticles aggregation process in respect to time with three constant masses defined by the number of particles in clusters. The percentage of the aggregated nanoparticles is denoted by Φint. Therefore, Φint = 0 stands for a fully dispersed system and Φint = 1 shows a fully aggregated nanofluid. In this section, this trend is

Fig. 3. (a) A 3D representation of the constructed stochastic clusters with the number of particles of Nint = 30 and the number of backbone particles of (i) Nc=10, (ii) Nc=15, (iii) Nc=22. As the number of backbone particles increases, the aggregate changes to a chain-like cluster. (b) A 3D representation of the stochastic clusters with a smaller number of particles (Nint = 20) and the number of backbone particles of (i) Nc=10, (ii) Nc=14, (iii) Nc=18. (c) A 3D representation of the cluster distribution regime in constant nanoparticles volume fraction (Φp): (i) Four aggregates with Nint = 10 and Nc = 7, (ii) two aggregates with Nint = 20 and Nc = 14, and (iii) one aggregate with Nint = 40 and Nc = 28. 7

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Fig. 4. The generated 2D microstructure for Al2O3/water nanofluid with Φp= 0.1, Nint = 6 and Nc = 5. The evolution of the aggregation process from (a) fully dispersed nanofluid with the zero percentage of the aggregated nanoparticles, Φint=0 to (i) fully aggregated nanofluid with Φint=1.

which all nanoparticles are in the backbone. In this trend, the effect of nanoparticles aggregations shape on nanofluids thermal conductivity depends on the number of particles joining cluster's backbone.

5. Results and discussion A very important point regarding microstructure generation includes mimicking the real behavior of suspended nanoparticles. In fact, the nanoparticles aggregation is a dynamic process that needs to be investigated not only with respect to time, but also with the consideration of its random behavior. To cover the time-lapse effect, for each of investigations in the last three sections, more than five intermediate aggregation states have been defined, so that the ETC behavior versus aggregation effect (Keff versus Nint) can be observed precisely. To model the random nature of aggregation, the microstructure generation and the heat transfer simulations have been iterated for five rounds (five distinct microstructures of a specific aggregation state) after which the overall behavior showed to be comprehensive enough to cover the deviations from the expected value. Therefore, quantities reported in this section are the expected values. Generally, it can be said that more 180 rounds of microstructure generations and heat transfer simulations have been conducted in this project. Through precise geometrical modeling, LBM is capable of simulating heat conduction phenomenon in a two-component system (a solid-fluid system) through a mesoscopic approach in which the microscopic nature of heat conduction is considered. Thermal properties of Al2O3-water nanofluid, as one of the most referred nanofluid in literature (e.g., [26,88–93]), is selected for the LBM simulations in this study. The large difference between Al2O3 and water thermal

4.3. Changes of clusters' mass in fully aggregated system Considering the elapsed-time (i.e., the amount of time after preparation of nanofluid), a few researchers [8,12,107,108] have declared that aggregation phenomenon has a negative effect on nanofluids effective thermal conductivity (ETC) because they have observed formation of concentrated clusters. So, if nanoparticles do no aggregate, ETC will be enhanced. However, here we show that this is not conclusive as a linear behavior of ETC with respect to aggregation state (Keff versus Nint) that has been assumed in these studies. In this section, which is of great importance to nanofluids research, we demonstrate two regimes of aggregation between which the maximum effective thermal conductivity can be obtained. The first regime contains a high number of small aggregations, while the second one has a low number of large aggregations. Fig. 8 represents an evolution in the mass increase for generated aggregates from Nint = 6 to Nint = 52. In all aggregates in this figure df=1.8 and dl=1.4. The first figure presents a large number of relatively small aggregate (the first regime) which merge together to form a small number of large aggregates (the second regime). 8

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Fig. 5. The generated 2D microstructure for Al2O3/water nanofluid with Φp= 0.1, Nint = 11 and Nc = 7. The evolution of the aggregation process from (a) fully dispersed nanofluid with the zero percentage of the aggregated nanoparticles, Φint=0 to (i) fully aggregated nanofluid with Φint=1.

( )

( )

linearly with the solid volume fraction. The calculated LBM values are in great agreement with the Maxwell model (EMT) indicating the acceptability of our model to retrieve the theoretical limit (< 0.3% at Φp = 0.1). However for the experimental results, similar nanofluids have significantly different keff values. This approves the acceptability of the LBM solver and the microstructure generator paying the way for the next step investigations.

conductivities, i.e., 40 mK and 0.613 mK respectively, is in an acceptable range for BGK-LBM [109]. For significantly larger differences between thermal conductivities of two components, BGK collision operator loses its numerical stability. In the followings, ETC for non-aggregated Al2O3/water nanofluid is simulated for different solid volume fractions (Φp=2% to Φp=10%) to validate the accuracy of LBM in respect to theoretical models and experimental results in the literature. Then, ETC is calculated for the three aggregation scenarios presented in the previous section. W

W

5.2. Constant-mass aggregation growth Nanoparticles can aggregate into clusters with different characteristics. Indeed, aggregation process (from fully dispersed to fully aggregated, i.e., Φint=0 to Φint=1) contains both the formation of new aggregates and the changes in the mass of aggregates. To break down this process into a simpler mechanism, three aggregation processes with three constant masses have been devised. Therefore, we would be able to observe not only the effect of aggregation on ETC of nanofluids, but also the effect of size/mass of aggregates. Fig. 10 shows ETC of nanofluids modeled with Nint=6, 11 and 18. Aggregates modeled in all three scenarios follow the typical characteristics of aggregates found in nanofluids, df = 1.8 and dl = 1.4. The horizontal line in this figure indicates the effective thermal conductivity (ETC) of a fully dispersed system calculated by the Maxwell model [110]. The results show that an increase in the level of aggregation (from Φint=0 to Φint=1) enhances the ETC of nanofluids. At first, ETC increases rapidly, but after a

5.1. Fully dispersed nanofluids To analyze the effect of aggregation on the nanofluids effective thermal conductivity, the first step is to validate the LBM solver for a non-aggregated Al2O3/water nanofluid. The importance of this step is that the calculated values can be used as a baseline to visualize the effect of nanoparticles aggregation. As it shows in Figs. 4(a), 5(a), and 6(a), the microstructure for non-aggregated nanofluids (Φp=0.10) are used as the base configuration upon which the aggregation scenarios are generated. Therefore, Al2O3/water nanofluid in different solid volume fractions (Φp=0.03 to Φp=0.10) is modeled and the results are compared with the Maxwell model, effective medium theory (EMT) [110], and experimental data in the literature (Fig. 9). Fig. 9 indicates that the effective thermal conductivity increases 9

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Fig. 6. The generated 2D microstructure for Al2O3/water nanofluid with Φp= 0.1, Nint = 18 and Nc = 10. The evolution of the aggregation process from (a) fully dispersed nanofluid with the zero percentage of the aggregated nanoparticles, Φint=0 to (i) fully aggregated nanofluid with Φint=1.

certain level of aggregation (around Φint=0.5) this trend tends to slow down. Also, the comparison between two aggregation regimes (a small number of large aggregates (Nint= 18) vs. large number of small aggregates Nint=6) shows that thermal energy conduction is significantly higher for the case of a large number of small aggregates due to the percolating network of solid particles. Another important point is that, in all these aggregation styles, the effective thermal conductivity is larger than that of the fully dispersed nanofluid (i.e., obtained from the Maxwell model). The LBM results can be compared with the three-level homogenization theory developed by Evans et al. [57]. Their model is extensively studied for different parameters of Al2O3/water nanofluid and reported by Okeke et al. [115]. Although an exact validation is not possible due to discrepancies in definition of parameters (df/dl instead of Nc/Nint), an overall comparison would approve the possibility of reaching such a high limit for Keff because of nanoparticles aggregation. In the mentioned model, aggregates are embedded in an imaginary sphere of radius Rg. Then, aggregates aspect ratio is defined by Rg/(dp/ 2). The aspect ratio of aggregates with Nint=6 (highest Keff in Figs. 9 and 4) is 4.8. The Keff value calculated by Okeke et al. for aggregates with df = 1.8, dl = 1.4, Rg /

5.3. Aggregation morphology This section sheds light on the influence of the shape of random fractal aggregations on nanofluids thermal conductivity. This issue has been investigated by many researchers ([8,18,19,24,57,116–118]). However, in most previous studies, aggregates morphology is not geometrically modeled as in Fig. 7. Fig. 11 reports the value of the effective thermal conductivity with respect to the evolution of aggregates morphology in a fully aggregated nanofluid (Φint = 1). As the clusters turn into a chain-like shape (Fig. 7 (e), Nint= Nc = 18) from the concentrated mode (Fig. 7 (a), Nint = 18 and Nc = 10), ETC enhances up to 175%. The aggregates with the chain-like shape (Nint = Nc) would hardly ever take place passively. However, in magnetite nanofluids, where magnetic field forms highly linear-ordered clusters, as observed by [78], ETC can be enhanced up to 1.67 for the case of parallel magnetic field/temperature gradient compared to a fully dispersed nanofluid. The results of our simulations in Fig. 11 show similar values: an increase in ETC from 1.43 (obtained for the fully dispersed nanofluid with Φint=0) to 1.77 for chain-like shape clusters (for a cluster with Nint=18). Although the value of Keff is anomalously high, it is still lower than that of a relatively percolated aggregation regime shown in Fig. 4. The reason should be addressed to the fact that in a percolated regime, particle-free regions (high thermal resistance) are not critical as the way they are in a non-percolated regime. In other word, the aggregation

( ) = 5, and Φ = 0.1 is denoted in Fig. 9. dp 2

p

The effect of aspect ratio in this paper is extensively investigated in Section 5.3.

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Fig. 7. The generated microstructure for Al2O3/water nanofluid with Φp= 0.1 and the number of particles in an aggregate of Nint=18: The evolution of the aggregation shape from concentrated aggregates with the number of backbone particles of (a) Nc = 10, (b) Nc = 11, (c) Nc = 13, (d) Nc = 15, to chain-like aggregates with (e) Nc=18.

of nanoparticles would produce particle free zones that can have a negative effect on Keff if not distributed evenly in the base fluid (percolated regime) or not aligned with the heat conduction direction (parallel more conduction) [12]. The generated chain-like aggregates (Fig. 7(e)) are not necessarily alighted with the heat transfer direction. If that was the case, Keff could be enhanced up to the level of parallel conduction mode as indicated in [6]. It should be noted that these

effects of spatial distribution of aggregates cannot be investigated in analytical models or MD simulations. A more comprehensive study on the spatial distribution of aggregates can be found in [119]. 5.4. Aggregation mass growth Fig. 12 shows the calculated ETCs of the clustering states for which

Fig. 8. The generated microstructure for Al2O3/water nanofluid with Φp= 0.1 in different distribution regimes from a large number of small aggregates to a small number of large aggregates: (a) Nint=6, (b) Nint=11, (c) Nint=18, (d) Nint=25, (e) Nint=42, (f) Nint=52. 11

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Fig. 9. LBM simulations of the effective thermal conductivity ratio (Knanofluid / Kbasefluid) for Al2O3/water nanofluid as a function of solid volume fractions (Φp) compared to Maxwell model, Timofeeva [26], Patel [111], Zhao [112], Angayarkanni [113], and Pak [114].

Fig. 10. LBM simulations of the effective thermal conductivity ratio (Knanofluid / Kbasefluid) for Al2O3/water nanofluid (with Φp= 0.1) as a function of the aggregation level of nanoparticles (Φint=0 to Φint=1) for three different aggregate masses (Nint=6, Nint=11, Nint=18).

the evaluation of clusters mass/size are illustrated in Fig. 8. When Nint=6, thermal conductivity increases up to 2.58%. After the clustering state reaches Nint=18, thermal conductivity reduces (stays constant for larger clusters), but it is still higher than a fully dispersed state (Φint=0). These results indicate that clustering phenomenon has an optimum state at which ETC is maximum. This optimum clustering state stands for the regime in which a large number of small aggregates are formed. The trend shown in Fig. 12 is in agreement with the studies in [57,116], where ETC saturation, after a certain level of aggregation, and the optimum level of aggregation corresponding to a maximum value of ETC were shown. It should be noted that the quantities and the parameters used to define aggregations in these studies are different from what we have implemented here, but the overall behavior is the same. Also, considering the settling phenomena, it may be seen that ETC would even be lower than that of a fully dispersed system as the nanofluid is not stable anymore.

6. Conclusions The Lattice Boltzmann Method (LBM) was implemented to investigate the complicated thermal conduction behavior in nanofluids. The statistical nature of the lattice Boltzmann method, the capability of handling complicated geometries, and the low computational cost (due to the absence of inter-component boundary tracking) are the main parameters representing LBM as an efficient numerical tool for the case of nanofluids thermal conductivity simulations. The effect of nanoparticles aggregation on nanofluids thermal conductivity in terms of their shape, size and distribution regime was investigated through generation of complicated stochastic aggregates. The following concluding remarks are extracted from the numerical simulations in this paper:

• The nanofluids effective thermal conductivity was calculated in the 12

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Fig. 11. LBM simulations of the effective thermal conductivity ratio (Knanofluid / Kbasefluid) for Al2O3/water nanofluid with Φp= 0.1, and Φint=1 as a function of the number of aggregation's backbone-particles (Nc=10 to Nc=18) of a typical aggregate with Nint=18.

Fig. 12. LBM simulations of the effective thermal conductivity ratio (Knanofluid / Kbasefluid) for Al2O3/water nanofluid with Φp= 0.1, as a function of aggregate's mass. Fully aggregated nanofluid with increasing number of particles in an aggregate (Nint=6 to Nint=52).

• •

absence of all mechanisms (including aggregation), and the standard Maxwell limit was obtained using LBM. Putting aside the other probable mechanism known for anomalous enhancement of ETC in nanofluids (such as Brownian motion and Nano liquid layering), nanoparticles aggregation can enhance ETC up to about 3 (Knf/ Kbf=2.6) times of the standard Maxwell limit (for a relatively small aggregates with Nint=6), indicating the reason behind anomalous ETC enhancement. This status can be observed in magnetite nanofluids in which external magnetic field forces nanoparticles to form highly ordered aggregates. A fractal theory-based algorithm was implemented to form random aggregates equivalent to the real ones, and the effect of their shape was investigated on ETC of nanofluids. It was found that elongated chain-like types work better than concentrated aggregates. Neither very small nor concentrated large nanoaggregates are efficient types of aggregation when it comes to the conduction of thermal energy. In fact, a distributed network of a large number of

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