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Dynamic nanoparticle aggregation for a flowing colloidal suspension with nonuniform temperature field studied by a coupled LBM and PBE method Dongxing Song, Mohammad Hatami, Jiandong Zhou, and Dengwei Jing Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02261 • Publication Date (Web): 11 Sep 2017 Downloaded from http://pubs.acs.org on September 15, 2017
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Dynamic nanoparticle aggregation for a flowing colloidal suspension with nonuniform temperature field studied by a coupled LBM and PBE method Dongxing Song, Mohammad Hatami, Jiandong Zhou, Dengwei Jing* Corresponding Author's email:
[email protected] State Key Laboratory of Multiphase Flow in PowerEngineering&International Research Center for Renewable Energy, Xi’an Jiaotong University Xi’an 710049, China Abstract Due to the high surface energy, aggregation of particles in colloidal dispersions is almost inevitable. In present study, a comprehensive model was established by coupling a two-phase lattice Boltzmann method (LBM) and population balance equations (PBEs) to describe the dynamic particle aggregations in flowing and heated colloidal suspensions. Typically, the dynamic aggregations in suspensions under natural convection in a rectangular enclosure are investigated. The real-time particle size distribution (PSD) significantly depended on Ra (Rayleigh numbers), concentration and enclosure geometry, where increasing particle concentrations can inevitably increase the particle size, but the effect of Ra on PSD is different when enclosure geometry changes. The results of average Nusselt number (Nuave) suggested a significant negative impact of particle aggregation on the heat transfer, especially for high concentration and low Ra cases. Our work should be of value when the properties of practical colloidal dispersions are to be predicted and estimated. Keywords Colloidal suspension, Particle aggregation, Lattice Boltzmann method, Population balance equations
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1. Introduction Colloidal suspensions in which nano-scale and/or micro-scale particles are dispersed in pure or mixed liquids, are widely applied in many industrial processes, such as food engineering,1 medical treatment,2 material preparation,3 energy conversion,4 environment engineering,5 etc. They are named as nanofluid when used for heat transfer and cooling. For simplification however, we will call them all as colloid suspension in our introduction. As a complex system, the stability, particle size distribution, flowing property, and the inner particle behaviors of colloids will dramatically affect their performance, which have been paid extensive attentions.6-10 As for the particle behaviors in colloid suspension, it includes migrations, collisions, and aggregations of nanoparticles which are the most basic mechanism significantly influencing their other properties. Particles in colloidal suspensions suffer many forces from base fluid, such as drag force, buoyancy force, Brownian force, and from other particles, such as interaction potential force, etc. General theoretical methods to investigate particle motion and aggregation include Monte Carlo simulation,11 molecular and Brownian dynamic simulations,12-14 population balance equations (PBEs),15-16 etc. The first three simulation methods focus on the aggregation processes of primary particles and microscopic structure of aggregates, where the rotational radium, fractal dimension, shear strength etc. of aggregate can be obtained. However, they can hardly be employed to study a system more analogous to real suspension such as a colloid system with millions of primary particles and acquire the statistic information of aggregates within them, such as the average size, and particle size distribution (PSD), due to the very high computational requirement. As a contrast, PBEs, is based on the law of mass conservation and takes into account such mechanisms as nucleation, growth, aggregation, and breakage, etc., which makes them more suitable for the
prediction of the statistic information of aggregates in
colloidal suspension.17 Erabit et al.18 have investigated the thermally-induced aggregation of whey protein solutions based on PBEs. By comparing the calculation and experimental results for different particle concentrations and at various times, they found that PBEs have a good accuracy to predict the aggregation processes. Lazzari7 presented a mathematical framework to describe the transient colloid aggregation and deposition based on PBEs. Three different aggregation regimes are proposed to investigate the aggregation between free clusters and simple shear aggregation. Also based on this method, Sadeghy et al.19 investigated the size distribution, average size and settling rate of aggregates under different 2
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temperatures. Their study indicates the population balance model can successfully apply to describe particle aggregation in nanofluids which has been verified by dynamic light scattering (DLS) measurement. In spited of so many reports employing PBEs to investigate the aggregation in colloidal suspensions, rare studies considered by PBEs the particle behaviors in a colloid if it is in a flowing state and/or heated with temperature gradient, owing to the complex physical fields of temperature and flow in such cases. In fact, PBEs are a group of differential equations and the coefficients of equations are the function of temperature, shear rate, solid concentration, and particle size.15 When solving PBEs, the local information of flow and temperature fields should be given at first. However, when PBEs are solved for a given case, the results of PBEs, such as the particle size distribution, will lead to the change of the dominant forces within the colloid, such as Brownian force, drag force, on particles and liquid, which will, in turn, affect the distributions of velocity and temperature. In this respect, coupling the PBEs and a CFD method could be one of the solutions if one intends to accurately predict the particle behaviors in a colloidal suspension subjecting to changing flow and heating conditions. Lattice Boltzmann method (LBM) as a new CFD method have been widely employed for the numerical simulations and investigations of complex flows, including porous flows, thermal flows, reactant transport, turbulence flows, and multiphase flows.20-21 Compared to the macroscopic methods, such as finite-difference method, finite-volume method, finite-element method, spectral method, etc., LBM has many advantages. For instance, they have clear physical meaning, need simple boundary treatment, are easy for implementation of code and have good parallel processing and high robustness.22 In present work, a comprehensive model will be established by coupling a two-phase LBM with PBEs to describe the motion and aggregation of particles in a heated convection system where colloid is considered to be in a flow state and unevenly heated from one side. Specifically, the two-phase LBM is applied to simulate the flow and heat transfer and obtain the parameters of flow field, including distributions of shear rate, temperature, concentration, etc. The PBEs is used for the description of the aggregation processes and the particle size distribution (PSD) in colloid based on the flow field parameters obtained by LBM. Then, the results of PSD will be used in turn in LBM to amend the forces on particles and liquid which will result in the change in flow field and temperature distribution. The changing flow field will also affect the coefficient of PBEs and lead to a continuously changing PSD until the system is in a dynamic equilibrium. In our study, to quantify the effect of particle 3
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aggregation and verify our model, the Nusselt numbers considering and without considering PSD will be calculated and compared with the reported experimental results.
2. Theoretical model and validation 2.1 Mathematical models In this section will describe our theoretical model in three parts, i.e., two-phase Lattice Boltzmann method (LBM) for flow and heat transfer, population balance equations for particle size distribution (PSD), and the coupling of these two methods. 1) Lattice Boltzmann method LBM, a mesoscopic method, has been widely applied as a powerful alternative CFD method for studying fluid flow and transport processes. Both single-phase and two-phase lattice Boltzmann models have been employed to explore the flow and heat transfer of suspensions in previous studies.23-26 However, single-phase model fail to consider the inter-phase forces, such as drag force, Brownian force, etc. and particle migration. Two-phase model can avoid these simplifications and therefore is expected to achieve a better accuracy. Density evolution equation of a two-phase LB model can be written as:23-24 f + , + − f , = − f , − f , +
+
! .
(1)
In this equation, σ=1 and 2 expresses the liquid and solid phases in a colloid suspension, eα and α are lattice velocity vector and direction, r the lattice position, t and δt are time and time step, respectively, "# is the dimensionless collision-relaxation time of flow field, f , is the local equilibrium
distribution function of f , , c= δx/δt is the reference lattice velocity, Bα is weight coefficient to distribute the total interparticle interaction forces to lattice, and the values of Bα can be found in Ref.23
and
!
are interphase force and external force, respectively, where
drag force, interaction potential force, Brownian force, etc, and
!
includes buoyancy force, =$∙
& f , '
,
( = −)* − *+ , is the effect of buoyancy lift induced by density change. - is the macroscopic
velocity of a lattice, ) is the thermal expansion coefficient, T and T0 are local and reference
temperature, respectively.
For a D2Q9 model, there are nine discrete velocities eα, which are given at different direction α as follows: 4
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. = /
2345 6
√2 A345 6
7
B 7
0 7 8 , 59: 6 8;
+ 8 , 59: 6 7 C
B 7
1=0 1 = 1, 2, 3, 4.
+ 8D
1 = 5, 6, 7, 8
7 C
(2)
The density equilibrium distribution function can be written as:27 f
= I 6J + J+ 2
+
∙- K
∙- KL
−
- ∙- K
;8,
(3)
where cN = 3/√3 is sound velocity of lattice, J+ = ∑QR+ f , is density, J ≈ J+ for low
velocity flow (Ma
