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Colloids and Surfaces A: Physicochem. Eng. Aspects 345 (2009) 112–120

Contents lists available at ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

journa l homepage: www.e lsev ier .com/ locate /co lsur fa

novel surface-slip correction for microparticles motion

bouzar Moshfegh a, Mehrzad Shams a,∗, Goodarz Ahmadi b, Reza Ebrahimi c

Faculty of Mech. Engng., K.N. Toosi University of Tech., Pardis St., Vanak Sq., Tehran 19395-1999, IranDepartment of Mech. and Aeronautical Engng., Clarkson University, Potsdam, NY 13699-5725, USAFaculty of Aerospace Engng., K.N. Toosi University of Tech., East Vafadar St., Tehranpars Ave., Tehran 16765-3381, Iran

r t i c l e i n f o

rticle history:eceived 11 November 2008eceived in revised form 18 April 2009ccepted 23 April 2009vailable online 3 May 2009

a b s t r a c t

As the fine particles expose to the rarefied regimes, the solid-surface effects become prominent andchange the interfacial interactions between the gas and particle. The particle drag force, which is mainlyconcerned in aerosol researches, is affected through the changes observed in the way that fluid entitiesreflect from the particle surface. A 3D simulation study for an incompressible rarefied slip flow past aspherical aerosol particle was performed. The full Navier–Stokes equations were solved and the velocity

eywords:as–particle interfaceFDunningham correctionurface-slip correction

jump at the gas–particle interface was treated numerically by imposition of the slip boundary condition.Analytical solution to the Stokesian slip flow past a spherical particle was used as a benchmark for codeverification, and reasonable agreement was achieved. The simulation results showed that in addition tothe Knudsen number, the Reynolds number affects the slip correction factor. Thus, the Cunningham-basedslip corrections must be augmented by the inclusion of the effect of Reynolds number for application toLagrangian tracking of fine particles. A novel expression for the surface-slip correction factor as a functionof both Knudsen number (0.01 < Kn < 0.1) and Reynolds number (0 < Re < 1) was developed.

Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.

. Introduction

Gas–solid two-phase flows are encountered frequently in manynvironmental and pharmaceutical applications and in assess-ng human exposure to environmental pollutants. Many industrialpplications such as dry powder flows, fluidized bed, settling cham-ers, cyclones, electrostatic and electromagnetic precipitators, and

nertial dust collectors involve dispersed two-phase flows. Aerosolsn the nature typically have the size range from a few nanometerso several micrometers.

When the characteristic size of the particle decreases down tovalue comparable to the mean free path of the molecules, the

arefaction effects become important and significantly affect theow properties, wall shear stress and aerodynamic drag force. Inhis case, the continuum assumption fails and the Navier–Stokesquations with no-slip boundary conditions cannot be applied. Inuch situations, intermolecular collisions play a prominent role andhe flow properties will be affected by the Knudsen number, which

s a dimensionless measure of the relative magnitudes of the gas

ean free path and flow characteristic length. Schaaf and Chambre1] have proposed the following ranges to determine the degreef rarefaction flow regime based on the local Knudsen number,

∗ Corresponding author.E-mail address: shams@kntu.ac.ir (M. Shams).

927-7757/$ – see front matter. Crown Copyright © 2009 Published by Elsevier B.V. All rioi:10.1016/j.colsurfa.2009.04.042

Kn. For Kn < 0.01, the continuum hypothesis holds and fluid is inlocal thermodynamic equilibrium. Thus, the Navier–Stokes equa-tion in conjunction with no-slip boundary condition describes theflow behaviors. The rarefaction effects become noticeable when theKnudsen number is in the range of 0.01–0.1. This range is referredto as the slip flow regime, where the continuum hypothesis isstill valid, but the local thermodynamic equilibrium of near-wallgas is violated. That is, the conventional no-slip boundary condi-tion imposed at the gas–solid interface begins to break down butthe linear stress–strain relationship is still valid [2]. In this range,the Navier–Stokes equations can be utilized, but the slip bound-ary condition is used to account for the non-equilibrium effectsat the gas–solid interface. The range of Kn > 10 represents the freemolecular regime where the collisions among the molecules arenegligible and individual molecules colloid with the particle [58].Finally, in the transitional regime (0.1 < Kn < 10) the size of particleis comparable to the gas mean free path, and continuum assump-tion breaks down but the flow cannot be regarded as free molecularregime.

The ultra fine (nanometer) aerosols under normal condition typ-ically experience a rarefied flow regime, and therefore the drag

force acting on the particle reduces compared with what is pre-dicted by the Stokes’ law [3]. It has been customary to account forthe drag reduction by introducing a correction factor to considerthe rarefaction effects and velocity discontinuity at the gas–particleinterface.

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. Literature review

About a century ago, Cunningham [4] derived a correlation of theorm (1 + AKn) to consider the slip effects on the drag force of actingn a spherical particle in the Stokesian flow regime. An experimen-al study for Kn < 0.3 was carried out in the same year by Millikan5] who reported the linear dependency of Cunningham correc-ion factor on the gas mean free path. More rarefied flow regimesere experimentally investigated by Knudsen and Weber [6] and

he parameter A was expressed as a function of Knudsen number.hat is

C(Kn) = 1 + AKn

A = ˛ + ˇ exp(−�

Kn

)Kn = �

�

(1)

here � is the gas mean free path, and � is the flow referenceength-scale which has been taken equal to the spherical particleadius in the following results. The constants ˛, ˇ and � are deter-

ined experimentally. Cunningham suggested a value of A = 1.267] disregarding its dependence on Kn as indicated in Eq. (1). Untilecently, researchers have tried to propose slip correction factorsased upon Eq. (1). Millikan [8] tested various particle surfacessing his conventional oil drop method in air with the mean freeath of 0.094 �m, and different particle size, which generated aange of 0.5 < Kn < 134. In another experiment by Millikan [9], arst-order correction technique was proposed to account for thearefaction effects on an oil drop. He found that the Stokes law can-ot predict accurately the terminal velocity when the oil drop size iseduced to nanometer ranges. Polystyrene spherical microparticlesere used by Allen and Raabe [10] to measure their appropriate

lip correction factors over the range of Knudsen numbers from.03 to 7.2. Five years later, Rader [11] re-analyzed the slip correc-ion factor of the small particles in various gaseous mediums for theange of 0.2 < Kn < 0.95, and provided accurate values for ˛, ˇ and. Hutchins et al. [12], used modulated dynamic light scattering tond the slip correction factor of polystyrene and polyvinyl toluenepherical particles with the diameters between 1 and 2.12 �m. Theyeasured the drag force acting on the spherical particles suspended

n dry air to determine the diffusion coefficient of a single levitatedarticle from which the slip correction factor could be obtained.ecently, Kim et al. [3] have investigated the slip correction factort reduced pressure and high Knudsen numbers using polystyreneatex (PSL) particles. Nano-differential mobility analyzer (NDMA)

as used to determine the slip correction factor for 0.5 < Kn < 83.he data obtained for three particle sizes were fitted well by the

xpression given in Eq. (1). Table 1 includes the experimental sliporrection factors in the form of Eq. (1), which have been proposednd used by the researchers in the past several decades. All thesexpressions assume the correction factor to be as a function ofnudsen number based on the particles radius.

able 1unningham-based slip correction factors for air with mean free path of 67.3 nm at 101.3

o. Slip correction factor

1 + Kn[1.034 + 0.536 exp(−1.219/Kn)]1 + Kn[1.209 + 0.406 exp(−0.893/Kn)]1 + Kn[1.155 + 0.471 exp(−0.596/Kn)]1 + Kn[1.142 + 0.558 exp(−0.999/Kn)]1 + Kn[1.209 + 0.441 exp(−0.779/Kn)]1 + Kn[1.231 + 0.469 exp(−1.178/Kn)]1 + Kn[1.257 + 0.400 exp(−1.100/Kn)]1 + Kn[1.170 + 0.525 exp(−0.780/Kn)]1 + Kn[1.165 + 0.483 exp(−0.997/Kn)]

a They originally reported the slip correction factor as 1 + Kn[0.683 + 0.354 exp(−1.845/b The correlation was originally reported in the standard conditions and mean free path

cochem. Eng. Aspects 345 (2009) 112–120 113

For an incompressible fluid, an analytic solution to the Stoke-sian slip flow past a stationary, impermeable and solid microspherewas derived by Barber and Emerson [13]. Inertia terms of theNavier–Stokes equations were neglected in their analysis. The exis-tence of an additional drag force originating from the normalstress was confirmed. This additional drag is unlike what appearsin the continuum flow regime. In a related work, Barber andEmerson [14] also simulated the slip flow regime past a confinedmicrosphere within a circular pipe using a two-dimensional finite-volume Navier–Stokes solver that accounts for the slip boundarycondition. The confined sphere in the circular pipe was also stud-ied numerically over a wide range of Knudsen numbers by Liu etal. [15]. This configuration is used in macro-scale spinning rotorgauges to measure the pressure, viscosity and molecular weights inlow pressure gases [16,17].

In the research works cited above, all investigators tried to cor-relate the rarefaction effects on the particle drag force only to theKnudsen number, and the effect of Reynolds number on the slipcorrection factor was not directly reported. In addition, the earlierreported experiments did not cover the Knudsen numbers asso-ciated with the slip flow regime (0.01 < Kn < 0.1); therefore, theproposed correlations may not be accurately for predicting the rar-efaction effects in this range. Thus, the slip flow regime merits morerigorous studies for developing more accurate correlations. Manyresearchers [18–23] have used the Cunningham-based slip correc-tion factors for their Lagrangian simulations of gas–solid two-phaseflows. Therefore, availability of a more accurate expression wouldbe helpful in the related areas.

In the present work, the incompressible slip flow regime past anunconfined, stationary, impermeable, solid and spherical particle issimulated. The main goal of the study is to provide a better under-standing of dynamic effects originating from the slip regime for atypical aerosol particle. In particular, a more accurate slip correctionfactor as a function of both Reynolds and Knudsen numbers is pro-posed. Comparison of the new slip correction with those proposedby other researchers is also performed. The analytic expression forthe drag coefficient derived in the Stokesian slip flow regime byBarber and Emerson [13] is used to ensure the validity of the CFD(computational fluid dynamics) procedure that considers the veloc-ity jump (rarefaction effect) at the gas–particle surface interface.

3. Flow governing equations

For the steady and incompressible flow of a Newtonian fluidwhich is in local thermodynamic equilibrium, the Navier–Stokesequations governing in continuum flow regime can be written as

∂(uk)∂xk

= 0 (2)

�∂(ukui)

∂xk= − ∂p

∂xi+ ∂�ik

∂xk(3)

kPa and 23 ◦C.

Particle material Author

Glass Knudsen and Weber [6]a

Oil drops Millikan [8]b

Oil drops Allen and Raabe [64]PSL Allen and Raabe [10]Oil drops Rader [11]PSL Hutchins et al. [12]– Davies [65]– Crowe [58]PSL Kim et al. [3]

Kn)]using the mean free path of 100.65 nm at 101.3 kPa and 20.2 ◦C [3].of 94.17 nm [3].

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here x is the position in coordinate system, u is the velocity, p ishe pressure, � is the fluid density, and � is the stress tensor. For aewtonian and isotropic fluid the stress tensor is given by

ik = �

(∂ui

∂xk+ ∂uk

∂xi

)+ ıik

(∂uj

∂xj

)(4)

ere, � is the fluid dynamic viscosity, is the coefficient of bulk vis-osity and ıik is the Kronecker delta function. For monatomic gases,he Stokes’ hypothesis relates the coefficients � and through theollowing expression

+ 23

� = 0 (5)

.1. Slip boundary condition on the particle surface

To account for the effects of local non-equilibrium in the slipow regime, the conventional no-slip boundary condition must beeplaced appropriately. As postulated first by Navier in 1827 [24],he tangential slip velocity is proportional to the tangential wallhear. That is

t |wall ≈ ˛�t |wall (6)

ere, ˛ is a constant. Later, Maxwell in 1879 [24] showed from theinetic theory of gases that ˛ ≈ �/�. For gases, the mean free path�) is the average distance traveled by molecules between collisions.or a perfect gas, the mean free path is given as

= kT√2pd2

m

(7)

ere, k is the Boltzmann constant, T is the gas temperature, and dm ishe collision diameter of the molecules. To account for the portionf molecules reflected diffusely from the solid surface in the slipelocity formulation, Schaaf and Chambre [1] suggested that theonstant ˛ in Eq. (6) must be multiplied by a factor of ((2 − �)/�)here � is the tangential momentum accommodation coefficient

TMAC). Consequently, the slip velocity can be written as

t |wall ≈(

2 − �

�

)�

��t |wall (8)

q. (8) relates the tangential slip velocity to the tangential sheartress via the TMAC, gas mean free path and gas dynamic viscosity.egarding Eq. (1), the flow length-scale, �, is typically evaluated ashe ratio of a flow thermodynamic property (such as temperaturer density) to its spatial gradient. Alternatively, � may be estimateds a length-scale of the flow geometry. For the incompressible andsothermal flow simulated here, we used the particle radius (d/2)s the reference length-scale of the flow, which is consistent withhe definition used in aerosol communities. Using the definition ofnudsen number (Eq. (1)), the relation of tangential slip velocity on

he particle surface (Eq. (8)) becomes

t |wall ≈(

2 − �

�

)Kn

d

2��t |wall (9)

.2. Relationship between Reynolds, Knudsen and Mach numbers

Rarefied gas flows in low pressure environments and microflowsre dynamically similar, if the two flow conditions have similareometries and identical Kn, Re and Ma numbers [25]. The particleelative Reynolds and Mach numbers are defined by

f p

e = �|u − u |��

(10)

a = �u

c= |uf − up|√

�RT(11)

cochem. Eng. Aspects 345 (2009) 112–120

where �u is the local relative velocity between gas and particle,c is the speed of sound in the gas, uf is the gas local velocity atthe location of the particle, up is the particle velocity, � is specificheat ratio, R is the constant of gases and T is the temperature of thegas surrounding the particle. For the physical consistency betweenthe inertial and viscous forces and the slip rarefaction effects, samelength-scale (� = d/2) is also used for the Re number. It is well knownthat the Knudsen number, the Reynolds number and the Mach num-ber are related. Kinetic theory of gases relates the gas mean free pathto viscosity as follows

�

�= 1

2� vm = 1

2� c

√8

�(12)

Here, vm is the mean molecular speed, which is somewhat higherthan the speed of sound (c). Combining the definition of Kn, Re andMa numbers and Eq. (12), it follows that [2]

Ma =√

�

2Kn Re (13)

This relation reveals the interconnection between gas compressibil-ity and the rarefaction effects. For a definite value of Mach number,the Kn and Re numbers are inversely proportional. Thus, Reynoldsnumber cannot change arbitrarily for a given Ma in the slip flowregime stands (0.01 < Kn < 0.1).

It is well known [24,26–29] that the flow of a compressible fluidlike can be regarded as incompressible in the range of Ma < 0.3.It should be noted here that this criterion is necessary but notsufficient. There are flow conditions for which Mach number issmaller than 0.3, but the compressibility effects may be a promi-nent feature of the physics of the flow. For example, wall heatingor cooling may cause the density to change significantly, and thecompressibility may become important even at low Mach num-bers. Another example is for flows in micro-devices (especially theconfined microflows), where the pressure may change sharply dueto viscous effects [2].

Here, we are concerned with an isothermal and unconfinedincompressible slip flow past small spheres in the Stokesian slipflow whose incompressibility condition is clearly confirmed via Eq.(13).

4. Numerical simulation

4.1. Computational domain and imposed boundary conditions

GAMBIT version 2.0.0, the companion package of commercialsoftware FLUENTTM, was used to build the model geometry andthe computational grids. A multiblock structured and body-fittedgrid was considered for analyzing the flow over a single sphericalparticle with the diameter of d. The layout of multiblock domaindecomposition and tri-dimensional grid distribution are illustratedin Fig. 1. Such topology of computational domain was used ear-lier by Jones and Clarke [30] and Lee [31], respectively, for theturbulent and laminar unsteady flows over the sphere. Domaindecomposition around the particle to capture the velocity jump atthe gas–solid interface is shown in Fig. 1(a). The view of plane z–yand the grid geometry adjacent to the particle surface are depictedin Fig. 1(b). The view of plane x–y and the grid geometry on theparticle surface are also shown in Fig. 1(c).

Grid elements in the cubic block surrounding the particle areall hexahedral with linear stretching while the other blocks arefilled with rectangular elements with exponential stretching. The

geometric lengths of the computational domain depicted in Fig. 1were selected regarding the flow geometry at low Reynolds num-ber regimes. In the z–y view, the sides of the cube surrounding theparticle (LC) and the marginal lengths (LM) were respectively setto 3d and 2d. These values yield a side length of 7d for the square

A. Moshfegh et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 345 (2009) 112–120 115

Fig. 1. (a) Domain decomposition around the spherical particle to capture the veloc-ity jump. (b) Multiblock grid distribution around the particle, view of plane z–yappended with the grid geometry near the particle surface. (c) Multiblock grid dis-tribution around the particle, schematic view of plane x–y appended with the gridgeometry on the particle surface.

Table 2The results of grid-independency study carried out at Re = 0.125 and Kn = 0.05.

Total cellsnumber

Total dragcoefficient (CD)

Relative error vs. analyticsolution, Eq. (14) (%)

Case I 91,000 115.153 25.404Case II 258,904 104.990 14.336

Case III 350,148 97.875 6.587Case IV 439,085 92.533 0.769Case V 867,090 92.142 0.344

cross-section of the computational domain. Along the line of thework presented by Jones and Clarke [30] on turbulent flow over anunconfined sphere, we found the side length far enough to removethe outer boundary effects on the incompressible and steady flowsimulated here.

To minimize the outflow effects on the flow, the outflow wasplaced 10d farther from the particle center, 1d more than what wastaken for the turbulent flow by Jones and Clarke [30]. To ensure thatthe sphere does not affect the uniform inflow, we used the upstreamdistance from the particle center of 3d following the results of grid-independency study reported in the Lee [31]. Therefore, in the x–yview, the beginning length (LB), the wake length (LW) and the down-stream length (LD) are, respectively, equal to 1.5d, 3.5d and 5d. Fivedifferent grid resolutions (Cases I–V) were picked to perform thegrid-independency study by measuring the total drag force on theparticle at Re = 0.25 and Kn = 0.05. The results are given in Table 2.From the values given in the relative error column, it could be con-cluded that the Case IV is the best choice among others to achievea grid-independent solution.

Boundary conditions considered to solve the flow are intro-duced in Fig. 2. For a fine particle traveling in the gas flow,the gas approaches the particle with a roughly uniform velocitydistribution. The left, upper, lower, back and front faces of the com-putational domain were all treated as velocity inlet with a givenvelocity component along the x-direction. The distance from theparticle center to the outflow captures successfully the flow steadywake past the particle and also ensures the assumption of zero gra-dients of flow variables across the outlet boundary. On the particlesurface, the slip (velocity jump) boundary conditions were imposed(in three dimensions).

4.2. Numerical solution

The commercial CFD software FLUENTTM v6.0.12 was adoptedto solve the Navier–Stokes equations using the finite-volumemethod (FVM). The pressure-based segregated algorithm was

employed to discretize the 3D, incompressible, steady, laminarand Navier–Stokes equations (Eqs. (2)–(5)). Modified QuadraticUpstream Interpolation for Convective Kinematics (QUICK) scheme,which can be implemented on the multidimensional, nonuni-

Fig. 2. Schematic of the boundary conditions. (Each couple of circles identifies aboundary face.)

1 : Physicochem. Eng. Aspects 345 (2009) 112–120

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16 A. Moshfegh et al. / Colloids and Surfaces A

orm and unstructured grids [32–37], was selected to discretizehe convective and diffusive fluxes in the momentum equations.ressure–velocity coupling was treated by the SIMPLE-ConsistentSIMPLEC) algorithm on a colocated grid. A second-order pressurenterpolation method was also used to calculate the pressure atell faces from the neighboring nodes. Gradient reconstruction waserformed by Green-Gauss cell-based formulation [55].

FLUENTTM solves the linearized scalar system of equations forhe dependent variables in each cell using a point implicit (Gauss-eidel) block matrix linear equation solver in conjunction with anlgebraic multigrid (AMG) method. Flexible multigrid cycle washosen for the three momentum equations, while the V-cycle wasonsidered for continuity equation. To achieve a stable solutionhrough the iterations, appropriate under-relaxation factors wereevised for pressure and three components of velocity. Solution wasssumed to converge when the sum of the relative errors in consec-tive iteration was less than 10−4. (This is one order smaller thanhe default convergence criteria of the FLUENTTM code.) The capa-ility of foregoing set of schemes to simulate the slip flow regimesas been examined as reported by some researchers [38–41].

The implementation of slip boundary condition in FLUENTTM

as performed with the User Defined Functions (UDF) (FLUENTTM

6.0.21 User’s Guide) [60]. The slip boundary condition is dis-retized in a way that yields the wall tangential shear stress (�t|wall)s a function of the tangential velocity of each cell adjacent to theall. The effects of the surface curvature are considered automati-

ally through this method. This approach comes from the strategye followed in the finite volume method (FVM) where the com-utational molecules are designed as to incorporate the dependentariables of cell nodes. The UDF is called once per iteration at theeginning of the solution main loop. The relevant coefficients to thelip boundary condition are then introduced into the block matrixnd linearized scalar system of equations for the dependent vari-bles. The resultant system of equations is solved implicitly, andnally the velocity of near-wall computational cells is calculateder iteration. Additional details of this approach may be found inhe works of Barber and Emerson [42] and Morínigo et al. [43].

.3. Solution verification

To verify the accuracy of numerical approach in simulation ofo-slip continuum flow regime, the solution results are comparedgainst the experimental drag force values proposed by White [24]

n Fig. 3.

The White’s drag formula has been proposed for a wide rangef Re numbers from 0 to 2.0E+5. Its accuracy was reported about10% in this range [24]. Based on these points, the correspondencef numerical results in the range 0 < Re < 1.0 to experiment is rea-

ig. 3. Difference percentage of present numerical results from the drag coefficientf no-slip continuum regime.

Fig. 4. Comparison of total drag coefficient calculated by analytic solution with thatcomputed by present simulation at � = 1: (a) total drag coefficient vs. the relativeReynolds number; (b) total drag coefficient vs. the Knudsen number.

sonable as diagrammed above. The correspondence is excellent atnear-zero Reynolds numbers where the White’s curve-fit formulamust accurately fit the experiment as the starting point of the range0 < Re < 2.0E+5.

In next step, the slip flow solution verification is performed bycomparison against the analytic solution to the Stokesian slip flowpast an impermeable, stationary and solid microsphere as reportedby Barber and Emerson [13]. Accordingly, the total drag coefficientis given by

CD = FD

1/2 �U2A= 12

Re

(1 + 2((2 − �)/�)Kn

1 + 3((2 − �)/�)Kn

)(14)

Here, FD is the total drag force acting on the particle, U is the inletgas flow speed and A is the projected area of the spherical particle.

In Fig. 4, assuming TMAC = 1, the values of total drag coefficientcomputed by the present CFD simulation in the slip flow regimefor Re ≤ 1 are compared with those computed by the analytic solu-tion given by Eq. (14). The variations of total drag coefficient versusthe Reynolds and Knudsen numbers are illustrated, respectively, inFig. 3(a) and (b). A reasonable agreement of the present numeri-cal solutions with the analytical solution is observed. ComparingFig. 3(a) and (b) shows that the Reynolds number is more domi-nant than the rarefaction degree of gas or Knudsen number on thevariations of total drag coefficient.

5. Results and discussion

In this work, it is assumed that the particle is traveling in the dryair at standard ambient temperature and pressure (SATP), respec-tively, at 298.15 K and 101.3 kPa. The corresponding air mean freepath calculated from Eq. (7) is 69.2 nm. The air density and dynamic

: Physicochem. Eng. Aspects 345 (2009) 112–120 117

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A. Moshfegh et al. / Colloids and Surfaces A

iscosity are, respectively, 1.1686 kg/m3 and 1.832E−5 N s/m2. Inheory, TMAC, which accounts for the average tangential momen-um exchange across the gas–solid interface, can be assigned byhe values ranging from the zero (specular reflection) up to thenity (diffuse reflection). Thomas and Lord [44] and Arkilic et al.45] empirically determined the TMAC values of some materialsn the range of 0.2–1.0. This parameter depends on both gas andolid materials, and surface finish of the wall [45]. Accordingly, sili-on micro-machined components exhibit the TMAC values rangingrom the 0.8 to 1.0 [45]. Here, we assumed a value of unity forhe TMAC to study the slip correction factor for most common

aterials. Computer simulations are performed for five values ofn number and four values of Re number distributed uniformly,espectively within the slip flow range (0.01 < Kn < 0.1) and Stoke-ian regime (0 ≤ Re ≤ 1).

.1. Slip correction factor in Stokesian slip regimes

An important assumption in the Stokes drag force is that theelative velocity of the gas at the surface of the particle is zero.his assumption is not accurate for small particles whose diame-ers are of the order of or smaller than the mean free path of theurrounding gas. These particles settle faster than what is predictedy the Stokes law, because of the “slip” at the surface of the particle3,7] and smaller relaxation time or higher velocity. As noted before,unningham-based slip corrections were developed for the Stokesow regime. Hence in the present work, the new slip correction

actor is proposed in the range 0 < Re < 1 or the Stokes regime.To account for the slip effects on the particle drag force in the

arefied regimes, Cunningham slip correction factor (Cc) which isefined as a function of Knudsen number [11], is given by

lip correction factor (SCF) = Fcont.

FSlip(15)

here the numerator denotes the drag force experienced by thearticle in the continuum Stokes flow, and Fslip is the drag force inhe slip flow regime. This relation assumes that the particle dragorce (Fslip) is decreased by the rarefaction effects as a function ofnudsen number.

Fcont. is the particle drag force in the continuum no-slip regime.he experimental data are more realistic and sensible than theumerical ones to evaluate Fcont. Therefore, we use the curve fit for-ula proposed by Hinds [7] (modified for Reynolds number defined

ased on the particle radius) as given below

cont. = 12Re

[1 + 3

16Re

](12

� U2A)

(16)

ig. 5 shows the numerical slip correction factors evaluated usingqs. (15) and (16) in the Stokesian regime. As was noted before, there

s no available experimental data for Fslip in this range. Here, theumerical simulation data for the drag force in the slip flow regimere used. Fig. 5 shows that the resultant slip correction factor islways greater than unity and increases roughly linearly with Kn. Its also seen that the slip corrections varies significantly dependingn Reynolds number. The observed dependence on Re, suggests thathe correlation for slip correction should include variations of Re. Asas recognized by some researchers [24,46,47], the fluid inertia oreynolds number has an important role to mitigate and/or intensifyhe rarefaction effects in the slip flow regimes. Fig. 5 clearly showshat higher values of Re lead to higher rarefaction effects and con-equently higher fluid slip on the particle surface that yields higher

alues of SCF.

A comparison between the Cunningham-based correlationsTable 1) and the slip correction factor computed through theresent simulation using Eqs. (15) and (16) is also carried out inig. 5. The computed slip correction factor is under-predicted by

Fig. 5. Comparison of Cunningham-based correlations (Table 1) and computed slipcorrection (Eq. (15)).

the Cunningham-based corrections (Table 1). As one closes to theRe = 0, computed SCF approaches the corrections reported in Table 1that proves the accuracy of the computed data via Eq. (15) at thelimiting case of creeping (very low Reynolds number) flows. Inac-curacy of SCFs listed in Table 1 increase as the gas rarefaction effect(or Knudsen number) increases.

Barber and Emerson [48] studied the effects of Kn and Re num-bers on the hydrodynamic development length in circular andparallel plate ducts. Similar to Fig. 5, their results showed that bothReynolds and Knudsen numbers affect the slip flow physical param-eters in a similar manner (in some flow cases). The increment ofboth numbers (Kn, Re) intensifies the gas slip as recorded in the fig-ure. Zuppardi et al. [47] quantified the effects of rarefaction in highvelocity slip flow regimes, and reported that both slip velocity andtemperature jump are increased when the free stream velocity (orthe flow Reynolds number) increases.

Regarding such behaviors, the Cunningham-based slip correc-tions (Table 1) do not seem to be appropriate to describe completelythe rarefaction effects only through the Knudsen number. One maingoal in this study is to modify the slip correction factor based on theCunningham’s mathematical definition (Fcont. divided by Fslip). Inparticular, we plan to incorporate the dependence of slip correctionon the Reynolds number into the correlation. Therefore, a correctionas a function of both relative Re and Kn numbers has to be prepared.Curve-fit statistics were analyzed quantitatively and it was foundthat the original form of Cunningham correction (Eq. (1)) is justappropriate to correlate the Kn variations in the slip regime whenthe Re is considered to be fixed. The original form can be multipliedby an additional term to introduce the Re into the correction. Sev-eral additional functions of Reynolds number were tested and it wasrecognized that the basic proposed form of Cunningham correction(Eq. (1)) is not the best choice for 3D curve-fitting process involv-ing both Kn and Re numbers. Exponential curve fitting (Eq. (1)) iscommonly used when the rate of variations accelerates suddenly,so the Cunningham-based correlations are appropriate to correlatethe rarefaction effects over a wide range of Knudsen numbers, butthis is not the case in the slip flow regime. Moreover, if we couldfind an appropriate additional expression as a function of Re, thefinal slip correction would be too large in terms, and could not beutilized easily by the users during the particle tracking procedure.

Due to the nonlinear behavior of drag coefficient acting bythe slip flow regime on a suspending spherical particle, similarto what was recognized in the continuum regime, we used the

nonlinear least squares model [49–51] to propose the correla-tions herein. There are a variety of methods to converge to theminimum least-squares, that the Levenburg–Marquardt algorithm[49,50] was adopted here. There were no prominent outliers to be

118 A. Moshfegh et al. / Colloids and Surfaces A: Physi

Table 3Coefficients and statistics of curve-fitted correlation (Eq. (18)).

Coefficient Mean value Uncertainty (SD)

A 0.23994 0.193E−1B 2.0024 0.980E−3C 0.97832 0.397E−2

RS = 0.998; ARS = 0.998; SDF = 0.257E−02; AAR = 0.190E−02.

F

etwct

S

TTadh

bo

Fv

ig. 6. The 3D fitted curve for new proposed slip correction factor (SCF) at Eq. (18).

xcluded from the original computed data set, and consequently,he robust fitting process was not enabled. Convergence toleranceas also considered equal to 1E−6. Here, we propose a novel and

oncise expression (Eq. (18)) which was the best among the 480ested 3D functions

CF = A (Re) + B(Kn) + C for 0 < Re < 1 and 0.01 < Kn < 0.1

(18)

he calculated coefficients and the curve fit statistics are given inable 3. The mean values of computed coefficients, standard devi-tion (SD) or uncertainty of fitted coefficients, RS, ARS, standard

eviation of the fitting (SDF) and average absolute residual (AAR)ave been also reported.

The 3D fitted curve on the computed slip correction factorsased on Eq. (15) is illustrated in Fig. 6. Scatter data points placedn the solid lines represent the computed data, and the 3D curve

ig. 7. Fit residuals of 3D fitted curve (Fig. 6) considering both Re and Kn numbersariations.

cochem. Eng. Aspects 345 (2009) 112–120

denotes the fitted (predicted) data by Eq. (18). The fit residual ateach data point vs. the Knudsen and Reynolds numbers is alsodemonstrated in Fig. 7. Maximum error through the curve-fittingprocess amounts to less than 1.9% that shows an acceptable fit. Theproposed slip correction factor in Eq. (18) needs to be utilized alongwith Eqs. (15) and (16) to yield the microparticle drag force in theslip flow regime.

6. Conclusions

In the present work, the incompressible slip flow regime past anunconfined, stationary, impermeable, solid and spherical aerosolparticle was simulated. Gas slip on the particle surface was treatednumerically by the imposition of slip boundary condition. A novelformulation was proposed to correlate the surface-slip correctionfactor against both Knudsen and Reynolds numbers. Based on thepresented results, the following conclusions are drawn:

(1) A prominent difference between the slip corrections at differ-ent Reynolds numbers is readily discernable. This differencenecessitates the incorporation of Re number into the slip cor-rection which parameterizes the drag force in the slip regime.The Reynolds number has a profound effect on the slip correc-tion factor similar to the Knudsen number. The computed slipcorrection factors are all under-predicted by the Cunningham-based corrections (Table 1) in the range 0 � Re < 1. Higher valuesof Re lead to higher rarefaction effects and consequently highergas slip on the particle surface. In the other word, both Re andKn numbers affect the slip flow parameters such as slip correc-tion factor in a same manner. The increment of these numbers(Kn, Re) intensifies the gas slip on the particle.

(2) A novel two-variable slip correction factor as a function of bothReynolds and Knudsen numbers was proposed to account forthe flow inertia effect which had been neglected in the previ-ously suggested correlations.

Appendix A. Slip boundary condition

In order to extend the solver to the slip-flow regime0.01 < Kn < 0.1, additional subroutines have been developed toaccount for the tangential slip-velocity boundary condition at thesurface of the spherical particle. In particular, the slip-velocity con-straint requires the tangential shear stress to be calculated at everygrid node along a solid–fluid interface. Since the governing aerody-namic equations are solved using a non-orthogonal boundary-fittedcoordinate system, the tangential shear stress has to be calcu-lated using a generalised stress tensor transformation procedureto account for the non-orthogonality of the grid lines. Furthermore,the determination of the drag force on the microsphere requiresboth the tangential shear stress component and the normal stresscomponent to be computed around the perimeter. It is thereforenecessary to transform the axisymmetric (x,r) coordinate stresstensor into a localized coordinate reference frame based upon theorientation of the boundary surface.

Consider an arbitrary section along the boundary of the sphereas shown in Fig. A.1. Let n and

t denote the local unit normal and

tangential vectors of the surface as shown above:

ˆ

n = n1i + n2j and t = t1i + t2j (A.1)

It can be seen that the coefficients in Eq. (A.1) are related via

t1 = n2 and t2 = −n1 (A.2)

A. Moshfegh et al. / Colloids and Surfaces A: Physi

g

[

std

[

w

[

aq

[

w�[

Hg

�

w

�

st

u

u

[

[[

[

[

[[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

Fig. A.1. Definition sketch of normal and tangential base vectors.

In the (x,r) plane, the stress tensor for an incompressible flow isiven by

�] =[

�11 �12�21 �22

]=

⎡⎢⎢⎣

2�∂u

∂x�

(∂u

∂r+ ∂v

∂x

)

�

(∂v∂x

+ ∂u

∂r

)2�

∂u

∂x

⎤⎥⎥⎦ (A.3)

The problem is therefore to transform [�] into a generalisedtress tensor [� ′] related to the normal and tangential base vec-ors, n and

t. Lien [61] has demonstrated that in the transformed

omain, the stress tensor can be expressed as

� ′] = [G][�][G]T (A.4)

here the matrix [G] is given by

G] =[

t1 t2n1 n2

](A.5)

nd the superscript T denotes the transpose of a matrix. Conse-uently, the generalised stress tensor can be evaluated as

� ′] =[

� ′11 � ′

12� ′

21 � ′22

]=

[t1 t2n1 n2

][�11 �12�21 �22

][t1 n1t2 n2

](A.6)

hich yields after multiplication (and simplification using21 = �12):

� ′11 � ′

12� ′

21 � ′22

]

=[

t21 �11 + 2t1t2�12 + t2

2 �22 t1n1�11 + (t1n2 + t2n1) �12 + t2n2�22

t1n1�11 + (t1n2 + t2n1)�12 + t2n2�22 n21�11 + 2n1n2�12 + n2

2�22

](A.7)

ence, the tangential shear stress on the surface of the sphere isiven by′12 = t1n1�11 + (t1n2 + t2n1)�12 + t2n2�22 (A.8)

hile the normal stress is given by′22 = n2

1�11 + 2n1n2�12 + n22�22 (A.9)

The tangential shear stress is then used to evaluate thelip-velocity along the wall of the microsphere. Specifically, theangential slip-velocity ut is found using Eq. (9) of Section 3.1:

t |wall ≈(

2 − �

�

)Kn

d

2��t |wall =

(2 − �

�

)Kn

d

2�� ′

12 (A.10)

Substituting for the tangential shear stress from Eq. (8) yields

t |wall ≈(

2 − �

�

)Kn

d

2�(t1n1�11 + (t1n2 + t2n1) �12 + t2n2�22)

(A.11)

[

[

cochem. Eng. Aspects 345 (2009) 112–120 119

References

[1] S.A. Schaaf, P.L. Chambre, Flow of Rarefied Gases, Princeton University Press,1961.

[2] M. Gad-el-Hak, The fluid mechanics of microdevices—the Freeman Scholar Lec-ture, J. Fluids Eng. 121 (1999) 5–33.

[3] J.H. Kim, G.W. Mulholland, S.R. Kukuck, D.Y.H. Pui, Slip correction measure-ments of certified PSL nanoparticles using a nanometer differential mobilityanalyzer (Nano-DMA) for Knudsen number from 0.5 to 8, J. Res. Natl. Inst. Stand.Technol. 110 (2005) 31–54.

[4] E. Cunningham, On the velocity of steady fall of spherical particles through fluidmedium, Proc. R. Soc. Lond. A 83 (1910) 357–365.

[5] R.A. Millikan, The isolation of an ion, a precision measurement of its charge,and the correction of Stokes’s law, Science 32 (1910) 436–448.

[6] M. Knudsen, S. Weber, Ann. D. Phys. 36 (1911) 981.[7] W.C. Hinds, Aerosol Technology, Properties Behavior, and Measurement of Air-

borne Particles, 2nd ed., John Wiley and Sons, New York, 1982.[8] R.A. Millikan, The general law of fall of a small spherical body through a gas, and

its bearing upon the nature of molecular reflection from surfaces, Phys. Rev. 22(1923) 1–23.

[9] R.A. Millikan, The Electron: Its Isolation and Measurement and the Determina-tion of Some of Its Properties, 8th ed., University of Chicago Press, 1963.

10] M.D. Allen, O.G. Raabe, Slip correction measurements of. spherical solid aerosolparticles in an improved Millikan apparatus, Aerosol Sci. Technol. 4 (1985) 269.

[11] D.J. Rader, Momentum slip correction factor for small particles in nine commongases, J. Aerosol Sci. 21 (1990) 161–168.

12] D.K. Hutchins, M.H. Harper, R.L. Felder, Aerosol Sci. Tech. 22 (1995) 202.13] R.W. Barber, D.R. Emerson, Analytic Solution of Low Reynolds Number Slip Flow

Past a Sphere, Centre for Microfluidics, Department of Computational Scienceand Engineering, CLRC Daresbury Laboratory, Daresbury, Warrington, 2001,WA4 4AD.

14] R.W. Barber, D.R. Emerson, Numerical simulation of low Reynolds number slipflow past a confined microsphere, in: Proceedings of the 23rd InternationalSymposium on Rarefied Gas Dynamics, Whistler, Canada, 2002, pp. 20–25.

15] H-C.F. Liu, A. Beskok, N. Gatsonis, G.E. Karniadakis, Flow past a microsphere ina pipe: effects of rarefaction, DSC-Vol. 66, Micro-Electro-Mechanical Systems(MEMS), ASME (1998) 445–452.

16] J.K. Fremerey, Spinning rotor vacuum gauges, Vacuum 32 (1982) 685–690.17] G. Reich, Spinning rotor viscosity gauge: a transfer standard for the laboratory

or an accurate gauge for vacuum process control, J. Vac. Sci. Technol. 20 (1982)1148–1152.

18] A. Li, G. Ahmadi, R.G. Bayer, M.A. Gaynes, Aerosol particle deposition in anobstructed turbulent duct flow, J. Aerosol Sci. 25 (1994) 91–112.

19] M. Soltani, G. Ahmadi, H. Ounis, B. McLaughlin, Direct simulation of chargedparticle deposition in a turbulent flow, Int. J. Multiphase flow 24 (1998) 77–92.

20] S. Dhaniyala, P.O. Wennberg, R.C. Flagan, D.W. Fahey, M.J. Northway, R.S. Gao,T.P. Bui, Stratospheric aerosol sampling: effect of a blunt-body housing on inletsampling characteristics, J. Aerosol Sci. Technol. 38 (2004) 1080–1090.

21] T.M. Peters, D. Leith, Particle deposition in industrial duct bends, Ann. Occup.Hyg. (2004) 1–8.

22] Z. Zhang, C. Kleinstreuer, J.F. Donohue, C.S. Kim, Comparison of micro- and nano-size particle depositions in a human upper airway model, J. Aerosol Sci. 36(2005) 211–233.

23] X. Wang, A. Gidwani, S.L. Girshick, P.H. McMurry, Aerodynamic focusing ofnanoparticles: II. Numerical simulation of particle motion through aerody-namic lenses, J. Aerosol Sci. Technol. 39 (2005) 624–636.

24] F.M. White, Viscous Fluid Flow, Mc-Graw Hill, New York, 2006.25] E.R.G. Eckert, R.M. Drake Jr., Analysis of Heat & Mass Transfer, McGraw-Hill Inc.,

New York, 1972, pp. 467–486.26] A. Kawahara, P.M.-Y. Chung, M. Kawaji, Investigation of two-phase flow pattern,

void fraction and pressure drop in a microchannel, Int. J. Multiphase Flow 28(2002) 1411–1435.

27] A.J. Christlieb, W. Nicholas, G. Hitchon, I.D. Boyd, Q. Sun, Kinetic description offlow past a micro-plate, J. Comput. Phys. 195 (2004) 508–527.

28] S.F. Gimelshein, G.N. Markelov, T.C. Lilly, N.P. Selden, A.D. Ketsdever, Experimen-tal and numerical modeling of rarefied gas flows through orifices and shorttubes, in: Proceedings of the 24th International Symposium on Rarefied GasDynamics, 2005, pp. 437–443.

29] L. Shui, Eijkel, C.T. Jan, A. van den Berg, Multiphase flow in micro- and nanochan-nels, Sens. Actuator 121 (2007) 263–276.

30] D.A. Jones, D.B. Clarke, An Evaluation of the FIDAP Computational Fluid Dynam-ics Code for the Calculation of Hydrodynamic Forces on Underwater Platforms,Maritime Platforms Division, Platforms Sciences Laboratory, 2003, DSTO-TR-1494.

31] S. Lee, A numerical study of the unsteady wake behind a sphere in a uniformflow at moderate Reynolds numbers, J. Comput. Fluids 29 (2000) 639–667.

32] B.P. Leonard, Adjusted quadratic upstream algorithms for transient incompress-ible convection. A collection of technical papers, in: AIAA Computational FluidDynamics Conference, 1979, AIAA Paper 79-1469.

33] R.W. Davis, J. Noye, C. Fletcher (Eds.), Finite Difference Methods for Fluid Flow,

Computational Techniques and Applications, Elsevier, 1984, pp. 51–69.

34] C.J. Freitas, R.L. Street, A.N. Findikakis, J.R. Koseff, Numerical simulation of three-dimensional flow in a cavity, Int. J. Numer. Methods Fluids 5 (1985) 561–575.

35] W. Ji, P.K. Wang, Numerical simulation of three-dimensional unsteady viscousflow past fixed hexagonal ice crystals in the air—preliminary results, Atmos.Res. 25 (1990) 539–557.

1 : Physi

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[Ph.D. Thesis, University of Manchester Institute of Science and Technology,1992.

20 A. Moshfegh et al. / Colloids and Surfaces A

36] B.P. Leonard, S. Mokhtari, ULTRA-SHARP nonoscillatory convection schemes forhigh-speed steady multidimensional flow, NASA Lewis Research Center, 1990,NASA TM 1-2568 (ICOMP-90-12).

37] W. Ji, P.K. Wang, Numerical simulation of three-dimensional unsteady viscousflow past finite cylinders in an unbounded fluid at low intermediate Reynoldsnumbers, Theor. Comput. Fluid Dyn. 3 (1991) 43–59.

38] S. Giors, F. Subba, R. Zanino, Navier–Stokes modeling of a Gaede pump stage inthe viscous and transitional flow regimes using slip-flow boundary conditions,J. Vac. Sci. Technol. A 23 (2) (2005) 336–346.

39] V. Jain, C.X. Lin, Numerical modeling of three-dimensional compressible gasflow in microchannels, J. Micromech. Microeng. 16 (2006) 292–302.

40] E.O.B. Ogedengbe, G.F. Naterer, M.A. Rosen, Slip-flow irreversibility of dissi-pative kinetic and internal energy exchange in microchannels, J. Micromech.Microeng. 16 (2006) 2167–2176.

41] L. Zhuo, H. Ya-Ling, T. Gui-Hua, T. Wen-Quan, Experimental and numerical stud-ies of liquid flow and heat transfer in microtubes, Int. J. Heat Mass Transf. 50(2007) 3447–3460.

42] R.W. Barber, D.R. Emerson, Numerical Simulation of Low Reynolds NumberSlip Flow Past a Confined Microsphere, Centre for Microfluidics, Departmentof Computational Science and Engineering, CLRC Daresbury Laboratory, Dares-bury, Warrington, 2001, Technical Report DL-TR-01-001.

43] J.A. Morínigo, J.H. Quesada, F.C. Requena, Slip-model performance for underex-panded micro-scale rocket nozzle flows, in: Proceedings of the 8th InternationalSymposium on Experimental and Computational Aerothermodynamics ofInternal Flows Lyon, 2007, Paper reference: ISAIF8-0026.

44] L.B. Thomas, R.G. Lord, Comparative measurements of tangential momentum

and thermal accommodations on polished and on roughened steel spheres, in:K. Karamcheti (Ed.), Rarefied Gas Dynamics, 8th ed., Academic Press, New York,1974, pp. 405–412.

45] E.B. Arkilic, M.A. Schmidt, K.S. Breuer, TMAC measurement in silicon micro-machined channels, in: Rarefied Gas Dynamics 20, Beijing University Press,1997.

[

[

cochem. Eng. Aspects 345 (2009) 112–120

46] H.P. Kavehpour, M. Faghri, Y. Asako, Effects of compressibility and rarefactionon gaseous flows in microchannels, Numer. Heat Transf. A: Appl. 32 (1997)677–696.

47] G. Zuppardi, D. Paterna, A. Rega, Quantifying the effects of rarefaction in highvelocity, slip-flow regime (Published Conference Proceedings style), in: Rar-efied Gas Dynamics: 25-th International Symposium, Russia, Novosibirsk, 2007.

48] R.W. Barber, D.R. Emerson, A Numerical Study of Low Reynolds Number SlipFlow in the Hydrodynamic Development Region of Circular and Parallel PlateDucts, Centre for Microfluidics, Department of Computational Science and Engi-neering, CLRC Daresbury Laboratory, Daresbury, Warrington, 2002, WA4 4AD.

49] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences,McGraw-Hill, New York, 1969.

50] W.H. Press, H. William, et al., Numerical Recipes in C; The Art of ScientificComputing, Cambridge University Press, Cambridge, UK, 1988.

51] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes inC++; The Art of Scientific Computing, Cambridge University Press, Cambridge,UK, 2002, p. 691.

55] Fluent, Inc., Fluent 6.012 User’s Guide, Fluent, Inc., Centerra Resource Park, 10Cavendish Court, Lebanon, NH, 2001.

58] C.T. Crowe (Ed.), Multiphase Flow Handbook, 1st ed., Taylor and Francis, Florida,2006.

60] UDF (User Defined Functions) Manual, FLUENT 6.0.12, Fluent Inc., November2001.

61] F.-S. Lien, Computational modelling of 3D flow in complex ducts and passages,

64] M.D. Allen, O.G. Raabe, Re-evaluation of Millikan’s oil drop data for the motionof small particles in air, J. Aerosol Sci. 13 (1982) 537.

65] C.N. Davies, Definite equations for the fluid resistance of spheres, Proc. Phys.Soc. 57 (1945) 18.