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Numerical Study on the Deposition Rate of Hematite Particle on Polypropylene Walls: Role of Surface Roughness Christophe Henry,*,†,‡ Jean-Pierre Minier,*,† Gregory Lefevre,‡ and Olivier Hurisse† † ‡
Fluid Dynamics, Power Generation and Environment, EDF R&D, 6 quai Watier, Chatou 78401, France LECIME, CNRS-Chimie ParisTech, UMR 7575 11 rue Pierre et Marie Curie, Paris 75005, France
bS Supporting Information ABSTRACT: In this paper, we investigate the deposition of nanosized and microsized particles on rough surfaces under electrostatic repulsive conditions in an aqueous suspension. This issue arises in the general context of modeling particle deposition which, in the present work, is addressed as a two-step process: first particles are transported by the motions of the flow toward surfaces and, second, in the immediate vicinity of the walls, the forces between the incoming particles and the walls are determined using the classical DLVO theory. The interest of this approach is to take into account both hydrodynamical and physicochemical effects within a single model. Satisfactory results have been obtained in attractive conditions but some discrepancies have been revealed in the case of repulsive conditions, in line with other studies which have noted differences between predictions based on the DLVO theory and experimental measurements for similar repulsive conditions. Consequently, the aim of the present work is to focus on this particular range and, more specifically, to assess the influence of surface roughness on the DLVO potential energy. For this purpose, we introduce a new simplified model of surface roughness where spherical protruding asperities are placed randomly on a smooth plate. On the basis of this geometrical description, approximate DLVO expressions are used and numerical calculations are performed. We first highlight the existence of a critical asperity size which brings about the highest reduction of the DLVO interaction energy. Then, the influence of the surface covered by the asperities is investigated as well as retardation effects which can play a role in the reduction of the interaction energy. Finally, by considering the random distribution of the energy barrier of the DLVO potential due to the random geometrical configurations, the overall effect of surface roughness is demonstrated with one application of the complete deposition model in an industrial test case. These new numerical results show that nonzero deposition rates are now obtained even in repulsive conditions, which confirms that surface roughness is a relevant aspect to introduce in general approaches to deposition.
’ INTRODUCTION Fouling (i.e., the formation of deposits) of heat transfer surfaces is one of the most important issue in the design and operation of heat exchangers in many industries. For instance, in the automotive industry, the inefficiency of heat transfer resulting from fouling has a direct link to excess fuel consumption. Fouling of heat exchangers in a pressurised water reactor (PWR)1 reduces the heat transfer efficiency and, sometimes, coagulation can lead to a blockage of some channel cross sections. In PWR, fouling of heat exchangers in the secondary circuit results from the deposition of colloidal particles (mainly magnetite) produced by the corrosion of the pipes walls. With respect to these concerns, there is an ongoing need to improve our understanding of the basic mechanisms at play and to devise numerical models which can predict the possible onset of particle deposition for various physical (including fluid and chemical aspects) and geometrical situations. Deposition Process. Since hydrodynamic transport (interactions between particles and the fluid) and physicochemical interactions (between particles and surfaces) occur at two different r 2011 American Chemical Society
length scales, particle deposition can be considered as a two-step process:2 • a transport step, where particles are transported hydrodynamically to the surface; • an attachment step, in which the adhesion of particles is governed by interfacial forces. In the second case, interfacial interactions are generally well described by the DLVO (Derjaguin, Landau, Verwey, and Overbeek) theory,3 which defines the interaction between two surfaces as the sum of the Lifshitz-van der Waals (VDW) and the electrostatic double-layer (EDL) interactions. Since van der Waals interactions are always attractive (only a few cases exist where the VDW interaction is repulsive, but such cases are not considered in this study), when both surfaces are oppositely charged, the DLVO theory predicts an attraction between surfaces, leading to the so-called electrostatically attractive Received: November 10, 2010 Revised: February 17, 2011 Published: March 15, 2011 4603
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Figure 1. Interaction energy between a sphere and a plate for Rpart = 500 nm, ζpart = ζplate = 50 mV and κ1 = 3 nm: (---) VDW interaction energy; ( 3 3 ) EDL interaction energy; () DLVO interaction energy.
condition. Otherwise, for similarly charged surfaces, electrostatic repulsion occurs, leading to the so-called electrostatically repulsive condition. As an example, the interaction energy obtained using the classical DLVO theory under such repulsive conditions is displayed in Figure 1 (both surfaces have a potential equal to 50 mV) for a particle radius equal to 500 nm and a Debye length κ1 = 3 nm. It can be seen that, at small separations, the van der Waals attraction overwhelms the electrostatic repulsion. At larger separations, the van der Waals attraction is quickly balanced by the electrostatic repulsion which further dominates the interaction, leading to the so-called energy barrier (here close to 500 kT at 2 nm). At even larger separations (here larger than 9 nm), the DLVO interaction becomes attractive, and a secondary minimum exists at a separation distance equal to 15 nm. In the context of particle deposition, an incoming particle will be able to deposit on a surface under repulsive conditions if its kinetic energy is high enough to overcome the repulsive potential encountered (if the kinetic energy is not large enough, the particle may also be trapped in the secondary minimum). Why a Study on Surface Roughness?. In a previous paper from Guingo and Minier,4 a stochastic Lagrangian model of the hydrodynamic transport has been proposed, aiming mainly at reproducing the curve of the normalized deposition rate kþ p = mp/(C uτ) (where mp is the mass of deposited particles per unit area per unit time, C is the mass concentration of particles in the bulk, and uτ is the wall-friction velocity of the flow) with respect to the dimensionless relaxation time τsp = Dpart2uτ2Fp/(18νf2Ff) (with Dpart the particle diameter, Fp the particle density, Ff the fluid density, and νf the kinematic viscosity of the fluid). In this model, the interactions between particles and near-wall coherent structures have been taken into account. This model has been extended to simulate the deposition rate of colloidal particles in turbulent flows under variable pH conditions and, for this purpose, the hydrodynamic model has been coupled with a suitable model for particlewall interactions based on an energetic approach. In this energetic approach, the classical DLVO interaction energy is modeled as a step function with a height equal to the height of the energy barrier and particles are regarded as depositing if their incoming kinetic energy is larger than this energy barrier. This model has been shown to reproduce satisfactorily the deposition rate of particles in attractive conditions. However, important discrepancies have
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been revealed in repulsive conditions: no deposition is predicted numerically whereas a nonzero deposition rate is observed experimentally. Several previous studies have also reported similar significant discrepancies between experimental observations and predictions based on the classical DLVO theory under repulsive conditions.59 In the frame of this study, we concentrate on the deposition rate under repulsive conditions and try to investigate the origin of this discrepancy. Since the process of particle deposition is represented here as a two-step process, such a discrepancy can be thought to be due to various possibilities: • An underestimation of the particle kinetic energy (which affects the transport step) • An overestimation of the energy barrier which, among other possible effects, may stem from overestimation of electrostatic interactions underestimation of van der Waals interactions morphology of the surface surface charge heterogeneities deposition in the secondary minimum extended-DLVO theories, for instance taking into account acidbase interactions • A combination of the above-mentioned effects Yet, correct predictions have been obtained under attractive conditions where the deposition process is governed only by the hydrodynamic transport (the energy barrier is zero in that case). Under repulsive conditions, the deposition rate remains governed by the same hydrodynamic transport coupled now with the interfacial interaction and this indicates that discrepancies are more likely to be related to the specific details of the interfacial interaction itself. Moreover, the underestimation of the deposition rate cannot stem from the first two possibilities since they lead to only small changes in the energy barrier that governs the deposition mechanism. In the past few years, the role of surface heterogeneities in colloidal particle deposition has been investigated. On the one hand, surface roughness ranging from nanoscale to microscale was shown to have a pronounced impact on the deposition kinetics (either with experimental data10 or with numerical predictions1116). On the other hand, surface charge heterogeneities may provide attractive patches on surfaces even in repulsive conditions, enhancing the deposition rate.1723 In the present work, we are dealing with industrial surfaces that are known to be rough and, therefore, surface roughness appears as an important issue to consider. In the absence of specific chemical treatments leading to surface patches with different charges (a situation that is not considered here), it is not clear whether industrial surfaces can exhibit such patches of opposite charges and, due to the lack of experimental data on this issue, this effect will not be considered here. Moreover, deposition in the secondary minimum is not taken into account in this study, since particles in the secondary minimum are weakly bounded to the surface, allowing their reentrainment in the bulk fluid (the deposition in the secondary minimum is important in packedbed techniques, where the kinetic energy of particles is low enough to allow trapping of particles in the secondary minimum). Consequently, the specific aim of this study is to investigate the impact of surface roughness on the energy barrier and, as a result, its influence on a model system,24 where the deposition rate of hematite particles on polypropylene walls has been experimentally studied. This work concerns mainly industrial 4604
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surfaces but it is worth mentioning that the study of surface roughness is also related to environmental issues: for instance, micrometer to submicrometer roughness has been shown to enhance particle retention to mineral and rocks surfaces.2528
’ INFLUENCE OF SURFACE ROUGHNESS ON THE ENERGY BARRIER In this first section, the theoretical model used to generate a rough surface and to determine the interaction energy between a rough surface and a smooth particle is described. Then, some numerical applications are also discussed, and the main parameters influencing the energy barrier are analyzed. Description of the Model. The classical DLVO theory was originally designed for geometrically smooth surfaces and, since then, several studies have investigated the role of surface roughness on DLVO interactions.1116 Among these studies, two different approaches to calculate the interaction energy between rough surfaces can be distinguished. On the one hand, the approach proposed by Suresh and Walz11 consists of modeling a rough surface by a smooth wall covered by hemispherical asperities and to assume that the interaction energies are additive. On the other hand, another approach, known as the surface element integration or SEI-method,13,16,29 consists of calculating the energy barrier by integrating the interaction energy per unit area between two infinite half-spaces over the exact surface topography. The SEI approach appears as more elementary or microscopic than the former one and is less restrictive since the Derjaguin approximation (DA), which is valid provided that the particle radius is large enough, is not explicitly required. In that sense, the SEI is an interesting approach to investigate the impact of different shapes or surface morphologies. However, the SEI method is much more time-consuming than the former one since the interaction energy has to be integrated over the exact topography of the surface for each configuration. Furthermore, Huang et al.16 have shown that the modified Derjaguin approach remains an accurate approximation when the particle radius is larger than the asperity radius. In the situation considered in this work, another aspect must also be taken into account. Indeed, we are dealing with industrial rough surfaces that are known to have a rather complex morphology and for which not one but many different configurations can exist. Thus, the impact of surface roughness on the energy barrier cannot be evaluated for a single calculation but only makes sense by considering many possible arrangements since the explicit geometrical surface morphology is unknown (for example, in the present work 45 000 random configurations are considered to calculate the statistics of the energy barrier). Therefore, in our situation, the use of the SEI method would be far too computationally demanding. For these reasons, considering that we are dealing with particles larger than the surface asperities, we have chosen to develop a new model along the approach of Suresh and Walz.11 Model for Surface Roughness. The purpose of the present model is to reproduce the statistics of complex geometrical morphologies while retaining a model that is simple enough to allow tractable calculations to be performed. Thus, for the sake of simplicity (particularly for equations describing the interaction energy), asperities are supposed to be spherical, with half of their volume included in the smooth surface (the error made by this assumtion is negligible, as detailed in Supporting Information). The present model is extended to represent real rough surfaces, with a rather chaotic morphology, by placing asperities of various
Figure 2. Interacting disk and Rref eff .
Figure 3. Generation of a rough surface: the surface of interaction is first determined, and then asperities are placed on this surface.
sizes randomly on the surface (following a uniform distribution). Therefore, the complexity of surface morphology is modeled by random and possibly complex arrangements of simple elementary asperities. For each particle near a wall, a certain number of asperities is placed on the surface. This number is determined using the surface coverage Scov and, to account for surface randomness, these asperities are placed randomly on the surface with a uniform distribution. The asperities are placed within a cutoff radius of 10/κ (κ being the inverse Debye length), since the interaction energy between the particle and asperities located further away will be negligible (less than 103 times the particleplate energy barrier). Thus, for each particle, only a part of the surface is considered in the model, which will be later referred to as the interacting disk. By geometric considerations (see Figures 2 and 3a), when the particle is in contact with the plate, and the center-to-center distance between the particle and the furthest asperity is equal to Rpart þ Rasp þ 10/κ, the interacting disk has a radius Rref eff equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ref ¼ ð2Rpart þ Rasp þ 10=kÞðRasp þ 10=kÞ ð1Þ Reff where Rpart is the particle radius, and Rasp is the asperity radius. As a result, the surface of the interacting disk is equal to Sref eff = 2 π(Rref eff ) . To achieve a proper convergence, the effective interacting disk is greater and has a surface Seff = πReff2, which is taken here as equal to 2.5 Sref eff . The number of asperities to be placed is then given by a Poisson distribution with mean πR2eff Scov 2 π Rasp
ð2Þ
Asperities are also placed so that two asperities are not coincident (see Figure 3b); otherwise multiple interactions will lead to an overestimation of the energy barrier when the surface coverage is increased to values close to 0.5. Once all the asperities are placed, the contact point between the particle and the rough plate is determined by moving the particle perpendicularly to the smooth surface. Then, for each particle size, the surface roughness is generated 45 000 times, to be able to get a precise idea of the average impact of surface roughness, and not of the impact of one particular configuration, which may seldom occur in reality. 4605
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Calculation of Interaction Energies. The interaction energy between a particle and a rough plate is determined using the modified Derjaguin approach, first proposed by Suresh and Walz:11 the DLVO theory is still used and interaction energies are assumed to be additive. Therefore, the interaction energy between a rough plate and a particle (USR) is given by the weighted sum of the sphere-plate interaction (USP) and of sphere-asperity interactions (USA):16,30 USR ¼ ð1 Scov ÞUSP þ Scov USA
ð3Þ
In our case, the interaction energy between a rough plate and a smooth sphere is given by: USR ¼ ð1 Scov ÞUSP þ
∑
asperities
ð4Þ
USA
where U SP and U SA are determined according to the DLVO theory, i.e., the sum of the van der Waals interaction energy and the electrostatic interaction. In this case, the presence of asperities on the smooth plate is accounted for in an averaging manner, since asperities could be placed differently on the surface without changing the sphere-plate interaction energy. van der Waals Interactions. van der Waals interaction energies are evaluated by taking into account retardation effects.31 Indeed, van der Waals forces result from the correlated movements of charge. The electromagnetic fields emitted and received by the two interacting charges are fluctuating since charges are constantly trying to adapt to the surrounding electromagnetic field. Yet, due to the finite velocity of light, these electromagnetic fields will not travel instantaneously back and forth between the charges and will require a retardation time. At small separations, the motion of charges is strongly correlated and this retardation time is negligible but, at larger separations, the motion of charges becomes less and less correlated and the nonzero retardation time leads to the so-called retardation effect.31 For sphereplate interactions, the van der Waals contribution is determined using formulas from Gregory,32 valid at small separations, and Czarnecki, valid at larger separations, for retarded van der Waals interactions:33 VdW USP ¼
VdW USP
AHam Rpart 1 1 þ 14h=λ 6h
h
