research article an analysis of nanoparticle...

Research ArticleAn Analysis of Nanoparticle Settling Times in Liquids

D D Liyanage1 Rajika J K A Thamali1 A A K Kumbalatara1

J A Weliwita2 and S Witharana13

1Department of Mechanical and Manufacturing Engineering University of Ruhuna Galle Sri Lanka2Department of Mathematics University of Peradeniya Kandy Sri Lanka3Sri Lanka Institute of Nanotechnology Colombo Sri Lanka

Correspondence should be addressed to S Witharana switharanaieeeorg

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

Academic Editor P Davide Cozzoli

Copyright copy 2016 D D Liyanage et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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-sizeparticles there are fairly accurate tools to predict settling rates However for smaller particles in particular micro-to-nanosizesthere is a gap in knowledge This paper develops an analytical method to predict the settling rates of micro-to-nanosized particlesuspensions The method is a combination of classical equations and graphical methods Validated using the experimental datain literature it was found that the new method shows an order of magnitude accuracy A remarkable feature of this method is itsability 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 magneticoptical thermal and transport properties Among these is thethermal conductivity that has been extensively studied anddebated over the past two decades For instance the additionof traces of nanoparticles often less than 1 vol to a commonheat transfer liquid like water has demonstrated an increasein thermal conductivity by up to 40 [1ndash3] Despite addingto excitement such high degrees of enhancement surpass thepredictions by classical theories However there are a fewcontradictory observations too In the famous INPBE exper-iment [4] the measured thermal conductivity enhancementswere observed to be within the range predicted by the effec-tivemedium theoriesThese nanoparticle-liquid heat transferblends are popularly known as nanofluids

Particles suspended in liquids are prone to form aggre-gates that would finally lead to separation and settling due togravity On the other hand aggregation is contemplated as amajor mechanism responsible for the enhanced thermal con-ductivity demonstrated by nanofluids [5] A certain degree ofaggregation in a nanofluidmay therefore be beneficial but theultimate settling would limit its practical use In addition to

the deterioration of thermal conductivity separated largeaggregatesmay clog filters and block the flow in narrow chan-nels in heat transfer devices However for mineral extractionand effluent treatment industries separation and settling arebasic prerequisites of operation To welcome or to avoid itone may need to understand the particle settling processesand settling rates Having said so only a few experimentalstudies have been hitherto dedicated to investigate the aggre-gation and settling dynamics of nanoparticles [6ndash8] Even formicrosized particles only a fewmethods are available in liter-ature to calculate the settling velocities [9] Complex nature ofaggregating nanoparticulate systems and difficulties in takingaccurate measurements seem to be major challenges for theprogress of experimental work [6 10 11]

This paper puts forward analytical method to calculatethe nanoparticle aggregation and settling times in liquids Tostart the computation sequence one should know the particleconcentration in the nanofluid In a situation of unknownparticle concentration one couldmeasure the viscosity of thenanofluid and compute it as suggested by Chen et al [12 13]First step is to determine the aggregation time For this acorrelation is derived by taking into account all governingparameters Second step is to determine the settling time Forthis a new method is proposed which is a combination of

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2016 Article ID 7061838 7 pageshttpdxdoiorg10115520167061838

2 Journal of Nanomaterials

known equations graphs and estimations The total time forsettling should be the sum of the aggregating time and thesettling time determined this way

Lastly the proposed analytical method is validated withthe experimental data for nanoparticle settling available inliterature

2 Materials and Methods

In this section computational sequences are introduced toestimate the nanoparticle aggregation time and the aggregatesettling time

21 Method to Estimate the Nanoparticle Aggregation TimeConsider a colloidal system where nanoparticles are welldispersed in a liquid Gradually the nanoparticles will startto form aggregates driven by a number of parameters such asnanoparticles size and concentration solution temperaturestability ratio and the fractal dimensions of objects [15ndash18]At any given time 119905 these aggregates can be characterized byradius of gyration (119877

119886) as follows

119877119886

119903119901

= (1 +119905

119905119901

)

1119889119891

(1)

where 119889119891is the fractal dimension of the nanoparticle aggre-

gates which is practically found to be in the range of 25ndash175[10] And the aggregation time constant 119905

119901is defined as [11]

119905119901=

(120587120583119903119901

3

119882)

(1198961198611198790119901)

(2)

Here 119896119861 119879 0119901 and 120583 are respectively the Boltzmann con-

stant temperature nanoparticle volume fraction and viscos-ity of the liquid

The stability ratio119882 is defined by

119882 = 2119903119901int

infin

0

119861 (ℎ) exp (119881119877+ 119881119860) 119896119861119879

(ℎ + 2119903)2

119889ℎ (3)

where 119861(ℎ) is the parameter that captures the hydrodynamicinteraction

After Chen et al [12] for interparticle distance ℎ

119861 (ℎ) =

6 (ℎ119903119901)2

+ 13 (ℎ119886) + 2

6 (ℎ119903119901)2

+ 4 (ℎ119886)

(4)

Moreover 119881119877is the repulsive potential energy given by

119881119877= 2120587120598

1199031205980119903119901Ψ2 exp (minusΛℎ) (5)

for dielectric constant of free space 1205980 and 120577 potential Ψ

Note that this expression is valid when Λ119903119901lt 5

Debye parameter is given byΛ = 5023times101111986805(120598119903119879)05

where 120598119903is the relative dielectric constant of the liquid and 119868

is the concentration of ions in water In the absence of salts119868 and pH relate in the form of 119868 = 10minuspH for pH le7 and 119868 =10minus(14minuspH) for pH gt7

Also the attractive potential energy is defined as

119881119860= minus

119860

6

[

[

2119903119901

2

ℎ (ℎ + 4119903119901)

+

2119903119901

2

(ℎ + 2119903119901)2

+ ln(ℎ(ℎ + 4119903

119901)

(ℎ + 2119903119901)2)]

]

(6)

where 119860 is the Hamaker constantUsing above set of equations and calculation procedure

one could compute the time taken to form an aggregate (119905119901)

of a known radius (119877119886)

22Method to Estimate the Aggregate Settling Time Terminalsettling velocity of a particle in a fluid body is governed bymultiple factors such as fluid density and particle density andsize and shape and concentration and degree of turbulenceand solution temperature [19] The Stokes law of settlingwas originally defined for small mm or 120583m size sphericalparticles with low Reynolds Numbers (Re le 03) The dragforce of a creeping flow over a rigid sphere consists of twocomponents namely pressure drag (119865

119901= 120587120583119889119880) and the

shear stress drag (119865 = 2120587120583119889119880) [18 20 21]Thus the total dragbecomes 119865

119863= 3120587120583119889119880 Using the Stokes equation a spherical

particle moving under ideal conditions of infinite fluid vol-ume lamina flow and zero acceleration can be expressed inthe form

41205871198773

(120588119901minus 120588119891) 119892

3minus 6120587120583119880119877 = 0 (X)

where 119880 is terminal velocity and 119877 is the equivalent radiusof the aggregate Stokes drag (6120587120583119880119877) could be reexpressedas 119865SD = 119862

119863(1205881198911198802

2)119860 where 119862119863= 1198651015840

(1205881198911198802

2) for force1198651015840 per unit projected area and 119860 is the projected area of the

particle to incoming flow For a sphere119860 = 412058711987733 Value for119862119863can be found fromFigure 3Note that a particle in a creep-

ing flow where Reynolds Number is very small tends to facethe least projected area to the flow [14]

Thus (X) becomes 41205871198773(120588119901minus120588119891)1198923 minus119862

119863(1205881198911198802

2)119860 = 0

and hence

119880 = radic81205871198773

119886(120588119901minus 120588119891) 119892

3119862119863120588119891119860

(7)

However a nanoparticle will hardly qualify for the Stokesconditions because of its larger surface area-to-volume ratioIn these circumstances the surface forces dominate overgravitational forces Also for a nanoparticle dispersed in aliquid the intermolecular forces (Van der Waalrsquos iron-ironinteractions 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 settlingvelocity anymore Thus the nanoparticles in suspension will

Journal of Nanomaterials 3

have a random motion not only vertically downward Recallthat we assumed that the nanoparticles do not start noticeablesettling till they made aggregates of a sufficient mass Exper-imental data shows that the nanosize particles (1 nmndash20 nm)formmicrosize aggregates (01ndash15 120583m) [10]The shapes of theaggregates depend upon fractal dimension (119889

119891) that typically

varies between 15 and 25 in most cases As 119889119891gets closer

to 3 the shape of aggregates approaches a spherical shapeAlso when a colony of nanoparticles form one microsizeaggregate the size factor comes into effect the intermolecularforces disappear and the Newtonian forces begin to dom-inate on the aggregate [24] For instance mean free path(119871 = radic21198961198793120587119909120583119905) due to Brownian motion shortens by8644when nanoparticles of 23 nm come together and form25 120583m aggregate [25] Following assumptions are made forapplication of Stokes law for the present work

(a) Reynolds Number (Re) Aggregates have very low settlingvelocities and thus they give very small Reynolds Numbers

For example Re can be expected in the region of 0001sim00001[26]

Re =119863119886119880120588119891

120583 (8)

The diameters of settling aggregates were determined inSection 21 above With this information alone the derivationof Re of these objects is still not possible Their settlingvelocities too are required but not known Therefore aniterative method is proposed to estimate Re [27]

119862119863Re2 =

4119892 (120588119901minus 120588119891) 1205881198911198893

31205832

(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 theaggregate 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 (10)

Rhodes [20] developed graphs to correlate Re and 119862119863for

different values of sphericity (Ψ) These graphs are shown inFigure 2 However the aggregates of smaller sizes have verylow Reynolds Numbers in the order of 10minus4sim10minus6 as statedbefore In order to capture values of119862

119863for such lowReynolds

Numbers graphs were extended to the left-hand sideThe extended graphs are presented in Figure 3 Equationscorresponding to the set of graphs are listed in Table 1

To use Figure 3 one needs to know the sphericity (Ψ) ofaggregates This information is given in Table 2 for commonand general shapes [17 28 29]

(c) Density of the Aggregate Smoluchowski model states thatnanoparticles clustered together form a complete sphere withvoids inside [11 18] Moreover when fractural dimensiondecreases the aggregate geometry gets closer to a two-dimensional flat object In this case 119889

119891would approach a

value of 18 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 ofliquidmolecules Density of the settling aggregate can thus besafely assumed as equal to the density of the solid

(d) Density of Suspension When preparing a nanofluidthe suspension is thoroughly stirred for the particles to bedistributed evenly in the container Similarly it is assumedthat the aggregates too are evenly distributed throughoutthe liquid Thus this system portrays a homogenous flow ofaggregates The density of this solid-liquid mixture is deter-mined using

120588119898= 0119886120588119886+ (1 minus 0

119886) 120588119891 (11)

where 0119886is the aggregate volume fraction given by

0119886= 0119901(119877119886

119903)

3minus119889119891

(12)

(e) Viscosity of the Suspension Maxwell Garnett [25] putforward the following relationship to calculate the viscosityof suspensionswhere the particle volume concentration is lessthan 5

120583 = 1205830(1 + 250

119901) (13)

(f) Zero-Slip Condition and Smooth Surface When nanopar-ticles are dispersed in water the water molecules make anorderly layer around the nanoparticle a phenomena knownas liquid layering [12 30] The water layer directly in contactwith the nanoparticle gets denser than the bulk liquid furtheraway Due to this particle-water bond at the boundary it isreasonable to assume a no-slip region for water Further thesurface of the aggregate is smooth and therefore the drag dueto roughness of the aggregate may not come into effect [31]

119865119860= 6120587120583

119887119877119886119880119886(2120583119887+ 119877119886120573119886119887

3120583119887+ 119877119886120573119886119887

) (14)

where 120583119887and 120573

119886119887are viscosity of the liquid and the coefficient

of sliding friction When there is no tendency for slipping120573119886119887asymp infin and therefore the above expression retracts to the

Stokes law 119865119860= 6120587120583

119887119877119886119880119886 Hence (X) can be used

(g) Batch Settling Originally the Stokes law is for a spheretravelling in infinite medium at low Re However in theaggregation and settling systems studied in this work


Table 1 Equations of logarithm of drag coefficients (119862119863) with respect to logarithm of Re for different values of sphericity (Ψ) extracted from

graphs in Figure 2

Ψ Relationship of drag coefficient (119862119863) and low Reynolds Numbers (Re)

0125 log (119862119863) = 01202 log (Re)2 minus 06006 log (Re) + 2032

022 log (119862119863) = 00043161 log (Re)3 + 09725 log (Re)2 minus 068 log (Re) + 1937




10minus15

10minus10

10minus5

100

Reyn

olds

Num

ber (

Re)

10minus6 10minus4 10minus210minus8 102100

CDRe2

Figure 1 Re versus 119862119863Re2 for spherical particles

Ψ = 0125Ψ = 022Ψ = 06Ψ = 0806Ψ = 10

0001 1000001 10001 106104 105101

Single particle Reynolds Number Rep

01

1

10

1000

100

104

Dra

g co

effici

entCD

2468

020406

08

Figure 2 Drag coefficient (119862119863) and particle Reynolds Number

(Re119901) for different sphericity (Ψ) [14]

the particles are in large number in a finite volume of liquidThose close proximity aggregates are obviously influenced byeach other and may deviate from the Stokes law Richardsonand Zaki [32] defined the term batch settling velocity (119880

119901)

or particle superficial velocity for an event where there are anumber of particles in a finite volume When Re lt 03 119880

119901=

1198801198791205761015840465 where 1205761015840 is liquid void fraction expressed as 1205761015840 =

Voids VolumeTotal Volume Here 1205761015840 should be less than 01for the batch settling effect to have a significant effect on theStokes law As a result the solutionswith 1205761015840 over 01 are consid-ered to be governed by the Stokes law alone safely ignoring

minus4 minus2 0 2 4minus6log(Re)

minus5

0

5

10

15

log(CD

)

012502206

08061

Figure 3 Drag coefficient (119862119863) for low Re and for different Ψ

effect of close proximity particles In nanofluids the particleconcentrations are far smaller than this (hence 1205761015840 is muchlarger than 01) For example in Witharana et al [10] settlingexperiments 1205761015840 was 0723 (1205761015840 = 1 minus 0

119901(119877119886119903)3minus119889119891) Therefore

in the context of this work the batch settling scenario is veryweak


Table 2 Geometric details of different aggregate shapes 1198771 1198772 and 119877

3are respectively aggregates of radii 25 120583m 4 120583m and 5 120583m

Shape Volume (120583m3) Surface area(120583m2)

Projected crosssection area (120583m2) 119889

119891

Sphericity(Ψ)

1198771

1198772

1198773

65452680852360

78542010631416

196350267853

333

111

1198771

1198772

1198773

65442681052308

91622932438860

66112573318

18ndash2018ndash2018ndash20

085706860808

1198771

1198772

1198773

65432681052362

110512947641900

161472279256


07100682075

N = 256 df = 18

(a)

50120583m

(b)

Figure 4 (a) Structure of an aggregate [10] (b) Optical microscopy images of aqueous Al2O3aggregates [10]

3 Results and Discussion

31 Estimation of Aggregation and Settling Times from theProposed Model For the validation of this model the exper-imental data from Witharana et al [10] were recruitedTheir system was polydisperse spherical alumina (Al

2O3)

nanoparticles suspended in water at near-IEP The particlesizes were rangingwithin 10sim100 nm verified by TEM imageswith the average size of 46 nm From the optical microscopyimages aggregates were observed to have radius in the range

of 1 120583msim10 120583m as seen from Figure 4(b) For the purposeof validating this model the equivalent aggregate radiusis taken as 25 120583m and density of Al

2O3nanoparticles is

taken as 3970 kgm3 Based on the geometry of the aggregateshown on the microscopy images the fractal dimension (119889

119891)

is estimated to be 18 Witharana et alrsquos [10] nanoparticleconcentration was 05 wt which converts to equivalentvolume fraction (0

119901) of 0001 vol The height of the vials

where their samples were stored during the experiment was6 cm


311 Aggregation Time At IEP the repulsive and hydrody-namic forces become minimum and the value of119882 tends to1 Now using (2) and (1) first 119905

119901and then 119905 are calculated for

the following values120583 = 892lowast10minus4 kgms 119903119901= 23 nm120601

119901=

0001119879 = 293K and 119896119861= 138times10

minus23 Jm2K4sThis yields119905119901= 845 times 10

minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)

312 Settling Velocity Consider the following(i) Density of the aggregate (120588agg) is taken from Section 3

above which was 3970 kgm3(ii) Density of homogenous flow (120588

119898) is calculated from

(11) which is 1003 kgm3(iii) Aggregation fraction (0

119886) is calculated from (12)

which is 0277 and void fraction of the liquid (1205761015840) is0723

(iv) Viscosity of the liquid (120583) is calculated from (13)which is 892 times 10minus4 kgmminus1 sminus1

(v) Calculated value for 119862119863Re2 for a 5 120583m diameter

aggregate from (9) is 614 times 10minus3(vi) Now from Figure 1 the approximate Re is 3 times 10minus5(vii) Sphericity (Ψ) is taken as 0857 from Table 2 which is

in the interval of 1 and 0806(viii) From the line corresponding to 0806 in Figure 3 the

drag coefficient (119862119863) is found to be approximately 3 times

105 (indicated in Figure 3 by vertical and horizontallines)

(ix) From (7) and assuming equivalent radius as 25 120583mterminal velocity is calculated to be

437 times 10minus5ms (b)

(x) On the experimental study reported in [10] the actualsettling velocitywas stated as 666times10minus5msThis fallswithin the same order of magnitude as calculated inthe foregoing step (b)

313 Total Settling Time When the suspension is at near-IEPthe nanoparticle aggregation occurs rapidly Furthermore asmentioned in the Section 22 above it is assumed that thenanoparticles have no resultant downward motion till theyget fully aggregated This makes the total time for settlingequal to the sumof aggregation time and settling time Aggre-gation time was calculated in above (a) as 065mins Onceaggregates were formed assume they reached the terminalvelocity in negligible time Now the total time for settlingbecomes

Total time for settling

= aggregation time + settling time

= 065min + 6 times 10minus2

437 times 10minus5= 065 + 2288

= 2353min

(15)

Figure 3 in Witharana et al [10] provides the camera imagesof their settling nanofluid Close to 30mins after preparationtheir samples were completely settled The calculated totalsettling time presented above is therefore in good agreementwith 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 interestExperimentation with a real suspension would be the mostaccurate method to study these complex systems Howeverfor industrial applications the predictability of settling ratesis of utmost importance The equations available in literatureaddress the sizes of submillimeter or above leaving a gap fora predictive model that can cater to nano- and micrometersized particlesThe analytical method presented in this paperwas an effort to fill this gap To begin the procedure one needsto know the particle concentration in the liquid Once it isknown firstly the aggregation rates can be estimated usingthe modified classical correlations presented in this paperSettling rates were then determined from a combination ofequations and graphs Total settling time thus becomes thesum of aggregation and settling times To validate the newmethod experimental data for Al

2O3-water nanoparticulate

system was recruited from literature Their experimentallydetermined settling rates were 666 times 10minus5ms For the samesystem our model prediction was 437 times 10minus5ms Thus theexperimental and analytical schemes were in agreement withthe same order of magnitude

Furthermore this analytical model is able to accountfor roundness deviations and fractal dimensions Howevermore experimental data are required to further examine theresilience of the proposed model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] X-F Yang and Z-H Liu ldquoApplication of functionalizednanofluid in thermosyphonrdquo Nanoscale Research Letters vol 6no 1 article 494 2011

[2] H Kim ldquoEnhancement of critical heat flux in nucleate boilingof nanofluids a state-of-art reviewrdquo Nanoscale Research Lettersvol 6 no 1 article 415 2011

[3] S Witharana J A Weliwita H Chen and L Wang ldquoRecentadvances in thermal conductivity of nanofluidsrdquo Recent Patentson Nanotechnology vol 7 no 3 pp 198ndash207 2013

[4] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo Journal ofApplied Physics vol 106 no 9 Article ID 094312 2009

[5] S Lotfizadeh and T Matsoukas ldquoA continuum Maxwell theoryfor the thermal conductivity of clustered nanocolloidsrdquo Journalof Nanoparticle Research vol 17 no 6 article 262 2015

[6] Y T He J Wan and T Tokunaga ldquoKinetic stability of hematitenanoparticles the effect of particle sizesrdquo Journal of Nanoparti-cle Research vol 10 no 2 pp 321ndash332 2008


[7] Y R Chae Y J Yoon and K G Ryu ldquoDevelopment of modifiedstokes expression to model the behavior of expanded bedscontaining polydisperse resins for protein adsorptionrdquo KoreanJournal of Chemical Engineering vol 21 no 5 pp 999ndash10022004

[8] H Mann P Mueller T Hagemeier C Roloff D Theveninand J Tomas ldquoAnalytical description of the unsteady settlingof spherical particles in Stokes and Newton regimesrdquo GranularMatter vol 17 no 5 pp 629ndash644 2015

[9] S Zhiyao W Tingting X Fumin and L Ruijie ldquoA simpleformula for predicting settling velocity of sediment particlesrdquoWater Science and Engineering vol 1 no 1 pp 37ndash43 2008

[10] S Witharana C Hodges D Xu X Lai and Y Ding ldquoAggrega-tion and settling in aqueous polydisperse alumina nanoparticlesuspensionsrdquo Journal of Nanoparticle Research vol 14 article851 2012

[11] R Prasher P E Phelan and P Bhattacharya ldquoEffect of aggrega-tion kinetics on the thermal conductivity of nanoscale colloidalsolutions (nanofluid)rdquoNano Letters vol 6 no 7 pp 1529ndash15342006

[12] H Chen S Witharana Y Jin C Kim and Y Ding ldquoPredictingthermal conductivity of liquid suspensions of nanoparticles(nanofluids) based on rheologyrdquo Particuology vol 7 no 2 pp151ndash157 2009

[13] H Chen S Witharana Y Jin C Ding and Y Kim ldquoPredictingthe thermal conductivity of nanofluids based on suspensionviscosityrdquo in Proceedings of the 4th International Conference onInformation and Automation for Sustainability (ICIAFS rsquo08) pp234ndash239 IEEE Colombo Sri Lanka December 2008

[14] R L Hamilton and O K Crosser ldquoThermal conductivity ofheterogeneous two-component systemsrdquo Industrial and Engi-neering Chemistry Fundamentals vol 1 no 3 pp 187ndash191 1962

[15] J Israelachvili ldquoContrasts between intermolecular interpar-ticle and intersurface forcesrdquo in Intermolecular and SurfaceForces pp 205ndash222 Elsevier 3rd edition 2011

[16] L H Hanus R U Hartzler and N J Wagner ldquoElectrolyte-induced aggregation of acrylic latex 1 dilute particle concen-trationsrdquo Langmuir vol 17 no 11 pp 3136ndash3147 2001

[17] YMinMAkbulut KKristiansen YGolan and J IsraelachvilildquoThe role of interparticle and external forces in nanoparticleassemblyrdquo Nature Materials vol 7 no 7 pp 527ndash538 2008

[18] G Pranami ldquoUnderstanding nanoparticle aggregationrdquo 2009httplibdriastateeducgiviewcontentcgiarticle=1856ampcon-text=etd

[19] S P Jang and S U S Choi ldquoEffects of various parameters onnanofluid thermal conductivityrdquo Journal of Heat Transfer vol129 no 5 pp 617ndash623 2007

[20] M Rhodes Introduction to Particle Technology John Wiley ampSons Hoboken NJ USA 2013

[21] D E Walsh and P D Rao A Study of Factors Suspected ofInfluencing the Settling Velocity of Fine Gold Particles vol 76University of Alaska Fairbanks Fairbanks Alaska 1988

[22] J Israelachvili Intermolecular and Surface Forces Elsevier 3rdedition 2011

[23] T Brownian Brownian Motion no 1 Clarkson UniversityPotsdam NY USA 2011

[24] S Lee S U-S Choi S Li and J A Eastman ldquoMeasuring ther-mal conductivity of fluids containing oxide nanoparticlesrdquo Jour-nal of Heat Transfer vol 121 no 2 pp 280ndash288 1999

[25] J C Maxwell Garnett ldquoColours in metal glasses and in metallicfilmsrdquo Philosophical Transactions of The Royal Society Series A

Mathematical Physical and Engineering Sciences vol 203 pp385ndash420 1904

[26] G Bai W Jiang and L Chen ldquoEffect of interfacial thermalresistance on effective thermal conductivity of MoSi

2SiC com-

positesrdquo Materials Transactions vol 47 no 4 pp 1247ndash12492006

[27] D Walsh and P Rao A Study of Factors Suspected of Influencingthe Settling Velocity of Fine Gold Particles OrsquoNeill ResearchLaboratory Fairbanks Alaska USA 1988

[28] N Visaveliya and J M Kohler ldquoControl of shape and size ofpolymer nanoparticles aggregates in a single-step microcon-tinuous flow process a case of flower and spherical shapesrdquoLangmuir vol 30 no 41 pp 12180ndash12189 2014

[29] L Li Y Zhang H Ma and M Yang ldquoMolecular dynamicssimulation of effect of liquid layering around the nanoparticleon the enhanced thermal conductivity of nanofluidsrdquo Journal ofNanoparticle Research vol 12 no 3 pp 811ndash821 2010

[30] B M Haines and A L Mazzucato ldquoA proof of Einsteinrsquos effec-tiveviscosity for a dilute suspension of spheresrdquo httparxivorgabs11041102

[31] J A Molina-Bolıvar F Galisteo-Gonzalez and R Hidalgo-Alvarez ldquoCluster morphology of protein-coated polymer col-loidsrdquo Journal of Colloid and Interface Science vol 208 no 2 pp445ndash454 1998

[32] J F Richardson and W N Zaki ldquoSedimentation and fluidisa-tion part Irdquo Chemical Engineering Research and Design vol 75no 3 supplement pp S82ndashS100 1997

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known equations graphs and estimations The total time forsettling should be the sum of the aggregating time and thesettling time determined this way

Lastly the proposed analytical method is validated withthe experimental data for nanoparticle settling available inliterature

2 Materials and Methods

In this section computational sequences are introduced toestimate the nanoparticle aggregation time and the aggregatesettling time

21 Method to Estimate the Nanoparticle Aggregation TimeConsider a colloidal system where nanoparticles are welldispersed in a liquid Gradually the nanoparticles will startto form aggregates driven by a number of parameters such asnanoparticles size and concentration solution temperaturestability ratio and the fractal dimensions of objects [15ndash18]At any given time 119905 these aggregates can be characterized byradius of gyration (119877

119886) as follows

119877119886

119903119901

= (1 +119905

119905119901

)

1119889119891

(1)

where 119889119891is the fractal dimension of the nanoparticle aggre-

gates which is practically found to be in the range of 25ndash175[10] And the aggregation time constant 119905

119901is defined as [11]

119905119901=

(120587120583119903119901

3

119882)

(1198961198611198790119901)

(2)

Here 119896119861 119879 0119901 and 120583 are respectively the Boltzmann con-

stant temperature nanoparticle volume fraction and viscos-ity of the liquid

The stability ratio119882 is defined by

119882 = 2119903119901int

infin

0

119861 (ℎ) exp (119881119877+ 119881119860) 119896119861119879

(ℎ + 2119903)2

119889ℎ (3)

where 119861(ℎ) is the parameter that captures the hydrodynamicinteraction

After Chen et al [12] for interparticle distance ℎ

119861 (ℎ) =

6 (ℎ119903119901)2

+ 13 (ℎ119886) + 2

6 (ℎ119903119901)2

+ 4 (ℎ119886)

(4)

Moreover 119881119877is the repulsive potential energy given by

119881119877= 2120587120598

1199031205980119903119901Ψ2 exp (minusΛℎ) (5)

for dielectric constant of free space 1205980 and 120577 potential Ψ

Note that this expression is valid when Λ119903119901lt 5

Debye parameter is given byΛ = 5023times101111986805(120598119903119879)05

where 120598119903is the relative dielectric constant of the liquid and 119868

is the concentration of ions in water In the absence of salts119868 and pH relate in the form of 119868 = 10minuspH for pH le7 and 119868 =10minus(14minuspH) for pH gt7

Also the attractive potential energy is defined as

119881119860= minus

119860

6

[

[

2119903119901

2

ℎ (ℎ + 4119903119901)

+

2119903119901

2

(ℎ + 2119903119901)2

+ ln(ℎ(ℎ + 4119903

119901)

(ℎ + 2119903119901)2)]

]

(6)

where 119860 is the Hamaker constantUsing above set of equations and calculation procedure

one could compute the time taken to form an aggregate (119905119901)

of a known radius (119877119886)

22Method to Estimate the Aggregate Settling Time Terminalsettling velocity of a particle in a fluid body is governed bymultiple factors such as fluid density and particle density andsize and shape and concentration and degree of turbulenceand solution temperature [19] The Stokes law of settlingwas originally defined for small mm or 120583m size sphericalparticles with low Reynolds Numbers (Re le 03) The dragforce of a creeping flow over a rigid sphere consists of twocomponents namely pressure drag (119865

119901= 120587120583119889119880) and the

shear stress drag (119865 = 2120587120583119889119880) [18 20 21]Thus the total dragbecomes 119865

119863= 3120587120583119889119880 Using the Stokes equation a spherical

particle moving under ideal conditions of infinite fluid vol-ume lamina flow and zero acceleration can be expressed inthe form

41205871198773

(120588119901minus 120588119891) 119892

3minus 6120587120583119880119877 = 0 (X)

where 119880 is terminal velocity and 119877 is the equivalent radiusof the aggregate Stokes drag (6120587120583119880119877) could be reexpressedas 119865SD = 119862

119863(1205881198911198802

2)119860 where 119862119863= 1198651015840

(1205881198911198802

2) for force1198651015840 per unit projected area and 119860 is the projected area of the

particle to incoming flow For a sphere119860 = 412058711987733 Value for119862119863can be found fromFigure 3Note that a particle in a creep-

ing flow where Reynolds Number is very small tends to facethe least projected area to the flow [14]

Thus (X) becomes 41205871198773(120588119901minus120588119891)1198923 minus119862

119863(1205881198911198802

2)119860 = 0

and hence

119880 = radic81205871198773

119886(120588119901minus 120588119891) 119892

3119862119863120588119891119860

(7)

However a nanoparticle will hardly qualify for the Stokesconditions because of its larger surface area-to-volume ratioIn these circumstances the surface forces dominate overgravitational forces Also for a nanoparticle dispersed in aliquid the intermolecular forces (Van der Waalrsquos iron-ironinteractions 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 settlingvelocity anymore Thus the nanoparticles in suspension will








Re =119863119886119880120588119891

120583 (8)


119862119863Re2 =

4119892 (120588119901minus 120588119891) 1205881198911198893

31205832

(9)















120588119898= 0119886120588119886+ (1 minus 0

119886) 120588119891 (11)


0119886= 0119901(119877119886

119903)

3minus119889119891

(12)


120583 = 1205830(1 + 250

119901) (13)


119865119860= 6120587120583

119887119877119886119880119886(2120583119887+ 119877119886120573119886119887

3120583119887+ 119877119886120573119886119887

) (14)

where 120583119887and 120573



Stokes law 119865119860= 6120587120583





graphs in Figure 2







10minus15

10minus10

10minus5

100

Reyn

olds

Num

ber (

Re)


CDRe2


Ψ = 0125Ψ = 022Ψ = 06Ψ = 0806Ψ = 10

0001 1000001 10001 106104 105101


01

1

10

1000

100

104

Dra

g co

effici

entCD

2468

020406

08




119901)


119901=




minus5

0

5

10

15

log(CD

)

012502206

08061



119901(119877119886119903)3minus119889119891) Therefore







119891

Sphericity(Ψ)

1198771

1198772

1198773

65452680852360

78542010631416

196350267853

333

111

1198771

1198772

1198773

65442681052308

91622932438860

66112573318


085706860808

1198771

1198772

1198773

65432681052362

110512947641900

161472279256


07100682075

N = 256 df = 18

(a)

50120583m

(b)




2O3)



2O3nanoparticles is


119891)








119901=

0001119879 = 293K and 119896119861= 138times10


minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)




















437 times 10minus5= 065 + 2288

= 2353min

(15)


4 Conclusions







References





























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










Re =119863119886119880120588119891

120583 (8)


119862119863Re2 =

4119892 (120588119901minus 120588119891) 1205881198911198893

31205832

(9)















120588119898= 0119886120588119886+ (1 minus 0

119886) 120588119891 (11)


0119886= 0119901(119877119886

119903)

3minus119889119891

(12)


120583 = 1205830(1 + 250

119901) (13)


119865119860= 6120587120583

119887119877119886119880119886(2120583119887+ 119877119886120573119886119887

3120583119887+ 119877119886120573119886119887

) (14)

where 120583119887and 120573



Stokes law 119865119860= 6120587120583





graphs in Figure 2







10minus15

10minus10

10minus5

100

Reyn

olds

Num

ber (

Re)


CDRe2


Ψ = 0125Ψ = 022Ψ = 06Ψ = 0806Ψ = 10

0001 1000001 10001 106104 105101


01

1

10

1000

100

104

Dra

g co

effici

entCD

2468

020406

08




119901)


119901=




minus5

0

5

10

15

log(CD

)

012502206

08061



119901(119877119886119903)3minus119889119891) Therefore







119891

Sphericity(Ψ)

1198771

1198772

1198773

65452680852360

78542010631416

196350267853

333

111

1198771

1198772

1198773

65442681052308

91622932438860

66112573318


085706860808

1198771

1198772

1198773

65432681052362

110512947641900

161472279256


07100682075

N = 256 df = 18

(a)

50120583m

(b)




2O3)



2O3nanoparticles is


119891)








119901=

0001119879 = 293K and 119896119861= 138times10


minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)




















437 times 10minus5= 065 + 2288

= 2353min

(15)


4 Conclusions







References





























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





graphs in Figure 2







10minus15

10minus10

10minus5

100

Reyn

olds

Num

ber (

Re)


CDRe2


Ψ = 0125Ψ = 022Ψ = 06Ψ = 0806Ψ = 10

0001 1000001 10001 106104 105101


01

1

10

1000

100

104

Dra

g co

effici

entCD

2468

020406

08




119901)


119901=




minus5

0

5

10

15

log(CD

)

012502206

08061



119901(119877119886119903)3minus119889119891) Therefore







119891

Sphericity(Ψ)

1198771

1198772

1198773

65452680852360

78542010631416

196350267853

333

111

1198771

1198772

1198773

65442681052308

91622932438860

66112573318


085706860808

1198771

1198772

1198773

65432681052362

110512947641900

161472279256


07100682075

N = 256 df = 18

(a)

50120583m

(b)




2O3)



2O3nanoparticles is


119891)








119901=

0001119879 = 293K and 119896119861= 138times10


minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)




















437 times 10minus5= 065 + 2288

= 2353min

(15)


4 Conclusions







References





























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








119891

Sphericity(Ψ)

1198771

1198772

1198773

65452680852360

78542010631416

196350267853

333

111

1198771

1198772

1198773

65442681052308

91622932438860

66112573318


085706860808

1198771

1198772

1198773

65432681052362

110512947641900

161472279256


07100682075

N = 256 df = 18

(a)

50120583m

(b)




2O3)



2O3nanoparticles is


119891)








119901=

0001119879 = 293K and 119896119861= 138times10


minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)




















437 times 10minus5= 065 + 2288

= 2353min

(15)


4 Conclusions







References





























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119901=

0001119879 = 293K and 119896119861= 138times10


minus3 s For 119877119886= 25 120583m and 119889

119891= 18

119905 = 065mins (a)




















437 times 10minus5= 065 + 2288

= 2353min

(15)


4 Conclusions







References





























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials

























2SiC com-















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article an analysis of nanoparticle...

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