Journal of Nanomaterials

Volume 2016 (2016), Article ID 7061838, 7 pages

http://dx.doi.org/10.1155/2016/7061838

## An Analysis of Nanoparticle Settling Times in Liquids

^{1}Department of Mechanical and Manufacturing Engineering, University of Ruhuna, Galle, Sri Lanka^{2}Department of Mathematics, University of Peradeniya, Kandy, Sri Lanka^{3}Sri Lanka Institute of Nanotechnology, Colombo, Sri Lanka

Received 10 September 2015; Revised 28 December 2015; Accepted 6 January 2016

Academic Editor: P. Davide Cozzoli

Copyright © 2016 D. D. Liyanage et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aggregation and settling are crucial phenomena involving particle suspensions. For suspensions with larger than millimeter-size particles, there are fairly accurate tools to predict settling rates. However for smaller particles, in particular micro-to-nanosizes, there is a gap in knowledge. This paper develops an analytical method to predict the settling rates of micro-to-nanosized particle suspensions. The method is a combination of classical equations and graphical methods. Validated using the experimental data in literature, it was found that the new method shows an order of magnitude accuracy. A remarkable feature of this method is its ability to accommodate aggregates of nonspherical shapes and of different fractal dimensions.

#### 1. Introduction

Nanoparticles have drawn interest from scientific communities across a broad spectrum for their unusual magnetic, optical, thermal, and transport properties. Among these is the thermal conductivity that has been extensively studied and debated over the past two decades. For instance, the addition of traces of nanoparticles, often less than 1 vol%, to a common heat transfer liquid like water, has demonstrated an increase in thermal conductivity by up to 40% [1–3]. Despite adding to excitement, such high degrees of enhancement surpass the predictions by classical theories. However there are a few contradictory observations too. In the famous INPBE experiment [4], the measured thermal conductivity enhancements were observed to be within the range predicted by the effective medium theories. These nanoparticle-liquid heat transfer blends are popularly known as* nanofluids*.

Particles suspended in liquids are prone to form aggregates that would finally lead to separation and settling due to gravity. On the other hand, aggregation is contemplated as a major mechanism responsible for the enhanced thermal conductivity demonstrated by nanofluids [5]. A certain degree of aggregation in a nanofluid may therefore be beneficial but the ultimate settling would limit its practical use. In addition to the deterioration of thermal conductivity, separated large aggregates may clog filters and block the flow in narrow channels in heat transfer devices. However, for mineral extraction and effluent treatment industries, separation and settling are basic prerequisites of operation. To welcome or to avoid it, one may need to understand the particle settling processes and settling rates. Having said so, only a few experimental studies have been hitherto dedicated to investigate the aggregation and settling dynamics of nanoparticles [6–8]. Even for microsized particles, only a few methods are available in literature to calculate the settling velocities [9]. Complex nature of aggregating nanoparticulate systems and difficulties in taking accurate measurements seem to be major challenges for the progress of experimental work [6, 10, 11].

This paper puts forward analytical method to calculate the nanoparticle aggregation and settling times in liquids. To start the computation sequence, one should know the particle concentration in the nanofluid. In a situation of unknown particle concentration, one could measure the viscosity of the nanofluid and compute it as suggested by Chen et al. [12, 13]. First step is to determine the aggregation time. For this a correlation is derived by taking into account all governing parameters. Second step is to determine the settling time. For this a new method is proposed, which is a combination of known equations, graphs, and estimations. The total time for settling should be the sum of the aggregating time and the settling time determined this way.

Lastly the proposed analytical method is validated with the experimental data for nanoparticle settling available in literature.

#### 2. Materials and Methods

In this section, computational sequences are introduced to estimate the nanoparticle aggregation time and the aggregate settling time.

##### 2.1. Method to Estimate the Nanoparticle Aggregation Time

Consider a colloidal system where nanoparticles are well dispersed in a liquid. Gradually the nanoparticles will start to form aggregates driven by a number of parameters such as nanoparticles size and concentration, solution temperature, stability ratio, and the fractal dimensions of objects [15–18]. At any given time , these aggregates can be characterized by radius of gyration () as follows:where is the fractal dimension of the nanoparticle aggregates, which is practically found to be in the range of 2.5–1.75 [10]. And the aggregation time constant is defined as [11]Here , , , and are, respectively, the Boltzmann constant, temperature, nanoparticle volume fraction, and viscosity of the liquid.

The stability ratio is defined by where is the parameter that captures the hydrodynamic interaction.

After Chen et al. [12], for interparticle distance , Moreover, is the repulsive potential energy given by for dielectric constant of free space, , and potential, . Note that this expression is valid when .

Debye parameter is given by , where is the relative dielectric constant of the liquid and is the concentration of ions in water. In the absence of salts, and pH relate in the form of for pH ≤7 and for pH >7.

Also the attractive potential energy is defined aswhere is the Hamaker constant.

Using above set of equations and calculation procedure, one could compute the time taken to form an aggregate () of a known radius ().

##### 2.2. Method to Estimate the Aggregate Settling Time

Terminal settling velocity of a particle in a fluid body is governed by multiple factors such as fluid density and particle density and size and shape and concentration and degree of turbulence and solution temperature [19]. The Stokes law of settling was originally defined for small, mm, or *μ*m size spherical particles with low Reynolds Numbers The drag force of a creeping flow over a rigid sphere consists of two components, namely, pressure drag () and the shear stress drag [18, 20, 21]. Thus the total drag becomes . Using the Stokes equation, a spherical particle moving under ideal conditions of infinite fluid volume, lamina flow, and zero acceleration can be expressed in the form where is terminal velocity and is the equivalent radius of the aggregate. Stokes drag () could be reexpressed as , where for force per unit projected area and is the projected area of the particle to incoming flow. For a sphere, . Value for can be found from Figure 3. Note that a particle in a creeping flow where Reynolds Number is very small tends to face the least projected area to the flow [14].

Thus becomes and, hence, However, a nanoparticle will hardly qualify for the Stokes conditions because of its larger surface area-to-volume ratio. In these circumstances, the surface forces dominate over gravitational forces. Also, for a nanoparticle dispersed in a liquid, the intermolecular forces (Van der Waal’s, iron-iron interactions, iron-dipole interactions, dipole-dipole interactions, induced dipoles, dispersion forces, and overlap repulsion) along with the thermal vibrations (Brownian motion) and diffusivity will take over the Newtonian forces [22, 23]. Hence the gravitational force does not dominate the settling velocity anymore. Thus the nanoparticles in suspension will have a random motion, not only vertically downward. Recall that we assumed that the nanoparticles do not start noticeable settling till they made aggregates of a sufficient mass. Experimental data shows that the nanosize particles (1 nm–20 nm) form microsize aggregates (0.1–15 *μ*m) [10]. The shapes of the aggregates depend upon fractal dimension () that typically varies between 1.5 and 2.5 in most cases. As gets closer to 3, the shape of aggregates approaches a spherical shape. Also, when a colony of nanoparticles form one microsize aggregate, the size factor comes into effect, the intermolecular forces disappear, and the Newtonian forces begin to dominate on the aggregate [24]. For instance, mean free path () due to Brownian motion shortens by 86.44% when nanoparticles of 23 nm come together and form 2.5 *μ*m aggregate [25]. Following assumptions are made for application of Stokes law for the present work.

*(a) Reynolds Number (Re)*. Aggregates have very low settling velocities, and thus they give very small Reynolds Numbers. For example, can be expected in the region of 0.001~0.0001 [26]:The diameters of settling aggregates were determined in Section 2.1 above. With this information alone the derivation of of these objects is still not possible. Their settling velocities too are required but not known. Therefore an iterative method is proposed to estimate [27]:Figure 1 was constructed for (8) and (9). This enables determination of using the aggregate diameter.