Aerosol Science 39 (2008) 957 – 973 www.elsevier.com/locate/jaerosci
A new model for the simulation of particle resuspension by turbulent flows based on a stochastic description of wall roughness and adhesion forces Mathieu Guingoa, b,∗ , Jean-Pierre Minierb a LEMTA-UMR 7563 CNRS, ESSTIN, Henri Poincaré University-Nancy I, 2, rue Jean Lamour, 54500 Vandoeuvre-lès-Nancy, France b EDF, Research and Development Division, Department of MFEE, 6, Quai Watier, Chatou 78401, France
Received 7 March 2008; received in revised form 2 June 2008; accepted 26 June 2008
Abstract We propose in this paper a new model aiming at simulating particle reentrainment in turbulent flows using stochastic Lagrangian methods. The resuspension model presented here emphasizes the role played by surface roughness in the reentrainment process, both in the stochastic calculation of adhesion forces based on a random model of large-scale and fine-scale wall asperities and in a newly proposed kinetic scenario of resuspension. The whole model has been implemented in a dedicated code and statistics of interest are obtained through Monte Carlo simulations. A step-by-step validation process is carried out by first assessing the adhesion-force sub-model, before analyzing the ability of the model to predict particle onset along the wall as measured in recent experimental studies. The complete particle resuspension model is then validated by comparing numerical outcomes to experimental data, where it is seen that the model is able to capture the various phenomena quite well. The present work follows a precedent study devoted to the modeling of particle deposition [Guingo, M., & Minier, J.-P. (2007). A stochastic model of coherent structures in boundary layers for the simulation of particle deposition in turbulent flows. In: Proceedings of the 6th international conference on multiphase flow. Leipzig, Germany; Guingo, M., & Minier, J.-P. (2008). A stochastic model of coherent structures for particle deposition in turbulent flows. Physics of Fluids 20, 053303]. 䉷 2008 Elsevier Ltd. All rights reserved. Keywords: Particle resuspension; Reentrainment; Surface roughness; Stochastic model
1. Introduction Particle resuspension (also called particle reentrainment) refers to the process that leads to the separation between initially deposited particles and a wall. It occurs in many industrial processes involving dispersed two-phase turbulent flows and its prediction is of critical interest, for instance in the case of atmospheric pollution or of fouling of heat transfer surfaces in nuclear power plants. However, it is a very difficult issue to address because it covers fundamentally a wide range of complex and random sub-phenomena which, themselves, depend on several parameters. These subphenomena include near-wall turbulence, physico-chemical forces between particles and walls (electrostatic, van der ∗ Corresponding author at: LEMTA-UMR 7563 CNRS, ESSTIN, Henri Poincaré University-Nancy I, 2, rue Jean Lamour, 54500 Vandoeuvre-lès-
Nancy, France. Tel.: +33 1 30 87 83 72; fax: +33 1 30 87 79 16. E-mail addresses:
[email protected],
[email protected] (M. Guingo),
[email protected] (J.-P. Minier). 0021-8502/$ - see front matter 䉷 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2008.06.007
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Waals forces, chemical bonds) as well as particle and wall material characteristics (roughness, elasticity, deformation, particle size, etc.). Over the years, several studies have been devoted to the identification of mechanisms responsible for particle resuspension and to the construction of theoretical models based on these mechanisms. These models have been divided in two categories in the comprehensive review of Ziskind, Fichman, and Gutfinger (1995). On the one hand, a class of models estimates particle resuspension rate by measuring the balance between hydrodynamic efforts and particle–wall adhesion bonds (Cleaver & Yates, 1973; Reeks & Hall, 2001) (the so-called “quasi-static” or “force-and moment- balance models”). On the other hand, a second class of models is based on finding conditions for a resonance in the particle–wall system that could lead to the breaking of adhesion bonds (Reeks, Reed, & Hall, 1988; Ziskind, Fichman, & Gutfinger, 2000), and is sometimes called “energetic” or “energy-balance models”. Besides, near-wall coherent structures such as “bursts” have also been proposed to explain resuspension process. According to some research teams (Braaten, Paw, & Shaw, 1990), these violent and periodic ejections of fluid from the wall toward the core flow could directly remove particles from the wall surface. Experimental programs have also been carried out in the past decades, mainly to measure the flow velocity necessary to bring about particle resuspension with respect to several parameters (particle diameter, material properties, etc.). Among recent works, we can cite the STORM program (Hontañón, de los Reyes, & Capitão, 2000), the experimental study of Reeks and Hall (2001) and the experiments of Ibrahim, Dunn, and Brach (2003) where special care has been taken to control critical parameters. Indeed, in particle resuspension studies, the main issue, when trying to obtain reliable experimental results, seems to be the number of factors to control at the same time so as to ensure an acceptable repeatability between trials. In this paper, we propose a new stochastic Lagrangian model to simulate particle reentrainment based on a scenario that stresses the role of surface roughness, which aims at being implemented in a full Lagrangian approach to model particulate flows (Guingo & Minier, 2007, 2008; Minier & Peirano, 2001). It may be worth underlining that the objective is not to come up with a very detailed model limited to specific surfaces (whose surface roughness is known beforehand to be in a given range) but rather to develop a general model that can be applied for various surfaces. The paper is organised as follows: first, the resuspension scenario and the modeling issues are exposed. Then, the stochastic description of wall roughness is presented in Section 4 as well as the resulting model retained for adhesion forces. The hydrodynamic forces responsible for possible particle motion along the wall are developed in Section 5 while the complete resuspension model is put together in Section 6. The model has then been numerically implemented and a progressive validation methodology has been followed, starting from the most basic component (the adhesion submodel) up to the whole resuspension model, as presented in Section 7.4. As a first step, calculations of adhesion forces are compared to available experimental data. Then, we assess numerical results against the recent experimental study of Ibrahim et al. (2003) that focuses on the flow velocity level that causes particle to start moving on the wall. Finally, particle resuspension itself is studied by comparing our numerical results to different reentrainment experiments. 2. Overview of the proposed resuspension scenario Due to the generally high intensity of adhesion forces and to the weakness of wall-normal hydrodynamic efforts in the immediate vicinity of the wall, direct pull-off (or direct removal) by wall-normal hydrodynamic forces is hardly probable (Ziskind et al., 1995). Consequently, other processes have to be at play and thus have been investigated. For modeling purposes, two main conclusions can be drawn from the existing literature: • Recent experimental evidence (Ibrahim et al., 2003) tends to show that rolling or sliding motion on the wall are generally the main mechanisms that cause particle resuspension. • Surface roughness plays an important role in the reentrainment process, by reducing the adhesion forces (Greenwood & Williamson, 1966) and because of the particular geometry of particle/asperity contact (Ziskind, Fichman, & Gutfinger, 1997). Existing models such as the Rock’n Roll model of Reeks and Hall (2001) or the model of Ziskind et al. (1997) regard particles as being reentrained once there are set in motion by hydrodynamic efforts. Nevertheless, to our knowledge, particles which are set in motion may simply roll on the wall, thus avoiding an actual separation from the surface. Therefore, we propose here another scenario based on a two-step mechanism: first, due to longitudinal hydrodynamical
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Fig. 1. Proposed kinetic mechanism of particle reentrainment stressing the role of surface roughness.
forces, particles can start to move on the wall; then, depending on the following interactions with the asperities that make up wall roughness, actual separation from the wall may, or may not, occur. This representation therefore builds upon the mechanism underlying the Rock’n Roll model. However, it not only focuses on the first part of this process, but also tries to describe in more detail the influence of wall roughness. This complete reentrainment scenario is illustrated in Fig. 1 in three stages: • Stage 1: The particle starts to roll, since the moment of the hydrodynamic forces is greater than the moment created by the adhesion forces. • Stage 2: The particle moves on the wall. As adhesion forces fluctuate all along particle motion on the wall due to the stochastic nature of the particle–wall contact, it is possible that, at a later moment, they become sufficient to force the particle to halt. However, supposing here that it keeps rolling, the contacts with asperities are not considered as violent enough to trigger particle resuspension. • Stage 3: The particle may reach an interval between two asperities which is larger than the particle diameter. In such a case, the particle will hit the surface of the next asperity and resuspension occurs if its kinetic energy is sufficient, or it keeps rolling otherwise. More precisely, rocking on an asperity, a particle can leave the wall if its kinetic energy is greater than the van der Waals potential responsible for particle adhesion. We thus consider that the particle kinetic energy is fully converted from the streamwise to the wall-normal direction during the rocking stage.
3. Modeling issues Different modeling challenges arise at each stage of this scenario. At stage 1, we must properly evaluate the adhesion forces and the hydrodynamic forces applied to a deposited particle on a rough wall, as well as the corresponding moments exerted on the particle. Stage 2 requires the simulation of the rolling (or sliding) motion on a rough wall. Finally, at stage 3, we have to model the rocking of the particle on an asperity which could cause resuspension. It is worth noting that the model focuses only on the longitudinal direction in which, in our opinion, the mechanisms leading to resuspension are the most likely to occur. Furthermore, the model must remain sufficiently tractable to be implemented in a full stochastic Lagrangian approach to model polydispersed particulate flows (Minier & Peirano, 2001), and to enable computations on industrial
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configurations. The ambition is thus different from fine-grained existing approaches describing precisely particle motion on walls but for surfaces whose roughness does not exceed some nanometers (Ibrahim et al., 2003; Ibrahim, Dunn, & Brach, 2004). The limited available information, particularly for industrial surfaces, calls for a stochastic approach, in line with what is being done for particle deposition (Guingo & Minier, 2008), which aims at capturing mostly the main statistical features by making choices between microscopic fine details and macroscopic outcomes. 4. A stochastic model of adhesion forces In the present study, adhesion forces are assumed to be caused only by van der Waals forces between particles and surfaces. The case of atomically smooth surfaces and elastically deformable, smooth and spherical particles can be addressed by the JKR theory (Johnson, Kendall, & Roberts, 1971). However, the vast majority of industrial surfaces are rough and often present different roughness scales, which have long been known to strongly influence adhesion forces (Greenwood & Williamson, 1966). Therefore, different fine-grained adhesion models have been proposed to take roughness effects into account (Cheng, Dunn, & Brach, 2002; Eichenlaub, Gelb, & Beaudoin, 2004). However, since they involve relatively complex calculations, these approaches are, in our opinion, ill-adapted to our final goal, i.e. developing a resuspension model to simulate particle resuspension to be used in industrial computations. Moreover, we have to choose a representation of the wall roughness both compatible with the scenario described in the previous section and enabling fast calculations of van der Waals forces. Hence, the model we propose follows the simplified approach initiated by Rumpf (1990) and developed by Rabinovich, Adler, Ata, Singh, and Moudgil (2000) and Katainen, Paajanen, Ahtola, Pore, and Lahtinen (2006). It consists of considering wall roughness as a collection of hemispherical asperities. This geometry of surface roughness makes it possible to evaluate adhesion forces (and their moments) using analytical formulas believed to provide reasonable assessments of adhesion forces between particles and industrial surfaces. The general form of the adhesion force, retained in the present context, is therefore expressed by (Israelachvili, 1991) Rp R A12 (1) Fa = 6z 02 Rp + R where A12 is the effective Hamaker constant, Rp the particle radius, R the diameter of the hemisphere considered and z 0 = 0.3 nm the contact distance corresponding to an atomic diameter. However, in order to obtain a precise expression in our case, the stochastic description of the surface asperities must be detailed. 4.1. Chosen description of surface roughness Industrial surfaces often present roughness distributed over a wide range of scales, from nanometers up to several microns. The smallest scale of wall-roughness is significant for the calculation of adhesion forces: due the short range of van der Waals interactions, it ultimately determines their intensity. Besides, in the newly proposed resuspension scenario, the largest scale of wall-roughness plays also an important role, since rocking on an asperity is the event that can trigger particle resuspension. For these reasons, these two scales appear as the most significant for modeling purposes. Hence, we propose to model surface roughness as the superimposition of two roughness scales: one micrometric, referred to as the “large scale”, and the other nanometric, referred to as the “fine scale” and characterized by its mean asperity radius rfine . Moreover, Robbe-Valloire (2001) has especially stressed that surfaces obtained from different machining surfaces present asperities with a large variability of size, which could be modeled with a lognormal distribution with standard deviation of the order of the mean value. Therefore, to take this effect into account, the fine-scale asperity radius is taken as a realization of a lognormal random variable whose standard deviation is equal to the mean and denoted L(rfine ). The resuspension scenario deals with the longitudinal motion on the wall of a deposited particle: we are thus primarily interested in the longitudinal distribution of large-scale asperities. Considering that they are randomly distributed on the surface, it is reasonable to regard the longitudinal distance between two successive asperities as a random variable which follows a Poisson law, whose mean is denoted L large (it may be worth recalling that the statistics of random points uniformly distributed on an interval follow a Poisson distribution; Papoulis, 1991). Furthermore, the fine-scale roughness is characterized by the asperity radius rfine and the surface density denoted fine . These asperities are also
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Fig. 2. Scheme of the chosen simplified geometry of wall roughness, consisting in two scales of hemispherical asperities.
Fig. 3. Adhesion model, cases A and B: small and mid-size particle in contact with a surface whose roughness is represented by hemispherical asperities.
regarded as randomly distributed on the wall: noting Ap the contact area between a particle and the wall, the number of fine-scale asperities in contact with the particle is then a random variable following a Poisson law of mean fine × Ap . To sum up, our modeling of surface roughness consists in three successive steps: first, considering two roughness scales (the smallest and the largest); second, assuming the asperity size of the fine-scale to be lognormally distributed; thirdly, assuming the spatial distribution of the asperities of the two scales to be Poissonnian. This representation of surface roughness is presented in Fig. 2. 4.2. Calculation of adhesion forces and moments The present adhesion-force model depends on the ratio between the particle diameter and the peak-to-peak distance between asperities on the wall. With respect to this ratio, three cases are possible: • Case A: The particle is sufficiently small to be located between two fine-scale asperities: it “sees” a smooth surface. This case is represented in Fig. 3. • Case B: The particle is large enough to be in contact with several fine-scale asperities but sufficiently small to settle in a hollow space between two large-scale asperities. This case is also displayed in Fig. 3. • Case C: The particle is sufficiently large to be in contact with one or several large-scale asperities. This case is displayed in Fig. 4. 4.2.1. Case A: small particles Case A is the only case where the wall is regarded as smooth: consequently, we apply the JKR theory, where the adhesion force is given by the equation: Fa = 3Rp where stands for the surface energy.
(2)
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Fig. 4. Adhesion model, case C: large particle in contact with a wall whose roughness is represented by hemispherical asperities.
4.2.2. Case B: mid-size particles To evaluate the number of fine-scale asperities in contact with the particle in case B, we first have to estimate the area covered by the particle on the wall, i.e. the contact radius a0 . For this purpose, we use the contact radius under zero load given by the JKR theory, regarded as an acceptable approximation: 1/3 12Rp2 (3) a0 = E where E is the composite Young’s modulus defined by (Ibrahim et al., 2003): −1 1 − 22 4 1 − 21 E= + 3 E1 E2
(4)
where E 1 and E 2 are the values of Young’s modulus and 1 and 2 are the values of Poisson’s ratio for the two spheres considered. Once the contact area Ap = a02 is calculated, we compute the number of fine-scale asperities in contact with the particle. As stated above, this number is modeled by a random variable following a Poisson law of mean fine × Ap and is denoted P(fine × Ap ). Supposing Rp ?rfine , the final expression of the adhesion force for case B is then Fa = P(fine × Ap )
A12 × L(rfine ) 6z 02
(5)
4.2.3. Case C: large particles Case C involves an additional step, since we have in this case to determine first the number of large-scale asperities in contact with the particle. As these asperities are also randomly distributed, this number can be estimated by a random variable that follows a Poisson law of mean Dp /L large . Then we calculate the number of fine-scale asperities per large-scale asperity similarly to case B, but considering now the particle and the large-scale asperity as two spheres of different diameters in the computation of the contact radius a0 . The total adhesion force is then given by the formula P(Dp /L large )
Fa =
Fa,i
i=1 P(Dp /L large )
=
Pi (fine × Ap )
i=1
(6) A12 × Li (rfine ) 6z 02
(7)
4.2.4. Calculation of the adhesion moment We have now completely presented the method for the calculation of wall-normal adhesion forces. We are also interested in calculating the moment of the forces about point O (given the flow direction, this point is the farthest contact point in the downstream direction between the particle and the wall surface), around which the particle will rotate if the moment of the hydrodynamic forces exerted on the particle overcomes the moment of the adhesion forces. For cases A and B, the adhesion moment about O is then MO (Fa ) = Fa × a0
(8)
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For case C, it is given by P(Dp /L large )
MO (Fa ) =
Fa,i × di
(9)
i=1
where di are distances between point O and the other asperities. 4.3. Summary of the adhesion-force model To sum up, the calculation of the adhesion forces is based on the following procedure: first, the number P(Dp /L large ) of large asperities in contact with the particle is calculated. If this number is greater than zero, we are in case C: then, the number of fine-scale asperities par large-scale asperity in contact with the particle is evaluated, from which the adhesion force in this case is computed with Eq. (7). Otherwise, if P(Dp /L large ) = 0, we are either in case A or B. In these cases, we calculate P(fine × Ap ): if P(fine × Ap ) = 0, we are actually in case A and the JKR theory applies where the adhesion force is given by Eq. (2). Otherwise, we are in case B and the adhesion force is computed with Eq. (5). 5. Description of the hydrodynamics 5.1. Hydrodynamic efforts Even though lift forces have been taken into account by different teams (Ibrahim et al., 2003; Reeks & Hall, 2001) drag forces are believed to be predominant in the resuspension scenario presented in the first section. In this study, we consider particles sufficiently small to be completely embedded in the viscous sublayer when deposited on the wall. Hence, the drag force applied to the particles is taken as the Stokes drag. In the streamwise direction, it is given by FD,x = 6f Rp Us,x × f
(10)
where f the fluid density, the kinematic viscosity, Us,x the streamwise velocity of the flow seen by the particle and f = 1.7 is a correcting factor accounting for the presence of the wall (O’Neill, 1968). Due to its non-uniformity, the flow exerts a moment on the particle around O roughly equal to (O’Neill, 1968): MO (FD,x ) ≈ 1.4 × Rp × FD,x
(11)
Similar relations exist in the wall-normal direction for the drag force and its moment: FD,y = 6f Rp Us,y × g
(12)
where Us,y is the normal velocity of the flow seen by the particle and g = 3.39 a correcting factor due to the presence of the wall (Maude, 1963), while the moment is MO (FD,y ) = lp × FD,y
(13)
where lp represents the distance of point O to the vertical axis going through the particle center of mass. 5.2. Modeling of the flow seen In the current state of the model, we have so far implemented a simple representation of the velocity of the flow seen, that takes into account its stochastic nature and is based on Ornstein–Uhlenbeck processes: ⎧ U + (t) = U + + u + x x ⎪ ⎪ ⎪ + ⎨ 2u 2 + u + + du = − + dt + dWx+ (14) + T T ⎪ ⎪ + + 2 ⎪ 2v v ⎩ + dv = − + dt + + dW y+ T T+
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Fig. 5. Force balance for a large particle (case C) in contact with a rough wall in the proposed simplified description.
with Ux + = Rp+ , dWx and dW y standing for the increments of two independent Wiener processes. Considering the particle motionless on the wall, T corresponds to the Eulerian correlation timescale of the flow and can be taken as + approximately equal to T 5 in the viscous sublayer. To model the average streamwise and normal fluctuation intensities u 2 + and v 2 + , the following correlations are used (Matida, Nishino, & Torii, 2000):
0.4Rp+ u 2 + = (15) 1 + 0.00239(Rp+ )1.496
v 2 + =
0.0116(Rp+ )2 1 + 0.203Rp+ + 0.00140(Rp+ )2.421
(16)
The above stochastic model for the velocity components is actually a simple one, yet that already reproduces typical fluctuation intensities. In this first version, the two components are basically treated as independent and, thus, the normalized correlation or fluid shear is zero: u + v + = 0. Yet, a more precise estimation (Pope, 2000) yields that the fluid shear scales as u + v + ∼ (y + )3 and is thus, in the present case, considered as negligible. Due to the very nature of the processes involved (diffusion processes), the potential effects of near-wall coherent structures are not explicitly taken into account in the current state of the model. The reason of this choice is twofold: first, we wanted to keep a reasonably low degree of complexity while reproducing realistic fluctuations of the flow velocity. Second, as some experimental studies tend to show (Yung, Merry, & Bott, 1989), few deposited microparticles are directly removed by coherent, ‘bursts’ events. However, for the sake of completeness, we may propose here guidelines for a future modeling of coherent structures in the immediate vicinity of the wall: as the flow in the viscous sublayer may be considered as a succession of low-speed streaks and “high-speed streaks”, its longitudinal velocity may be modeled by using a Markovian jump process (in the same fashion as in Guingo & Minier, 2008) that takes its values in {Ulow (Rp+ ), Uhigh (Rp+ )}, where Ulow and Uhigh are typical values of the flow velocity within the streaks. 6. Modeling particle motion on the wall and resuspension The previous two sections have provided models for the efforts applied to a deposited particle in a turbulent flow and we can now consider particle motion along the wall. The efforts exerted on a large deposited particle are summarized in Fig. 5. In a word, following Ibrahim et al. (2003), a particle can be set in motion if: • A sufficient wall-normal removal force is applied: FD,y > Fa . • The tangential drag force is greater than the tangential component of adhesion force: the particle starts to slide: FD,x > s Fa .
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• The moment of the hydrodynamic forces at point O overcome the moment of adhesion forces: the particle starts to roll. Concerning sliding motion, particle acceleration is determined by the equation dUp,x (17) = FD,x − d ∗ Fa dt where m p is the particle mass and d the dynamic fiction coefficient. In the case of rolling motion, the time-evolution equation of the angular velocity around point O is given by mp
d (18) ≈ MO (FD,x ) + MO (FD,y ) − MO (Fa ) dt with the angular velocity around point O and I the moment of inertia of the particle around point O. For a solid sphere, it is given by the relation: I
I = 75 m p Rp2
(19)
The streamwise particle velocity is then estimated by Up,x ≈ Rp ∗
(20)
Once the particle is either sliding or rolling on the wall, the number of asperities in contact with the particle change. With regard to particle size and to the random distribution of large-scale asperities on the wall, the particle may reach a hollow space between two asperities. Then, following the scenario presented at the beginning of the paper, the particle will hit the surface of the next asperity. Using a straightforward kinetic energy balance, we consider that the particle leaves the wall provided that, at this moment, we have 1 2
2 m p Up,x > VA
(21)
where VA = Fa,k ∗ z 0 is the van der Waals interaction potential between the particle and the rocked asperity k. 7. Numerical results and discussion In this section, we present numerical results obtained from the practical implementation of the model. Details of the numerical procedure are first given and computations are then compared to experiments in a three-step approach which corresponds to different stages of the model: adhesion force; particle dynamics along the wall; and, finally, the complete resuspension model. However, as particle resuspension is a complex phenomenon involving numerous parameters, existing experiments often lack data related to one or several parameters that would be necessary for a rigorous and complete validation of every aspect of the model. Consequently, in the following, we sometimes had to estimate parameters, such as surface roughness or Hamaker constants. Although we are therefore not in the ideal configuration to evaluate the model in a comprehensive manner, this approach is still a validation process that can, given the whole range of experimental situations considered, bring out the actual capacity of the model. This situation is also not far from industrial applications, where these parameters are difficult to know precisely and have thus to be estimated. 7.1. Numerical procedure For every calculation, thousands of numerical particles with different diameters are deposited on a wall. For each particle, we first calculate the number N (0) of large-scale asperities in contact with the particle using a generator of Poisson random variables. The probability for N (0) to be equal to n is given by the formula (Dp /L large )n (22) n! Supposing N (0) > 0 (case C), the contact C j (with j ∈ [1, N (0)]) is characterized by two parameters: its position under the particle (which can thus vary between 0 and Dp ) and the number of fine-scale asperities in contact with the P(N (0) = n) = e−Dp /L large
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particle. For each particle, the initial position of each large-scale asperity relatively to the particle is calculated by a uniform random variable: X c j (0) = U([0, Dp ])
(23)
By convention, the most downstream asperity is taken as the one whose position is minimal. Then, the number N c j of fine-scale asperities per large-scale asperity is computed for each large-scale asperity. This number is also given by a Poisson variable generator parameterized by fine ∗ Ap , as detailed in the previous part of the article devoted to the adhesion submodel. In a word, in the model, a deposited particle is represented by its streamwise coordinate X p , its streamwise velocity Up , the two components of the velocity of the flow seen Us,x and Us,y , and by the list of the contacts with large-scale asperities. At time t ⎛ ⎞ X p (t) ⎜ Up (t) ⎟ ⎜ ⎟ ⎜ Us,x (t) ⎟ ⎜ ⎟ X c j (t) (t) U ⎜ ⎟ s,y A deposited particle ≡ ⎜ ⎟ with C j (t) ≡ Ncj ⎜ C1 (t) ⎟ ⎜ . ⎟ ⎝ .. ⎠ C N (t) If the particle moves on the wall, at the next timestep, the list of contacts between the particle will possibly evolve. Indeed, if X c j (t + t) > Dp the asperity is not in contact any more with the particle. The first substep is then to identify these asperities and to eliminate them from the list: afterwards, we have N1 < N (t) remaining asperities. The second substep is to calculate if the particle has encountered other large-scale asperities during its motion from time t to time t + t: for this purpose, we use a Poisson random generator, and the probability for the number of additional asperities N2 to be equal to n is given by • If X p (t + t) − X p (t) < Dp : P(N2 = n) = exp(− X p (t + t) − X p (t) /L large )
( X p (t + t) − X p (t) /L large )n n!
(24)
• If X p (t + t) − X p (t) > Dp , there is therefore a complete renewal of the list of contacts: P(N2 = n) = exp(−Dp /L large )
(Dp /L large )n n!
(25)
The position of the new asperities is calculated following the procedure previously described, as well as the number of fine-scale asperities per new contact. The total number of large-scale asperities in contact with the particle at time t + t is therefore N (t + t) = N1 + N2 , and we can then calculate the adhesion forces and moments exerted on the particle. As stated in the first section presenting the resuspension scenario, as long as the number N (t) of contacts is greater than zero, the particle is considered as rolling or sliding on the wall in a way which is regular enough to neglect the shocks with the encountered asperities. However, if the number N (t) falls to zero, the next contact with an asperity will be regarded as a shock, and the particle will leave the wall if its kinetic energy if sufficient. Concerning particle motion on wall with only nanometric roughness (our case B), tracking exactly the position of all fine-scale asperities in contact with the particle is too computationally expensive, and we use a coarser description: the number of fine-scale asperities in contact with the particle is frozen during the time taken by the particle to cover a distance equal to its diameter, and then calculated again. This method makes it possible to carry calculations using reasonable timesteps. 7.2. Numerical calculation of adhesion forces Experimental validation of the adhesion submodel is a difficult issue for several reasons. First, technological advances (with the development of Atomic Force Microscopy) have only recently enabled precise and repeatable measurements of adhesion forces between micrometric particles and rough surfaces. This difficulty is further compounded by the
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1
0.8
CDF
0.6
0.4
0.2
0 0
50
100 adhesion force (nN)
150
200
Fig. 6. Adhesion forces: comparison between numerical results (solid lines) and experimental data (symbols) of Zhou et al. (2003) for different substrates: () smooth substrate; (䊏) polished aluminum; (•) PVD-coated aluminum.
question of the statistical treatment of the adhesion measurement which is the subject of current research (Götzinger & Peukert, 2003). Moreover, available experimental data concern surfaces presenting nanometric roughness, thus much smoother than the majority of industrial surfaces. Interesting experimental results have been recently presented by Zhou, Götzinger, and Peukert (2003), who considered smooth, spherical particles having a 10-m diameter, different substrates of different roughness (although all of the order of the nanometer), and measured the cumulative density function (CDF) of the adhesion force. The two rough substrates used in the experiments presented different peak-to-peak distances. For the first one (named “PVD-coated” aluminum), asperities were relatively closely packed. For the second one (polished aluminum), the average interval between asperities was larger and this, according to the authors, enabled particles to be located between asperities, seeing therefore an almost smooth surface. Consequently, we can test case A and B in order to check not only the average intensity of the computed adhesion-force, but also to see if the model can retrieve the influence of roughness profiles on the adhesion-force distribution, which would constitute a justification of the present approach. With a smooth substrate (corresponding to our case A), the experimental results indicate a small dispersion of the adhesion force around 160 nN. Using the JKR theory, this corresponds to a surface energy roughly equal to 3.5 mJ m−2 , which is a realistic value for a surface contaminated with adsorbed water layers (Kendall, 2001). This value of the surface energy allows the computation of the contact area between particles and the surface using Eq. (3) and, given the density fine , the computation of adhesion forces for the rough substrates. The results of the adhesion submodel are displayed in Fig. 6 and compared to experimental values. Concerning the rough substrates, the influence of the average distance between asperities can be observed: with polished aluminum, the model can retrieve the clear-cut division between particles that see a smooth surface (our case A) and particles in contact with several fine-scale asperities (our case B), for which the adhesion forces are weaker. Particles on PVD-coated alumina present adhesion forces which are more randomly distributed, nearly over a continuous range and where the division observed for polished alumina is less present in this case. The tendency observed from the experiments is thus reproduced by the adhesion submodel, even though, quantitatively, the calculated adhesion forces are slightly weaker than in the experiments. Hence, the stochastic representation of the surface roughness makes it possible to retrieve the main characteristics of the CDF of adhesion forces of the experiments of Zhou et al. (2003). 7.3. Start of the rolling motion As the second step of the numerical validation procedure, we present in this section numerical results concerning the flow velocity needed to set particles in motion. Results from the model are assessed against the recent experiments of Ibrahim et al. (2003), who performed experiments to measure the flow velocity necessary to make deposited particles
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Table 1 Particle parameters in the experiments of Ibrahim et al. (2003)
Diameter (m) Density (kg m−3 ) Young’s modulus (GPa) Poisson’s ratio Surface energy on glass (J m−2 )
GL72
GL32
72 2420 80.1 0.27 0.2
32 2420 80.1 0.27 0.2
1
Detached fraction
0.8
0.6
0.4
0.2
0 0
2
4
6
8
10 12 14 Flow velocity
16
18
20
22
24
Fig. 7. Comparison between model results (solid lines) and experimental date of Ibrahim et al. (2003): (•) GL72 particles; (䊏) GL32 particles.
start rolling on a surface presenting only nanometric roughness. It is worth noting that as soon as the particles are set in motion, they are labeled as detached and are not tracked any more: thus, the results of this section do not involve the full resuspension scenario. This experimental study makes it possible to assess simultaneously two aspects the model: case B of the adhesion submodel (micrometric particles in contact with nanometric roughness) as well as the calculation of the adhesive and hydrodynamic moments in this case. In these experiments, great care was taken to obtain repeatable results, which indicate that the critical parameters were properly controlled. The deposited particles (of different diameters) are spherical, atomically smooth, and made of glass. Their characteristics are defined in Table 1. The glass substrate presents roughness of approximately 2 nm. It is properly prepared to prevent water layers adsorption, and for this reason we use a Hamaker constant around 1.5 × 10−19 J. As to our knowledge, no information is provided concerning the distance between asperities that could be used in the model, and we chose the value fine = 5 × 1015 , which corresponds to an average distance between two asperities of the order of 15 nm. The results obtained from both classes of particles are represented in Fig. 7. It can be seen that the model is able to retrieve the main features of the experimental data: the velocity U1/2 necessary to set half the particles into motion is correctly reproduced, as well as the dispersion around U1/2 . Another interesting phenomenon that is reproduced by the model is the increasing dispersion around U1/2 while reducing particle diameter. 7.4. Simulation of particle resuspension The previous sections were the first stages of the validation procedure of the model, and made it possible to test rather static aspects of the model, namely the adhesion submodel and the calculation of the moments exerted on a deposited particle. The final step consists of calculating particle resuspension using the whole model, including the modeling of particle motion on the wall. To assess the numerical results, we compare them to the experiments of Reeks and Hall (2001), and we study afterwards the influence of several significant numerical and physical parameters on these results.
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Table 2 Particle parameters in the experiments of Reeks and Hall (2001)
Diameter (m) Density (kg m−3 ) Young’s modulus (GPa) Poisson’s ratio Surface energy on stainless steel (J m−2 )
1
Alumina
10 2300 20 0.3 0.075
10 and 20 1600 350 0.3 0.28
1 Calculations Experimental data
Experimental data Calculations
0.8 Remaining fraction
0.8 Remaining fraction
Graphite
0.6 0.4 0.2
0.6 0.4 0.2
0
0 0.1
1 Friction velocity
10
0.1
1 Friction velocity
10
1 Experimental data Calculations
Remaining fraction
0.8
0.6 0.4 0.2 0 0.1
1 Friction velocity
10
Fig. 8. Particle resuspension: comparison between calculations from the model and the experiment of Reeks and Hall. (a) 10 m-diameter alumina; (b) 20 m-diameter alumina and (c) 13 m-diameter graphite.
7.4.1. Comparison with the experiments of Reeks and Hall (2001) Reeks and Hall (2001) studied particle resuspension experimentally using the following protocol: first, they allowed particles of different sizes and different materials to deposit on a substrate, then, they counted particles remaining on the substrate after 1 s in a flow, whose velocity can be adjusted. The statistic of interest is thus here the particle remaining fraction against the flow velocity. The particles are made of graphite and alumina, with the alumina particles almost monodispersed in size, whereas the graphite particles present larger diameter dispersion. The detailed particle properties are reported in Table 2. The substrate is made of stainless steel, and polished using diamond paste. No
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1 Δt = 1.D-3 Δt = 1.D-4 Δt = 1.D-5
Remaining fraction
0.8
0.6
0.4
0.2
0 0.1
1 Friction velocity
10
Fig. 9. Check of the timestep independence in the calculations of particle resuspension.
1
Remaining fraction
0.8
0.6
0.4
0.2
0 0.1
1 Friction velocity
10
Fig. 10. Sensitivity of the results to the form of the model: (•) basic version of the model; ( ) basic model + stochastic velocity of the flow; () basic model + lognormal adhesion forces; (䊏) full model.
quantitative data on surface roughness is reported, so we assumed in our calculations a large-scale roughness of 2 m, and a fine-scale roughness of 2 nm. The average distance between two large-scale asperities was chosen to be 30 m. It is worth emphasising that these parameters are related to the nature of the wall surface. Thus, once they are evaluated, they are used for the three sets of particles, regardless of their size or nature. As such, this suggests that this test case can really be regarded as a validation case. In Fig. 8 the remaining particle fraction on the substrate is plotted against the friction velocity of the flow, for both experiments and calculations with the new model for the three classes of particles. The timestep was chosen equal to 10−4 s. The diameter of the alumina particles (Fig. 8(a) and (b)) was chosen to be equal to 10 and 20 m for all the particles with respect the case studied. It can be seen that the numerical results are in good agreement with the experiments for the two sizes of particles: the calculations show that the remaining fraction after one second falls abruptly from 1 to 0.1 for a friction velocity level approximately equal to 1 m s−1 , with a velocity slightly smaller for 20 m alumina particle, and with a slope that compares quite well with the experiments. Results with graphite particles are represented on Fig. 8(c). For this calculation, we chose a Hamaker constant greater than the one we used for alumina particle (A12 = 1.5 × 10−19 J), and the particle diameter was taken as a random
M. Guingo, J.-P. Minier / Aerosol Science 39 (2008) 957 – 973
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1
Remaining fraction
0.8
0.6
0.4 1 μm 10 μm 70 μm
0.2
0 0.1
1 Friction velocity
10
Fig. 11. Results from the model of particle resuspension concerning three particle classes of different diameters.
1
Remaining fraction
0.8
0.6
0.4 Llarge = 30 μm Llarge = 3 μm Llarge = 100 μm
0.2
0 0.1
1 Friction velocity
10
Fig. 12. Study of the evolution of the results with the respect to variation in the mean peak-to-peak distance.
variable following a lognormal distribution to reproduce the dispersion in size of the particle. The results compare well with the experiments: especially, the slope in less significant than for the alumina particle, thus suggesting the dispersion in size is well taken into account. 7.4.2. Sensitivity analysis of the model to different parameters In this part, a sensitivity study of the results with respect to the choice of different parameters is proposed. First, the numerical methods used in the simulations are validated by checking the timestep independence of the calculations, which is a critical step especially for stochastic models. For this purpose, we performed the same calculations with three different timesteps ranging from 10−3 to 10−5 s, represented in Fig. 9. Timestep independence is seen to be reached for a timestep of 10−4 s. Moreover, the results obtained with a timestep of 10−3 s differ only slightly from the other two. Fig. 10 displays results obtained with simpler versions of the model used with 10 m alumina particles. The basic version makes use of the mean values of the flow seen and of the size of the fine-scale asperities. The results shows that on the average, these versions give a higher velocity necessary to trigger particle resuspension than the complete version that uses a stochastic velocity of the flow seen and a lognormal distribution of the adhesion forces. The effects of the stochastic representation of the flow seen and of the lognormal adhesion forces are approximately of the same
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order. Additional results for alumina particles with 70 and 1 m diameter are represented in Fig. 11 which show that, for this range of diameter, the larger the particle, the smaller the predicted resuspension velocity, which corresponds to the tendency observed in the experiments. Finally, we have carried out calculations with different average peak-to-peak distance L large between the large-scale asperities while keeping 10 m alumina particles, and the results are presented in Fig. 12. For L large = 3 m, particles resuspension occurs for much higher velocity: this can be explained by the fact that, in this case, particles have to cover a greater average distance to find a hollow space since the asperities are then closely packed. With L large = 100 m, particles have now to cover more distance to rock an asperity so as to trigger their resuspension. In our model, the resuspension rate is thus not monotonic with respect to L large , and seems to reach its maximum for L large of the order of the particle diameter. 8. Conclusion In this paper, we have presented a new Lagrangian approach to simulate particle reentrainment in a turbulent flow, which relies mainly on a stochastic and simplified description of wall-surface roughness. This representation of the wall allows calculations of adhesion forces and moments to be performed accurately following recently developed methods (Katainen et al., 2006). A new kinetic scenario of particle resuspension has also been proposed: it is based on the interactions between particles and surface asperities, and considers that particles are first set in motion due to hydrodynamic forces, then start rolling or sliding on the wall, and finally can encounter a large-scale asperity where the collision may trigger its separation from the surface. The model has been implemented in a dedicated numerical code and Monte Carlo simulations have been performed and compared to experimental data. Due to the complexity of the resuspension phenomenon, not all physical constants or parameters were always given or measured in these experiments. As a consequence, in the calculations, we had sometimes to estimate parameters which are inputs of our model. This concerns especially surface roughness and Hamaker constants. However, it is worth noting that these parameters are more than just free parameters which are only meaningful within a model: they do represent physical quantities (for the Hamaker constant) or material properties (for surface roughness data) and have therefore strong intrinsic physical meaning. Then, setting what we believed are physical values for the non-reported experimental parameters, numerical results obtained with the new model compare quite well with experiments, thus suggesting that the most important aspects of particle reentrainment are correctly captured. The test cases considered for the model validation were in air flow, in controlled conditions of humidity. Should water flow be studied, the adhesion model would have to be modified, since in this case other forces are applied to a particle (for instance, electrostatic forces). In high humidity conditions, capillary effects would also have to be taken into account. In our opinion, these modifications would only affect the adhesion submodel and the proposed scenario emphasizing the role of surface roughness is likely to remain valid. In a near future, the present model will be coupled with a previously developed Lagrangian model for particle deposition in turbulent flows, so as to simulate the resulting effect from the competition of the two processes (particle deposition toward the wall and particle resuspension from the wall) and retrieve particle behavior observed in experimental studies (Tanière, Oesterlé, & Monnier, 1997). These respective approaches constitute the first two steps of a program devoted to model and simulate the complete particle–wall interaction process in representative industrial situations, including also particle agglomeration and fouling. References Braaten, D., Paw, K., & Shaw, R. (1990). Particle resuspension in a turbulent boundary layer—observed and modeled. Journal of Aerosol Science, 21(5), 613–628. Cheng, W., Dunn, P., & Brach, R. (2002). Surface roughness effects on microparticle adhesion. The Journal of Adhesion, 78, 929–965. Cleaver, J., & Yates, B. (1973). Mechanism of detachment of colloidal particles from a flat substrate in a turbulent flow. Journal of Colloid and Interface Science, 44(3), 464–474. Eichenlaub, S., Gelb, A., & Beaudoin, S. (2004). Roughness models for particle adhesion. Journal of Colloid and Interface Science, 280, 289–298. Götzinger, M., & Peukert, W. (2003). Dispersive forces of particle–surface interaction: Direct AFM measurements and modelling. Power Technology, 130, 102–109. Greenwood, J., & Williamson, J. (1966). Proceedings of the Royal Society A, 295, 300–319.
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