Hindawi Publishing Corporation
`Journal of Nanomaterials
`Volume 2016, Article ID 7061838, 7 pages
`http://dx.doi.org/10.1155/2016/7061838
`
`Research Article
`An Analysis of Nanoparticle Settling Times in Liquids
`
`D. D. Liyanage,1 Rajika J. K. A. Thamali,1 A. A. K. Kumbalatara,1
`J. A. Weliwita,2 and S. Witharana1,3
`1Department of Mechanical and Manufacturing Engineering, University of Ruhuna, Galle, Sri Lanka
`2Department of Mathematics, University of Peradeniya, Kandy, Sri Lanka
`3Sri Lanka Institute of Nanotechnology, Colombo, Sri Lanka
`
`Correspondence should be addressed to S. Witharana; switharana@ieee.org
`
`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.
`
`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 commu-
`nities 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 exper-
`iment [4], the measured thermal conductivity enhancements
`were observed to be within the range predicted by the effec-
`tive medium theories. These nanoparticle-liquid heat transfer
`blends are popularly known as nanofluids.
`Particles suspended in liquids are prone to form aggre-
`gates 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 con-
`ductivity 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 chan-
`nels 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 aggre-
`gation and settling dynamics of nanoparticles [6–8]. Even for
`microsized particles, only a few methods are available in liter-
`ature 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
`
`

`

`2
`
`Journal of Nanomaterials
`
`Also the attractive potential energy is defined as
`
`(6)
`
`2
`
`2𝑟
`𝑝
`
`2
`
`2𝑟
`𝑝
`
`+
`
`ℎ (ℎ + 4𝑟
`𝑝
`
`)
`
`2
`
`(ℎ + 2𝑟
`𝑝
`
`)
`
`[ [
`
`𝐴 6
`
`𝑉
`𝐴
`
`= −
`
`+ ln (
`
`ℎ (ℎ + 4𝑟
`𝑝
`
`)
`
`2
`
`(ℎ + 2𝑟
`𝑝
`
`)
`
`)]
`
`,
`
`]
`
`where 𝐴 is the Hamaker constant.
`Using above set of equations and calculation procedure,
`one could compute the time taken to form an aggregate (𝑡
`𝑝)
`𝑎).
`of a known radius (𝑅
`
`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 𝜇m size spherical
`particles with low Reynolds Numbers (Re ≤ 0.3). The drag
`force of a creeping flow over a rigid sphere consists of two
`= 𝜋𝜇𝑑𝑈) and the
`components, namely, pressure drag (𝐹
`shear stress drag (𝐹 = 2𝜋𝜇𝑑𝑈) [18, 20, 21]. Thus the total drag
`= 3𝜋𝜇𝑑𝑈. Using the Stokes equation, a spherical
`becomes 𝐹
`particle moving under ideal conditions of infinite fluid vol-
`ume, lamina flow, and zero acceleration can be expressed in
`the form
`
`𝐷
`
`𝑝
`
`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 𝑡, these aggregates can be characterized by
`𝑎) as follows:
`radius of gyration (𝑅
`
`𝑅
`𝑎
`
`𝑟
`𝑝
`
`1/𝑑𝑓
`
`𝑡
`
`= (1 +
`
`)
`
`,
`
`𝑡
`𝑝
`
`(1)
`
`where 𝑑
`𝑓 is the fractal dimension of the nanoparticle aggre-
`gates, which is practically found to be in the range of 2.5–1.75
`𝑝 is defined as [11]
`[10]. And the aggregation time constant 𝑡
`
`𝑡
`𝑝
`
`=
`
`(𝜋𝜇𝑟
`𝑝
`
`3𝑊)
`
`.
`
`(𝑘
`𝐵
`
`𝑇0
`𝑝
`
`)
`
`(2)
`
`4𝜋𝑅3 (𝜌
`
`𝑝
`
`− 𝜌
`𝑓
`
`) 𝑔
`
`3
`
`− 6𝜋𝜇𝑈𝑅 = 0,
`
`(X)
`
`Here 𝑘
`𝐵, 𝑇, 0𝑝, and 𝜇 are, respectively, the Boltzmann con-
`
`stant, temperature, nanoparticle volume fraction, and viscos-
`ity of the liquid.
`The stability ratio 𝑊 is defined by
`
`∞
`
`𝑊 = 2𝑟
`𝑝
`
`∫
`
`0
`
`𝐵 (ℎ) exp {(𝑉
`
`𝑅
`
`+ 𝑉
`𝐴
`
`) /𝑘
`𝐵
`
`𝑇}
`
`(ℎ + 2𝑟)
`
`2
`
`𝑑ℎ,
`
`(3)
`
`where 𝐵(ℎ) is the parameter that captures the hydrodynamic
`interaction.
`After Chen et al. [12], for interparticle distance ℎ,
`
`6 (ℎ/𝑟
`𝑝
`
`2
`)
`
`+ 13 (ℎ/𝑎) + 2
`
`𝐵 (ℎ) =
`
`.
`
`6 (ℎ/𝑟
`𝑝
`
`2
`)
`
`+ 4 (ℎ/𝑎)
`
`𝑅 is the repulsive potential energy given by
`Moreover, 𝑉
`
`𝑉
`𝑅
`
`= 2𝜋𝜖
`𝑟
`
`𝜖
`0
`
`𝑟
`𝑝
`
`Ψ
`
`2 exp (−Λℎ)
`
`(4)
`
`(5)
`
`0, and 𝜁 potential, Ψ.
`for dielectric constant of free space, 𝜖
`< 5.
`Note that this expression is valid when Λ𝑟
`Debye parameter is given by Λ = 5.023 × 1011𝐼0.5/(𝜖
`0.5,
`𝑟 is the relative dielectric constant of the liquid and 𝐼
`where 𝜖
`is the concentration of ions in water. In the absence of salts,
`𝐼 and pH relate in the form of 𝐼 = 10−pH for pH ≤7 and 𝐼 =
`10−(14−pH) for pH >7.
`
`𝑇)
`
`𝑟
`
`𝑝
`
`(𝜌
`𝑓
`
`𝐷
`
`𝐷
`
`𝑓
`
`𝐶
`
`where 𝑈 is terminal velocity and 𝑅 is the equivalent radius
`of the aggregate. Stokes drag (6𝜋𝜇𝑈𝑅) could be reexpressed
`𝑈2/2)𝐴, where 𝐶
`𝑈2/2) for force
`as 𝐹SD = 𝐶
`= 𝐹󸀠/(𝜌
`𝐹󸀠 per unit projected area and 𝐴 is the projected area of the
`particle to incoming flow. For a sphere, 𝐴 = 4𝜋𝑅3/3. Value for
`𝐷 can be found from Figure 3. Note that a particle in a creep-
`ing flow where Reynolds Number is very small tends to face
`the least projected area to the flow [14].
`Thus (X) becomes 4𝜋𝑅3(𝜌
`and, hence,
`
`− 𝜌
`𝑓
`
`)𝑔/3 − 𝐶
`𝐷
`
`(𝜌
`𝑓
`
`𝑝
`
`𝑈2/2)𝐴 = 0
`
`𝑈 = √
`
`8𝜋𝑅3
`
`𝑎
`
`(𝜌
`𝑝
`
`− 𝜌
`𝑓
`
`) 𝑔
`
`.
`
`3𝐶
`𝐷
`
`𝜌
`𝑓
`
`𝐴
`
`(7)
`
`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 interac-
`tions, induced dipoles, dispersion forces, and overlap repul-
`sion) 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
`
`

`

`Journal of Nanomaterials
`
`3
`
`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. Exper-
`imental data shows that the nanosize particles (1 nm–20 nm)
`form microsize aggregates (0.1–15 𝜇m) [10]. The shapes of the
`𝑓) that typically
`aggregates depend upon fractal dimension (𝑑
`𝑓 gets closer
`varies between 1.5 and 2.5 in most cases. As 𝑑
`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 dom-
`inate on the aggregate [24]. For instance, mean free path
`(𝐿 = √2𝑘𝑇/3𝜋𝑥𝜇𝑡) due to Brownian motion shortens by
`86.44% when nanoparticles of 23 nm come together and form
`2.5 𝜇m 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, Re can be expected in the region of 0.001∼0.0001
`[26]:
`
`Re =
`
`𝐷
`𝑎𝑈
`
`𝜌
`𝑓
`
`.
`
`𝜇
`
`(8)
`
`The diameters of settling aggregates were determined in
`Section 2.1 above. With this information alone the derivation
`of Re of these objects is still not possible. Their settling
`velocities too are required but not known. Therefore an
`iterative method is proposed to estimate Re [27]:
`
`𝐷Re2
`
`𝐶
`
`=
`
`4𝑔 (𝜌
`𝑝
`
`− 𝜌
`𝑓
`
`) 𝜌
`𝑓
`
`𝑑3
`
`3𝜇2
`
`(9)
`
`Figure 1 was constructed for (8) and (9). This enables deter-
`mination of Re using the aggregate diameter.
`
`(b) Sphericity (Ψ). Sphericity (Ψ) indicates how spherical the
`aggregate is. This is defined by
`
`Sphericity (Ψ) =
`
`surfce area of a sphere
`surface area of the aggregate which has the same volume of the sphere .
`
`Rhodes [20] developed graphs to correlate Re and 𝐶𝐷 for
`
`different values of sphericity (Ψ). These graphs are shown in
`Figure 2. However the aggregates of smaller sizes have very
`low Reynolds Numbers in the order of 10−4∼10−6 as stated
`𝐷 for such low Reynolds
`before. In order to capture values of 𝐶
`Numbers, graphs were extended to the left-hand side.
`The extended graphs are presented in Figure 3. Equations
`corresponding to the set of graphs are listed in Table 1.
`To use Figure 3, one needs to know the sphericity (Ψ) of
`aggregates. This information is given in Table 2 for common
`and general shapes [17, 28, 29].
`
`(c) Density of the Aggregate. Smoluchowski model states that
`nanoparticles clustered together form a complete sphere with
`voids inside [11, 18]. Moreover, when fractural dimension
`decreases, the aggregate geometry gets closer to a two-
`𝑓 would approach a
`dimensional flat object. In this case 𝑑
`value of 1.8 and appears like the shape in Figure 4(a).
`Now consider the density of a settling aggregate. This flat
`object, shown in Figure 4(a), is surrounded by a thin layer of
`liquid molecules. Density of the settling aggregate can thus be
`safely assumed as equal to the density of the solid.
`
`(d) Density of Suspension. When preparing a nanofluid,
`the suspension is thoroughly stirred for the particles to be
`distributed evenly in the container. Similarly it is assumed
`that the aggregates too are evenly distributed throughout
`the liquid. Thus this system portrays a homogenous flow of
`aggregates. The density of this solid-liquid mixture is deter-
`mined using
`
`𝜌
`𝑚
`
`= 0
`𝑎
`
`𝜌
`𝑎
`
`+ (1 − 0
`𝑎
`
`) 𝜌
`𝑓
`
`,
`
`(11)
`
`(10)
`
`(12)
`
`𝑎 is the aggregate volume fraction given by
`where 0
`
`0
`𝑎
`
`= 0
`𝑝
`
`(
`
`3−𝑑𝑓
`
`)
`
`.
`
`𝑅
`𝑎
`
`𝑟
`
`(e) Viscosity of the Suspension. Maxwell Garnett [25] put
`forward the following relationship to calculate the viscosity
`of suspensions where the particle volume concentration is less
`than 5%:
`
`𝜇 = 𝜇
`0
`
`(1 + 2.50
`𝑝
`
`) .
`
`(13)
`
`(f) Zero-Slip Condition and Smooth Surface. When nanopar-
`ticles are dispersed in water, the water molecules make an
`orderly layer around the nanoparticle, a phenomena known
`as liquid layering [12, 30]. The water layer directly in contact
`with the nanoparticle gets denser than the bulk liquid further
`away. Due to this particle-water bond at the boundary, it is
`reasonable to assume a no-slip region for water. Further, the
`surface of the aggregate is smooth and therefore the drag due
`to roughness of the aggregate may not come into effect [31]:
`
`𝐹
`𝐴
`
`= 6𝜋𝜇
`𝑏
`
`𝑅
`𝑎
`
`𝑈
`𝑎
`
`(
`
`2𝜇
`𝑏
`
`+ 𝑅
`𝑎
`
`𝛽
`𝑎𝑏
`
`3𝜇
`𝑏
`
`+ 𝑅
`𝑎
`
`𝛽
`𝑎𝑏
`
`) ,
`
`(14)
`
`where 𝜇
`𝑏 and 𝛽𝑎𝑏 are viscosity of the liquid and the coefficient
`
`of sliding friction. When there is no tendency for slipping
`≈ ∞ and therefore the above expression retracts to the
`𝑎. Hence (X) can be used.
`Stokes law, 𝐹
`
`= 6𝜋𝜇
`𝑏
`
`𝑅
`𝑎
`
`𝑈
`
`𝛽
`𝑎𝑏
`
`𝐴
`
`(g) Batch Settling. Originally the Stokes law is for a sphere
`travelling in infinite medium at low Re. However, in the
`aggregation and settling systems studied in this work,
`
`

`

`4
`
`Journal of Nanomaterials
`
`Table 1: Equations of logarithm of drag coefficients (𝐶𝐷) with respect to logarithm of Re for different values of sphericity (Ψ) extracted from
`
`
`graphs in Figure 2.
`
`Relationship of drag coefficient (𝐶𝐷) and low Reynolds Numbers (Re)
`
`log (𝐶
`) = 0.1202 log (Re)
`− 0.6006 log (Re) + 2.032
`) = 0.0043161 log (Re)
`+ 0.9725 log (Re)
`− 0.68 log (Re) + 1.937
`log (𝐶
`) = 0.0067 log (Re)
`+ 0.083 log (Re)
`− 0.73 log (Re) + 1.8
`log (𝐶
`) = 0.0099 log (Re)
`+ 0.0697 log (Re)
`− 0.7906 log (Re) + 1.7225
`log (𝐶
`) = 0.0116 log (Re)
`+ 0.05793 log (Re)
`− 0.8866 log (Re) + 1.443
`log (𝐶
`
`𝐷
`
`𝐷
`
`𝐷
`
`𝐷
`
`100
`
`10−5
`
`10−10
`
`10−15
`
`Reynolds Number (Re)
`
`Ψ
`
`0.125
`0.22
`0.6
`0.806
`1
`
`𝐷
`
`2
`
`3
`
`3
`
`3
`
`3
`
`2
`
`2
`
`2
`
`2
`
`10−8
`
`10−6
`
`10−4
`
`10−2
`
`100
`
`102
`
`CDRe2
`
`
`
`Figure 1: Re versus 𝐶𝐷Re2 for spherical particles.
`
`−6
`
`−4
`
`−2
`
`0
`
`2
`
`4
`
`log(Re)
`
`15
`
`10
`
`5
`
`0
`
`−5
`
`log(CD)
`
`0.125
`0.22
`0.6
`
`
`Figure 3: Drag coefficient (𝐶𝐷) for low Re and for different Ψ.
`
`0.806
`1
`
`effect of close proximity particles. In nanofluids the particle
`concentrations are far smaller than this (hence 𝜀󸀠 is much
`larger than 0.1). For example, in Witharana et al. [10] settling
`3−𝑑𝑓). Therefore,
`experiments, 𝜀󸀠 was 0.723 (𝜀󸀠 = 1 − 0
`in the context of this work, the batch settling scenario is very
`weak.
`
`(𝑅
`𝑎
`
`𝑝
`
`/𝑟)
`
`Ψ = 0.125
`Ψ = 0.22
`Ψ = 0.6
`Ψ = 0.806
`Ψ = 1.0
`
`0.1
`
`10
`
`100
`
`1000
`
`104
`
`105
`
`106
`
`1
`0.4
`0.8
`0.6
`0.2
`Single particle Reynolds Number, Rep
`
`104
`
`1000
`
`100
`
`10
`
`1
`
`8
`6
`
`24
`
`Drag coefficient,CD
`
`0.1
`0.001
`
`0.01
`
`Figure 2: Drag coefficient (𝐶𝐷) and particle Reynolds Number
`
`(Re𝑝) for different sphericity (Ψ) [14].
`
`the particles are in large number in a finite volume of liquid.
`Those close proximity aggregates are obviously influenced by
`each other and may deviate from the Stokes law. Richardson
`𝑝)
`and Zaki [32] defined the term batch settling velocity (𝑈
`or particle superficial velocity for an event where there are a
`number of particles in a finite volume. When Re < 0.3, 𝑈
`𝜀󸀠4.65, where 𝜀󸀠 is liquid void fraction expressed as 𝜀󸀠 =
`Voids Volume/Total Volume. Here 𝜀󸀠 should be less than 0.1
`for the batch settling effect to have a significant effect on the
`Stokes law. As a result the solutions with 𝜀󸀠 over 0.1 are consid-
`ered to be governed by the Stokes law alone, safely ignoring
`
`=
`
`𝑝
`
`𝑈
`𝑇
`
`

`

`Journal of Nanomaterials
`
`5
`
`
`
`
`
`
`
`Table 2: Geometric details of different aggregate shapes. 𝑅1, 𝑅2, and 𝑅3 are, respectively, aggregates of radii 2.5 𝜇m, 4 𝜇m, and 5 𝜇m.
`
`Shape
`
`Volume (𝜇m3)
`
`Surface area
`(𝜇m2)
`
`Projected cross
`section area (𝜇m2)
`
`Sphericity
`(Ψ)
`
`111
`
`𝑑
`𝑓
`
`333
`
`1.8–2.0
`1.8–2.0
`1.8–2.0
`
`0.857
`0.686
`0.808
`
`65.45
`268.08
`523.60
`
`78.54
`201.06
`314.16
`
`65.44
`268.10
`523.08
`
`91.62
`293.24
`388.60
`
`19.63
`50.26
`78.53
`
`6.61
`12.57
`33.18
`
`𝑅
`1
`
`𝑅
`2
`
`𝑅
`3
`
`𝑅
`1
`
`𝑅
`2
`
`𝑅
`3
`
`𝑅
`1
`
`𝑅
`2
`
`𝑅
`3
`
`65.43
`268.10
`523.62
`
`110.51
`294.76
`419.00
`
`16.14
`72.27
`92.56
`
`2.0–2.5
`2.0–2.5
`2.0–2.5
`
`0.710
`0.682
`0.75
`
`N = 256, df = 1.8
`
`(a)
`
`50 𝜇m
`
`(b)
`
`Figure 4: (a) Structure of an aggregate [10]. (b) Optical microscopy images of aqueous Al2O3 aggregates [10].
`
`3. Results and Discussion
`
`3.1. Estimation of Aggregation and Settling Times from the
`Proposed Model. For the validation of this model, the exper-
`imental data from Witharana et al. [10] were recruited.
`Their system was polydisperse spherical alumina (Al2O3)
`nanoparticles suspended in water at near-IEP. The particle
`sizes were ranging within 10∼100 nm verified by TEM images,
`with the average size of 46 nm. From the optical microscopy
`images aggregates were observed to have radius in the range
`
`of 1 𝜇m∼10 𝜇m, as seen from Figure 4(b). For the purpose
`of validating this model, the equivalent aggregate radius
`is taken as 2.5 𝜇m, and density of Al2O3 nanoparticles is
`taken as 3970 kg/m3. Based on the geometry of the aggregate
`𝑓)
`shown on the microscopy images, the fractal dimension (𝑑
`is estimated to be 1.8. Witharana et al.’s [10] nanoparticle
`concentration was 0.5 wt% which converts to equivalent
`) of 0.001 vol%. The height of the vials
`volume fraction (0
`where their samples were stored during the experiment was
`6 cm.
`
`𝑝
`
`

`

`6
`
`Journal of Nanomaterials
`
`3.1.1. Aggregation Time. At IEP, the repulsive and hydrody-
`namic forces become minimum and the value of 𝑊 tends to
`𝑝 and then 𝑡 are calculated for
`1. Now, using (2) and (1), first 𝑡
`the following values: 𝜇 = 8.92∗10−4 kg/m/s, 𝑟
`= 23 nm, 𝜙
`0.001, 𝑇 = 293 K, and 𝑘
`= 1.38 × 10−23 J/m2/K4/s. This yields
`= 8.45 × 10−3 s. For 𝑅
`= 2.5 𝜇m and 𝑑
`= 1.8,
`𝑡 = 0.65 mins.
`
`𝑡
`𝑝
`
`𝐵
`
`𝑎
`
`𝑓
`
`𝑝
`
`=
`
`𝑝
`
`(a)
`
`𝑎
`
`3.1.2. Settling Velocity. Consider the following:
`(i) Density of the aggregate (𝜌agg) is taken from Section 3
`above, which was 3970 kg/m3.
`(ii) Density of homogenous flow (𝜌
`𝑚) is calculated from
`(11), which is 1003 kg/m3.
`(iii) Aggregation fraction (0
`) is calculated from (12)
`which is 0.277 and void fraction of the liquid (𝜀󸀠) is
`0.723.
`(iv) Viscosity of the liquid (𝜇) is calculated from (13),
`which is 8.92 × 10−4 kg m−1 s−1.
`(v) Calculated value for 𝐶
`𝐷Re2 for a 5 𝜇m diameter
`aggregate from (9) is 6.14 × 10−3.
`(vi) Now, from Figure 1, the approximate Re is 3 × 10−5.
`(vii) Sphericity (Ψ) is taken as 0.857 from Table 2, which is
`in the interval of 1 and 0.806.
`(viii) From the line corresponding to 0.806 in Figure 3, the
`𝐷) is found to be approximately 3 ×
`drag coefficient (𝐶
`105 (indicated in Figure 3 by vertical and horizontal
`lines).
`(ix) From (7) and assuming equivalent radius as 2.5 𝜇m,
`terminal velocity is calculated to be
`−5 m/s.
`
`4.37 × 10
`
`(b)
`
`(x) On the experimental study reported in [10], the actual
`settling velocity was stated as 6.66×10−5 m/s. This falls
`within the same order of magnitude as calculated in
`the foregoing step (b).
`
`3.1.3. Total Settling Time. When the suspension is at near-IEP,
`the nanoparticle aggregation occurs rapidly. Furthermore, as
`mentioned in the Section 2.2 above, it is assumed that the
`nanoparticles have no resultant downward motion till they
`get fully aggregated. This makes the total time for settling
`equal to the sum of aggregation time and settling time. Aggre-
`gation time was calculated in above (a) as 0.65 mins. Once
`aggregates were formed, assume they reached the terminal
`velocity in negligible time. Now the total time for settling
`becomes
`Total time for settling
`= aggregation time + settling time
`
`= 0.65 min +
`
`6 × 10−2
`
`4.37 × 10−5
`
`= 0.65 + 22.88
`
`(15)
`
`= 23.53 min.
`
`Figure 3 in Witharana et al. [10] provides the camera images
`of their settling nanofluid. Close to 30 mins after preparation,
`their samples were completely settled. The calculated total
`settling time presented above is therefore in good agreement
`with the actual experiment reported in literature.
`
`4. Conclusions
`Determination of the settling rates of nano- and micropar-
`ticulate systems are of both academic and industrial interest.
`Experimentation with a real suspension would be the most
`accurate method to study these complex systems. However,
`for industrial applications, the predictability of settling rates
`is of utmost importance. The equations available in literature
`address the sizes of submillimeter or above, leaving a gap for
`a predictive model that can cater to nano- and micrometer
`sized particles. The analytical method presented in this paper
`was an effort to fill this gap. To begin the procedure, one needs
`to know the particle concentration in the liquid. Once it is
`known, firstly the aggregation rates can be estimated using
`the modified classical correlations presented in this paper.
`Settling rates were then determined from a combination of
`equations and graphs. Total settling time thus becomes the
`sum of aggregation and settling times. To validate the new
`method, experimental data for Al2O3-water nanoparticulate
`system was recruited from literature. Their experimentally
`determined settling rates were 6.66 × 10−5 m/s. For the same
`system, our model prediction was 4.37 × 10−5 m/s. Thus the
`experimental and analytical schemes were in agreement with
`the same order of magnitude.
`Furthermore, this analytical model is able to account
`for roundness deviations and fractal dimensions. However
`more experimental data are required to further examine the
`resilience of the proposed model.
`
`Conflict of Interests
`The authors declare that there is no conflict of interests
`regarding the publication of this paper.
`
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