Nanofluidics and Nanofluids

Lead Guest Editor: Jianzhong Lin Guest Editors: Mingzhou Yu, Martin Seipenbusch, Xiaoke Ku, and Yu Feng Nanofluidics and Nanofluids Journal of Nanotechnology

This is a special issue published in “Journal of Nanotechnology.” All articles are open access articles distributed under the Creative Com- mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Simon Joseph Antony, UK Joanna Kargul, Poland Ramakrishna Podila, USA Raul Arenal, Spain Valery Khabashesku, USA Paresh Chandra Ray, USA Thierry Baron, France María J. Lázaro, Spain Marco Rossi, Italy Carlos R. Cabrera, Puerto Rico Eduard Llobet, Spain Jorge M. Seminario, USA Enkeleda Dervishi, USA Oleg Lupan, Moldova Xiaowei Sun, Singapore Dmitriy A. Dikin, USA Abdel Salam H. Makhlouf, USA Boris I. Yakobson, USA Dimitris Drikakis, UK Paolo Milani, Italy Yoke K. Yap, USA Thomas Fischer, Germany A. E. Miroshnichenko, Australia ChuanJianZhong,USA Noritada Kaji, Japan Tomonori Ohba, Japan Contents

Nanofluidics and Nanofluids Jianzhong Lin , Mingzhou Yu, Martin Seipenbusch, Xiaoke Ku ,andYuFeng Editorial (2 pages), Article ID 8767624, Volume 2019 (2019)

Shape Oscillation of a Single Microbubble in an Ultrasound Field Xian Li ,FubingBao ,andYuebingWang Research Article (6 pages), Article ID 3701047, Volume 2018 (2019)

A New Spray Approach to Produce Uniform Ultrafine Coatings Lijuan Qian ,ShaoboSong ,andXiaoluLi Research Article (8 pages), Article ID 8978541, Volume 2018 (2019)

Molecular Dynamics Simulation of Nanoscale Channel Flows with Rough Wall Using the Virtual-Wall Model Xiaohui Lin, Fu-bing Bao , Xiaoyan Gao, and Jiemin Chen Research Article (7 pages), Article ID 4631253, Volume 2018 (2019)

Experimental Views of Tran-Bend Particle Deposition in Turbulent Flow with Nanoscale Effect Kun Zhou ,KeSun , Xiao Jiang, Shaojie Liu, Zhu He ,andZhouDing Review Article (10 pages), Article ID 7025458, Volume 2018 (2019)

Study on the Interaction between Modes of a Nanoparticle-Laden Aerosol System Yueyan Liu , Kai Zhang ,andShunaYang Research Article (7 pages), Article ID 6374394, Volume 2018 (2019)

Experimental Study on Expansion Characteristics of Core-Shell and Polymeric Microspheres Pengxiang Diwu, Baoyi Jiang, Jirui Hou, Zhenjiang You , Jia Wang, Liangliang Sun, Ye Ju, Yunbao Zhang, and Tongjing Liu Research Article (9 pages), Article ID 7602982, Volume 2018 (2019)

Two-Dimensional Numerical Study on the Migration of Particle in a Serpentine Channel Yi Liu, Qucheng Li, and Deming Nie Research Article (10 pages), Article ID 2615404, Volume 2018 (2019)

Iterative Dipole Moment Method for the Dielectrophoretic Particle-Particle Interaction in a DC Electric Field Qing Zhang and Kai Zhang Research Article (7 pages), Article ID 3539075, Volume 2018 (2019)

Stochastic Simulation of Soot Formation Evolution in Counterflow Diffusion Flames Xiao Jiang, Kun Zhou , Ming Xiao ,KeSun,andYuWang Research Article (8 pages), Article ID 9479582, Volume 2018 (2019)

Design and Numerical Study of Micropump Based on Induced Electroosmotic Flow Kai Zhang , Lengjun Jiang, Zhihan Gao, Changxiu Zhai, Weiwei Yan, and Shuxing Wu Research Article (6 pages), Article ID 4018503, Volume 2018 (2019) Simulation of Motion of Long Flexible Fibers with Different Linear Densities in Jet Flow Peifeng Lin ,WenqianXu,YuzhenJin,andZefeiZhu Research Article (10 pages), Article ID 8672106, Volume 2018 (2019)

Simulation and Visualization of Flows Laden with Cylindrical Nanoparticles in a Mixing Layer Wenqian Lin and Peijie Zhang Research Article (6 pages), Article ID 6548689, Volume 2018 (2019)

Modeling of Scattering Cross Section for Mineral Aerosol with a Gaussian Beam Wenbin Zheng and Hong Tang Research Article (7 pages), Article ID 6513634, Volume 2018 (2019)

Numerical Research on Convective Heat Transfer and Resistance Characteristics of Turbulent Duct Flow Containing Nanorod-Based Nanofluids Fangyang Yuan ,JianzhongLin ,andJianfengYu Research Article (9 pages), Article ID 4349572, Volume 2018 (2019)

The Asymptotic Behavior of Particle Size Distribution Undergoing Brownian Coagulation Based on the Spline-Based Method and TEMOM Model Qing He and Mingliang Xie Research Article (7 pages), Article ID 1579431, Volume 2018 (2019) Hindawi Journal of Nanotechnology Volume 2019, Article ID 8767624, 2 pages https://doi.org/10.1155/2019/8767624

Editorial Nanofluidics and Nanofluids

Jianzhong Lin ,1 Mingzhou Yu,2 Martin Seipenbusch,3 Xiaoke Ku ,1 and Yu Feng 4

1Department of Engineering Mechanics, Zhejiang University, Hangzhou, China 2College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou, China 3Institute of Chemical Process Engineering, University of Stuttgart, Stuttgart, Germany 4School of Chemical Engineering, Oklahoma State University, Stillwater, OK, USA

Correspondence should be addressed to Jianzhong Lin; [email protected]

Received 5 November 2018; Accepted 5 November 2018; Published 2 May 2019

Copyright © 2019 Jianzhong Lin et al. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction behavior were also investigated. Liu et al. mathematically investigated the interaction of nanoparticle dynamics be- Nanofluidics is the research of the behavior, manipulation, tween modes by establishing two joint population balance and control of fluids which are confined to nanometer-sized equations (PBEs) in the paper entitled “Study on the In- structures, while nanofluids are a class of fluids which teraction between Modes of a Nanoparticle-Laden Aerosol contain nanoparticles. Both nanofluidics and nanofluids can System.” In another paper entitled “Modeling of Scattering exhibit novel physical behaviors not observed in larger Cross Section for Mineral Aerosol with a Gaussian Beam,” structures (e.g., increased viscosity, enhanced thermal Zheng and Tang studied the scattering cross section of conductivity, and special rheological and acoustical prop- nonspherical mineral particles within the Gaussian beam erties), which make them potentially useful in many ap- based on the Generalized Lorenz–Mie &eory (GLMT). A plications. Recently, the advancement in nanotechnology review covering particle flow measurement and nano- and has triggered a greater motivation to investigate nanofluidics microaerosol distribution and deposition was presented by and nanofluids in detail. Zhou et al. in their paper entitled “Experimental Views of Tran-Bend Particle Deposition in Turbulent Flow with 2. Overview of the Works Published in This Nanoscale Effect.” &e effect of environmental humidity, Special Issue particle and surface properties, nanoparticle formation, coagulation, or evolution phenomena on particle deposition &e papers published in this special issue cover a wide range was discussed. of research topics from fundamental physical concepts to &e second group consists of 9 papers and covers a wide applied technologies in the field of nanofluidics and range of micro/nanoscale particulate matter types (e.g., solid nanofluids. Both experimental and modeling studies are particles, droplets, bubbles, and flexible fibers) used in included, and they might be broadly categorized into three different applications. To understand the expansion char- groups. acteristic differences between polymeric and core-shell &e focus of the first group which consists of 4 papers is microspheres, Diwu et al. experimentally and mathemati- on aerosol systems. Particle size distribution (PSD) is one of cally investigated the expansion behaviors of these two types the most important properties of aerosol particles. In the of microspheres and provided insightful discussions towards paper entitled “&e Asymptotic Behavior of Particle Size the advantages of using those microspheres on oil recovery. Distribution Undergoing Brownian Coagulation Based on Details can be found in the paper entitled “Experimental the Spline-Based Method and TEMOM Model” by He and Study on Expansion Characteristics of Core-Shell and Xie, the PSD was reconstructed using finite moments based Polymeric Microspheres.” Soot formation and evolution are on a converted spline-based method, and the evolution of constantly investigated issues in combustion processes. Jiang PSD undergoing Brownian coagulation and its asymptotic et al. presented the paper entitled “Stochastic Simulation of 2 Journal of Nanotechnology

Soot Formation Evolution in Counterflow Diffusion Particle Interaction in a DC Electric Field,” an iterative Flames,” in which soot formation and evolution in coun- dipole moment (IDM) method was applied to study the terflow diffusion flames were investigated. Moreover, two dielectrophoretic (DEP) forces of particle-particle in- detailed gas kinetic mechanisms (ABF and KM2) were also teractions in a two-dimensional DC electric field. &e re- compared with each other. Li et al. employed the volume-of- lationship between the chain patterns and the DEP fluid (VOF) method to investigate microbubble transport properties was found by tracking each particle movement. patterns in an ultrasound field in the paper entitled “Shape Oscillation of a Single Microbubble in an Ultrasound Field.” 3. Conclusions In another paper entitled “Molecular Dynamics Simulation of Nanoscale Channel Flows with Rough Wall Using the &is special issue documents some new applications and Virtual-Wall Model,” Lin et al. applied molecular dynamics challenges in the area of nanofluidics and nanofluids and the simulation to study the nanoscale gas flow characteristics in accepted papers show a diversity of new findings and rough channels. &ey mainly concerned the effect of overviews of the recent research and development. However, roughness element geometry on flow behaviors and found this special issue is far from an exhaustive survey of all the that the fluid velocity decreased with increasing roughness current topics and trends in nanofluidics and nanofluids element height. Qian et al. proposed a new design of spray- research. Many additional important research issues of coating nozzle and quantitatively evaluated its enhanced nanofluidics and nanofluids still remain to be explored in performance of nanoparticle-coating uniformity using both more depth. numerical and experimental methods. Specific discussions can be found in the paper entitled “A New Spray Approach Conflicts of Interest to Produce Uniform Ultrafine Coatings.” Using the lattice &e authors declare that they have no conflicts of interest. Boltzmann method (LBM) in the paper entitled “Two- Dimensional Numerical Study on the Migration of Parti- Acknowledgments cle in a Serpentine Channel,” Liu et al. numerically studied the migration of particles in a serpentine channel with &e editors are grateful to the participants of this special parametric analyses of how Reynolds number and initial issue for their inspiring contributions and the anonymous particle positions could impact particle migration behaviors. reviewers for their helpful and constructive comments which &ey also found that there existed a critical solid-to-fluid greatly improved the contents of the papers published in this density ratio at which the particle traveled fastest in the special issue. channel. Lin et al. utilized the flexible fiber model and large eddy simulation (LES) to simulate the transport and de- Jianzhong Lin formation of a micro/nanoscale fiber in a jet flow field, in the Mingzhou Yu paper entitled “Simulation of Motion of Long Flexible Fibers Martin Seipenbusch with Different Linear Densities in Jet Flow.” Yuan et al. Xiaoke Ku constructed a coupled numerical model for nanorod-based Yu Feng suspension flow and investigated the convective heat transfer and resistance characteristics of the nanofluid duct flow in the paper entitled “Numerical Research on Con- vective Heat Transfer and Resistance Characteristics of Turbulent Duct Flow Containing Nanorod-Based Nano- fluids.” Lin and Zhang studied the motion of cylindrical nanoparticles in a mixing layer using the pseudospectral method and discrete particle model in their paper entitled “Simulation and Visualization of Flows Laden with Cylin- drical Nanoparticles in a Mixing Layer.” &e effect of Stokes number and particle aspect ratio on the mixing and orientation distribution of cylindrical particles was analyzed. &e third group is related to flow and transport behaviors with electrical issues. For the purpose of investigating electric driving mechanism due to an induced charge electro-osmotic flow, Zhang et al. applied the finite volume method to solve the induced charge electro-osmotic flow in the paper entitled “Design and Numerical Study of Micropump Based on Induced Electroosmotic Flow.” &ey mainly studied vortices in the flow as well as their effect on the electric double layer, and the findings might be useful for the practical application of induced electro-osmotic flow in a micropump. In Zhang and Zhang’s work entitled “Iterative Dipole Moment Method for the Dielectrophoretic Particle- Hindawi Journal of Nanotechnology Volume 2018, Article ID 3701047, 6 pages https://doi.org/10.1155/2018/3701047

Research Article Shape Oscillation of a Single Microbubble in an Ultrasound Field

Xian Li , Fubing Bao , and Yuebing Wang

Institute of Fluid Measurement and Simulation, China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Fubing Bao; [email protected]

Received 2 February 2018; Accepted 25 July 2018; Published 15 August 2018

Academic Editor: Xiaoke Ku

Copyright © 2018 Xian Li et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

,e shape oscillation of a single two-dimensional nitrogen microbubble in an ultrasound field is numerically investigated. ,e Navier–Stokes equations are solved by using the finite-volume method combined with the volume-of-fluid model. ,e numerical results are in good accordance with experimental and theoretical results reported. According to the analyses of the shape oscillation process, the bubble deformation period is twice the driving acoustic pressure period and the shape oscillation is mainly caused by the change of interface velocity. ,e vortexes produced due to velocity variations lead to the deformation of the bubble interface.

1. Introduction So far, the characteristics of a single microbubble in the ultrasound field are mostly studied through experiments Two-phase flow occurs in a wide range of natural phenomena [13]. For the first time, Benjamin et al. [14] used an and engineering applications [1–4]. Among them, micro- ultrahigh-speed imaging technique to systematically study bubbles have received much attention in the past decades, aspheric oscillations and surface modes of bubbles. It was either for the energy that bubbles can release by cavitation pointed out that the nonspherical deformation was caused [5–7] or for their utility as ultrasound contrast agents in by the parameter instability driven by radial oscillations. medicine [8]. Recently, shape oscillations of a single micro- And the relationship between the driving frequency and the bubble in microfluidic chips in an ultrasound field have been resonance frequency was concluded. In the study of the reported by many researchers. Rabaud et al. [9] found that the impact of bubble oscillation on the ultrasound cleaning, Kim bubble tended to form periodic crystal-like lattices within and Kim [15] found that bubble behaviors in the ultrasound a finite interbubble distance when excited in an external field could be divided into four types: volume oscillation, acoustic field. From the figures and videos they provided, shape oscillation, collapse, and chaos oscillation. ,e shape apparent shape oscillations of a single microbubble can be oscillation was the manifestation of the bubble surface in- found, although the authors did not mention this. Liu et al. stability. In addition, they also gave a bubble behavior [10] found that the surface instability of a microbubble can classification map, which is very helpful for studying bubble lead to the bubble rupture, thus shortening the microbubble surface instability. Versluis et al. [16] experimentally studied residence time. However, in gene therapy, the ultrasound- the response characteristics of air bubbles in water. ,ey targeted microbubble destruction technique (UTMD) needs to found that there was a linear relationship between oscillation induce the instability of the bubbles. A certain intensity of mode n and the bubble radius, while the mode n had no ultrasound is applied on specific microbubbles for carrying the certain relationship with the amplitude of the driving target gene in the vicinity of the cells, causing microbubble pressure. McDougald and Leal [17] numerically simulated cavitation collapse, then formatting holes in the cell mem- the nonlinear mode vibration of spherical bubbles under brane, finally promoting target gene into the cell, to achieve the nonviscous conditions by using the boundary integral purpose of gene transfection [11, 12]. ,erefore, it is important method. Recently, Liu et al. [10, 18, 19] numerically sim- to analyze the stability of a single microbubble in an ultra- ulated the shape oscillation of the encapsulated microbubble sound field. under viscous conditions. Mekki-Berrada et al. [20] carried 2 Journal of Nanotechnology out a detailed investigation on the dynamics of two- A D dimensional (2D) bubbles at the wall interface of a microfluidic channel. By studying the response of the nitrogen bubble in the surfactant-added liquid to ultrasound, it was demonstrated that the wall deformation had no significant effect on the bubble dynamics characteristics. Furthermore, Microbubble R 0 they found that, above a critical pressure threshold, the bubble exhibited a two-dimensional shape oscillation around its periphery with a period doubling characteristic of a parametric instability. ,ese phenomena are interesting. However, it is difficult to obtain the detailed flow behaviors inside and around the oscillating bubble experimentally. ,erefore, the main objective of this work is to study the Water shape oscillations of a single microbubble in an ultrasound field based on numerical simulation, and the relationship between the bubble shape variation period and the driving B C pressure period is investigated. L Figure 1: Simulation domain of a single microbubble in an ul- 2. Numerical Methods trasound field. In this study, a nitrogen microbubble in water in an ultrasound field is investigated numerically to analyze the where pg0 is the initial gas pressure inside the bubble; pl0 is characteristics of the bubble shape oscillation. ,e the initial liquid pressure; c is the surface tension coefficient, Navier–Stokes equations [21] are used, and the volume-of- and c � 0.0735 N/m is used in this study; and R0 is the initial fluid (VOF) model [22] is introduced to track the interface bubble radius. between gas and liquid. ,e following assumptions are To solve the time-dependent Navier–Stokes equations, adopted to simplify the model: the finite-volume method [24] is applied. ,e volume fraction equation is solved through explicit time dis- (1) ,e liquid is considered to be incompressible, while cretization. ,e PISO algorithm [25] is used for pressure- the gas inside the bubble is compressible, con- velocity coupling, and implicit time discretization is used. forming to the ideal gas law. Numerical criteria are equally computed in double precision (2) ,e gas phase is immiscible with the liquid phase, with a segregated solver. For all cases, the convergence and the mass transfer between two phases is criteria for continuity and momentum equations are set to neglected. be 10−5 and for energy equation, are set to be 10−7. ,e time (3) ,e effect of gravity is ignored [23]. step is chosen depending on the frequency of ultrasound. In the present study, a 2D simulation is carried out. ,e 2D microbubble is corresponding to the pancake bubble 3. Results and Discussions confined between two walls in the microchannel, as de- 3.1. Model Validation. ,e simulation results of different scribed in [20]. ,e schematic of the simulation domain is shape oscillation modes are first compared with experi- shown in Figure 1. In the simulation, the side length of the mental results of references [20] (as shown in Figure 2) and square domain is 500 μm. ,e 2D bubble is positioned in the [26] to verify the model. middle of the domain, and the bubble radius is variable. ,e ,e dynamic behaviors of a microbubble at specified orthogonal quadrilateral grid is adopted, and mesh re- ultrasound frequency and acoustic pressure amplitude are finement is applied near the bubble. ,e grid independence first investigated. ,e ultrasound frequency is set to be is first carried out, and a total grid number of 150,000 is 130 kHz, and the acoustic pressure amplitude is 42 kPa. ,e adopted in the simulation. shape oscillation of microbubbles at different initial bubble In all simulations, the operating pressure is set to be radii is shown in Figure 3. In these snapshots of phase 1 atm and the initial velocity is zero. All boundaries are set as concentration contours, color blue stands for gas, while the pressure inlet. ,e inlet pressures follow the ultrasound color red stands for liquid. ,ese bubble shapes agree well pressure equation: with those experimental results of Mekki-Berrada et al. [20], p � Pd · cos(2πft), (1) as shown in Figure 2. ,e experimental figures presented in Figure 2 are carried out in a channel confined between two where f is the driving frequency and Pd is the acoustic walls 25 μm apart. It is a pancake bubble. ,e boundaries of pressure amplitude. ,e initial pressure inside the bubble the bubble in the experiment are not flat. It is different from follows the equation as the 2D bubble in the simulation. So the numerical results 2c seem to give smoother boundaries of bubbles than that of the pg0 � pl0 + , (2) R0 reference. Journal of Nanotechnology 3

Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10

100 µm

Figure 2: Snapshots of a bubble oscillation under high amplitude of ultrasound [20].

(a) (b) 6

(n) 5

4 (c) (d)

2 10 20 30 40 50 60 70

R0 (µm) (e) (f) n =3 n =7 Figure 3: Shape oscillation of a microbubble at different oscillation n =4 n =8 n =5 Equation (3) modes, when f � 130 kHz and Pd � 42 kPa. ,e corresponding initial radii are 22 μm, 30 μm, 36 μm, 42 μm, 50 μm, and 56 μm. (a) n � 3; n =6 Equation (4) � � � � � (b) n 4; (c) n 5; (d) n 6; (e) n 7; (f) n 8. Figure 4: ,e pattern distribution of the bubble shape oscillation mode, when f � 130 kHz and Pd � 42 kPa. ,e natural frequency of two-dimensional bubble shape oscillation was studied by Mekki-Berrada et al. [20]. ,ey reliability of the simulation results. In our present simula- presented a theoretical formula: tions, when the bubble’s initial radius ranges from 20 μm to �� 60 μm, the bubble’s shape oscillation mode increases with an gas c initial radius. It can be found from Figures 4 and 5 that the ωn � n(n − 1)(n + 1) 3. (3) ρR0 simulation results are in good agreement with the theoretical formula [20, 26]. In general, the bubble oscillation mode increased mo- notonously with the radius within a certain frequency. As 4. Numerical Simulation of Shape Oscillation shown in Figure 4, the relation between bubble modes and radius at 130 kHz is in good agreement with (3) [20]. Lamb ,e dynamic behaviors of bubble oscillation in the ultra- [27] investigated 3D bubble oscillations and found the sound field at different modes are similar. Without loss of following equation: generality, the shape oscillation processes of a single bubble �� at the fourth mode are discussed in detail. gas c A dimensionless time is first defined, t∗ � t/T. Here, T is ωn � (n − 1)(n + 1)(n + 2) 3. (4) ρR0 the period of the driving ultrasound wave, T �1/f. ,e bubble shapes in an oscillation cycle are shown in Figure 6 at ,e shape oscillation modes of a 2D bubble in simu- t∗ � iT,(i + 0.5)T,(i + 1)T,(i + 1.5)T, and (i + 2)T. Here, we lations are slightly different with that of a 3D bubble. choose a circular bubble shape as the start of a shape os- However, the overall trend remains the same. It can be found cillation period, as shown in the figure. During a whole shape from the figure that the same oscillation mode occurs over oscillation circle, the microbubble changes from its initial a certain bubble radius range. For example, when f � 130 kHz circular shape into the four-sided cross shape in the first half ∗ and Pd � 42 kPa, the oscillation mode is 4 when the bubble driving ultrasound period (0 < t < 0.5T). ,en, the bubble initial radius is in the range of 26 to 30 μm. returns to the circular shape in the next half driving ul- ,e simulation results and theoretical results at four trasound period (0.5T < t∗ < T). Later, it changes to a four- frequencies are displayed in Figure 5 to further verify the sided fork shape in the third half driving ultrasound period 4 Journal of Nanotechnology

160 variations and the velocity vector near the gas-liquid interface are analyzed. ,e external liquid pressure is probed near the boundary, while the internal gas pressure is probed 140 at the bubble center. ,e variations of liquid and gas pressures from t∗ � iT to t∗ � (i + 2)T are shown in Figure 7. ,e amplitude of the liquid pressure is approximate 42 kPa, 120 which is the same as the driving ultrasound pressure. ,e amplitude of the gas pressure is approximately 18 kPa, which is much smaller than that of the liquid pressure. We can also f (kHz) 100 find from Figure 7 that the fluctuating frequency of the gas pressure is exactly the same as that of the liquid pressure. However, there is a hysteresis in the gas pressure. ,e 80 hysteresis time is about 0.45T. According to the extreme outline of the bubble shape deformation, we presented the contrast chart of the instantaneous bubble morphology under the extreme state. As 60 ∗ 0 102030405060shown in Figure 8, with the bubble center as the origin, ρ(θ, t ) ∗ R0 (µm) is the distance from the bubble center to interface at time t . n =0 n =5 Collapse Distinguishing bubble shape deformation is mainly based n =2 n =6 Equation (3), n =3 on the angle corresponding to the maximum value of ρ. ,e n =3 n =7 Equation (3), n =4 fork-shaped bubble appears when ρmax corresponds to θ1 � n =4 n =8 Equation (3), n =5 ((2i − 1)/4)π, and when ρmax corresponds to θ2 � (1/2)π, the Figure cross-shaped bubble appears. 5: Mode distribution at different frequencies, when ,e instantaneous velocity vectors and bubble shapes in P � 42 kPa. d a single shape oscillation period are shown in Figure 9. ,e shape oscillation period is exactly twice of the driving ultrasound period. Let’s first take a look at the time t∗ � (i + 1.75)T. ,e bubble shape is approximately a square, as shown in the figure. ,e gage liquid pressure is at its maximum point, and the inner gas pressure is near the minimum, as can be found in Figure 7. ,ere exists a pressure difference between the liquid and the gas across the interface, and therefore, the interface moves towards the (a) (b) bubble from four corners along the direction of θ1. Eight vortexes appear at the gas-liquid surface. As a result, the interface in the θ1 direction moves inward, and the interface in the θ2 direction moves outward. Later, the external liquid pressure starts to decrease, and the internal gas pressure starts to increase. But the liquid pressure is still larger than the gas pressure, so the inward (c) (d) flow velocity in the θ1 direction keeps increasing. ,is velocity reaches its maximum at t∗ � iT, as shown in Figure 9. At this moment, the internal pressure equals roughly to the external pressure. ,e bubble shape is approximately a circle, which is the first stage in Figure 6. After this moment, the liquid pressure keeps decreasing and it is smaller than the gas pressure. ,e velocity in the θ1 direction begins to decrease. ,e interface in the θ1 direction keeps moving inward, and (e) the interface in the θ2 direction keeps moving outward. ,e cross-shaped bubble begins to appear. ,e liquid pressure Figure 6: Shape oscillation of a microbubble in a single cycle at reaches its minimum, and the inner gas pressure is near the n � 4, when R � 30 μm, f � 130 kHz, and P � 42 kPa. (a) t∗ � iT; (b) 0 d maximum at t∗ � (i + 0.25)T. ,e bubble begins to expand t∗ � (i + 0.5)T; (c) t∗ � (i + 1)T; (d) t∗ � (i + 1.5)T; (e) t∗ � (i + 2)T. because of its high pressure. ,e velocity magnitude in the θ2 direction is larger than that in the θ1 direction. (T < t∗ < 1.5T) and returns to the circular shape when ,e bubble reaches its extreme cross-shaped state at ∗ ∗ 1.5T < t < 2T. ,e bubble’s shape variation period is exactly t � (i + 0.5)T. At this moment, the inward velocity in the θ1 2 times of the driving ultrasound pressure period. direction vanishes and these eight vortexes near the gas- In order to further explore the shape variation during the liquid interface disappear. ,e gas and liquid pressures are oscillation process, the internal and external gage pressure nearly the same. ,e bubble reaches its maximum volume. Journal of Nanotechnology 5

45 0.45T Velocity Velocity (m/s) (m/s) 2.4 2.4 2.2 2.2 30 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 15 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 0.2 0.2

p (kPa) (a) (b) –15 Velocity Velocity (m/s) (m/s) 2.4 2.4 2.2 2.2 –30 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 –45 1 1 iT (i + 0.5)T (i +1)T (i + 1.5)T (i +2)T 0.8 0.8 0.6 0.6 t∗ 0.4 0.4 0.2 0.2 PL (c) (d) PG Velocity Velocity Figure 7: ,e liquid and gas pressure at n � 4, when R � 30 μm, (m/s) (m/s) 0 2.4 2.4 f � 130 kHz, and Pd � 42 kPa. 2.2 2.2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 ρ(θ, t∗) 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

(e) (f) θ Velocity Velocity (m/s) (m/s) 2.4 2.4 2.2 2.2 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 Figure � 8: Schematic diagram of the bubble’s shape at n 4. ,e (g) (h) solid and dashed lines correspond to oscillation’s extreme shape. Figure 9: Instantaneous velocity vectors and bubble shapes during a single bubble oscillation period when n � 4. Black solid lines After this moment, the outside liquid pressure is larger than indicate the gas-liquid interface. (a) t∗ � iT; (b) t∗ � (i + 0.25)T; (c) the inner gas pressure. ,e bubble enters the compression t∗ � (i + 0.5)T; (d) t∗ � (i + 0.75)T; (e) t∗ � (i + 1)T; (f) t∗ � (i + 1.25) stage. ,e interface in the θ2 direction begins to flow inward. T; (g) t∗ � (i + 1.5)T; (h) t∗ � (i + 1.75)T. As a result, it returns to the square shape at t∗ � (i + 0.75)T but with an angle of π/4 different from the bubble shape at t∗ � (i + 1.75)T. instability of the bubble surface and the oscillation mode are Up to now, an ultrasound vibration period has finished. combined effects of liquid viscosity, bubble diameter, and ,e bubble undergoes a compression and expansion process. driving ultrasound parameters. In the next ultrasound vibration, the bubble undergoes another compression and expansion process and the bubble 5. Conclusions varies from circle to cross-shaped. ,e shape oscillation period is exactly twice the driving ultrasound period. In the present paper, the response of a single 2D It can be concluded from Figure 9 that the shape os- microbubble in an ultrasound field was investigated nu- cillation of the bubble in an ultrasound field is mainly in- merically. ,e Navier–Stokes equations are used, and the duced by the interface velocity and vortex, which is an VOF mode was applied to capture the bubble interface intuitive result of the pressure difference between the outside between gas nitrogen and liquid water. Bubble’s shape liquid and inner gas across the gas-liquid interface. ,e instability was investigated, and the effects of the initial 6 Journal of Nanotechnology radius and ultrasound frequency on the unstable mode of [8] J. N. De, F. J. Tencate, C. T. Lancee´ et al., “Principles and the microbubble were discussed. It can be found from the recent developments in ultrasound contrast agents,” Ultra- simulations that numerical results are in good accordance sonics, vol. 29, pp. 324–330, 1991. with experimental and theoretical results. ,e shape os- [9] D. Rabaud, P. ,ibault, M. Mathieu, and P. Marmottant, cillation mode increases with the increase of the bubble’s “Acoustically bound microfluidic bubble crystals,” Physical Review Letters, vol. 106, no. 13, pp. 186–197, 2011. initial radius and driving ultrasound frequency. ,e same [10] Y. Q. Liu, K. Sugiyama, S. Takagi et al., “Numerical study on oscillation mode occurs over a range of the bubble radius. the shape oscillation of an encapsulated microbubble in ul- In order to explore the mechanism of bubble shape os- trasound field,” Physics of Fluids, vol. 23, no. 4, article 041904, cillation in the ultrasound field, the detailed velocity 2011. variation of the bubble in a single shape oscillation period [11] P. Marmottant and S. Hilgenfeldt, “Controlled vesicle de- was presented. ,e bubble oscillation period is exactly formation and lysis by single oscillating bubbles,” Nature, twice the driving ultrasound pressure period, and the vol. 423, no. 6936, pp. 153–156, 2003. shape oscillation of the bubble is mainly caused by the [12] M. L. Noble, C. S. Kuhr, S. S. Graves et al., “Ultrasound- change of interface velocity. Variations in the velocity targeted microbubble destruction-mediated gene delivery into result in vortexes, which will lead to the deformation of canine livers,” Molecular 4erapy, vol. 21, no. 9, pp. 1687– the bubble interface. 1694, 2013. [13] O. Kanae, E. Manabu, and H. Naoya, “Numerical simulation for cavitation bubble near free surface and rigid boundary,” Data Availability Materials Transactions, vol. 56, pp. 534–538, 2015. [14] D. Benjamin, M. Sander, V. D. Meer et al., “Nonspherical ,e data used to support the findings of this study are oscillations of ultrasound contrast agent microbubbles,” Ul- available from the corresponding author upon request. trasound in Medicine and Biology, vol. 34, pp. 1465–1473, 2008. [15] T. H. Kim and H. Y. Kim, “Disruptive bubble behaviour leading to microstructure damage in an ultrasonic field,” Conflicts of Interest Journal of Fluid Mechanics, vol. 750, pp. 355–371, 2014. [16] M. Versluis, D. E. Goertz, P. Palanchon et al., “Microbubble ,e authors declare that there are no conflicts of interest shape oscillations excited through ultrasonic parametric regarding the publication of this paper. driving,” Physical Review E, vol. 82, no. 2, article 026321, 2010. [17] N. K. McDougald and L. G. Leal, “Numerical study of the oscillations of a non-spherical bubble in an inviscid, in- Acknowledgments compressible liquid. Part II: the response to an impulsive decrease in pressure,” International Journal of Multiphase ,is work was supported by the National Natural Science Flow, vol. 25, no. 5, pp. 921–924, 1999. Foundation of China (Grant nos. 11672284 and 11474259) [18] Y. Q. Liu and Q. X. Wang, “Stability and natural frequency of and the National Key R&D Program of China (Grant nos. nonspherical mode of an encapsulated microbubble in a vis- 2017YFB0603701 and 2016YFF0203302). cous liquid,” Physics of Fluids, vol. 28, no. 6, article 062102, 2016. [19] Y. Q. Liu, M. L. Calvisi, and Q. X. Wang, “Nonlinear oscil- References lation and interfacial stability of an encapsulated microbubble under dual-frequency ultrasound,” Fluid Dynamics Research, [1] M. Z. Yu, X. T. Zhang, G. D. Jin, J. Z. Lin, and M. Seipenbusch, vol. 49, no. 2, article 025518, 2017. “A new moment method for solving the coagulation equation [20] F. Mekki-Berrada, P. ,ibault, and P. Marmottant, “Acoustic for particles in Brownian motion,” Aerosol Science and pulsation of a microbubble confined between elastic walls,” Technology, vol. 42, no. 9, pp. 705–707, 2008. Journal of Computational Physics, vol. 39, pp. 201–225, 2016. [2] J. Z. Lin, P. F. Lin, and H. J. Chen, “Research on the transport [21] F. M. White, Fluid Mechanics, McGraw-Hill, New York, NY, and deposition of nanoparticles in a rotating curved pipe,” USA, 7th edition, 2011. Physics of Fluids, vol. 21, no. 12, article 122001, 2009. [22] C. W. Hirt and B. D. Nichols, “Volume of fluid (VOF) method [3] M. Z. Yu and J. Z. Lin, “Taylor-expansion moment method for for the dynamics of free boundaries,” Journal of Computa- agglomerate coagulation due to Brownian motion in the tional Physics, vol. 39, no. 1, pp. 201–225, 1981. entire size regime,” Journal of Aerosol Science, vol. 40, no. 6, [23] J. H. Ferziger, M. Peric, and A. Leonard, “Computational pp. 549–562, 2009. methods for fluid dynamics,” Physics Today, vol. 50, no. 3, [4] M. Z. Yu, J. Z. Lin, and T. L. Chan, “Effect of precursor loading pp. 80–84, 1997. on non-spherical TiO2 nanoparticle synthesis in a diffusion [24] S. Patankar, K. Karki, and K. Kelkar, Finite-Volume Method: flame reactor,” Chemical Engineering Science, vol. 63, no. 9, Handbook of Fluid Dynamics, CRC Press, Boca Raton, FL, pp. 2317–2329, 2008. USA, 2nd edition, 2016. [5] C. E. Brennen, Cavitation and Bubble Dynamics, Oxford [25] M. S. Seif, A. Asnaghi, and E. Jahanbakhsh, “Implementation University Press, Oxford, UK, 1995. of PISO algorithm for simulating unsteady cavitating flows,” [6] R. P. Tong, W. P. Schiffers, S. J. Shaw, J. R. Blake, and Ocean Engineering, vol. 37, no. 14-15, pp. 1321–1336, 2010. D. C. Emmony, “,e role of ‘splashing’ in the collapse of laser- [26] A. Prosperetti, “Viscous effects on perturbed spheres flows,” generated cavity near a rigid boundary,” Journal of Fluid Quarterly of Applied Mathematics, vol. 34, no. 4, pp. 339–352, Mechanics, vol. 380, pp. 339–361, 1999. 1977. [7] A. Osterman, M. Dular, and B. Sirok, “Numerical simulation [27] H. Lamb, Hydrodynamics, Cambridge University Press, of a near-wall bubble collapse in an ultrasonic field,” Journal of Cambridge, UK, 1932. Fluid Science and Technology, vol. 4, no. 1, pp. 210–221, 2009. Hindawi Journal of Nanotechnology Volume 2018, Article ID 8978541, 8 pages https://doi.org/10.1155/2018/8978541

Research Article A New Spray Approach to Produce Uniform Ultrafine Coatings

Lijuan Qian , Shaobo Song , and Xiaolu Li

China Jiliang University, Hangzhou, China

Correspondence should be addressed to Lijuan Qian; [email protected]

Received 4 February 2018; Accepted 2 April 2018; Published 3 July 2018

Academic Editor: Xiaoke Ku

Copyright © 2018 Lijuan Qian et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new spray approach is proposed to overcome the disadvantages of the traditional single-orifice nozzle, such as uneven coatings, overspray, and low efficiency. Both the experimental measurements and numerical simulation are used to investigate the spray characteristics of the multiorifice nozzle. +e results show the new nozzle structure is able to disperse the particles in a wider regime and reduce the central pressure. It is an effective way to produce uniform ultrafine coatings.

1. Introduction approaches can alleviate the uneven spray pattern, whereas the effect is limited and not essential to solve the above Nanoparticles with size between a few nanometers and problems. 100 nm have great potential for use in various industry In this paper, a new spray approach with nozzle structure applications such as nanostructured film deposition [1], of multiple gas orifices is introduced to greatly improve the drug carriers [2], superconductors [3], and catalyst. +ere uniformity of the ultrafine coatings. Both the diagnostic are a large number of techniques for the preparations of measurements and numerical modeling are used to in- ultrafine particles [4]. Among them, spray technology using vestigate the differences in flow behavior of multiorifice suspension or solution precursors has been most widely used spray and traditional single-hole spray. For the experimental [5]. +e suspension or solution feeding is prepared by the setup, hot-wire anemometer is used to measure the distri- coating material dispersing or dissolving in a liquid carrier, butions of gas velocity, and laser particle analyzer is in- which allows the controlled injection of much finer particle troduced to obtain the data of the particle size distribution. than conventional thermal spraying [6, 7]. For the numerical calculation, based on the commercial One of the largest challenges of spray technology is how software, discrete particle model (DPM) is adopted to to control the spray pattern and droplet size distribution. As simulate the gas-liquid two-phase flow. Taylor analogy concern to the traditional spray technology, such as round breakup (TAB) model is used to describe the droplet jet and annular spray, there are some significant defects. For breakup. +e calculation results are compared by the ex- example, firstly, the droplets tend to concentrate in the perimental data. After validation, spatial distribution of center axis which gives rise to the highly uneven distribution pressure and velocity field is analyzed in detail to reveal the of liquid fragments. Second, the pressure is too high in the spray pattern characteristics of the multiorifice nozzle. +en, spray center, which always results in overspray. +ird, the the particle size and spatial distributions are discussed, and spray efficiency is low since the spatial distribution is nar- the advantages of the new spray approaches are concluded. row. +ese problems are particularly important for ultrafine coating due to its strict requirements. In order to solve these 2. Numerical Setup problems, the common practices are optimization design of the operating parameters such as adjustment of pressure, 2.1. Multiorifice Nozzle Structure. Figure 1 illustrates the nozzle orifice, and gas-to-liquid flow ratio [8–11]. +ese structure of multiorifice nozzle. +ere are three types of 2 Journal of Nanotechnology

Central oriﬁce 400 mm B C

Fan gas oriﬁce Auxiliary gas oriﬁce A Plane X-Z D 200 mm

z X-axis Y-axis Plane X-Y y F G x Z-axis

E H 200 mm

Figure 2: Schematic of computational domain.

drag coefficient model which is incorporated in FLUENT software has been used in Δ. τr is the discrete phase particle relaxation time. +e droplet distortion equation of TAB model for droplet breakup can be expressed as below: dx d2x F − kx − d � m , (2) dt dt2 where x is the displacement of the droplet equator from its spherical position. +e coefficients of (2) are determined by 2 3 Taylor analogy [16]: F/m � CFρgu /ρpr, k/m � Ckσ/ρpr , Figure 1: Multiorifice nozzle structure. 2 and d/m � Cdμp/ρpr . Figure 2 shows the computational regime of the nu- orifices, which include central orifice, auxiliary gas orifices, merical model. +e nozzle is located in a cuboid compu- and fan gas orifices. Based on an extremely large relative tational domain with size of 200 mm∗200 mm∗400 mm. 21 velocity between gas and liquid (more than 100 m/s), the million computational cells were used in the whole domain. central orifice can be able to facilitate primary atomization. Finer meshes are used for the core region of the spray. +e +e auxiliary gas orifices are used to carry out secondary grid dependency has been validated, and the computational atomization and alleviate the phenomenon of entrainment. results are convergence under the present mesh size. For the fan gas orifices, gas was provided at a constant slope Pressure inlet boundary conditions are applied on the ori- angle. By controlling gas flux and pressures, the gas flow fices of the nozzle. +e injection pressures of central and from fan orifice can be used to adjust the spray shape. auxiliary gas orifices are 120 KPa and 50 KPa for the fan gas orifices. +e wall boundary condition is used at the plane ADHE. Pressure outlet boundary conditions are applied on 2.2. Computing Model. Using the commercial software, the the planes ABCD, EFGH, DCGH, ABFE, and BCGF. For discrete particle model (DPM) was adopted to describe the discrete particles, the parcels are introduced in the system at gas-liquid two-phase flow. DPM is a multiphase flow model the central orifice exit, which has the original size of 65 μm based on the Eulerian–Lagrangian method, in which the gas with Rosin–Rammler distribution and original velocity of governing equation is solved as a continuous phase in the 50 m/s. Usually, a total number of 10,000 computational Euler coordinate system [12, 13]. +e particles are consid- parcels are injected into the flow for each case. +ese initial ered as a discrete phase in the Lagrangian coordinate system. conditions are provided and validated by the experimental Initially, the gas flow field was numerically calculated by k-ε data. turbulent model and pressure-based solver. +en, the droplet particle was added subsequently by using solid-cone injection type, dynamic drag model, and Taylor analogy 3. Experimental Setup breakup (TAB) model [14, 15]. +e equation of motion of particles is shown below: 3.1. Velocity Measurements. +e schematic of hot-wire anemometry measuring system is shown in Figure 3. +is du ug − up g�ρp − ρg � system can be used to obtain the velocity data of gas flow p � + + , (1) Δ filed. +e gas source is provided by an air pump which can dt τr ρp satisfy actual working conditions. +e airflow pressure can where ρp is the density of the particle, ρg is the density of the be measured by the piezometer. Before measurement, the fluid, up � (up, vp, wp) and ug � (ug, vg, wg) are, respectively, velocity calibration should be conducted. +en, a traverse the velocity vectors of discrete phase particle and continuous measurement system is applied to acquire velocity data by phase fluid, g is the gravitational acceleration, and Δ is an a hot-wire probe. +e measuring results include the velocity additional acceleration term for unit particle mass. +e dynamic data on planes X-Y and X-Z (shown in Figure 2). Journal of Nanotechnology 3

Air pump Piezometer Valve

Air flow meter Spray gun

Traverse system Probe

Support

Dantec streamline Terminal (a) (b) Figure 3: Schematic of hot-wire anemometry measuring system.

Air pump Valve Valve

Piezometer Piezometer Diaphragm pump Liquid flow meter Air flow meter Spray gun Tank

Emitter Receiver

Terminal (a) (b) Figure 4: Schematic of laser particle analyzer measuring system.

3.2. Particle Size Measurements. Figure 4 illustrates the laser distributions of the multiorifice nozzle. Both the experi- particle analyzer measurement system. +e laser particle ments and calculations are conducted under the same op- analyzer device includes the emitter and receiver part. +ese erating conditions that are presented in Section 2.2. +e two parts are set upon the different position of spray gun to agreement between the predictions and measurements is ensure laser traversing measuring area. +e receiver device qualitatively achieved. Results show the gas velocity reaches must be calibrated before the first measurement. +e lid of highest in the center of the spray core at the nozzle exit. With the receiver is used to ensure that no natural light distorts the the evolution of the spray, gas velocity declined dramatically accuracy of measurements. A traverse coordinate system is from 70 m/s to lower than 10 m/s. Due to the effects of fan used to measure particle size data from various cross sec- gas orifice, the spray angle on the X-Y plane is much larger tions. Sources of gas and liquid are provided with air pump than that on the X-Z plane, which is totally different from the and diaphragm pump, respectively. +e pressure and flow single round or annular orifice jet. rate can be measured by the piezometer and flow meter, Figure 6 illustrates the quantitative comparison of the respectively. +e gas media is air, and the liquid media is velocity evolution along with the Y-axis and X-axis. In water. Figure 6(a), d represents the distance between the cross section and the nozzle orifice. As the d increases, the velocity curve of the gas filed along with the Y-axis tends to be flat 4. Result and Discussion progressively, which means the velocity field approaches to 4.1. Single-Phase Flow Field homogenization. As Figure 6(b) indicates, the gas velocity decays at the downstream of the spray due to the entrain- 4.1.1. Validation. Figure 5 shows a comparison of experiment effects. In Figure 6, the calculated results agreed with mental data and model predictions for the gas velocity experimental data well. 4 Journal of Nanotechnology

Velocity (m/s) Velocity (m/s) 200 200

70 70

60 60 100 100

50 50

40 0 40 0 -axis (mm) -axis (mm) Y Y 30 30

–100 20 –100 20

10 10

–200 0 –200 0 50 100 150 200 50 100 150 200 X-axis (mm) X-axis (mm) (a) Velocity (m/s) Velocity (m/s) 100 70 100 70 60 60 50 50 50 50 40 0 40 0 30 -axis (mm) -axis (mm) 30 Z Z

–50 20 20 –50 10 10

–100 0 –100 0 50 100 150 200 50 100 150 200 X-axis (mm) X-axis (mm) (b)

Figure 5: Comparison of computational and experimental gas velocity distribution. (a) Contours on the X-Y plane (left: calculation; right: experiment); (b) contours on the X-Z plane (left: calculation; right: experiment).

4.1.2. Spatial Distribution of Pressure and Velocity Field. the single-nozzle case, the jet pattern is similar to a cylinder. Among the investigations of the spray filed, the distribution +e spray angles for both X-Y and X-Z planes are small. of pressure and velocity is the primary interest. Figures 7 and While for the multiorifice case, the spray range becomes 8 compare the velocity and pressure contours between larger on the X-Y plane after leaving the orifice for a dis- single-hole nozzle spray and multiorifice nozzle spray, re- tance. On the contrary, the spray angle becomes smaller on spectively. In Figures 7(a) and 8(a), for the single-orifice the X-Z plane beyond 15 mm from the nozzle exit. Second, at case, the velocity and pressure contours are almost sym- the downstream of the spray, the central pressure of the metrical at the X-Y plane and X-Z plane, whereas, for the multiorifice jet is lower than that of the single-orifice jet, multiorifice case, with the help of auxiliary gas orifices and which can relieve the phenomenon of overspray in the fan gas orifices, the spray pattern becomes totally different. center. Finally, due to the effect of spray entrainment, the +e differences can be concluded in three points: firstly, in nozzle exits are used to be polluted by the whirling gas and Journal of Nanotechnology 5

80 80 70 70 60 60 50 50 40 30 40 Velocity (m/s)

Velocity (m/s) 20 30 10 20 0 10 –200 –100 0 100 200 50 100 150 200 Y-axis (mm) X-axis (mm) d =40mm d =80mm Calculation d =40mm d = 140mm Experiment d =80mm d = 140mm Experiment Calculation

(a) (b)

Figure 6: Comparison of computational and experimental gas velocity evolution. (a) Gas velocity along with the Y-axis for diﬀerent measuring points; (b) gas velocity along with the X-axis.

Velocity (m/s) Velocity (m/s) 10 440 10 440 400 400 360 360 5 5 320 320 280 280 240 240 0 0 200 200 -axis (mm) Z -axis (mm) Y 160 160 120 –5 120 –5 80 80 40 40 –10 0 –10 0 0 5 10 15 20 0 5 10 15 20 X-axis (mm) X-axis (mm) (a) Velocity (m/s) Velocity (m/s) 10 10 440 440 400 400 5 360 5 360 320 320 280 280 0 240 0 240 200 200 -axis (mm) Z -axis (mm) Y 160 160 –5 120 –5 120 80 80 40 40 –10 0 –10 0 05101520 0 5 10 15 20 X-axis (mm) X-axis (mm) (b)

Figure 7: Comparison of calculated spatial distribution of velocity ﬁeld. (a) Velocity contours for single-hole nozzle: X-Y plane (left); X-Z plane (right). (b) Velocity contours for multioriﬁce nozzle: X-Y plane (left); X-Z plane (right). 6 Journal of Nanotechnology

Total pressure (Pa) Total pressure (Pa) 10 118500 10 118500 105000 105000 91500 91500 5 5 78000 78000 64500 64500 51000 51000 0 0 37500 37500 Y -axis (mm) 24000 Z -axis (mm) 24000 –5 10500 –5 10500 –3000 –3000 –16500 –16500 –10 –30000 –10 –30000 0 5 10 15 20 0 5 10 15 20 X-axis (mm) X-axis (mm) (a) Total pressure (Pa) Total pressure (Pa) 10 118500 10 118500 105000 105000 91500 91500 5 5 78000 78000 64500 64500 51000 51000 0 0 37500 37500 Y -axis (mm) 24000 Z -axis (mm) 24000 10500 –5 –5 10500 –3000 –3000 –16500 –16500 –10 –30000 –10 –30000 0 5 10 15 20 0 5 10 15 20 X-axis (mm) X-axis (mm) (b)

Figure 8: Comparison of spatial distribution of pressure field. (a) Pressure contours for single-hole nozzle: X-Y plane (left); X-Z plane (right). (b) Pressure contours for multiorifice nozzle: X-Y plane (left); X-Z plane (right). small droplets. +is problem can be improved by the in- have the Sauter mean diameter (defined as the ratio of volume- troduction of auxiliary gas orifices. surface mean diameter) around 55 μm.

4.2. Gas-Liquid Two-Phase Flow Field 4.2.2. Droplet Spatial and Size Distributions. In Figure 10, the horizontal coordinate presents the location of the droplet 4.2.1. Droplet Size Distribution. Figure 9 indicates the his- particles, and the vertical coordinate presents the diameter of togram of experimental and calculated drop size distributions. the particles. As we can see from Figure 10(a), according to Figure 9(a) shows the experimental size distribution of the the particle spatial distribution on the plate, the shape of the atomized droplets with the size range mainly between 2.31 μm multiorifice spray is oval. From the Y-axis, the droplets are and 74.82 μm. Figure 9(b) presents the calculated results based dispersed between −25 and 25 mm. From the Z-axis, the on the DPM model, in which the droplet size range is mostly droplets are concentrated between −10 and 10 mm. In order between 26.6 μm and 70.13 μm. Due to the droplet collision to compare, a further simulation analysis for the spray with and coalescent effect, both the experiment and simulation have single orifice is conducted. In Figure 10(b), without the larger particles with a diameter larger than 150 μm. Although auxiliary gas orifices and fan gas orifices, the atomization the calculated results match the experimental data well, the effect deteriorates, the larger droplet increases, and most of significant divergence also exists for the smaller particles. +e the droplets are concentrated in a circular region with radius finer droplets with size between 2.31 μm and 26.6 μm take about 10 mm. certain percentage for experiments. +is is because some droplets dissolved in the spray filed in the experimental setup. 5. Conclusions +ese phenomena can not be calculated by the present model. Generally, as indicated in Figure 9(c), particle size distributions A new nozzle structure which can improve the spray coating of the experiment and simulation are consistent. Both of them uniformity was proposed in the present paper. Both the Journal of Nanotechnology 7

Experiment Calculation 30 30 20 20 10 10 Percent (%) Percent 0 (%) Percent 0 0 50 100 150 200 0 50 100 150 200 Diameter (μm) Diameter (μm) Histogram Histogram Fitting curve Fitting curve (a) (b) 30 20 10

Percent (%) Percent 0 0 50 100 150 200 Diameter (μm) Calculation Experiment (c)

Figure 9: Droplet size distribution. (a) Histogram of experimental drop size distribution, (b) histogram of calculated drop size distribution, and (c) comparison of experimental data with calculated results.

300 300 250 200 200 μ m) μ m) 300 200 100

Diameter ( Diameter 100 ( Diameter 0 0 –100 –50 0 50 100 –100 –50 0 50 100 Y-axis (mm) Z-axis (mm)

(a) 600 600 500 500 400 400 μ m) μ m) 300 300 200 200 100 100 Diameter ( Diameter Diameter ( Diameter 0 0 –100 –50 0 50 100 –100 –50 0 50 100 Y-axis (mm) Z-axis (mm)

(b)

Figure 10: Droplet size and spatial distributions. (a) Droplet spatial and size distributions for multiorifice nozzle: Y-axis (left); Z-axis (right). (b) Droplet spatial and size distributions for single-hole nozzle: Y-axis (left); Z-axis (right). experimental measurements and simulation were in- phenomenon in the central spray, and improvement of the troduced to present the effects and characteristics of the new spray efficiency with wider spatial distribution. nozzle. +e spatial distribution of pressure and velocity field and the droplet spatial and size distributions have been Data Availability discussed in detail. With the auxiliary gas orifices and fan gas orifices, the multiorifice nozzle can provide the advantages +e data used to support the findings of this study are of even distribution of droplets, alleviation of the overspray available from the corresponding author upon request. 8 Journal of Nanotechnology

Conflicts of Interest [15] N. Ashgriz, Handbook of Atomization and Sprays: ;eory and Applications, Springer, New York, NY, USA, 2011. +e authors declare that there are no conﬂicts of interest [16] P. K. Senecal, D. P. Schmidt, I. Nouar, C. J. Rutland, regarding the publication of this paper. R. D. Reitz, and M. L. Corradini, “Modeling high-speed viscous liquid sheet atomization,” International Journal of Multiphase Flow, vol. 25, no. 6-7, pp. 1073–1097, 1999. Acknowledgments +is work was supported by the National Natural Science Foundation (Grant nos. 11632016 and 11372006), Zhejiang Provincial Natural Science Foundation (LY17A020008), and Young Talent Cultivation Project of Zhejiang Association for Science and Technology (2016YCGC013).

References [1] M. Bastwros and G. YongKim, “Ultrasonic spray deposition of SiC nanoparticles for laminate metal composite fabrication,” Powder Technology, vol. 288, pp. 279–285, 2016. [2] S. Melzig, D. Niedbalka, C. Schilde, and A. Kwade, “Spray drying of amorphous ibuprofen nanoparticles for the production of granules with enhanced drug release,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 536, pp. 133–141, 2018. [3] K. Okuyama and I. W. Lenggoro, “Preparation of nanoparticles via spray route,” Chemical Engineering Science, vol. 58, no. 3–6, pp. 537–547, 2003. [4] L. J. Qian, J. Z. Lin, and H. B. Xiong, “Modeling of non- Newtonian suspension plasma spraying in an inductively coupled plasma torch,” International Journal of ;ermal Sciences, vol. 50, no. 8, pp. 1417–1427, 2011. [5] N. P. Padture, M. Gell, and E. H. Jordan, “+ermal barrier coatings for gas-turbine engine applications,” Science, vol. 296, no. 5566, pp. 280–284, 2002. [6] L. J. Qian, J. Z. Lin, and M. Z. Yu, “Parametric study on suspension behavior in an inductively coupled plasma,” Journal of ;ermal Spray Technology, vol. 22, no. 6, pp. 1024–1034, 2013. [7] S. Lee and S. Park, “Spray atomization characteristic of a GDI injector equipped with a group hole nozzle,” Fuel, vol. 137, pp. 50–59, 2014. [8] J. C. Lasheras, E. Villermaux, and E. J. Hopfinger, “Break-up and atomization of a round water jet by a high-speed annular air jet,” Journal of Fluid Mechanics, vol. 357, pp. 351–379, 1998. [9] L. J. Qian, J. Z. Lin, H. B. Xiong, and T. L. Chan, “Effects of operating conditions on droplet deposition onto surface of atomization impinging spray,” Surface and Coatings Tech- nology, vol. 203, no. 12, pp. 1733–1740, 2009. [10] L. J. Qian and X. Y. Chu, “Particle properties for suspension plasma spray,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24, no. 6, pp. 1378–1388, 2014. [11] A. Aliseda, E. J. Hopfinger, J. C. Lasheras, D. M. Kremer, A. Berchielli, and E. K. Connolly, “Atomization of viscous and non-Newtonian liquids by a coaxial, high-speed gas jet: Ex- periments and droplet size modeling,” International Journal of Multiphase Flow, vol. 34, no. 2, pp. 161–175, 2008. [12] L. J. Qian, J. Z. Lin, and F. B. Bao, “Numerical models for viscoelastic liquid atomization spray,” Energies, vol. 9, no. 12, p. 1079, 2016. [13] C. L. Rivera, “Secondary breakup of inelastic non-Newtonian liquid drops,” Ph.D. thesis, Purdue University, West Lafayette, IN, USA, 2010. [14] H. B. Xiong, J. Z. Lin, and Z. F. Zhu, “+ree-dimensional simulation of effervescent atomization spray,” Atomization and Sprays, vol. 19, no. 1, pp. 75–90, 2009. Hindawi Journal of Nanotechnology Volume 2018, Article ID 4631253, 7 pages https://doi.org/10.1155/2018/4631253

Research Article Molecular Dynamics Simulation of Nanoscale Channel Flows with Rough Wall Using the Virtual-Wall Model

Xiaohui Lin, Fu-bing Bao , Xiaoyan Gao, and Jiemin Chen

Institute of Fluid Measurement and Simulation, China Jiliang University, Hangzhou, 310018, China

Correspondence should be addressed to Fu-bing Bao; [email protected]

Received 1 February 2018; Accepted 10 May 2018; Published 24 June 2018

Academic Editor: Xiaoke Ku

Copyright © 2018 Xiaohui Lin et al. -is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Molecular dynamics simulation is adopted in the present study to investigate the nanoscale gas flow characteristics in rough channels. -e virtual-wall model for the rough wall is proposed and validated. -e computational efficiency can be improved greatly by using this model, especially for the low-density gas flow in nanoscale channels. -e effect of roughness element geometry on flow behaviors is then studied in detail. -e fluid velocity decreases with the increase of roughness element height, while it increases with the increases of element width and spacing.

1. Introduction adopted to investigate the gaseous flow in nanoscale-confined channels [11, 17–19]. Barisik and Beskok [11, 17] investigated Micro/nano-electromechanical systems (MEMS/NEMS) have shear-driven gas flows in nanoscale channels to reveal the gas- received considerable attentions over the past two decades. wall interaction effects for flows in the transition and free Fluid flows are usually encountered in these systems [1–3]. molecular regimes. Hui and Chao [18] studied the gas flows in Fluid transport and interaction with these systems serve an nanochannels with the Janus interface and found that the important function in system operations [4]. Understanding temperature has a significant influence on the particle number the behaviors and manipulations of fluids within nanoscale near the hydrophilic surface. Recently, Babac and Reese [19] confinements is significant for a large number of applications investigated classical thermosize effects by applying a tem- [5–7]. perature gradient within the different-sized domains. -e effect of the wall serves as a distinct feature of fluid In some MD simulations, idealized-wall models are con- flow in micro/nanoscale-confined devices [8–10]. -e wall sidered. -e interactions of fluid-wall atoms are usually con- plays an increasing role in fluid flow when decreasing the sidered as functions, for example, the diffuse and specular flow characteristic length scale. Barisik and Beskok found reflections, Maxwell’s scattering kernel [20], or Cercignani– that, in a channel with 5 nm in height, 40% of the channel is Lampis model [21]. -ese idealized-wall models are feasible in immersed in the wall force field [11]. -erefore, the fluid some specific situations. However, when we study the detailed transport characteristics, such as momentum and energy, flow behaviors in the rear-wall region, the atomic-wall model significantly deviate from predictions of kinetic theory [11]. must be considered. But the atomic-wall model is expensive -erefore, the effect of this near-wall force field on the both in computational time and memory. In confined channel nanoscale channel flow must be understood and evaluated. flows, most atoms are requisite to describe the atomic wall. -e Molecular dynamics simulation (MD) investigates the number of wall atoms is much larger than that of fluid mol- interactions and movements of atoms and molecules, using ecules. -is drawback is particularly fatal for the gas flow. For N-body simulation [12]. -is method has been employed by example, Barisik et al. [22] studied a nanoscale Couette flow at many researchers in the past to study the liquid flow in Kn � 10. -e simulation box is 162 nm × 3.24 nm × 162 nm. In nanochannels [13–16]. Recently, the MD simulation is also their study, the number of gas molecule is 4900, while the 2 Journal of Nanotechnology number of wall atom is 903003. As a result, most of the -e virtual-wall model for the smooth wall is first ex- computational resource is consumed on the computation of amined. Without losing generality, gas argon flow confined wall atoms. between FCC platinum walls is considered. -e walls are Recently, Qian et al. [23] proposed a virtual-wall model for along the xz plane and the simulation box are periodic in the MD simulation to reduce the computing time. -e unit cell both x and z directions. For argon-argon interactions, σAr and structures are infinite repetitive in the atomic wall. As a result, εAr are 0.3405 nm and 119.8kB, respectively. For argon-platinum the force on a fluid molecule from wall molecules is periodical. interactions, σAr-Pt is 0.3085 nm and εAr-Pt is 64.8kB, according -is force was first calculated and stored in memory. During to the Lorentz–Berthelot mixing rule [24]. In this study, rc is the simulation, when a fluid molecule moves into the near-wall set as 0.851 nm, which is approximately equal to 2.5σAr. -e region, the force on this fluid molecule from wall molecules masses of argon and platinum atoms are 6.64 ×10−26 kg and can be determined directly, according to the position of the 3.24 ×10−25 kg, respectively. -ese parameters have been vali- molecule relative to the wall. -e near-wall region here refers dated in previous studies [25, 26]. to the region near the wall with distance smaller than the cutoff -e simulation box is set to be 40.9 nm × 17.1 nm × 40.9 nm radius. Excessive calculations of fluid-wall interactions can be in x, y, and z directions. A force of 0.008εAr/σAr is acted on each avoided, and the computing time can be reduced drastically. gas molecule as an external force [27] to drive the gas to flow in -e time reduction is more significant for lower fluid density the nanoscale channel. -e atomic-wall model is also carried in nanoscale channels. out here to make a comparison. -e thickness of the wall is In present study, the virtual-wall model is adopted to 1.18 nm, which is larger than the cutoff radius. -e lattice describe the rough wall. -e remainder of this paper is constant of the FCC platinum lattice is 0.393 nm. organized as follows. Section 2 introduces the MD simu- In the MD simulation, the neighbor-list method is used to lation and the virtual-wall model. Section 3 describes the calculate the force between atoms while the velocity-Verlet application of this model to the rough wall. Finally, Section 4 algorithm is adopted to integrate the equations of motion [28]. elaborates the conclusions of the study. -e timestep in the simulation is set to be 10.8 fs. -e first 1 million steps are used to equilibrate the system, and another 5 million steps are used to accumulate properties in the y 2. MD Simulation and Virtual-Wall Model direction, with the bin size to be 0.0614 nm. -e Langevin thermostat method [29] is employed to control the gas tem- In the present MD simulation, interactions between fluid- perature before equilibrium. Only thermal velocities are used to fluid atoms and fluid-wall atoms are both described using compute the temperature and pressure. -e above parameters the truncated and shifted Lennard–Jones (LJ) 12-6 potential and techniques are adopted in all simulations. given as follows: -e open-source MD code called large-scale atomic/molecular ⎧⎪ 12 6 12 6 massively parallel simulator (LAMMPS) [30], developed by ⎪ 4ε��σ � − �σ � − �σ � + �σ � �, r ≤ r , ⎨⎪ rij rij rc rc ij c Sandia National Laboratories, is adopted to carry out the V� rij � � ⎪ ⎪ MD simulations. ⎩⎪ 0, r > r , -e density and velocity profiles across the nanoscale ij c channel calculated using the atomic- and virtual-wall models (1) are compared in Figure 1. Perfect agreement between these where r is the intermolecular distance between atoms i and two models can be found, which indicates that the virtual- ij wall model works well in the MD simulation. j, ε is the potential well depth, σ is the atomic diameter, and rc is the cutoff radius. Lorentz–Berthelot mixing rule [24] is In order to compare the computational time, these two employed to calculate the LJ parameters between fluid-wall simulations are performed on a single Inter i7-4790K CPU atoms. processor. -e computational time for the virtual-wall In the virtual-wall model, the force on a fluid atom from model is 0.4 h, while for the atomic-wall model, the time wall atoms can be expressed as is 67.5 h. -e virtual-wall model is much more efficient in the present case. N F � �−∇Vi �, (2) i 3. Rough Wall Simulations where N is the number of wall atoms which interact with the 3.1. Virtual-Wall Model for the Rough Wall. From the fluid atom. -e atomic wall is composed of FCC lattices and micropoint of view, all walls are rough. Surface roughness the unit cell structures in repetition. When wall atoms are plays an important role in fluid flow and heat transfer [31]. fixed to their lattice point, the force on the fluid atom is So, in the present study, the virtual-wall model is adopted to periodic in both x and z directions. For example, the force of describe the rough wall. a fluid molecule located at x, y, and z is exactly the same as In the present study, platinum atom cuboids on the the force of the same molecule located at x + iL, y, and z + kL, smooth atomic wall are used to represent the roughness where i and k are integers and L is the lattice constant. If the element, as illustrated in Figure 2. -e roughness element is force distributions in the unit cuboid domain (L × rc × L) are periodic in both x and z directions. -e geometry of the known, then the force can be determined anywhere else. -is roughness element is shown in Figure 2(b). -e height of the is the core concept of the virtual-wall model. roughness element is h, and the widths in x and z directions Journal of Nanotechnology 3

160 150

120 100 ) ) –3 −1 80 (m·s x ρ (kg∙m v 50

0 0 04812160 4 8 12 16 y (nm) y (nm) Atomic wall Atomic wall Virtual wall Virtual wall (a) (b)

Figure 1: Comparisons between the atomic- and virtual-wall models for the smooth wall: (a) density proﬁle; (b) velocity proﬁle.

Y L Y

Z X rc Z X l H

(a) (b)

Figure 2: Schematics of the rough wall and the unit cuboid domain: (a) axonometric view; (b) side view. are both l. -e spaces between two elements in x and z then the force on a molecule anywhere else can be deduced. directions are both L. -e unit cuboid domain is then divided into MX × MY × MZ In order to perform the virtual-wall model, a unit cuboid bins, and the forces in each bin are calculated and stored in is first introduced, as shown in Figure 2. -e rough wall can the memory [23]. During the simulation, the corresponding be considered as the close-packed array of this unit cuboid. force of a fluid molecule located in the near-wall region is -e size of the unit cuboid is L × H × L, where H � h + rc. called directly from memory according to its position. Fluid molecules interact with wall atoms only when they are -e virtual-wall model for the rough wall is first vali- located within these cuboids. When fluid molecules are dated. Argon molecules are supposed to flow between outside these cuboids, the distances are larger than rc and no nanoscale rough platinum walls. -e simulation setup is the interactions between fluid and wall atoms are needed. same as in Section 2. For the roughness element, h � l � 2a -e cuboid is periodic in both x and z directions. and L � 4a, where a is the lattice constant of the FCC -erefore, the force of a fluid molecule located at x, y, and z platinum lattice, which is 0.393 nm. In the simulation, gas is exactly the same as the force of the same molecule located density is set to be 7.17 kg/m3. -e Knudsen number, which at x + iL, y, and z + kL, where i and k are integers. If the force is defined as the ratio of gas mean free path to the channel distribution in the unit cuboid domain (L × H × L) is known, height, is 0.95, and the flow is in transition regime. In order 4 Journal of Nanotechnology

18 50

12 30 ) ) −3 −1 (m·s x ρ (kg·m

v 20 6

0 0 0 0.5 1 1.5 2 2.5 0481216 y (nm) y (nm)

Atomic wall Atomic wall Virtual wall Virtual wall

(a) (b)

Figure 3: Comparisons between the atomic- and virtual-wall models for the rough wall: (a) density proﬁle; (b) velocity proﬁle.

120

90 H′ H′ –2h

h ) −1 60 (m·s x v

0 0 4 8 12 16 y (nm) Outer channel Inner channel Rough channel

(a) (b)

Figure 4: Comparison of gas flows in nanoscale smooth and rough channels. to make a comparison, the atomic-wall model is also carried indicates that the virtual-wall model works well for the gas out here. In the simulation, 3087 gas argon atoms and flows in rough wall channels. 218406 wall platinum atoms are used. -e gaseous flows in nanoscale channels with smooth -e density and velocity profiles of the virtual-wall and rough walls are first compared. -e schematic diagram model are shown in Figure 3. -ese profiles are compared of channel geometry is shown in Figure 4(a). -ree channels with the corresponding atomic wall simulation. Perfect are investigated. -e outer channel and the inner channel are agreement between these two models can be found, which both smooth, with channel heights equal to H′ and H′ − 2h, Journal of Nanotechnology 5

60 60

45 45 ) ) −1

−1 30 30 (m·s x (m·s v x v

15 15

0 0 0481216 0 4 8 12 16 y (nm) y (nm) l=a l =3a h = a h = 3a l =2a Fitting curve h =2a Fitting curve Figure 6: Velocity profiles at different roughness element widths. Figure 5: Velocity profiles at different roughness element heights. Fitting curves are obtained for each velocity profile at different roughness element heights, based on the gas ve- respectively. Here, h is the height of the roughness element. locity in the central part of the channel. From the fitting -e third channel is rough, with the channel height equal to curves, we can deduce the slip velocity on the wall conve- H′, and the roughness element height is equal to h. In the niently. It can be found from Figure 5 that the slip velocity simulation, H′ is 15.35 nm and h is 0.786 nm. So, the height also decreases with the increase of element height. of the inner channel is 13.67 nm. -e other parameters are kept the same as in Section 2. -e velocity profiles for these three channels are shown 3.3. Roughness Element with Different Widths. Roughness in Figure 4(b). It can be found that the velocity of the rough elements with different widths are then studied. -e height channel is much smaller than those of smooth channels. It is h and spacing L of the roughness element are kept the same, well known that, in nanoscale channel flows, the wall plays while the width l is variable. -ree roughness element widths an extremely important role in the fluid flow. Here, in the (l � a, 2a, and 3a) are considered. rough channel, the total surface area is much larger than those -e velocity profiles at different roughness element in smooth channels because of the existence of roughness widths are shown in Figure 6. It can be found from the figure elements. As a result, the collision probability between fluid- that the element width has a great influence on the velocity wall atoms is larger and more fluid molecules are affected by profile. -e fluid velocity increases with the increase of the wall in the rough channel. So, the fluid velocity of gas in the element width. -e total surface areas are the same in these rough channel is smaller. -e effect of roughness is of great three cases, so are the wall effects, according to Section 3.1. importance to nanoscale channel flows. However, at large roughness width, for example, l � 3a, the gap between two roughness elements is small. As a result, it is hard for the gas molecules to enter into the gap, because of 3.2. Roughness Element with Different Heights. -e influences the repulsive force between fluid-wall atoms, according to of roughness element geometry on flow behaviors are then (1). -at is to say, the effective surface area diminishes. So, studied. Roughness elements with different heights are first the fluid velocity increases in the rough channel with the studied. -e widths l and the spacing L of the roughness increase of the element width. element are kept the same, while the element height h is -e fitting curves obtained for each velocity profile at variable. -ree element heights (h � a, 2a, and 3a) are different roughness element widths are also shown in Fig- considered. ure 6. It can be found that the slip velocity increases with the -e velocity profiles of the rough wall with different increase of the element width. element heights are shown in Figure 5. It can be found from the figure that the fluid velocity decreases with the increase of element height. -is is because the total surface area is 3.4. Roughness Element with Different Spacings. Roughness larger at higher element height. According to the explana- elements with different spacings are studied at last. -e tion in Section 3.1, the wall effect is larger at higher element height h and width l of the roughness element are kept the height. So, the fluid velocity is smaller. same, while the spacing L is variable. -ree roughness 6 Journal of Nanotechnology

60 Conflicts of Interest -e authors declare that there are no conﬂicts of interest regarding the publication of this paper. 45 Acknowledgments

) -is work was supported by the National Key R&D Pro- –1 30 gram of China (Grant no. 2017YFB0603701) and National Natural Science Foundation of China (Grant nos. 11672284 (m×s x v and 11602266).

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[18] X. Hui and L. Chao, “Molecular dynamics simulations of gas flow in nanochannel with a Janus interface,” AIP Advances, vol. 2, no. 4, article 042126, 2012. [19] G. Babac and J. M. Reese, “Molecular dynamics simulation of classical thermosize effects,” Nanoscale and Microscale ?ermophysical Engineering, vol. 18, no. 1, pp. 39–53, 2014. [20] G. Karniadakis, A. Beskok, and N. R. Aluru, Microflows and Nanoflows: Fundamentals and Simulation, Springer, New York, NY, USA, 2006. [21] C. Cercignani and M. Lampis, “Kinetic models for gas-surface interactions,” Transport ?eory and Statistical Physics, vol. 1, pp. 101–114, 1971. [22] M. Barisik, B. Kim, and A. Beskok, “Smart wall model for molecular dynamics simulations of nanoscale gas flows,” Computer Physics Communications, vol. 7, pp. 977–993, 2010. [23] L. J. Qian, C. X. Tu, and F. B. Bao, “Virtual-wall model for molecular dynamics simulation,” Molecules, vol. 21, no. 12, p. 1678, 2016. [24] J. Delhommelle and P. Millie,´ “In adequacy of the Lorentz- Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation,” Molecular Physics, vol. 99, no. 8, pp. 619–625, 2001. [25] J. Sun and Z. X. Li, “Effect of gas adsorption on momentum accommodation coefficients in microgas flows using molecular dynamic simulations,” Molecular Physics, vol. 106, no. 19, pp. 2325–2332, 2008. [26] B. Y. Cao, M. Chen, and Z. Y. Guo, “Temperature dependence of the tangential momentum accommodation coefficient for gases,” Applied Physics Letters, vol. 86, no. 9, article 091905, 2005. [27] J. Koplik, J. R. Banavar, and J. F. Willemsen, “Molecular dynamics of Poiseuille flow and moving contact lines,” Physical Review Letters, vol. 60, p. 1282, 1988. [28] D. C. Rapaport, ?e Art of Molecular Dynamics Simulation, Cambridge University Press, New York, NY, USA, 2004. [29] S. Richardson, “On the no-slip boundary condition,” Journal of Fluid Mechanics, vol. 59, pp. 707–719, 1973. [30] S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” Journal of Computational Physics, vol. 117, pp. 1–19, 1995. [31] Q. D. To, C. Bercegeay, G. Lauriat et al., “A slip model for micro/nano gas flows induced by body forces,” Microfluidics and Nanofluidics, vol. 8, pp. 417–422, 2010. Hindawi Journal of Nanotechnology Volume 2018, Article ID 7025458, 10 pages https://doi.org/10.1155/2018/7025458

Review Article Experimental Views of Tran-Bend Particle Deposition in Turbulent Flow with Nanoscale Effect

Kun Zhou ,1 Ke Sun ,1,2 Xiao Jiang,1,3 Shaojie Liu,1,2 Zhu He ,1,2 and Zhou Ding1,2

1 e State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China 2National-Provincial Joint Engineering Research Center of High Temperature Materials and Lining Technology, Wuhan University of Science and Technology, Wuhan 430081, China 3Department of Mechanical Engineering, e Hong Kong Polytechnic University, Kowloon, Hong Kong

Correspondence should be addressed to Ke Sun; [email protected]

Received 1 February 2018; Accepted 5 May 2018; Published 7 June 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Kun Zhou et al. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

'is paper presented experimental views of nano- and microaerosol distribution and deposition in turbulent tran-bend flows. 'ese views included the particle flow measurement and particle depositions through individual bends, bifurcation bends, and those behind bends. Selected experiments were summarized and compared according to the gas flow, the bend geometry, and the particle flow properties. Based on recent studies, the influencing factors of environmental humidity, particle and surface properties, nanoparticle formation, coagulation, or evolution phenomena were discussed, and then research suggestions were given for future research and applications. It is specially mentioned that the new particle formation and nanoparticle growth affect its deposition under environmental contaminant conditions; nanoscale particle dynamics and transport have a growing trend on attracting the research and industry attentions.

1. Introduction to analyse influencing factors; to summarize recent findings; and to give future research and application recommenda- 'e living environment is filled with suspended aerosol tions. To summarize, particle distribution and deposition in particles such as nanoparticles and PM2.5 (airborne particles a single bend, behind a bend, and through bifurcation bends with an aerodynamic diameter less than or equal to 2.5 μm) are reviewed. Selected or potential influencing parameters [1, 2]. 'ese nano- and microparticles will commonly flow are discussed, including the Dean number, curvature ratio, into enclosed places like window cracks, sampling pipes, or nonspherical particle, particle evolution, particle surface ventilation ducts [3]. Generally, these enclosed places own effect, roughness, and environment humidity. certain bends or curved ducts, which play a significant role of changing the air and particle flow directions [4, 5]. 2. Basic Definitions Accurate measurements of the background gas flow properties, particle properties, particle concentrations, and Previous studies of aerosol flow in bends focus mainly on curved flow line geometries can accurately predict particle averaged deposition and penetration. 'e basic aerosol flow distribution, deposition, and accumulation status [6, 7]. deposition or loss efficiency can be expressed as follows:

'ese kinds of measurements include but not limit to gas Ci − Co flow velocity, wall surface roughness, particle size distri- η �(1 − P)· 100% � , (1) Ci bution and evolution, deposition amount, concentration distribution and evolution, and so on [8]. where P is the particle penetration ratio or efficiency, Co and Hence, this article aims to review the experimental bend Ci are, respectively, the mean particle mass/number con- nano- and microparticle flow, distribution, and deposition; centration at the cross sections of bend exit and entrance for 2 Journal of Nanotechnology a steady or periodic measurement condition. Apparently, P ] 3π]μd Sc � � p, (6) is reversely proportional to η, which is determined by the D kTC concentration status and related measurement technique. f c 'e commonly adopted averaged concentration Cave is where ] is gas kinematic viscosity; Df is particle diffusivity; a convenient statistic parameter of the particle distribution k means the Boltzmann constant; and T stands for the description and flow status. Meantime, on the view of the temperature [10]. statistic theory, there are also fluctuating components varying Along with the particle deposition expressions, the with the measurement location and the time slot. 'e con- penetration efficiency P is usually depicted against the centration changes with the time is generally controlled by the particle Stokes number St or particle aerodynamic diameter measurement method. While for location-dependent dp. St can be formulated as changes, there exist particle concentration distributions, τpUave which will affect the local deposition velocity, related particle St � , (7) D / deposition distribution, and accumulation status. h 2 Particle deposition velocity has the definition as below: where Uave is the average air speed in the flow line like pipe, J channel, or duct; and Dh is the hydraulic diameter of the Vd � , (2) flow line, given by Cave 4Ac where J stands for a deposition flux onto a specific surface Dh � , (8) pc and Cave denotes a mean particle concentration near a surface. An effective approach to interpret the particle de- where Ac is the cross-sectional area of the flow line and pc is position is to build up the relationship between the the perimeter of a cross section. + dimensionless deposition velocity Vd and the dimensionless For bend particle flows, additionally, the Reynolds + relaxation time τp. 'e former parameter is defined as number (Re) in flow lines is determined as follows: ρ U D Re � a ave h, (9) μ + V J V � d � , (3) d u C u w ave w where ρa stands for background gas density. Based on the flow line Reynolds number, the bend Dean number can be where u is the airflow friction velocity, which is determined w calculated by by the averaged or bulk flow line velocity and the friction Re condition of the flow lines. 'e background flow velocity De � ��, (10) could be easily measured by the modern apparatus. From Ro this equation, V+ is found also to be influenced by the d R friction condition of the flow lines, for example, the wall where o means the curved flow line curvature ratio. It is surface roughness. computed by (r1 + r2)/D, where the parameters r1 and r2 are + inner and outside radii of the curved flow line wall, re- 'e dimensionless relaxation time τp mentioned above is determined as spectively. Both the Dean number and Reynolds number are useful to nondimensionally depict the bend particle flow and τ deposition phenomena, and they can be determined by + � p, τp (4) accurate measurement of the bend geometry, background τe gas property, and flow velocity. where τe represents the eddy lifetime, which could be computed by the background flow properties. τp means the particle relaxation time defined by the following equation: 3. Particle Flow Measurement through Bends 2 Ccρpd Some experiments of particle flow focused on the velocity τ � p, (5) p 18μ investigation of gas and particle phases [11–13]. Kliafas and Holt studied the average radial and streamwise velocities and ° where Cc stands for Cunningham slip correction factor for related turbulent stresses in a 90 vertical to horizontal bend particles; ρp and dp are, respectively, particle density and with a square section [11]. 'ey adopted the laser Doppler diameter; and μ means the gas dynamic viscosity. Cun- velocimetry (LDV) and analysed the effects of Reynolds ningham coefficient Cc caused by slippage is determined by number, mass ratio, microparticle diameter, particle-wall the Knudsen number Kn, which is defined as the ratio collision, and bend deflection angles. Later, Yang and Kuan between the mean free length of the air molecules and conducted similar turbulent experiments of dilute (<1% by particle diameter [9]. 'ese particle parameters can be de- mass loading) microsphere particle (77 μm in average) flow termined from the preknown or measured background flow through a 90° horizontal to vertical bend with square section and particle properties. by using 2D LDV [13]. Both average and fluctuating ve- For diffusion dominated nanoparticles, Schmidt number locities of the air and particle phases were obtained under Sc is a crucial parameter to describe the effect of viscosity and a Reynolds number of 102,000 (bulk velocity 10 m/s). Ob- diffusivity. It can be expressed as vious air-particle separation was observed around the bend Journal of Nanotechnology 3

Table 1: Selected experimental studies with turbulent flow for particle deposition through bends (1986–2006). Investigators Pui et al. [18] McFarland et al. [15]a Peters and Leith [16, 17] Sippola and Nazaroff [19, 20]c Duct bends Indoor rectangular Bend type Small tube Small tube Industrial exhaust pipe duct ventilation duct Deposition surface Round stainless steel Round wax tube Bend interior coated with Bare galvanized steel and material and glass tube bends bends petroleum jelly internally insulated bends Upwards vertical to Horizontal to Horizontal to horizontal and Orientation NRd horizontal and horizontal to downwards vertical horizontal to vertical downwards vertical Smooth, gored, and Construction technique NRd NRd NRd segmented Hydraulic diameter, 0.501 and 0.851 1.6 15.2 and 20.3 15.2 Dh (cm) Curvature ratio, Ro (–) 5.7 1–20 1.7–12 3.01 Bend angle, θ (deg) 90 45–135 45, 90, 180 90 Air flow Reynolds numberb1, Re 6 and 10 3.2–19.8 203 and 368 21.6–88.3 (×103) Bulk velocity, U ave 18 and 31 7.7 and 18.6 20 and 27.1 2.2, 5.3, and 9.0 (8.8) (m·s−1) Particles Aerosol type and Monodisperse liquid Monodisperse liquid Polydisperse glass spheres Monodisperse fluorescent material (density) oleic acid (0.89) oleic acid (0.89) (2.45) particles (1.15) (g/cm3) Diameter, dp (μm) 1.1–6.6 10 5–150 1–16 Stokes number, St (–) 0.03–1.35 0.07–0.7 0.08–16 0.00013–0.081 Reynolds numberb2, 1.3–12.7 0.05–1.5 10–200 NRd Rep (–) Dimensionless relaxation d + 0.4–27 0.4–23 NR 0.0046–12 time, τp (–) Assumption of no Assumption of no No particle rebounding System method to reduce the Comments particle rebounding particle rebounding using petroleum jelly effect of particle rebounding a b1 b2 c Only experimental data [15]; Reynolds number: Re � ρaUaveDh/μ; particle Reynolds number: Rep � CcρpdpUave/μ; bend 6 in [20] is not included, and bend 5 is the downstream half of a 180° quasi-bend; dNR: not reported. outer wall and so was the slip velocity. 'e fluctuating ve- was distinct from that in laminar fluid flows. McFarland et al. locities of particle flow were found to be higher than those of reported that the increased curvature ratio led the deposition airflow at the inlet of the bend. efficiency to decrease. 'is behaviour revealed a similar trend with that in laminar flows [15]. Wilson et al. examined higher 4. Particle Deposition through Bends Reynolds number (Re � 10,250–30,750) and found that the Reynolds number did not obviously change deposition effi- 4.1. Aerosol Deposition in Individual Bends. Particle de- ciency trends for St > 0.4 [21]. However, a remarkable increase position in pipe or duct bend sections is of potential sig- in deposition efficiency was found for 0.1 < St < 0.4 when the nificance, but this behavior has not been fully studied by Reynolds number increases. experimental methods under turbulent flow conditions. Recently, nanoparticle deposition in bend flows has 'ere are limited but growing experimental investigations attracted a lot of research. For a tube bend flow, as demon bend aerosol deposition, especially in recent years as onstrated in Figure 1, Ghaffarpasand et al. measured and shown in Tables 1 and 2 [10, 14–24]. 'e scarcity of ex- quantified the nanoparticle penetration efficiency in 90° perimental work during earlier years might be attributed to bends for different Reynolds numbers (Re � 4500–10,500), the reason that both the particle deposition and distribution Dean numbers (De � 1426–2885), Stokes numbers, and are comprehensive and can be varied by many factors even curvature ratios [24]. Agreements with Pui’s empirical in straight pipe/channel/ducts [25–28]. model were found for 12 nm and larger particles, while Some existing literature of aerosol deposition from tur- deviations were observed for smaller ones [18, 24]. As il- bulent flow within bends is associated with suspended droplet lustrated in Figure 2, nanoparticle deposition rates in two studies in small diameter tubes [15, 18, 21]. Pui et al. experi- bends are generally higher than those in a single bend. Lin mentally studied the droplet flow and deposition in circular et al. investigated the penetration efficiency of nanoparticles cross-sectional bends with Reynolds number Re � 6000 and with diameters of 8–550 nm under different laminar and 10,000 [18]. No dependence on the Reynolds number for the turbulent flows, as shown in Figure 3 [10]. 'e major studied particle deposition efficiency was discovered. 'is phenomenon parameters included the Dean number (De � 370–950 under 4 Journal of Nanotechnology

Table 2: Selected experimental studies with turbulent flow for particle deposition through bends (2007–2018). Investigators Wilson et al. [21] Ghaffarpasand et al. [24] Sun et al. [22, 23] Lin et al. [10] Duct bends Indoor rectangular Bend type Small tube Small tube Small tube ventilation duct Deposition surface Standard stainless steel Stainless steel, hydraulically Galvanized steel and acrylic Plexiglass material (grade 304) smooth glass Downwards vertical to Orientation NRd Horizontal to horizontal NRd horizontal Made by a standard tube Made by university Construction technique NRd NRd bender industrial center Hydraulic diameter, 1.02 0.48 10 1.2 Dh (cm) d Curvature ratio, Ro (–) 7.4 13.25–54 3.4 NR Bend angle, θ (deg) 90 90 90 NRd Airflow Reynolds number, 10.25, 20.5, and 30.75 4.5–10.5 De � 1426–2885e 17.9, 35.6 De � 370–950e Re (×103) Bulk velocity, d d −1 15.4, 30.8, and 46.2 NR 2.58, 5.14 NR Uave (m·s ) Particles Aerosol type and Arizona standard test Vitamin E, that is, alpha- Tungsten oxide (10.8) and Polydisperse particles material (density) particle, basically SiO2, and 3 tocopheryl acetate (0.91) ammonium nitrate (1.725) with vegetable oil (g/cm ) Al2O3 (2.65) Diameter, dp (μm) 2.2–11 0.001–0.02 (monodisperse) 0.7–100 (polydisperse) 0.008–0.55 Stokes number, St (–) 0.12–1.08 0.001–0.03 5.2 ×10−4–0.55 NRd Reynolds numberb2, NRd NRd NRd NRd Rep (–) Dimensionless d f f + NR Sc � 15–820 0.34–27.6 Sc � 186–268,819 relaxation time, τp (-) Nanoparticle Assumption of no particle Nanoparticle, neglected Comments With particle-wall collision mechanism rebounding coagulation considered e �� f Dean number, De � Re/ Ro; Schmidt number, Sc � ]/Df , where ] is the dynamic viscosity. turbulent flow), Schmidt number (Sc � 186–268,819), and Aerosol deposition in commonly used ventilation system dimensionless bend length (l � 2–10). Combined with in commercial or public buildings has been tested mainly in Taylor-series expansion method of moment (TEMOM), an straight pipe/duct flows, considering resuspension dynamics empirical relationship between nanoparticle penetration or the whole system performance [30–32]. 'ere are only efficiency and the above parameters was given. For nano- limited experiments conducted on tran-bend particle flow particle deposition, turbulent and Brownian diffusions behavior. Sippola and Nazaroff experimentally investigated controlled particle diffusion and thus enhanced the particle two kinds of bends in building the ventilation system [20]. deposition [29]. Nanoparticle penetration efficiency firstly 'e aerosol deposition in the bend was found to be larger than increased and then decreased with the increasing of the Dean that in straight ducts with fully developed turbulent flows. Sun number where the penetration rate maximized at a critical et al. designed laboratory experiments to test the particle flow Dean number. 'e deposition efficiency decreases with the through bends made of different materials in the ventilation increasing Schmidt number. system [22, 23]. 'ey discovered particle concentration dis- In large industrial pipe bends with the diameters of tribution, deposition, and penetration phenomena and de- 15.2 cm and 20.3 cm, Peters and Leith studied the developed models on deposition velocity and penetration. 'ese positions of 5–150 μm polydisperse glass particles with investigations above have revealed the significance of venti- particularly high Reynolds numbers (203,000 and 368,000) lation bends with noncircular cross sections considering [16, 17]. 'e tested results were found to agree roughly with aerosol pollution. Although there are some reported in- previous results in the literature of small tubes, and the vestigations, studies on particle deposition through bends discrepancies may come from the differences among their with rectangular cross sections which are common in ven- test conditions which included different flow conditions, tilation duct systems are far from fully understood. configurations, and other uncertain influencing factors. In realistic experiment or application, most of these factors are strongly coupled; hence, further measurement and analysis 4.2. Aerosol Deposition behind Bends. Existing studies of of the particle flow and deposition characteristics in duct aerosol deposition characteristics behind bends mainly fo- bends are needed. cused on the particle penetration, deposition, and the Journal of Nanotechnology 5

18 35

16 30 14

12 25

10 20 8

6 15 Particle deposition (%) Particle deposition (%) 4 10 2

0 5 0 50 100 150 200 250 300 350 400 450 0 100 200 300 400 500 600 Schmidt number Particle diameter (nm) De = 630 Ro =54 De = 554 Ro = 31.8 De = 150 Ro =13.25 Figure 3: Nano- and microparticle deposition rate against particle Figure 1: Nanoparticle deposition rate varying with the Schmidt diameter under diﬀerent Dean numbers [10]. number and bend curvature ratio (Ro) [24].

60 20

18 50

16 40 14 30 12

10 20

8 Particle deposition (%) 10 Particle deposition (%) 6

4 0 10–6 10–5 10–4 10–3 10–2 10–1 100 2 50 100 150 200 250 300 St Schmidt number 1st bifurcation Two bends 2nd bifurcation Single bend Entire duct

Figure 2: Nanoparticle deposition rate varying with the Schmidt Figure 4: Deposition rate of aerosol particles varying with the Stokes number and bend number [24]. number (St) in different locations of the respiratory tract: 1st bifurcation [34], 2nd bifurcation [35], and entire duct [36]. turbulent induced enhancement effects. Sippola and Naz- aroff experimentally investigated two typical combined 4.3. Aerosol Deposition through Bifurcation Bends. One bends under various ventilation system conditions [19, 20]. typical and useful application of these bend particle flows is 'e research results revealed that the particle flow and the flow through the bifurcating system, for example, human deposition behaviour behind bends were important for breathing system and building ventilation system. As il- contaminant dispersion and fate. With experimental vali- lustrated in Figure 4, Miguel et al. studied particle deposition dation, the particle deposition phenomenon varying with the in a rigid double bifurcation airway in conditions of hu- Stokes number and dimensionless distance was numerically midity near saturation, that is, 95% (±3%) [35]. It was found studied behind a bend [33]. In this study, new integrated that deposition increased with the particle size in the range bend-induced deposition models with both those within of 0.1–9 μm, especially for coarse aerosols, which deposited bends and behind bends were proposed. 'ese similar more severe at the 1st bifurcation and for the obstructive- phenomena were also observed in bifurcation bend de- disease patient. Miguel et al. conducted another experiment position in Miguel’s experimental reports [34]. to study the aerosol deposition for particles ranging from 8.1 6 Journal of Nanotechnology to 23.2 μm [34, 36]. Significant factors affecting bend de- laboratory and numerical investigations on bioaerosol deposition were identified as particle size, Reynolds number, position within a ventilation chamber [43]. Better com- bend angle, and curvature ratio. Depositions on bifurcation parisons between experiment and numerical results were segments increased with the Reynolds number. For all achieved when the nonspherical aerosol shape was con- segments, the deposition increased with the humidity sidered. 'ese findings are valuable to the future bend (66–95%) as well. particle flow research. Rissler et al. conducted experimental determination of diesel exhaust particle (DEP) deposition in the human respiratory tract [37]. 'ey found that the dominating de- 5.2. Particle Evolution Effect. When considering nano- position mechanism of particles below 500 nm was diffusion, particle concentration effect to deposition distribution, the and the diffusivity was independent of the particle effective particle evolution process is one influencing factor in certain density and a function of mobility diameter. Penconek and conditions with particle nucleation, coagulation, and growth Moskal experimentally studied the nanosized-aggregate [29, 44, 45]. For example, Koivisto et al. and their later study DEP depositions in a cast of the human respiratory sys- conducted size-dependent analysis of the indoor suspended tem under two breathing patterns for three typical kinds of particle coagulation effect on deposition behaviour [46]. It diesel fuels [38]. 'e results reflected that the DEP de- was found that this effect might vary several orders of positions varied largely with the DEP source or fuels. De- magnitude, and it was deeply associated with the aerosol positions of particles generally increased with the tidal concentration and average diameter. Vohra et al. observed volume and breaths per minute. similar significance of including the coagulation effect in nanoparticle depositions [47]. 'erefore, they recommended the consideration of the coagulation contribution in 4.4. Particle Distribution and Deposition through Bends. the aerosol concentration spectra change and deposition Particle deposition is affected by its concentration, which process. Due to the nanoparticle evolution mechanism, the could be seen in (2). For the convenience and simplicity, the particle size distribution changes and so is the size- averaged or uniform particle concentration is generally dependent particle concentration distribution. 'e exist- assumed in previous experimental, analytical, or semi- ing particle evolution studies are helpful for further in- empirical investigations. However, when the fully developed vestigation of the particle deposition throughout bends. turbulent flow is disturbed by some obstacles, such as bends, then flow vortex and uneven particle concentration will form accordingly [27, 28, 39]. In contrast to the averaged de- 5.3. Particle-Wall Interaction Effect. No particle re-rebound position behaviour, the new localized deposition velocity or resuspension from pipe/channel/duct surfaces is assumed will be important, and there will be different deposition in a large amount of previous experimental reports in the distribution patterns. For instance, Miguel et al. demon- literature, but this assumption may not be true for solid strated that the main particle accumulation happened near aerosols [11, 12, 48]. One of the major mechanisms to the carinal ridge of each bend and the bifurcation not far control particle penetration/deposition is particle-wall col- from the bend [34, 35]. lision [11, 48–50], which is of importance to the particle flow To analyse the deposition distribution, the particle flow in two-dimensional bends. 'e flow differences between the and concentration behaviour through bends are important. particle and gas phases near the bend outer wall are caused Due to the bend effect, the flow streamlines are changed, and by the particle-wall collision [13, 33]. 'e collision, for flow patterns, eddies, and particle pathlines are varied. A instance, contributes the slip velocity between the particle typical phenomenon is the formation of the “particle-free and gas flow, especially the enhanced transverse particle zone” and the “particle rope” throughout the bend. For velocity behind bends. 'is enhanced transverse velocity may example, Yang and Kuan showed there were very low or be one reason to strengthen the deposition velocity behind even void particle concentration around the bend inner wall bends as described by Sippola and Nazaroff [20, 33]. As [13, 22, 23]. 'ese “particle-free zone” and “particle rope” a result, there exist bend-induced enhancement deposition confirm the above different concentration distribution, ratio and deposition velocity enhancement factor. It could consequent different deposition distribution, and particle also be inferred that the bend-induced developing turbulence accumulation [40]. downwards would strengthen the deposition process.

5. Influencing Analysis of Bend- 5.4. Surface Material Effect. Furthermore, investigating the Induced Deposition surface material effect on particle deposition in ventilation bends is important as well for understanding the particle 5.1. Nonspherical Particle Effect. With the assumption of transport in piping systems or ventilation systems spherical aerosols, a large number of particle deposition [20, 22, 33]. Previous measurements with different testing experiments are conducted and analysed in the literature. materials to estimate particle penetration or deposition However, realistic aerosols are nonspherical, which would through bends are summarized in Tables 1 and 2. In these influence the aerosol concentration, orientation and dis- tables, both the bend type and particle type are listed to tribution, the particle deposition speed, and settling be- compare the commonly adopted bends and particles to haviour [41, 42]. For instance, Lai et al. reported both investigate particle deposition in experimental studies. In Journal of Nanotechnology 7

Table 3: Construction materials [51] applied in ventilation system. Duct for grease-laden Application Commercial and public Residential duct Industrial duct and moisture-laden Plastic duct field buildings vapors Galvanized steel, fiberglass insulated duct, corrugated aluminum, or flexible Steel, aluminum, or Galvanized steel, 'ermoplastic and Duct spiral-wound Mylar Carbon steel, type a material with a UL uncoated carbon steel, fiberglass reinforced materials [15]; iron, steel, 304/430 stainless steel Standard 181 listing. or aluminum thermosetting ducts aluminum, concrete, masonry, clay tile, fibrous glass, or G90- coated galvanized ducts detail, Sippola and Nazaroff measured the deposition direction is changed, developing turbulence, secondary flow, through 90° ventilation bends associated with typical and vortexes are formed within and behind the bend commercial or public mechanical ventilation systems [20]. [53, 54]. When vortex and turbulence are formed, different Peters and Leith investigated industrial large pipe bends with particles will have different streamlines, and uneven particle grease coating which was considered to capture all the concentration, deposition, and accumulation will appear. A approaching particles [16, 17]. typical phenomenon is the formation of the “particle-free Nowadays, to provide a comfortable building and distinct zone” and the “particle rope” throughout the bend environment, a good central ventilation system is commonly [13, 22, 23]. From the view point of turbulent Reynolds integrated in modern institutional, public, and/or commercial number (Re), different experimental reports have partial buildings, and the ventilation ducts have become one critical consistency on the dependence between deposition effi- way to transport particle-laden fresh air. To construct the ciency and Re [18, 21, 24]. 'e major deviations are observed required ventilation ducts, many types of materials are for smaller particles. For nanoparticle deposition, turbulent adopted. Examples of construction materials employed in diffusion is one of the main factors to control particle dif- different ventilation systems are summarized in Table 3 fusion and thus to enhance the particle deposition [29]. according to ASHRAE handbooks [51] and the dissertation of Sippola [19, 51]. Several types of material steel and aluminum with different surface coatings are adopted widely, 5.7. Moisture Effect. Moisture content in air has a positive and materials with specific usage such as plastic, glass, and effect on particle growth and deposition. With the increase masonry could be seen in the table as well. Construction of air humidity, particles become bigger, and depositions are materials of different function ducts may play important roles increased. For example, Miguel et al. found that depositions in ventilation systems, and material influences on the aerosol increased with the particle size increasing from 0.1 to deposition and distribution are suggested to be further 23.2 μm under the humidity 66–95% (near saturation) for all studied in detail as recommended by Sippola [19]. segments, especially for coarse aerosols, under larger Rey- nolds number (e.g., Re � 1066–3151) and in the 2nd bifurcation of the respiration system [34, 35]. 'e deposition 5.5. Roughness Effect. Roughness is a factor to influence the efficiency enhancement ranges from 10 to 35% with the near-wall flow and the particle-wall collision phenomena humidity rising from 66% to 95%. One explanation of this [52]. In fluid dynamics, roughness degree dominates the phenomenon is that particles adhesion to a surface is background friction flow or pressure loss of the pipe, duct, strengthened due to humidity rise [55]. Similarly, in future and channel flow [31]. For particle flow and deposition near study, the outdoor and indoor moisture effects on bend a surface, it affects the near-wall turbulence, the flow field, aerosol (e.g., PM2.5) flows are suggested to be further in- and the particle-wall contact behaviour. Vohra et al. vestigated for deeper understanding of the influencing designed series of laboratory experiments to study the mechanism and applications, such as the control of adverse nanoparticle (14.3 nm–697.8 nm) deposition for different impact of nanoparticles on human health [56]. roughness surfaces under different chamber conditions [47]. 'ey pointed out that the roughness and related turbulent 6. Conclusions and Suggestions flow enhanced the deposition cumulatively. It is thus inferred that the bend roughness will affect the particle 'is paper reviewed and analysed the investigations of distribution and deposition in the tran-bend flow process. particle distribution and deposition throughout the bend turbulent flow. Basic particle deposition and carrying gas flow theories were introduced at the beginning of this review 5.6. Turbulence Effect. Generally, the background fluid flow work. Particle deposition and penetration formula were is fully developed before entering into the bend. With the given to connect with the particle properties, concentrations, bend effect as described by previous researchers, the flow gas properties, flow velocities, and bend geometry and 8 Journal of Nanotechnology properties. 'is article compared selected experimental particle evolution, particle deposition, and accu- studies on particle deposition throughout individual labo- mulation processes. ratory bends. Particle deposition through bifurcation bends (6) Environmental humidity effect on nano- and mi- was presented as well. To figure out recent and future re- croparticle deposition and coagulation is significant search areas, certain influencing factors were discussed, but far from fully understood. Connecting with tran- including nonspherical particle deposition, nanoparticle bend flow, the humidity influence is suggested to be evolution like coagulation, particle-wall interaction, wall identified. surface properties, and atmospheric humidity. 'e study can be summarized as follows: Conflicts of Interest (1) Large amount of existing research focused on average aerosol penetration and deposition efficiency through 'e authors declare that there are no conflicts of interest 90° small tube bends. A wide range of curvature ratio regarding the publication of this paper. (e.g., 1–54), Reynolds number (e.g., 3,200–368,000), particle diameter (1 nm–150 μm), particle dimension- Acknowledgments less relaxation time τ+ (e.g., 0.0046–27.6), and Schmidt p 'is work was financially supported by the National Natural number (e.g., 15–268,819) were investigated. Science Foundation of China (nos. 11602179 and 11572274). 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Research Article Study on the Interaction between Modes of a Nanoparticle-Laden Aerosol System

Yueyan Liu ,1 Kai Zhang ,2 and Shuna Yang 3

1College of Modern Science and Technology, China Jiliang University, Hangzhou 310018, China 2Institute of Fluid Engineering, China Jiliang University, Hangzhou 310018, China 3School of Communication Engineering, Hangzhou Dianzi University, Hangzhou 310018, China

Correspondence should be addressed to Shuna Yang; [email protected]

Received 31 January 2018; Revised 27 February 2018; Accepted 30 April 2018; Published 5 June 2018

Academic Editor: Paresh Chandra Ray

Copyright © 2018 Yueyan Liu et al. )is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nanoparticle-laden two-phase flow systems, especially atmospheric aerosols, are usually found with several modes for particle size distribution (PSD). For the first time, a mathematical method is proposed to study the interaction of nanoparticle dynamics between modes by establishing two joint population balance equations (PBEs). )e PBEs are solved using the sectional method, which divides the PSD into discrete bins. )e nanoparticle-laden system involves Brownian coagulation, ventilation, and injection. )e interaction between modes within a size distribution is studied quantitatively with and without injection and ventilation. )e study shows that particles with smaller size are easier to be removed by background particles, but the lag time to be removed is affected by not only the total number concentration of small particles but also their sizes. Background particles play an important role in determining the evolution of small particle system, whose presence makes the secondary model absent for the small particles.

1. Introduction size distribution rather than a special part of the whole size distribution. )us, the traditional PBE cannot meet the Nanoparticle-laden two-phase flows exist in lots of natural requirement of distinguishing modes within a system. Re- and chemical engineering processes, such as atmospheric cently, Yu and Chan proposed a method to establish two aerosols [1], gels in colloid [2], nanoparticle synthesis [3–5], joint PBEs, each of PBEs accounting for one mode [6]. In this and combustion reactors [6, 7]. In these processes, the method, the system is artificially divided into two sub- evolution of system, especially the particle size distribution systems; the transfer of mass between modes is reserved. (PSD), is determined by more than one dynamics [3]. Re- Using this way, they studied the evolution of a bimodal size- gardless of a closed or an open nanoparticle-laden system, distributed system and claimed that this method has ca- multiple dynamics lead to more than one mode for the PSD pability to study the interaction of modes. Unfortunately, in [8, 9]. Each of these modes mainly comes or is determined by their study, the PBEs are solved using the Taylor-series one dynamical process; the interaction among these modes expansion method of moments (TEMOM). All methods dominates the evolution of the whole system. )erefore, the of moments (MOM) have a common shortcoming; that is, study on the interaction of dynamics between modes rather the size-resolved information of PSD cannot be resolved or than focusing on the whole system becomes necessary. cannot be grasped. )us, in Yu and Chan’s study, some Since first proposed by von Smoluchowski in 1917 [4], important information about the interaction between the the Smoluchowski mean-field theory has become the main modes might be lost. Similar studies can be also found in theory to investigate the evolution of nanoparticle size [7, 10]. distribution. In this theory, the key is to establish the In our surroundings, atmospheric aerosol is one of the population balance equation (PBE) according to particular most common nanoparticle-laden two-phase systems. Lots dynamics [5]. Traditionally, PBE is established for the whole of works have been carried out for the study of nanoparticle 2 Journal of Nanotechnology evolution of PSD determined by various dynamical pro- Yu and Chan in the implementation of the Taylor-series cesses, including both experiments and numerical simula- expansion method of moments [6], which has been verified tions [11–13]. Depending on the number of dynamical in their work. processes involved, atmospheric aerosols usually exhibit two )e remainder of this paper is organized as follows. )e or more modes. For atmospheric aerosols, three key pro- mathematical forms of two joint PBEs as well as their nu- cesses, namely, Brownian coagulation, ventilation, and in- merical solutions using the SM are analyzed in Section 2. )e jection, are usually involved. In mathematics, these three specifications of the initial conditions for comparison are processes can be easily represented by existing models. )us, outlined in Section 3. )e results and discussion are pre- the atmospheric aerosol, especially in an open homogeneous sented in Section 4. isotropic chamber, is usually selected by scientists as an ideal system for the comparative study of PSD [8, 13]. By selecting 2. Theory and Model an aerosol inside a homogeneous chamber as investigated objective, Liu et al. studied the evolution of nanoparticle size In this work, only two modes are considered in a size distribution due to continuous injection using the sectional distribution for simplification, in which the nanoparticle- method (SM) [8]. However, in their study, the interaction of laden system is artificially separated into two subsystems. dynamics between modes is not involved. Similar to Liu et al.’s work [8], in which one accounts for PBE is a nonlinear integral-differential equation, which small particles (SPs) originating from source or for small has no analytical solutions as real coagulation kernels are particle mode and another for background particles (BPs). selected [9]. Until now, three main methods have been For each subsystem, it needs to establish its corresponding proposed and used for the solution of PBE, namely, the SM, particle general dynamic equation based on the Smo- MOM [11, 14–18], and Monte Carlo (MC) method. )ese luchowski mean-field theory. In order to make the math- methods have both advantages and disadvantages regarding ematical equations approach to real physical phenomenon as accuracy and efficiency and are used in PBE studies much as possible, the transfer of particle between two according to specific requirements [12]. Among all the systems is considered, which is similar to the work [6]. numerical methods, the SM is regarded as the most suitable Because the particle size distribution of BP system is little method for studying PSD without losing size-distributed influenced by the existence of small particles and especially information but consumes much computational costs. In there is very large difference between background and source this work, we applied the SM to the solution of PBE in- particles in their sizes, a novel disposition is proposed here in volving two modes mainly because in some aerosol systems which particles belonging to SP system are considered to there are usually two or more modes, and thus, the in- become those belonging to BP system once these small teraction between modes inevitably exists. We separate the particles are attached to background particles. )erefore, in aerosol system into two subsystems each representing one theory, the mass of SP system always decreases with time mode; therefore, we can investigate the interaction between while the total mass of the whole system is conserved if there modes. We used the SM to solve population balance is no ventilation and no source. equation, and thus, the size-resolved particle distribution By including coagulation, injection, and ventilation, the can be exactly traced with time. In fact, the scheme sepa- particle general dynamic equation for SP system is repre- rating the size distribution into two parts has been used by sented as follows:

zn(v, t) 1 v ∞ � � βv − v′, v′ �n v − v′, t �n v′, t �dv′ − n(v, t) � βv, v′ �n v′, t �dv′ zt �√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√�2 0 0 Coagualtion within SP mode (1) ∞ F f(v, t)F −n(v, t) � cv, v′ �p v′, t �dv′ – out n(v, t) + in , �√√√√√√√√√√√√√��√√√√√√√√√√√√√�0 �√√√√√��√√√√√�V �√√√√√��√√√√√�V SP mode attached to BP mode Ventilation Injection where n(v, t)dv is the particle number of SP system whose BP system. )e last two terms account for particle loss and volume is between v and v + dv at time t and β(v, v′)is the increase due to ventilation and source production, re- collision kernel for two particles of volumes v and v′. On the spectively. p(v′, t)dv′ is the particle number between v′ and right hand of (1), the ﬁrst two terms account for particle v′ + dv′ for BP system. f(v, t) is the source number con- number increase and decrease due to interparticle collision centration intensity with volume v and time t. and join together within the SP system, while the third term Similarly, the particle general dynamic equation for BP serves as particle loss due to collision between SP system and system is as follows: Journal of Nanotechnology 3

zp(v, t) 1 v ∞ � � βv − v′, v′ �p v − v′, t �p v′, t �dv′ − p(v, t) � βv, v′ �p v′, t �dv′ zt �√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√�2 0 0 Coagulation within BP mode

v ∞ +� cv − v′, v′ �n v − v′, t �p v′, t �dv′ − p(v, t) � cv, v′ �n v′, t �dv′ (2) �√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√��√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√�0 0 Interaction between SP and BP mode F − out p(v, t) . �√√√√√��√√√√√�V Ventilation

Equations (1) and (2) comprise a governing equation 3. Computations describing nanoparticle evolution due to coagulation, ventilation, and injection for the first time. By introducing )e numerical computations were all performed on an Intel sectional method, the above two equations can be written as (R) Pentium 4 CPU 3.00 GHz computer with memory 4 GB. )e 4-order Runge–Kutta method with fixed time step was dN F F used to solve the set of ordinary differential equations, and k � ω − N � c P − outN + F k, dt k k ik i V k in V the error function in the SM model was computed using the i�1 incomplete gamma function method. In all the computa- ( ) dP F 3 tions, the time step for the SM model was set to be 27 s. All k � ϵ + � η c N P − P � c N − outP , dt k ijk ij i j k ik i V k the programs were written by the C Programming language i�1 i�1 j�1 and were performed on Microsoft Visual C++ 6.0 compiler. In all the simulations, the temperature and pressure of the with surrounding air were assumed to be 300 K and 1.013 ×105 Pa, respectively. At the condition, the viscosity and the mean free ⎧⎪ −N � β N , k � 1, path of gas molecules were 1.85 ×10−5 Pa·s and 68.41 nm, ⎪ 1 i1 i ⎪ i�1 respectively. )e morphology of the nanoparticle was as- ⎪ ⎨ sumed to be spherical or fractal-like and its bulk density is ω � k ⎪ 1 1,000 kg/m3. ⎪ � N N N � N , k , ⎪ χijkβij i j − k βik i > 1 ⎩⎪ 2 i�1 i�1 j�1 4. Analysis and Results ⎧⎪ −P � β N , k � 1, ⎪ 1 i1 i 4.1. Evolution of NSD of a Bimodal Aerosol. In theory, large ⎪ i�1 particles have an ability to scavenge SP with large efficiency ⎪ ⎨ [19]. In this section, the interaction between BP and SP ϵ � k ⎪ 1 modes is investigated as only coagulation is considered. )e ⎪ � P P P � P , k , ⎪ χijkβij i j − k βik i > 1 (4) characteristics of particles at BP and SP modes are shown in ⎩⎪ 2 i�1 i�1 j�1 Table 1 where the BP mode is fixed, but the SP mode is changed in its GMD. ⎧⎪ v v + v In Figure 1, two aerosols are considered: one is com- ⎪ k+1 − �i j � ⎪ , if vk ≤ vi + vj ≤ vk+ , posed by the BP mode and SP mode (1), while the second is ⎪ v − v 1 ⎪ k+1 k composed by the BP mode and SP mode (2). In the evo- ⎪ ⎨⎪ lution of NSD, it shows that the BP mode changes negli- χijk �⎪ �v + v � − v gibly, but SP modes change largely, and especially, the ⎪ i j k−1, v v + v v , ⎪ if k−1 ≤ i j ≤ k particle size in the SP mode moves in the large size di- ⎪ vk − vk−1 ⎪ rection. Due to the scavenge effect of the BP mode on the SP ⎪ ⎩⎪ mode, particles with smaller size (SP mode (1)) in the SP 0, else. mode are found to be easier to be removed. It needs to be noted here larger difference of two modes in SP mode (1) In order to simplify the program code, the above two and in SP mode (2). From the comparison of two different systems use the same division of particle size. So, the co- aerosols, it is found that the larger the difference of two agulation rate cij � βij, and also the collision coefficient modes in their size, the easier to be scavenged for particles χijk � ηijk. at the SP mode. )is finding is further validated in Figure 2 In this study, the particle loss due to deposition was where three aerosols with different sources (1–3) and with described by Lai and Nazaroff’s model [13], and the co- the same BP mode are investigated. In Figure 2, it shows agulation kernel was from Fuchs model [19]. that the total particle number concentration decreases 4 Journal of Nanotechnology

Table 1: Characteristics of both BP and SP modes. 3 Mode TPN (#/m ) GMD (nm) GSD Df Fout (lpm) Fin (lpm) Deposition (friction velocity, cm/s) BP 4.34 ×1010 500 1.32 3.0 0.0 0.0 0.0 SP (1) 4.34 ×1012 100 1.32 3.0 0.0 0.0 0.0 SP (2) 4.34 ×1012 50 1.32 3.0 0.0 0.0 0.0 SP (3) 4.34 ×1012 24 1.32 3.0 0.0 0.0 0.0

1.6E + 0.6 106 SP mode (2) SP mode (1) 1.4E + 0.6 105 1.2E + 0.6 ) ) 3 3 1.0E + 0.6 104 (#/cm p E

d 8.0 + 0.5 log 6.0E + 0.5 103 dw / d 4.0E + 0.5 BP mode 102 2.0E + 0.5 Total number conc. (#/cm

E 0.0 +00 101 101 102 103 Diameter (nm) Figure 1: Evolution of NSD with calculated interval time of 0123 11.25 min. Red lines represent particle size distribution of aerosols Time (hour) composed by the BP mode and SP mode (1), and blue lines rep- SP (1) resent particle size distribution of aerosols composed by the BP SP (2) mode and SP mode (2). SP (3) Figure 2: Change of total particle number concentration of small more quickly for SP modes with smaller size than those modes in the presence of large mode. At the same large mode, the total particle number concentration decreases more quickly for with larger size. small modes with smaller size than those with larger size by BP mode and SP mode (2). 4.2. A Source Injected into a Chamber Filled with Background Particles. )ere have been some important results available in experimental studies of references for a source also has an effect on the appearance of the secondary mode. In injected continuously to background particles [20, 21]. this study, it shows that in the presence of background Here, the evolution of NSD in the presence of background particles, the secondary mode does not appear, while the particles and the interaction between modes are further secondary mode emerges in the case without background investigated based on the simulation method. In this particles. )e conclusion cannot be obtained using the bi- study, both initial background particles and sources take modal TEMOM, in which the details of PSD are lost when lognormal distributions, and the background particle size calculating [19]. distribution is characterized by its GMD of 500 nm, TPN In order to study the evolution of NSD for sources with of 4.34 ×1010 #/m3, and GSD of 1.32. )e source acts as an different GMD at the same background particles, the injection to the background particles with constant flow calculation for sources 1, 2, and 3 is performed, which is rate of 4.6 lpm, and the source parameters are shown in shown in Figure 4. From the comparison, it is found that Table 2. In this study, only coagulation and the injection of the source GMD is nearer to the size of BP, and the particle the source are considered. number concentration for the SP mode is higher. )is )e comparison of NSD with and without background should attribute to the coagulation rate which is increased particles is conducted in this work. Figure 3 shows the with the increased difference between particles. )e above evolution of particle size distribution with interval time of conclusion can be further validated in Figure 5 where the 225 min for source (1) with and without background particles. comparison of TPN of SP mode is represented for sources It shows that the presence of background particles reduces the 1, 2, and 3. It shows that the TPN increases with the in- particle number concentration in the SP mode but has crease of source GMD. )is validates the conclusion that a negligible effect on the particle size where the SP mode peak the background particles have larger scavenge effect on is located. In addition, the presence of background particles smaller particles. Journal of Nanotechnology 5

Table 2: Parameters used in (3) for sources. 3 Source TPN (#/m ) GMD (nm) GSD Df Fout (lpm) Fin (lpm) Deposition (friction velocity, cm/s) Source (1) 4.34 ×1012 100 1.32 3.0 0.0 4.6 0.0 Source (2) 4.34 ×1012 50 1.32 3.0 0.0 4.6 0.0 Source (3) 4.34 ×1012 24 1.32 3.0 0.0 4.6 0.0

E 1.0 +06 8.0E +05 ) 3 8.0E +05 Interval time: 225 min 6.0E +05 ) 3 6.0E +05 (#/cm p

d 4.0E +05

log 4.0E +05 dw / d 2.0E +05 2.0E +05 Total particle number conc. (#/cm

E 0.0E +00 0.0 +00 0102030 101 102 103 Time (hour) Diameter (nm) Source (1) Figure 3: Comparison of NSD with interval time of 225 min for source (1) in Table 2 with (blue) and without background particles Source (2) (purple). In the calculation, only coagulation is considered. Source (3) Figure 5: Comparison of total particle number concentration of the SP mode for sources (1–3). )e total particle number con- 1.2E +06 centration of the SP mode is increased with the increase of source Interval time: 225 min GMD.

1.0E +06

) particles. )is is especially important in the studies on the 3 8.0E +05 establishment of environmental standards. In this study, the

(#/cm BP mode takes a lognormal distribution with GMD of p 11 3 d 500 nm, TPN of 4.34 ×1.0 #/m , and GSD of 1.32, and SP 6.0E +05 log modes also take lognormal distributions whose parameters are listed in Table 3. In this study, we study the effect of SP dw / d 4.0E +05 total particle number concentration and GMD on the lag time to be removed. Once the SP total particle number 3 3 2.0E +05 concentration is below 1.00 ×10 #/cm , the SP nanoparticles are considered to be absolutely removed. 0.0E + 0.0 Table 3 shows the lag time for the SP mode to achieve its 3 3 101 102 103 total particle number concentration of 1.00 ×10 #/cm . Diameter (nm) Even at the same SP initial total particle number concen- Figure tration, there is a large difference in lag time to achieve the 4: Comparison of NSD with interval time of 225 min for 3 3 sources (1, 2, and 3) (black, blue, and purple) at the same back- number concentration of 1.00 ×10 #/cm . When the SP ground particles with its GMD of 500 nm, TPN of 4.34 ×1010 #/m3, initial total particle number concentration is increased from 12 13 3 and GSD of 1.32. Although all three sources have the same TPN, the 4.34 ×1.0 to 4.34 ×1.0 #/cm , the lag time also increases particle number concentration for SP mode at steady state is in- by a factor of 2.38, 3.91, and 3.84 when GMD is 100, 50, and creased with the increase of source GMD. 24 nm. )erefore, the lag time to be removed for small particles is not directly proportional to the initial SP number concentration. )e size of small particles is also an important 4.3. Scavenging Effect of Background Particles on Small indicator to evaluate how far small particles are removed by Particles. Although it is easy to conclude that background background particles. particles have larger scavenge effect on SP from the above Figure 6 shows the change of total particle number calculation, and it is also necessary to know how much time concentration of SP mode at different initial SP mode and it takes for small particles to be removed by background the same BP mode. At the same total source particle number 6 Journal of Nanotechnology

Table 3: Lag time for SP mode to achieve its total particle number concentration of 1.03 #/cm3. 3 Mode TPN (#/m ) GMD (nm) GSD Df Fout (lpm) Fin (lpm) Lag time BP 4.34 ×1011 500 1.32 3.0 0.0 0.0 — SP (1) 4.34 ×1012 100 1.32 3.0 0.0 0.0 9.00 SP (2) 4.34 ×1012 50 1.32 3.0 0.0 0.0 2.34 SP (3) 4.34 ×1012 24 1.32 3.0 0.0 0.0 0.44 SP (4) 4.34 ×1013 100 1.32 3.0 0.0 0.0 21.44 SP (5) 4.34 ×1013 50 1.32 3.0 0.0 0.0 9.17 SP (6) 4.34 ×1013 24 1.32 3.0 0.0 0.0 1.69

Abbreviations

107 NSD: Nanoparticle size distribution PBE: Population balance equation

) GSD: Geometric standard deviation 3 GMD: Geometric mean diameter 106 PSD: Particle size distribution NGDE: Nanoparticle general dynamics equation TPN: Total particle number concentration 105 SM: Sectional method ODE: Ordinary diﬀerential equation BPs: Background particles Total number conc. (#/cm conc. number Total SPs: Small particles. 104 Data Availability

103 )e data used to support the ﬁndings of this study are 0 5 10 15 20 available from the corresponding author upon request. Time (hour) SP (1) SP (4) Conflicts of Interest SP (2) SP (5) SP (3) SP (6) )e authors declare that they have no conﬂicts of interest.

Figure 6: Change of total particle number concentration of SP mode at different initial SP mode and the same BP mode. Acknowledgments )e authors thank the Zhejiang Provincial Natural Science Foundation of China (LQ15A020002 and LR16A020002), concentration, the small particles need less time to be re- the National Key Research and Development Program of moved with smaller size. For the same GMD, higher initial China (2017YFF0205501), and the National Natural Science SP total particle number concentration needs longer time to Foundation of China (11372299 and 11632016). be removed. References 5. Conclusions [1] M. Kulmala, “Atmospheric science. How particles nucleate and grow,” Science, vol. 302, no. 5647, pp. 1000-1001, 2003. In this article, two joint population balance equations are [2] Z. Varga, G. Wang, and J. Swan, “)e hydrodynamics of first established for investigating the interaction between colloidal gelation,” Soft Matter, vol. 11, no. 46, pp. 9009–9019, modes within nanoparticle-laden aerosol systems. In the 2015. joint equations, mass transfer between modes is reserved. [3] R. Zhang, G. Wang, S. Guo et al., “Formation of urban fine )e population balance equations are solved by the sectional particulate matter,” Chemical Reviews, vol. 115, no. 10, method. )e study mainly involves the effect of background pp. 3803–3855, 2015. particles on the evolution of small particles and the lag time [4] M. von Smoluchowski, “Versuch einer mathematischen for the small particles to be removed. It shows that the )eorie der Koagulationskinetik kolloider Losungen,”¨ Zeits- chrift fürPhysikalische Chemie, vol. 92, no. 9, pp. 129–168, particles with smaller size are easier to be removed by 1917. background particles, and especially, the lag time to be [5] M. Yu and J. Lin, “Nanoparticle-laden flows via moment removed is related not only to the number concentration of method: a review,” International Journal of Multiphase Flow, small particles but also to their sizes. In the presence of vol. 36, no. 2, pp. 144–151, 2010. background particles, no secondary mode is found for the [6] M. Yu and T. L. Chan, “A bimodal moment method model for small particle system during its evolution. submicron fractal-like agglomerates undergoing Brownian Journal of Nanotechnology 7

coagulation,” Journal of Aerosol Science, vol. 88, pp. 19–34, 2015. [7] J. I. Jeong and M. Choi, “A bimodal particle dynamics model considering coagulation, coalescence and surface growth, and its application to the growth of titania aggregates,” Journal of Colloid and Interface Science, vol. 281, no. 2, pp. 351–359, 2005. [8] Y. H. Liu and H. Gu, “)e Taylor-expansion method of moments for the particle system with bimodal distribution,” :ermal Science, vol. 17, no. 5, pp. 1542–1545, 2013. [9] H. Liu, M. Yu, Z. Yin, Y. Jiang, and M. Chen, “Study on the evolution of nanoparticle size distribution due to continuous injection using the sectional method,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 24, no. 8, pp. 1803–1812, 2014. [10] M. Yu, Y. Liu, and A. J. Koivisto, “An efficient algorithm scheme for implementing the TEMOM for resolving aerosol dynamics,” Aerosol Science and Engineering, vol. 1, no. 3, pp. 119–137, 2017. [11] H. M. Hulburt and S. Katz, “Some problems in particle technology: a statistical mechanical formulation,” Chemical Engineering Science, vol. 19, no. 8, pp. 555–574, 1964. [12] M. Frenklach, “Method of moments with interpolative closure,” Chemical Engineering Science, vol. 57, no. 12, pp. 2229–2239, 2002. [13] R. McGraw, “Description of aerosol dynamics by the quadrature method of moments,” Aerosol Science and Technology, vol. 27, no. 2, pp. 255–265, 1997. [14] K. W. Lee, H. Chen, and J. A. Gieseke, “Log-normally preserving size distribution for Brownian coagulation in the free- molecule regime,” Aerosol Science and Technology, vol. 3, no. 1, pp. 53–62, 1984. [15] D. L. Marchisio, R. D. Vigil, and R. O. Fox, “Quadrature method of moments for aggregation-breakage processes,” Journal of Colloid and Interface Science, vol. 258, no. 2, pp. 322–334, 2003. [16] M. Yu, J. Lin, and T. Chan, “A new moment method for solving the coagulation equation for particles in Brownian motion,” Aerosol Science and Technology, vol. 42, no. 9, pp. 705–713, 2008. [17] M. Kraft, “Modelling of particulate processes,” KONA Powder and Particle Journal, vol. 23, no. 23, pp. 18–35, 2005. [18] A. C. K. Lai and W. W. Nazaroff, “Modeling indoor particle deposition from turbulent flow onto smooth surfaces,” Journal of Aerosol Science, vol. 31, no. 4, pp. 463–476, 2000. [19] N. A. Fuchs, :e Mechanics of Aerosols, Dover Publications, Mineola, NY, USA, 1964. [20] A. Joonas Koivisto, M. Yu, K. Hameri,¨ and M. Seipenbusch, “Size resolved particle emission rates from an evolving indoor aerosol system,” Journal of Aerosol Science, vol. 47, pp. 58–69, 2012. [21] M. Seipenbusch, A. Binder, and G. Kasper, “Temporal evolution of nanoparticle aerosols in workplace exposure,” An- nals of Occupational Hygiene, vol. 52, no. 8, pp. 707–716, 2008. Hindawi Journal of Nanotechnology Volume 2018, Article ID 7602982, 9 pages https://doi.org/10.1155/2018/7602982

Research Article Experimental Study on Expansion Characteristics of Core-Shell and Polymeric Microspheres

Pengxiang Diwu,1 Baoyi Jiang,2 Jirui Hou,1 Zhenjiang You ,3 Jia Wang,1 Liangliang Sun,1 Ye Ju,4 Yunbao Zhang,5 and Tongjing Liu 1

1Enhanced Oil Recovery Institute, China University of Petroleum, Beijing 102249, China 2China Huadian Institute of Science and Technology, Beijing 102249, China 3Australian School of Petroleum, 'e University of Adelaide, Adelaide, SA 5005, Australia 4China Oilﬁeld Services Limited, Tianjin 300452, China 5CNOOC China Limited, Tianjin Branch, Tianjin 300452, China

Correspondence should be addressed to Tongjing Liu; [email protected]

Received 16 February 2018; Revised 30 March 2018; Accepted 4 April 2018; Published 21 May 2018

Academic Editor: Mingzhou Yu

Copyright © 2018 Pengxiang Diwu et al. /is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Traditional polymeric microsphere has several technical advantages in enhancing oil recovery. Nevertheless, its performance in some field application is unsatisfactory due to limited blockage strength. Since the last decade, novel core-shell microsphere has been developed as the next-generation profile control agent. To understand the expansion characteristic differences between these two types of microspheres, we conduct size measurement experiments on the polymeric and core-shell microspheres, respectively. /e experimental results show two main differences between them. First, the core-shell microsphere exhibits a unimodal distribution, compared to multimodal distribution of the polymeric microsphere. Second, the average diameter of the core-shell microsphere increases faster than that of the polymeric microsphere in the early stage of swelling, that is, 0–3 days. /ese two main differences both result from the electrostatic attraction between core-shell microspheres with different hydration degrees. Based on the experimental results, the core-shell microsphere is suitable for injection in the early stage to block the near-wellbore zone, and the polymeric microsphere is suitable for subsequent injection to block the formation away from the well. A simple mathematical model is proposed for size evolution of the polymeric and core-shell microspheres.

1. Introduction been performed in Shengli, Jidong, and Dagang oilfields in China and Mannville Pools in Canada [6], where effec- Waterflooding is the most common technique in oil res- tive water control and enhanced oil recovery were observed ervoir development. In high-permeability reservoirs, such as [7–10]. In other cases, transport and retention of micro- Bohai oil field in eastern China, the heterogeneity usually spheres or particles may lead to well productivity impair- deteriorates sharply in the late stage of waterflooding [1, 2]. ment [11–13]. In low-permeability reservoirs, such as Changqing oil field /ere have been many types of polymeric microspheres, in western China, injection water channeling occurs by such as colloidal dispersion gel, gel microsphere (PPG), bright the interwell fracture network. /erefore, the polymeric water, and pH-sensitive cross-linked polymers [14–18]. /e microsphere, a profile control agent, is developed to improve major difference among these microspheres is the expansion the water swept volume [3]. When it meets water in the size versus swelling time [19, 20]. Formation temperature, reservoir, the polymeric microsphere swells and can be salinity, and swelling time affect the expansion performance of migrated, or retained. Following elastic deformation, it may microspheres. /e expansion factor of grain diameter grad- be remigrated and recaptured [4]. Moreover, this profile ually increases with the increasing temperature and swelling control agent can adapt to the characteristics of pore throat, time, whereas it decreases with the increasing salinity [21–24]. avoiding the injection difficulty [5]. Some pilot tests have However, some field applications in high-heterogeneity 2 Journal of Nanotechnology reservoirs are unsuccessful because it is fragile after swelling – and is prone to deformation under certain pressure [25]. To overcome the disadvantages of the polymeric microsphere, a novel core-shell microsphere is developed as the + next-generation profile control agent [26–29], which has a core-shell double-layered structure. It consists of an inner layer and an outer layer. /e inner layer has positive charge, whereas the outer layer has negative charge [30, 31]. /e – + + – core-shell microsphere swells with water, in which the core part swells fast, whereas the shell part expands relatively slowly. /erefore, the core-shell microsphere can easily become a bipolar microsphere with positive and negative + charges simultaneously [24, 32, 33]. It is quite difficult to precisely determine the local structure of microspheres using the conventional diffraction – techniques because of the nanoscale grain size and surface modification [34]. Research on expansion characteristic Figure 1: Schematic of the core-shell microsphere, which has differences between the core-shell and polymeric micro- a core-shell double-layered structure. sphere is not available in the literature. In the present work, we investigate the size distribution characteristics and average diameter evolution during expansion of the two types of microspheres, that is, the polymeric microsphere and core-shell microsphere. In Section 2, we analyze the main feature of the core-shell microsphere expansion mechanism. In Section 3, laboratory devices and materials are introduced, and the experimental procedure to measure the expansion size variation of the microsphere is presented. In Section 4, size distribution and average diameter evolution of two types of microspheres are analyzed. Section 5 presents two mathematical models for average diameter evolution. Conclusions in Section 6 finalize the paper. Figure 2: MICROTRAC S3500 laser particle size analyzer.

2. Core-Shell Microsphere Expansion Analyses influence on pore blocking [8, 13, 31]. To find out the 2.1. Core-Shell Microsphere Expansion Mechanism. Figure 1 difference of hydrated sizes between novel core-shell mi- illustrates the structure of a core-shell microsphere, which crosphere and traditional polymeric microsphere, we design consists of an outer layer and an inner layer which is gel core. and perform the experimental studies on the size distri- Since the rock surface usually has negative charge, it is bution evolution of these two types in Section 3. designed on purpose that the outer layer has negative charge, whereas the inner layer has positive charge [30, 31]. /e 2.2. Experiment Principle. /e particle size analyzer, traditional polymeric microsphere, which is mainly made of MICROTRAC S3500, is the key equipment employed in gel, does not have an outer layer. Polymeric microspheres or this study. It uses three precisely placed red laser diodes to the gel core is a viscoelastic plugging agent with 3D structure accurately characterize particle sizes. /e patented Tri- which can absorb much more water if compared to its own Laser System provides us accurate, reliable, and repeat- mass [35–37]. able particle size analysis for a diverse range of applications. /e core-shell microsphere has a core-shell double-layered It utilizes the proven theory of Mie compensation for structure. It swells with water, in which the core part swells fast, spherical particles and the proprietary principle of modi- whereas the shell part expands relatively slowly. /e core-shell fied Mie calculations for nonspherical particles, re- microsphere can become a bipolar microsphere easily with spectively. /e particle size analyzer can measure particle positive and negative charges simultaneously by hydration sizes from 0.02 to 2800 μm. In the experiments, polymeric [24, 32]. Consequently, multiple microspheres attract to each and core-shell microspheres are baked at 70°C to simulate other, agglomerate, and gradually form into a string or group formation condition, and then their size distributions and [24, 32]. /erefore, the size expansion mechanisms of the core- variations are measured by the particle size analyzer. shell microsphere are agglomeration caused by electrostatic attraction and gel swelling, while mainly, the later one applies 3. Laboratory Study to the traditional polymeric microsphere. In the profile control process, the initial and expanded In this section, we describe laboratory setup and materials sizes of core-shell microspheres need to match the pore (Section 3.1) and the procedure of microsphere size mea- throat sizes of porous media, in order to have remarkable surement (Section 3.2). Journal of Nanotechnology 3

12 100 10 100 9 10 80 8 80 8 7 60 6 60 6 5 40 4 40

Probability 4 Probability 3 2 20 2 20 Cumulative probability Cumulative Cumulative probability Cumulative 1 0 0 0 0 0 5 1015202530 0 10 20 30 40 50 60 Microsphere size (μm) Microsphere size (μm)

Probability Probability Cumulative probability Cumulative probability

(a) (b) 7 100 5 100 6 80 4 80 5 4 60 3 60

3 40 2 40

Probability 2 Probability 20 1 20 1 Cumulative probability Cumulative Cumulative probability Cumulative 0 0 0 0 0 25 50 75 100 125 150 0 30 60 90 120 150 180 Microsphere size (μm) Microsphere size (μm)

Probability Probability Cumulative probability Cumulative probability

5 80 4 60 3 40

Probability 2

1 20 Cumulative probability Cumulative 0 0 0 55 110 165 220 275 330 Microsphere size (μm)

Probability Cumulative probability

(e)

Figure 3: Polymeric microsphere size distribution at diﬀerent periods of hydration: (a) 0 days, (b) 3 days, (c) 7 days, (d) 14 days, and (e) 21 days.

3.1. Laboratory Setup and Materials. In the experiments, the of 8.13 μm. /e total salinity of formation water in the test is main equipment is the MICROTRAC S3500 laser particle 5863 mg/L. size analyzer (Figure 2). /e other tools include a separating funnel, a magnetic stirrer, analytical balance, and 3.2. Procedure of Laboratory Study. To measure size distri- thermostat. butions and their variation of the two different microspheres, /e particles used in the experiments are polymeric and we design the procedure of the laboratory study as follows. core-shell microspheres, respectively. /e polymeric microsphere is originated from polymeric nanoparticles with sizes around 300 nm. Because of initial agglomeration, the poly- 3.2.1. Microsphere-Dispersive Liquid Preparation. To pre- meric microsphere is formed with the average diameter pare the particle-dispersive liquid with concentration 0.2%, 4.67 μm. /e core-shell microsphere has the average diameter the polymeric or core-shell microspheres are added into the 4 Journal of Nanotechnology formation water gradually, keeping the rotating speed of 45 magnetic stirrer at 500 rpm for 30 minutes. 40 35 30 3.2.2. Microsphere-Degreasing Treatment. Due to the limitation of synthesis technology, there are usually some oil and 25 surfactants in the dispersive liquid. In order to observe and 20 measure the microsphere size accurately, we propose the 15 following degreasing operations: 10 Average diameter ( μ m) diameter Average 5 (1) Mix 600 mL of N-hexane into 300 mL particle- 0 dispersive liquid. /e volume ratio of N-hexane to 0 5 10152025 microsphere-dispersive liquid is 2 :1 Hydration time (days) (2) Put the mixed solution into a conical flask airtight Figure 4: Average diameter of polymeric microsphere versus and then stir it for two hours by using the magnetic hydration time. stirrer at 700 rpm (3) Move the stirred mixed solution into the separating funnel and leave it until there appear obvious two 5.41 times. Since then, the polymeric microsphere keeps layers in the solution constant expansion velocity (Figure 3(d)). On hydration (4) Collect the lower layer solution, which is microsphere- time of 21 days, the average diameter continues to increase to dispersive liquid 40.33 μm by 8.6 times, compared with the initial size (5) Repeat steps (1)–(4) twice. (Figure 3(e)). /e size distribution of polymeric microspheres may be used to calculate the fractal dimension [38, 39], which can then be applied to evaluate the ag- 3.2.3. Initial Microsphere Size Measurement. Use a dispos- glomeration degree at different periods of hydration [40]. able pipette to aspirate a small amount of degreased /e average diameter of polymeric microsphere in- microsphere-dispersive liquid and then measure the initial creases monotonically (Figure 4). /e polymeric micro- microsphere size by the particle size analyzer. sphere sizes increase relatively slowly in the early stage, that is, 0–3 days, which coincides with the results from previous research [24]. /is is because the polymeric microsphere 3.2.4. Microsphere Expansion Size Measurement needs time to unfold the polymer structure before the ag- (1) Place the degreased microsphere-dispersive liquid glomeration. From the point of view of profile control, the into a thermostat at 70°C, bake it, and then take polymeric microsphere can be injected into low permeability samples at various times: 3 days, 7 days, 14 days, and formation easily because of its small initial size. Moreover, 21 days the polymeric microsphere is able to filtrate into deep (2) Put the samples on the magnetic stirrer and stir it formation because of its slow expansion speed in the early continuously for 5 minutes stage. (3) Use a disposable pipette to aspirate a small amount of degreased microsphere-dispersive liquid and then measure the microsphere expansion size distribution 4.2. Core-Shell Microsphere Size Distribution. Following the by the particle size analyzer. experimental procedure in Section 3, we also obtained the core-shell microsphere size distribution at different periods 4. Experimental Results of hydration, that is, 0 days, 3 days, 7 days, 14 days, and 21 days, as shown in Figure 5. 4.1. Polymeric Microsphere Size Distribution. Following Under the conditions of constant salinity of 5863.27 mg/L the experimental procedure in Section 3, we obtained the and temperature of 70°C, we observed the unimodal distri- polymeric microsphere size distribution at different periods bution behavior in the core-shell microsphere expansion of hydration, that is, 0 days, 3 days, 7 days, 14 days, and process (Figure 5). /e maximal probability of microsphere 21 days, as shown in Figure 3. size increases with time, which is different from the polymeric Under the conditions of constant salinity of 5863.27 mg/L microsphere behavior shown in Figure 3. /is difference and temperature of 70°C, we observed the multimodal dis- indicates that the agglomeration effect of core-shell micro- tribution behavior in polymeric microsphere swelling process sphere is stronger than that of the polymeric microsphere. (Figure 3). /e maximal probability of microsphere size is less /e initial average diameter is 8.13 μm (Figure 5(a)) and than the initial value, that is, 10%. then expands 4.00 times to 32.48 μm after hydration for 3 /e initial average diameter of the polymeric micro- days (Figure 5(b)). It increases to 53.00 μm after 7 days sphere is 4.67 μm (Figure 3(a)). It results from agglomera- (Figure 5(c)), expanding by 6.50 times. Before hydration tion of polymeric nanoparticles [6, 9, 37]. After hydration for time of 7 days, the core-shell microsphere has con- 3 days, it increases 2.04 times to 9.41 μm (Figure 3(b)). After stant expansion velocity, which is higher than that dur- 7 days, it increases to 25.57 μm (Figure 3(c)), expanding by ing 7–21 days. On hydration time of 21 days, the average Journal of Nanotechnology 5

7 100 10 100 9 6 80 8 80 5 7 4 60 6 60 5 3 40 4 40

Probability 2 Probability 3 20 2 20 1 Cumulative probability Cumulative 1 probability Cumulative 0 0 0 0 0 7 1421283542 0 369121518 Microsphere size (μm) Microsphere size (μm)

Probability Probability Cumulative probability Cumulative probability

(a) (b) 10 100 18 100 9 16 8 80 14 80 7 12 6 60 60 10 5 8 4 40 40

Probability 3 Probability 6 2 20 4 20

1 probability Cumulative 2 probability Cumulative 0 0 0 0 0 369121518 0 25 50 75 100 125 150 Microsphere size (μm) Microsphere size (μm)

Probability Probability Cumulative probability Cumulative probability

6 40

Probability 4 20 2 Cumulative probability 0 0 0 35 70 105 140 175 210 Microsphere size (μm)

Probability Cumulative probability

(e)

Figure 5: Core-shell microsphere size distribution at different periods of hydration: (a) 0 days, (b) 3 days, (c) 7 days, (d) 14 days, and (e) 21 days. diameter continues to increase to 63.81 μm by 7.8 times, that we can use the core-shell microsphere for near- compared with the initial size (Figure 5(e)). wellbore zone blockage, whereas the polymeric micro- /e average diameter of the core-shell microsphere sphere for deep formation blockage. Based on the X-ray increases monotonically (Figure 6). As shown in Figures 5 computed tomography (CT), 3D digital core structure and 6, the core-shell microsphere has different swelling models [41] can be developed to evaluate the applicability velocities and average diameter from the polymeric mi- of different types of microspheres. crosphere. First, the core-shell microsphere has higher expansion velocity than the polymeric microsphere in the early stage, that is, 0–3 days. Second, the core-shell mi- 4.3. Comparison between Two Types of Microspheres. Both crosphere has relatively larger average diameter than the the polymeric and core-shell microspheres have micron- polymeric microsphere. /ese two key features indicate scale diameters. With the increasing hydration time, the 6 Journal of Nanotechnology

70 Table 1: Coeﬃcients by matching average diameters. 60 Particles di0 (μm) dimax (μm) ash c0 c1 c2 n Polymeric 50 4.67 56 0.10 18 8 5 3 microspheres 40 Core-shell 8.13 81 0.18 −19 46 −0.1 5 30 microspheres

Average diameter ( μ m) diameter Average 10 hydration degrees. /erefore, we conclude that the electrostatic interaction mainly acts in the early stage, that is, 0–3 0 0 510152025days for the core-shell microsphere. Hydration time (days) Apparently, the average diameter calculation plays an important role on microsphere optimization and ﬁeld ap- Figure 6: Average diameters of core-shell microspheres versus plication design. In reservoir-scale numerical simulation, hydration time. a mathematical model is necessary to describe the evolution of the microsphere size. In the next section, we apply a simple model to match the curves in Figure 7 and evaluate 70 the model feasibility on two types of microspheres.

50 5. Mathematical Models for Average Diameter Evolution 40

30 Based on the experimental results, we apply the traditional mathematical model (1) to describe average diameter vari- 20 ation versus hydration time. If the hydration time does not

Average diameter ( μ m) diameter Average 10 exceed the critical time twc, the average diameter increases with hydration time. /e average diameter is calculated as 0 0 5 10152025follows: Hydration time (days) a t d � d +d − d � sh w , (1) i i0 imax i0 1 + a t Core-shell microsphere sh w Polymeric microsphere where di is the microsphere average diameter at time tw, di0 Figure 7: Comparison of average diameter evolution between is the initial average diameter, dimax is the maximal average polymeric and core-shell microspheres. diameter, tw is the hydration time, and ash is the coefficient based on experimental data. Apparently, there is accelerated size evolution in the sizes of two types of microspheres increase gradually. hydration process of polymeric microspheres (Figure 7). However, there are two main differences between them. Accordingly, we present a new mathematical model (2) to Compared with the size distribution of polymeric mi- describe this feature: crosphere (Figure 3), the size distribution of core-shell 1/n di � c0 − c1c2 − tw � , (2) microsphere is narrower (Figure 5) due to positive and negative electrostatic interactions between different micro- where c0, c1, and c2 are coefficients based on experimental spheres. In synthesis of core-shell microspheres, the shell data. part is mainly made from acrylamide and anion monomer, By matching the curves in Figure 7 with (1) and (2), whereas the core part is mainly made from acrylamide and respectively, we obtain the corresponding coefficients, which cation monomer [42]. /e 3D microscopic graphs show that are shown in Table 1. the core-shell microspheres swell to different sizes, and the Figure 8(a) indicates that the traditional mathematical smaller ones will be attached around the bigger ones [43], model (1) yields a good fitting with measured data for the which indicate that there is an electrostatic interaction be- core-shell microsphere throughout the expansion process. tween different microspheres. /e slow-expansion feature of the polymeric microsphere in /e average diameter of the core-shell microsphere the early stage (0–3 days) cannot be captured by the model. increases faster than that of the polymeric microsphere in /erefore, in the average diameter calculation, the tradi- the hydration time of 0–3 days (Figure 7). /e expansion tional model (1) is applicable to the core-shell microsphere velocity of the polymeric microsphere mainly shows the but not to the polymeric microsphere. original swelling velocity which is caused by polymer gel In comparison, Figure 8(b) illustrates that the pro- swelling. /e higher expansion velocity of the core-shell posed model (2) can capture the size evolution behavior microsphere in 0–3 days results from the electrostatic at- not only for core-shell microspheres but also for poly- traction between core-shell microspheres with different meric microspheres. Journal of Nanotechnology 7

70 (3) /e traditional mathematical model is applicable to 60 average diameter evolution of the core-shell microsphere but not to the polymeric microsphere in 50 the early stage of hydration. /e proposed model 40 can be applied for both core-shell and polymeric microspheres. 30 (4) /e core-shell microsphere has larger initial size and 20 expands faster than the polymeric microsphere.

Average diameter ( μ m) diameter Average 10 /erefore, the core-shell microsphere is suitable for injection in the early stage to block the near-wellbore 0 0 5 10152025 zone, and the polymeric microsphere is suitable for Hydration time (days) subsequent injection to block the formation away from the well. Polymeric Core-shell microsphere-exp. microsphere-cal. Polymeric Core-shell Data Availability microsphere-cal. microsphere-exp. /e datasets used to support this study are currently under (a) embargo while the research ﬁndings are commercialized. 70 Requests for data, 12 months after initial publication, will be 60 considered by the corresponding author. 50 Conflicts of Interest 40 /e authors declare no conﬂicts of interest regarding the 30 publication of this paper. 20

Average diameter ( μ m) diameter Average 10 Acknowledgments

0 /is work was supported by the National Science 0 5 10152025and Technology Major Projects (2017ZX05009004 and Hydration time (days) 2016ZX05058003) and the Beijing Natural Science Foundation Polymeric Core-shell (2173061). microsphere-exp. microsphere-cal. Polymeric Core-shell References microsphere-cal. microsphere-exp. [1] J. M. Perez, S. W. Poston, and C. M. Edwards, “/e effect of (b) micro-fractures on oil recovery from low-permeability reservoirs,” in Proceedings of Low Permeability Reservoirs Figure 8: Matching curves of microsphere average diameters by Symposium, vol. 1993, Denver, CO, USA, April 1993. (a) traditional model (1) and (b) proposed model (2). [2] D.-F. Song, Y.-P. Jia, Y. Li et al., “Further enhanced oil recovery by using polymer minispheres at Gudao oil field after polymer flood,” Oilfield Chemistry, vol. 25, no. 2, pp. 165–169, 2008. 6. Conclusions [3] C. Yi, X.-A. Yue, and S.-R. Yang, “Experimental study on polymer microsphere emulsion profile control and flooding in Experimental study on expansion characteristics of two heterogeneity of reservoir,” Advanced Materials Research, types of microspheres leads to the following conclusions: vol. 361-363, pp. 437–440, 2011. [4] Y.-B. Jin, “Polymer microspheres reservoir adaptability (1) /e size distribution evolution of the core-shell evaluation method and the mechanism of oil displace- microsphere is different from that of the poly- ment,” Petrochemical Technology, vol. 46, no. 7, pp. 925– meric microsphere. /e core-shell microsphere ex- 933, 2017. hibits unimodal distribution, dominated by the [5] C.-J. Liu, “Research and application of polymer microspheres agglomeration effect. /e polymeric microsphere deep profile control technology,” Drilling and Production exhibits multimodal distribution, resulting from the Technology, vol. 33, no. 3, pp. 62–64, 2010. swelling effect. [6] R. Irvine, J. Davidson, M. Baker et al., “Nano spherical polymer pilot in a mature 18° API sandstone reservoir water (2) /e expansion process of polymeric microspheres flood in Alberta, Canada,” in Proceedings of SPE Asia Pacific can be divided into two stages. In the early stage, that Enhanced Oil Recovery Conference, vol. 2015, Kuala Lumpur, is, 0–3 days, it swells slowly as it needs time to unfold Malaysia, August 2015. the polymer structures. In the late stage, that is, 7–21 [7] T. Wan, R. Huang, Q. Zhao et al., “Synthesis and swelling days, it swells faster because of the agglomeration properties of corn stalk-composite super-absorbent,” Journal of between different microspheres. Applied Polymer Science, vol. 130, no. 1, pp. 698–703, 2013. 8 Journal of Nanotechnology

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Zhang, Charge-Stabilized Dispersion Polymerization to Control Technology of Functional Polymer Microspheres, Prepare Polymer Particles, Beijing University of Chemical Shandong University, Jinan, China, 2013. Technology, Beijing, China, 2010. [31] H.-B. Wang, “Research and application of adjust profile [15] B. Wang, G. Zhang, L. Huang et al., “Oilwell cement com- system of new polymer microballoons,” Inner Mongolia position YH98-07 for water control in high water cut pro- Petrochemical Industry, vol. 37, no. 23, pp. 133–135, 2011. duction wells,” Oilfield Chemistry, vol. 19, no. 3, pp. 230–232, [32] Q.-T. Zhou, Study on Depth Profile Control Technology by 2002. Core-Shell Polymeric Microspheress, China University of Pe- [16] X.-Z. Xiu, Y. Wen, B.-H. Yang et al., “Screening and demulsi- troleum, Beijing, China, 2011. fication capability of demulsifying bacterial strain TR-1 for crude [33] I. Odler and M. Gasser, “Mechanism of sulfate expansion in oil dehydration,” Oilfield Chemistry, vol. 23, no. 2, pp. 162–165, hydrated portland cement,” Journal of the American Ceramic 2006. Society, vol. 71, no. 11, pp. 1015–1020, 1988. [17] H.-Z. Zhao, X.-Y. Zheng, M.-Q. Lin et al., “Preparation and [34] H. Zhu, Q. Li, Y. Ren et al., “Hydration and thermal expansion performance research of water-soluble crosslinked poly- in anatase nano-particles,” Advanced Material, vol. 28, no. 32, mer microspheres,” Fine Chemicals, vol. 22, pp. 62–65, pp. 6894–6899, 2016. 2005. [35] H. Zhao, M. Lin, J. Guo, F. Xu, Z. Li, and M. Li, “Study on [18] H.-X. Ma and M.-Q. Lin, “Study of property of linked polymer plugging performance of cross-linked polymer microspheres micro-ball,” Applied Chemical Industry, vol. 35, no. 6, with reservoir pores,” Journal of Petroleum Science and En- pp. 453-454, 2006. gineering, vol. 105, no. 3, pp. 70–75, 2013. [19] X.-C. Chen, “Mechanistic modeling of gel microsphere sur- [36] H.-B. Yang, W. Kang, X. Yin et al., “Research on matching factant displacement for enhanced oil recovery after polymer mechanism between polymer microspheres with different flooding,” in Proceedings of SPE/IATMI Asia Pacific Oil and storage modulus and pore throats in the reservoir,” Powder Gas Conference and Exhibition, vol. 2015, Bali, Indonesia, Technology, vol. 313, pp. 191–200, 2017. 2015. [37] D.-M. Zhu, F. Wang, Z. Xu et al., “A facile method for [20] Y.-P. Sun, C. Dai, Y. Fang et al., “Imaging of oil/water fabrication of core-shell particles and hollow spheres by using migration in tightsand with nuclear magnetic resonance hydrophobic interaction and the thermo-sensitive phase and microscope during dynamic surfactant imbibition,” in separation,” Journal of Nanjing University, vol. 43, no. 5, Proceedings of SPE/IATMI Asia Pacific Oil and Gas Con- pp. 483–488, 2007. ference and Exhibition, vol. 2017, Jakarta, Indonesia, October [38] Y.-X. Xia, J.-C. Cai, W. 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characterization,” Advances in Geo-Energy Research, vol. 1, no. 1, pp. 18–30, 2017. [42] Y.-Q. Zhang, R.-Y. Ma, Q.-B. Wei et al., “Fabrication of core- shell polymer microspheres of P(St-AM) and the corresponding photonic crystals ﬁlms,” Acta Polymerica Sinica, vol. 12, no. 6, pp. 648–652, 2012. [43] Y.-Q. Jia, L. Zheng, and W. Chen, “Preparation of core-shell polymer microsphere and its laboratory evaluation,” Journal of Yangtze University, vol. 12, no. 28, pp. 38–42, 2015. Hindawi Journal of Nanotechnology Volume 2018, Article ID 2615404, 10 pages https://doi.org/10.1155/2018/2615404

Research Article Two-Dimensional Numerical Study on the Migration of Particle in a Serpentine Channel

Yi Liu, Qucheng Li, and Deming Nie

Institute of Fluid Mechanics, China Jiliang University, Hangzhou, China

Correspondence should be addressed to Deming Nie; [email protected]

Received 2 February 2018; Accepted 4 April 2018; Published 17 May 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Yi Liu et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this work, the momentum exchange scheme-based lattice Boltzmann method is adopted to numerically study the migration of a circular particle in a serpentine channel for the range of 20 ≤ Re ≤ 120. *e effects of the Reynolds number, particle density, and the initial particle position are taken into account. Numerical results include the streamlines, particle trajectories, and final equilibrium positions. Close attention is also paid to the time it takes for the particle to travel in the channel. It has been found that the particle is likely to migrate to a similar equilibrium position irrespective of its initial position when Re is large. Furthermore, there exists a critical solid-to-fluid density ratio for which the particle travels fastest in the channel.

1. Introduction it does not require external forces or multiple streams to focus particles. *e most famous phenomena of inertial Solid particles immersed in a viscous fluid lead to a two-phase focusing may be the Segre–Silberberg´ effect [2], which is flow problem, which is very common in nature and in many a fluid dynamic separation effect where a dilute suspension industrial processes, including atmospheric currents, aerosol of neutrally buoyant particles flowing in a tube equilibrates deposition, fluidized beds, and so on. *e motion and dy- at a distance ∼0.6R (tube radius) from the tube center. Later namics of particles suspended in a fluid is fundamental to a full analytical solution of the forces that dominate particles understanding suspension hydrodynamics. Over the past in Poiseuille flow was provided by Ho and Leal [3]. *ey [3] decade, great progress has been made for microfluidic devices showed that particles migrate from the center of a channel because of several benefits over conventionally sized systems, towards the wall due to shear-induced lift forces and are such as small volume of sample and reagent, low energy rejected from the channel perimeter by wall-induced lift consumption, high efficiency, and enhanced analytical sen- forces creating a stable equilibrium at a distance of 0.6R from sitivity. In microfluidic devices, manipulation and separation the center of the channel. Feng et al. [4] were the first to of particles are usually necessary in the processes of enzymatic observe the Segre–Silberberg´ effect for the motion of a single analysis, DNA analysis, and sample separation. However, it is circular particle in plane Poiseuille flow by using the finite- necessary to focus the samples in a tight stream before sepa- element method. Pan and Glowinski [5] simulated the ration, sorting, or analysis in order to ensure that these samples motion of multiple circular particles in plane Poiseuille flow. are passing through the microchannels quickly. *erefore, *eir results [5] showed that the collisions between particles understanding the behavior and characteristics of particle have a significant influence on the inertial migration of suspensions in microfluidics is helpful to provide insight into particles. Chun and Ladd [6] investigated the inertial mi- the design of microfluidic channels [1]. gration of neutrally buoyant particles in a square duct. For *ere are several methods of particle focusing that have the case of elliptical particles, Yi et al. [7] reported that the been developed and used in microfluidic systems. Inertial particles fluctuate about an averaged position at low Rey- focusing is usually adopted to align the particles along a tight nolds numbers while they converge to an equilibrium po- stream, which has the advantage over other methods because sition on each side of the channel center at high Reynolds 2 Journal of Nanotechnology numbers. Chen et al. [8] studied the motion of two spherical which presents some primitive results involving particle particles in tube flow through the lattice Boltzmann method migration in a serpentine channel. On basis of this, we aim (LBM). *ey reported an oscillatory state of motion for two to further present a thorough two-dimensional numerical spherical particles with different radii in opposite sides. study of a single particle migrating in a serpentine channel Similarly, Abbas et al. [9] simulated the motion of a spherical through direct numerical simulations (DNS). *e effects of particle in a square channel flow. *ey demonstrated that the Reynolds number as well as the initial position of particle there exist two states for the migration of particle which are on the particle migration are studied. Numerical results in- cross-streamline stage and cross-lateral stage, respectively. clude the streamline, particle trajectory, and the equilibrium In addition, they [9] showed that the former stage is much position of particle. Close attention will be paid to the time it faster than the latter one. Recently, Jiang et al. [10] in- takes for the particle to travel in the serpentine channel. We vestigated the migration of particles in a symmetrical ser- hope the simulation results would be helpful for the designing pentine channel through a three-dimensional LBM. *ey of microfluidic devices. focused on the influence of the Dean flow on particle focusing in a serpentine channel. *eir results showed that the 2. Numerical Model alternation of the Dean flow direction has special hydrodynamic effects to focus or separate particles of different 2.1. Lattice Boltzmann Method. In this work, the lattice sizes as the flow intensity becomes stronger. Bhatnagar–Gross–Krook (LBGK) Boltzmann method pro- So far, the inertial focusing has numerous applications in posed is used to solve the fluid flow [17]: microparticle manipulation ranging from microfluidic cell 1 (eq) sorting to particle separation and ordering [11–13]. How- f x + c Δt, t + Δt� − f (x, t) � − �f (x, t) − f (x, t)�, i i i τ i i ever, due to the compact size of microfluidic devices, the ( ) majority of the biomedical processes of inertial focusing are 1 carried out in the curved channels, such as expansion- (eq) where fi(x, t) and fi (x, t) are the distribution functions contraction array channel, spiral channel, and serpentine and corresponding equilibrium distribution functions as- channel. In comparison with the straight channel, the flow sociated with the ith discrete velocity direction ci. Δt is the conditions are very different in the curved channels. For time step and τ is the relaxation time, respectively. In the instance, it is naturally expected that the boundary layer two-dimensional nine-velocity lattice (D2Q9), the model separation will take place in the corners of a serpentine proposed by Qian et al. [17] is adopted here, channel, which usually affects the flow structure and of 0 1 0 −1 0 1 −1 −1 1 course the motion of particles. *e centrifugal force in the c � c� �. (2) curved channels is also a key factor which may dominate the 0 0 1 0 −1 1 1 −1 −1 migration of particles and should be taken into account [14]. *e lattice speed is c � Δx/Δt, where Δx is the lattice Furthermore, it is possible that the hydrodynamic in- (eq) teractions between particles in the curved channels are more spacing. fi (x, t) for this lattice is 2 2 complex which are usually responsible for the aggregation of ( ) c · u c · u � u f eq � w ρ �1 + i + i − �, (3) particles in the finite-Reynolds-number regime. Unlike the i i f c2 2c4 2c2 straight channel, it is very difficult to derive a detailed s s s mathematical description of the forces that dominate par- where ρf is the fluid density, cs is the sound speed, and wi is ticles in the curved channels due to the complex nature of the weight coefficient given by flow. At present, much of the development in the curved 2 ⎪⎧ 4/9, ci � 0 ⎪⎫ channels has followed an empirical approach which usually ⎪ ⎪ ⎪ ⎪ fails to predict the equilibrium position of particles. ⎨⎪ ⎬⎪ *erefore, a more complete understanding of the migration w � 1/9, c2 � c2 , i ⎪ i ⎪ of particles in these channels is needed to provide help with ⎪ ⎪ ⎪ ⎪ ( ) the design of microfluidic channels and to further enhance ⎩⎪ ⎭⎪ 4 1/36, c2 � 2c2 the focusing of particles. However, attempts to study the i flow characteristics as well as the motion of particles in c a curved channel are rarely reported in the past, most of √� cs � . which are involving experimental work. Little effort has 3 been paid to the study of the migration of particles in *e fluid density ρ and velocity u can be calculated by a curved channel from a numerical aspect. *is motivates f the following formula: the present work. Among all the types of curved channel, the serpentine ρf � � fi, i channel with linear structure is an optimal choice due to its (5) small footprint and easy parallelization. Furthermore, ex- ρf u � � cifi. i perimental work has shown that the serpentine channel can achieve focusing and separation within a much shorter *e Navier–Stokes equations can be obtained from the length due to the assistance of secondary flow [15]. *e lattice Boltzmann equation (LBE) through a Chapman– similar behavior was also demonstrated in our recent work [16], Enskog expansion proposed by He and Luo [18]. Journal of Nanotechnology 3

L1 =5hL2 =5h L3 =5h

y H =3h

x h

Figure 1: Physical model of the present work.

2.2. Problem Description. In this work, we aim to numeri- 0.33 cally study the migration of a circular particle in a serpentine channel. *e physical model is shown in Figure 1. *e width 0.32 of the channel is denoted as h. Other parameters such as L1, L2, L3, and H are shown in Figure 1, which are set to be 0.31 L1 � L2 � L3 � 5h and H � 3h. *e diameter and density of the particle are expressed by d and ρp, respectively. At the inlet, 0.3 a parabolic flow with the maximum velocity of U0 is applied, eq while the fully developed condition is applied at the outlet. No Y 0.29 slip boundary is used on all the channel walls. In the simulations, the parameters are chosen as follows: ρf � 1, d � 0.28 10, h � 8d, and U0 � 0.05 (in lattice unit). *e channel Reynolds number Re in this work is defined as 0.27 U h Re � 0 . (6) 0.26 ] 20 40 60 80 100 120 140 160 180 200 Re In order to avoid the influence of the inlet on the particle Chen's solution motion, the particle is always placed 20 d from the inlet, which Present result is released after the flow is fully developed. In this work, we focus on the effect of the Reynolds number, the particle initial Figure 2: Comparison of the equilibrium position of a particle position, as well as the density of particle on the final equi- migrating in a straight channel. librium position of particle, and its migration trajectory.

3. Validation Table 1: Relative errors of different Reynolds numbers. In order to validate the computational model in this work, Re (%) the migration of a circular particle in a straight channel is 20 1.14 numerically tested. We compare the present results with the 40 3.36 previous results proposed by Chen et al. [8], as shown in 100 0.35 Figure 2. *e parameters used here are the same as those in 200 6.77 [7]. Explicitly, the diameter of the circular particle is d � 22 and the width of the straight channel is H � 200, which leads to the size ratio K � d/H � 0.11. *e equilibrium position of illustrate the flow field in the serpentine channel, we present the particle Yeq, which is the vertical position of the particle the steady streamlines at different Reynolds numbers to the width of the channel, at different Reynolds numbers (Re � 20, 70, and 120) in Figure 3. Only the local enlarge- (Re � 20, 40, 60, 80, 100, 120, 160, 180, and 200) is shown in ment of steady flow field is shown because of large com- Figure 2. *e Segre–Silberberg´ effect is realized, with the putational domain. equilibrium position a little outside the midpoint between It is hard to observe the recirculation zones when the the wall and the channel centerline. In addition, the final Reynolds number is small, such as Re � 20, as shown in equilibrium position of particle is closer to the channel Figure 3(a). As a result, the distortion of the streamlines is centerline when increasing the Reynolds number. not significant. However, the recirculation zones are becoming In Table 1, we present the relative errors between our more notable for larger Re, as one can see in Figures 3(b) and results and those in [11] at different Reynolds numbers, 3(c). Totally speaking, there are two types of corners in the which shows a good agreement. present serpentine channel: corners with inwards right angle and those with outwards right angle, which lead to two types 4. Results and Discussion of recirculation zone. It is observed that there always exists a recirculation zone for each corner when Re is large. Fur- 4.1. Steady Flow Field. In the simulations, the particle is thermore, the rotation of recirculation zones on the upper released after the flow field is fully developed. To better wall is always counterclockwise while the opposite is true for 4 Journal of Nanotechnology

300

150 y

0 2100 2250 2400 2550 2700x 2850 3000 3150 3300 3450 3600

(a) 300

150 y

0 2100 2250 2400 2550 2700x 2850 3000 3150 3300 3450 3600

(b) 300

150 y

0 2100 2250 2400 2550 2700x 2850 3000 3150 3300 3450 3600

(c)

Figure 3: Streamlines of steady flow field at (a) Re � 20, (b) Re � 70, and (c) Re � 120. those on the bottom wall. Due to the recirculation zones, the as shown in Figure 4. *is is due to the fact that the larger the distortion of the streamlines becomes significant when the Reynolds number, the larger the centrifugal force experi- Reynolds number is large, such as Re � 120, which is expected enced by the particle. to considerately influence the migration of particle in the *e effect of q on the particle trajectories is shown in channel. Figure 5. *e particle trajectories of different q are almost parallel to each other for each Reynolds number. It is observed that the dependence of particle migration on the 4.2. .e Effect of the Reynolds Number. In this section, we value of q is stronger when Re is smaller. In other words, the study the effect of the Reynolds number on the particle final equilibrium position of particle is more sensitive to migration in the serpentine channel shown in Figure 1. *e the value of q for smaller Reynolds number. For instance, no density of the particle is fixed at ρp � 1.0, suggesting that the visible difference is observed for all results except that of particle is neutrally suspended. Here, a parameter q is in- q � 0.5 when Re � 120, the largest Reynolds number studied troduced to describe the initial lateral position of the particle in the work, as one can see in Figure 5(c). In order to provide in the channel which is defined by the distance of particle to a better understanding, we summarize the dependence of the the upper channel wall normalized by the channel width h. final equilibrium positions of particle on Re as well as q in As a result, the value of q � 0.5 indicates that the particle is Table 2, which further demonstrates that the effect of q on initially placed on the channel centerline. In what follows, we the particle migration is more significant at smaller Re. It is focus on the particle trajectory as well as the final equilib- also clear that the particle eventually stays on the channel rium position of the particle under different channel Rey- centerline for the range of Re studied when q � 0.5. nolds numbers. High efficiency is occasionally required in the micro- Figure 4 shows the particle trajectories at Re � 20, 70, and fluidic devices such as fast separation. In these cases, the 120 for different initial particle positions. As one can see, the particles are expected to travel fast in the serpentine channel. effect of Re on the particle migration in the channel is As a result, much attention should be paid to the time significant. To some extent, the particle trajectories are needed for the particles to travel in the channel. We carried similar to the streamlines shown in Figure 3. In comparison out a preliminary study upon this issue. *e dependence of with the result of Re � 20, the particle is driven farther away T∗ on the value of q is shown in Figure 6 for different Re. It ∗ from the channel wall after passing through each bend when should be stated that T is normalized by T0 which is de- increasing Re, which is more significant for small values of q, termined through T0 � d/U0. As shown in Figure 6, all the ora fNanotechnology of Journal Figure y y y y 150 300 150 300 150 300 150 300 1025 4025 7025 0035 3035 3600 3450 3300 3150 3000 2850 2700 2550 2400 2250 2100 0 0 0035 3035 6035 9045 2045 4500 4350 4200 4050 3900 3750 3600 3450 3300 3150 3000 0 1025 4025 7025 0035 3035 3600 3450 3300 3150 3000 2850 2700 2550 2400 2250 2100 1025 4025 7025 0035 3035 3600 3450 3300 3150 3000 2850 2700 2550 2400 2250 2100 0 :Pril rjcoisa Re at trajectories Particle 4: e=120 = Re Re=70 Re=20 q q q Re=20 e=120 = Re Re=70 Re=20 e=120 = Re Re=70 0.3 = 0.2 = 0.1 = � 0 0 n 2 o different for 120 and 70, 20, Figure :Continued. 5: x x x x (b) (a) (a) (c) q q 0.5 = 0.4 = (a) q: q � .,(b) 0.1, q � .,ad(c) and 0.3, q � 0.5. 5 6 Journal of Nanotechnology

300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 q = 0.1 q = 0.4 q = 0.2 q = 0.5 q = 0.3

(b) 300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 q = 0.1 q = 0.4 q = 0.2 q = 0.5 q = 0.3

(c)

Figure 5: *e eﬀect of q on particle trajectories: (a) Re � 20, (b) Re � 70, and (c) Re � 120.

Table 2: Dependence of the ﬁnal equilibrium position on Re as well 750 as q. 700 Re 20 70 120 q � 0.1 0.265 0.271 0.305 650 q � 0.2 0.268 0.273 0.308 q � 0.3 0.280 0.278 0.310 600

q � 0.4 0.328 0.308 0.314 ∗ T 550 q � 0.5 0.509 0.500 0.500 500

450 results decrease as q increases when q < 0.5, while the opposite is true when q > 0.5. *is leads to the fact that the time 400 T∗ reaches its minimum value when q � 0.5 for all cases, suggesting that the particle will travel fastest if it is initially 350 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 placed on the channel centerline. In addition, the results are q more likely to be symmetrical when Re is smaller, such as Re � 20, which is due to the symmetrical flow conditions. Re=20 Re=70 However, the recirculation zones resulting from corners Fitting curve Re = 120 become significant as increasing Re, which destroys the flow Figure 6: Time needed for the particle to travel in the channel for symmetry. Furthermore, it is found that the time needed for different values of q. the particle to travel in the channel has a quadratic-like relationship with the value of q for small Re (Re � 20): T∗ � 1632q2 − 1629q + 859. (7) Figure 7, the particle always rotates counterclockwise when traveling in the channel if q < 0.5 irrespective of Re. In To further study the migration behavior of the particle in addition, the smaller the value of q is, the faster the particle the serpentine channel, we also pay close attention to the rotates. *is is because the particle experiences larger gra- evolution of particle orientation θ which has an initial value dient of fluid velocity if it is closer to the channel wall. of π/2, as shown in Figures 7 and 8, which present the Figure 8 shows a different pattern of particle motion in corresponding results of q � 0.1–0.3 and q � 0.5 at different the serpentine channel when q � 0.5. Instead of rotating Reynolds numbers, respectively. As we can observe in counterclockwise, the particle will oscillate if it is initially Journal of Nanotechnology 7

35 35

30 30

25 25

20 20 θ ( π ) θ ( π ) 15 15

10 10

5 5

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 500 1000 1500 2000 2500 3000 3500 4000 4500 x x q = 0.1 q = 0.1 q = 0.2 q = 0.2 q = 0.3 q = 0.3

(a) (b) 35

20 θ ( π ) 15

0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 x q = 0.1 q = 0.2 q = 0.3

(c)

Figure 7: *e orientation of the particle for q � 0.1, 0.2, and0.3 at different Reynolds numbers: (a) Re � 20, (b) Re � 70, and (c) Re � 120 (also appeared in [16]). placed on the channel centerline for the range of Re in this 4.3. .e Effect of Particle Density. *e particle inertia is work. Especially, it is observed that the particle is oscillating central to the behavior of the particle suspended in fluids around θ � π/2 when traveling in the channel for large because it determines how the velocity of the particle decays Reynolds numbers, such as Re � 70 and 120, as shown in due to fluid drag. As is known to all, the particle inertia Figure 8. strongly depends on its density. Consequently, we focus on 8 Journal of Nanotechnology

0.7

0.6

0.5

0.4

0.3 θ ( π )

0.2

0.1

–0.1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 x Re=20 Re=70 Re = 120

Figure 8: *e orientation of the particle for q � 0.5 at diﬀerent Reynolds numbers (also appeared in [16]).

300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 α = 0.2 α = 0.4 α = 0.3 α = 0.5

(a) 300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 α = 0.5 α = 0.8 α = 0.6 α = 0.9 α = 0.7

(b)

Figure 9: Particle trajectories for different density ratios when α < 1: (a) α � 0.2–0.5 and (b) α � 0.5–0.9. the effect of the particle density on the migration behavior of We first consider the cases for which the fluid is heavier the particle in the serpentine channel in this section. *e than the particle, that is, α < 1. Figure 9 shows the particle solid-to-fluid density ratio is defined as α � ρp/ρf . All the trajectories for different values of α. It is interesting to find results in the previous section treat the particle density ρp that no visible difference is observed for all the results equal to the fluid density ρf , that is, α � 1. To save the shown, suggesting that the effect of the density ratio on the computational resources, some parameters are fixed at migration behavior of the particle in the serpentine channel ρf � 1, Re � 20, and q � 0.3 in this section. is negligible for α < 1. However, things are very different for Journal of Nanotechnology 9

300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 α =1 α =4 α =2 α =5 α =3

(a) 300

y 150

0 3000 3150 3300 3450 3600x 3750 3900 4050 4200 4350 4500 α =1 α =10 α =6 α =20 α =7

(b)

Figure 10: Particle trajectories for diﬀerent density ratios when α > 1: (a) α � 1–5 and (b) α � 1–20.

580 may take the shortest time for the particle to travel in the channel if it has a particular density. It is also interesting to 560 ﬁnd that for α < 1, the value of T∗ decreases as α increases despite that the particle trajectories (Figure 9) are almost 540 identical, as shown in Figure 11. Finally, it is found that a quintic polynomial function like (8) ﬁts the results quite ∗ 520 T well, 500 T∗ � −0.0006123α5 + 0.03686α4 − 0.8384α3 (8) 2 480 + 9.486α − 51.62α + 571.7.

460 Finally, it should be stated that the present work only 0 2 4 6 8 101214161820 involves two-dimensional numerical simulation, which is α different from the existing experimental work. For instance, Fitting curve the secondary flow may play an important role on the migration of particles as well as the interaction between particles Figure 11: Time needed for the particle to travel in the channel for different values of α. in channel flows when the fluid inertia is significant, which cannot be taken into account in the present two-dimensional study. However, it is generally agreed that the two-dimensional α > 1, as shown in Figure 10. *e particle is more likely to simulations can be served as a tool for studying three- maintain its motion for larger density ratio, leading to the dimensional flow characteristics when the Reynolds number fact that the particle approaches closer to the channel wall is not large. As mentioned above, this work is the first step after traveling in the straight section of channel. *is trend is of investigation. We hope it casts a light on some features of very significant when the particle is heavy such as particle migration in a serpentine channel. Certainly, the motion α � 10 and 20, as shown in Figure 10(b). of spherical particles as well as their interactions in a three- Similarly, we also present the time needed for the particle dimensional channel flow would be our future work. to travel in the channel upon different values of α in Figure 11. *e effect of density ratio on T∗ is significant. It is obvious 5. Conclusion that T∗ decreases as α increases initially and then increases afterwards. In addition, T∗ reaches its minimum value at In this work, the lattice Boltzmann method based on the α ≈ 7, as we can observe from Figure 11. *is suggests that it momentum exchange scheme has been adopted to numerically 10 Journal of Nanotechnology study the migration of a particle in a serpentine channel. We a Newtonian fluid. Part 2. Couette and Poiseuille flows,” focus on the effects of the Reynolds number (Re) as well as the Journal of Fluid Mechanics, vol. 277, no. 1, pp. 271–301, 1994. initial position of the particle (q) on the migration behavior of [5] T. W. Pan and R. Glowinski, “Direct simulation of the motion the particle in the channel. *e Reynolds number ranges from of neutrally buoyant circular cylinders in plane Poiseuille flow,” 20 to 120. Journal of Computational Physics, vol. 181, no. 1, pp. 260–279, 2002. (1) *e effect of Re on the final equilibrium position of [6] B. Chun and A. J. C. Ladd, “Inertial migration of neutrally the particle is significant, which is found to be more buoyant particles in a square duct: an investigation of multiple sensitive to the initial position of the particle when Re equilibrium positions,” Physics of Fluids, vol. 18, no. 3, is small. For Re � 120, the largest Reynolds number p. 031704, 2006. studied in this work, the particle has a similar equi- [7] H. H. Yi, L. J. Fan, and Y. Y. Chen, “Lattice Boltzmann simulation of the motion of spherical particles in steady librium position irrespective of its initial position for Poiseuille flow,” International Journal of Modern Physics C, q < 0.5 or q > 0.5. Interestingly, results show that the vol. 20, no. 6, pp. 831–846, 2009. particle almost stays on the channel centerline for the [8] S. D. Chen, T. W. Pan, and C. C. Chang, “*e motion of a single range of Re studied when q � 0.5. and multiple neutrally buoyant elliptical cylinders in plane (2) It has been found that the particle is driven farther Poiseuille flow,” Physics of Fluids, vol. 24, no. 10, p. 103302, 2012. away from the channel wall after passing through [9] M. Abbas, P. Magaud, Y. Gao, and S. Geoffroy, “Migration of each bend while increasing Re, which is more sig- finite sized particles in a laminar square channel flow from low nificant for small values of q. *is is due to the fact to high Reynolds numbers,” Physics of Fluids, vol. 26, no. 12, p. 123301, 2014. that the larger the Reynolds number, the larger the [10] D. Jiang, W. Tang, N. Xiang, and Z. Ni, “Numerical simulation centrifugal force experienced by the particle. of particle focusing in a symmetrical serpentine microchannel,” (3) *e solid-to-fluid density ratio α is central to the RSC Advances, vol. 6, no. 62, pp. 57647–57657, 2016. migration of the particle in the channel. It has been [11] J. M. Martel and M. Toner, “Particle focusing in curved found that the time it takes for the particle to travel in microfluidic channels,” Scientific Reports, vol. 3, no. 1, p. 3340, the channel decreases as α increases initially and then 2013. increases afterwards. *ere exists a critical value of [12] L. L. Fan, Y. Han, X. K. He, L. Zhao, and J. Zhe, “High- throughput, single-stream microparticle focusing using density ratio for which the particle travels fastest in a microchannel with asymmetric sharp corners,” Microfluidics the channel. and Nanofluidics, vol. 17, no. 4, pp. 639–646, 2014. [13] S. S. Kuntaegowdanahalli, A. A. S. Bhagat, G. Kumar, and Data Availability I. Papautsky, “Inertial microfluidics for continuous particle separation in spiral microchannels,” Lab on a Chip, vol. 9, *e data used to support the findings of this study are no. 20, pp. 2973–2980, 2009. available from the corresponding author upon request. [14] T. Morijiri, M. Yamada, T. Hikida, and M. Seki, “Microfluidic counter flow centrifugal elutriation system for sedimentation- based cell separation,” Microfluidics and Nanofluidics, vol. 14, Conflicts of Interest no. 6, pp. 1049–1057, 2013. [15] J. Zhang, S. Yan, R. Sluyter, W. Li, G. Alici, and N.-T. Nguyen, *e authors declare that they have no conflicts of interest. “Inertial particle separation by differential equilibrium positions in a symmetrical serpentine micro-channel,” Scientific Acknowledgments Reports, vol. 4, no. 1, p. 4527, 2014. [16] Y. Liu and D. M. Nie, “Numerical simulation of particle *is work was supported by the Zhejiang Provincial Natural motion in a curved channel,” IOP Conference Series: Materials Science Foundation of China (LY15A020004) and the Na- Science and Engineering, vol. 301, p. 012088, 2018. tional Key Research and Development Program of China [17] Y. H. Qian, D. D’Humieres, and P. Lallemand, “Lattice BGK (2017YFB0603700). model for Navier–Stokes equation,” Europhysics Letters, vol. 17, no. 6, pp. 479–484, 1992. [18] X. He and L. S. Luo, “Lattice Boltzmann model for the in- References compressible Navier–Stokes equation,” Journal of Statistical Physics, vol. 88, no. 3-4, pp. 927–944, 1997. [1] C. D. Carlo, D. Irimia, R. G. Tompkins, and M. Toner, “Continuous inertial focusing, ordering, and separation of particles in microchannels,” Proceedings of the National Academy of Sciences of the United States of America, vol. 104, no. 48, pp. 18892–18897, 2007. [2] G. Segre´ and A. Silberberg, “Radial particle displacements in Poiseuille flow of suspensions,” Nature, vol. 189, no. 4790, pp. 209-210, 1961. [3] B. P. Ho and L. G. Leal, “Inertial migration of rigid spheres in two-dimensional unidirectional flows,” Journal of Fluid Me- chanics, vol. 65, no. 2, pp. 365–400, 1974. [4] J. Feng, H. H. Hu, and D. D. Joseph, “Direct simulation of initial value problems for the motion of solid bodies in Hindawi Journal of Nanotechnology Volume 2018, Article ID 3539075, 7 pages https://doi.org/10.1155/2018/3539075

Research Article Iterative Dipole Moment Method for the Dielectrophoretic Particle-Particle Interaction in a DC Electric Field

Qing Zhang1 and Kai Zhang 2

1China Tobacco Standardization Research Center, Zhengzhou, China 2Institute of Fluid Engineering of China Jiliang University, Hangzhou, China

Correspondence should be addressed to Kai Zhang; [email protected]

Received 1 February 2018; Accepted 12 March 2018; Published 9 May 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Qing Zhang and Kai Zhang. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Electric force is the most popular technique for bioparticle transportation and manipulation in microfluidic systems. In this paper, the iterative dipole moment (IDM) method was used to calculate the dielectrophoretic (DEP) forces of particle-particle interactions in a two-dimensional DC electric field, and the Lagrangian method was used to solve the transportation of particles. It was found that the DEP properties and whether the connection line between initial positions of particles perpendicular or parallel to the electric field greatly affect the chain patterns. In addition, the dependence of the DEP particle interaction upon the particle diameters, initial particle positions, and the DEP properties have been studied in detail. +e conclusions are advantageous in elelctrokinetic microfluidic systems where it may be desirable to control, manipulate, and assemble bioparticles.

1. Introduction usually been used to calculate DEP force for single particle but cannot be used for the interaction between neighboring Microfluidic systems are widely used for biochemical analysis particles in an electric field. Maxwell stress tensor (MST) over the past decade, and electric forces are often used as an method [10] is complicated for the calculation of a large efficient and effective transport mechanism, which does not number of particles’ interaction [11]. +e iterative dipole involve the intervention of moving parts and offers good moment (IDM) [12–14] method, which does not solve manipulation over sample handling. Deposition of bio- complicated differential equations of an electric field while the particles and colloids onto the surface [1] is very important for particle is moving, was used in this work for calculating the many biochemical processes, such as assembly of carbon DEP forces of particle interactions. nanotubes [2, 3] and trapping of nanoparticles and nano- To control the assembling of bioparticles with high particle synthesis [4–6]. In these processes, particles’ trans- accuracy, an in-depth study is necessary to the underlying lation and rotation should be controlled accurately for mechanisms of DEP particle-particle interaction in micro- counting and assembly of these particles [7, 8]. In microfluidic fluidic systems. +e impetus for the present study came from systems, electrical and hydrodynamic forces [9] dominate the importance in understanding the mechanics of particle particles’ transportation. In the bulk fluid, particle transport is assembling, which is often used to produce specialized ma- mainly affected by the hydrodynamic interactions and its terial in a lab on a chip. In the present study, the DEP in- neighboring particles. teraction of particle-particle [15] in a DC electric field will be As one of the important forces of particle manipulation, studied, and a mathematical model based on the Lagrangian dielectrophoretic (DEP) force has been drawing much at- method incorporating DEP particle-particle interactions [16] tention in recent years. Equivalent dipole moment (EDM) has will be presented to compute particle trajectories, and 2 Journal of Nanotechnology

P 2 P1 d2 d1

E θ1

θ0

θi O x

R di

Figure 1: Cylindrical particles are equispaced on a circle in a two-dimensional DC uniform electric ﬁeld E � 1 KV/m; here, red represents positive DEP and black represents negative DEP.

1 1

2 2

(a) (b) Figure 2: Two particles (red or black dashed line) are initially equispaced on a circle (blue dashed line) with θ0 � π/2 in a uniform DC electric field with E � 1 kV/m along x direction, and then the final particle chains (solid particles) are, respectively, shown after DEP interaction under different statuses: (a) two negative (black) DEP particle, (b) one positive (red) and one negative DEP particle.

−1 −1 consequently, the assembling will be calculated. Finally, the dielectrophoresis, εm � 2.5 ∗ 8.8541878176e − 12 CV · m −1 −1 dependence of particle assembling on the particle’s initial position and εp � 6.9e − 10 CV · m . and radius will be further concluded and discussed in detail. +e dielectrophoretic force, FDEP, acting on a spherical, homogeneous particle suspended in a local electric field 2. Computational Model gradient is given by the expression 3 2 As shown in Figure 1, in a two-dimensional incompressible FDEP � 2πrpεmRe[K(w)]∇E , (1) still media, DEP cylindrical particles are initially equispaced on a circle with radius R � 20 μm and an initial angle where rp is the particle radius, εm is the permittivity of the θ0 with respect to the applied electric field. Here, parti- suspending medium, ∇ is the Del vector (gradient) operator, cle diameter d � 5 μm, for negative DEP, the permittivity E is the electric field incorporating additional fields due to −1 −1 of the media and particles are εm � 6.9e − 10 CV ·m particle interactions, and Re[K(w)] is the real part of the −1 −1 and εp � 2.5 ∗ 8.8541878176e − 12 CV · m ; for positive Clausius–Mossotti factor, which is given by Journal of Nanotechnology 3

221 1 2 2 1 1

(a) (b)

212 1

(c) Figure 3: +e initial position of two particles are initially equispaced on a circle with θ0 � 0 in a uniform DC electric field with E � 1 kV/m along x direction, and the final particle chains after DEP interaction under different status: (a) two negative DEP particle, (b) two positive DEP particle, and (c) one positive and one negative DEP particle.

∗ ∗ And the dielectrophoretic force can be modified as �εp − εm � K(w) � , (2) follows: ∗ ∗ �εp + 2εm � �ε − ε � � r3 p m · . ( ) where ε∗ and ε∗ are the complex permittivities of the FDEP 4π pεm E ∇E 4 m p �εp + 2εm � medium and particle, respectively, and ε∗ � ε − (jσ/w) with σ is the conductivity, ε is the permittivity, and w is the Considering the influence of the dipole-induced field of angular frequency of the applied electric field. +e limiting other particles surrounding particle i, the modified electric direct current (DC) case of the equation is field near the particle located at ri is shown below: n �εp − εm � (1) (0) K(w � 0) � . (3) Ei (r) � E0(r) + � Ed �rj, r �, i � 1, 2, 3, ... , N, �εp + 2εm � j�1,j ≠ i (5) 4 Journal of Nanotechnology

1 1

2 2

(a) (b) Figure 4: +e initial position of two particles is initially equispaced on a circle with θ0 � π/2 in a uniform DC electric field with E � 1 kV/m along x direction, and the final particle chains after DEP interaction under different status: (a) two heterogeneous particles with different diameters, (b) two negative DEP particles with different diameters.

(1) where Ei (r)denotes the modified electric field and +ese particles are assumed to be far from boundaries. (0) Ed (rj, r) is the influence of the dipole-induced field of +e forces on particles are calculated by (1) while the dipole particle j; here, moment have been modified using IDM every time step, and zφ particle transportation can be easily solved for particle E � � − d d x zx trajectories and final particle chains.

� � �P � cos α 2Δx Δx cos α + Δy sin α � 3. Results and Discussion � − i ⎡⎣ i − i i i i i ⎤⎦, 2πε Δx2 + Δy2 2 2 2 m i i Δxi + Δyi � Imagine a particle suspended in a dielectric fluid and subjected to a uniform electric field, which will polarize the zφ dielectric particle and induce a dipole moment in it, then the E � � − d d y zy dipole moment will induce an electrostatic potential as shown below: � � � � Pi cos θ �Pi� sin α 2Δy Δx cos α + Δy sin α � φ � . (9) � − ⎡⎣ i − i i i i i ⎤⎦. d 4πε r2 x2 + y2 2 2 2 m 2πε0 Δ i Δ i Δxi + Δyi � (6) When particles are close to each other, one particle’s induced electrostatic potential will distort its neighbor’s +e modified electric field induces a new dipole moment, electric field and make it nonuniform, and then the DEP which again induces a new electric field as follows: force acting on its neighbor comes to a nonequilibrium state. (1) (1) (1) +e nonzero DEP forces on particles influencing each other E r � → P → E r , r �, i � 1, 2, 3, ... , N, (7) i i i d i suspended in a two-dimensional DC uniform electric field (1) (1) (1) can be calculated by the IDM method. where Ei (ri), Pi , and Ed (ri, r) are the modified electric field, the modified dipole moment, and the modified dipole- As shown in Figure 2, if two heterogeneous DEP par- induced electric field, respectively. +ese parameters can be ticles with same diameter are released at a small distance iteratively modified until a converged value of the electric from each other and perpendicular to the electric field, they (n) field Ei (ri) is obtained and the final dipole moment will attract each other and are likely to cluster at the center of considering other particles’ influence can be achieved: line of their initial position. However, if all parameters 3 except for the DEP property are the same, the results show 4πai εmεi − εm � (n) Pi � Ei ri �. (8) that two homogeneous DEP particles will repel each other to εi + 2εm move outwards symmetrically. Journal of Nanotechnology 5

2 112 221 1

(a) (b) Figure 5: +e initial position of two particles is initially equispaced on a circle with θ0 � 0 in a uniform DC electric field with E � 1 kV/m along x direction, and the final particle chains after DEP interaction under different status: (a) two negative DEP particle with different diameter and (b) two positive DEP particle with different diameters.

2 2

3 112 1 1

3 3

(a) (b) Figure 6: +e initial position of three particles is initially equispaced on a circle with θ0 � 0 in a uniform DC electric field with E � 1 kV/m along x direction, and the final particle chains after DEP interaction under different status: (a) three negative DEP particle and (b) two negative and one positive DEP particle.

From Figure 3, it can be seen that if two homogeneous Figure 4 shows the DEP interaction between two par- DEP particles with same diameter are released at a small ticles with different diameters. Figure 4(a) shows that when distance from each other and parallel to the electric field, two heterogeneous DEP particles are released at a small they will attract each other and cluster at the center of line of distance from each other and perpendicular to the electric their initial position. However, if all parameters except for field, they will attract and cluster. However, as the difference the DEP property are the same, the results show that two of two particles’ diameters becomes larger, the position of heterogeneous DEP particles will repel each other to move final particle chains deviates from the center of line of their outwards symmetrically. initial position, and the chains will move towards the side of 6 Journal of Nanotechnology

2 2

3 1 1 33 1 3 1

4 4

(a) (b)

Figure 7: +e initial position of two negative and two positive DEP particles are initially equispaced on a circle in a uniform DC electric field with E � 1 kV/m along x direction, and the final particle chains after DEP interaction under different status: (a) θ0 � 0, (b) θ0 � π/16.

the smaller particle. In addition, if all parameters except for the DEP property are the same, Figure 4(b) shows that the two homogeneous DEP particles with different diameters will repel each other to move outwards asymmetrically, and the smaller particle moves faster. Figure 5 shows that when two homogeneous DEP 2 particles with different diameters are released at a small 2 3 distance from each other and parallel to the electric field, 1 they will attract and cluster, and the chains will move to- 3 wards the side of the smaller particle. 1 From Figure 6(a), it can be seen that if three homoge- 4 neous dielectrophoretic particles with same diameters are 4 initially equispaced on a small circle, they will attract each 5 other and finally cluster parallel to the electric field. How- 5 ever, from Figure 6(b), it can found that when one of the particles changed its DEP property, they still clustered but the chains were perpendicular to the electric field. From Figure 7(a), it can be seen that due to the sym- metric configuration, homogeneous DEP particles 1 and 3 are attracted to move inwards because their connection line Figure is parallel to the electric field; at the same time, positive DEP 8: +e initial position of five heterogeneous DEP particles is particles 2 and 4 are attracted by the chains of negative DEP initially equispaced on a circle with θ0 � 0 in a uniform DC electric � particles 1 and 3 to move inward. While the connection line field with E 1 kV/m along x direction, and the final particle chains after DEP interaction. of the initial position of homogeneous particles 2 and 4 is perpendicular to the electric field, they cannot contact the chains of negative DEP particles 1 and 3. However, when the connection line is not perpendicular to the electric field will connection line between two homogeneous DEP particles 2 cluster parallel to the electric field, and homogeneous ad- and 4 has a little shift θ0 � π/16 from the line perpendicular jacent heterogeneous particles will cluster perpendicular to to the electric field, DEP interactions among particles cause the electric field, and eventually form the complex and four particles to cluster as shown in Figure 7(b). asymmetric structure as shown in Figure 8. As shown in Figure 8, three negative and two positive According to these aforementioned conclusions, the DEP particles are initially equispaced on a circle with θ0 � 0 regular aggregation patterns of large number of polarizable in a uniform DC electric field. From the above conclusions, it particles can also be well understood and can be used to can be found that the homogeneous adjacent particles whose control, manipulate, and assemble polarizable particles. Journal of Nanotechnology 7

4. Conclusions [9] D. Das and D. Saintillan, “Electrohydrodynamic interaction of spherical particles under Quincke rotation,” Physical Review +e IDM method was used to study multiple particle DEP E, vol. 87, no. 4, p. 043014, 2013. interactions in a uniform DC electric field, and the DEP [10] S. Kumar and P. J. Hesketh, “Interpretation of ac dielec- interactions among particles cause particles to cluster. It was trophoretic behavior of tin oxide nanobelts using Maxwell found that the homogeneous adjacent particles whose stress tensor approach modeling,” Sensors and Actuators B: connection line is not perpendicular to the electric field will Chemical, vol. 161, no. 1, pp. 1198–1208, 2012. [11] G. Liu, J. S. Marshall, S. Q. Li, and Q. Yao, “Discrete-element cluster parallel to the electric field, and homogeneous ad- method for particle capture by a body in an electrostatic field,” jacent heterogeneous particles will cluster perpendicular to International Journal for Numerical Methods in Engineering, the electric field; however, as the difference of two particles’ vol. 84, no. 13, pp. 1589–1612, 2010. diameters becomes larger, the position of final particle [12] L. Liu, C. Xie, B. Chen, and J. Wu, “Iterative dipole moment chains deviates from the center of line of their initial po- method for calculating dielectrophoretic forces of particle– sition, and the chains will move towards the side of the particle electric field interactions,” Applied Mathematics and smaller particle. In addition, the dependence of the DEP Mechanics, vol. 36, no. 11, pp. 1499–1512, 2015. particle interaction upon the particle diameters, initial [13] L. Liu, C. Xie, B. Chen, N. Chiu-On, and J. Wu, “A new particle positions, and the DEP properties have been studied method for the interaction between multiple DEP particles: in detail. +is can be advantageous in elelctrokinetic iterative dipole moment method,” Microsystem Technologies, vol. 22, no. 9, pp. 1–10, 2015. microfluidic systems where it may be desirable to control, [14] L. Liu, C. Xie, B. Chen, and J. Wu, “Numerical study of particle manipulate, and assemble cylindrical bioparticles. chains of a large number of randomly distributed DEP particles using iterative dipole moment method,” European Conflicts of Interest Journal of Mechanics B/Fluids, vol. 58, pp. 50–58, 2015. [15] H. Feng, T. N. Wong, and Marcos, “Pair interactions in in- +e authors declare that there are no conflicts of interest duced charge electrophoresis of conducting cylinders,” In- regarding the publication of this paper. ternational Journal of Heat and Mass Transfer, vol. 88, pp. 674–683, 2015. [16] N. Sun and J. Y. Walz, “A model for calculating electrostatic Funding interactions between colloidal particles of arbitrary surface topology,” Journal of Colloid and Interface Science, vol. 234, +e authors received funding from the National Natural no. 1, pp. 90–105, 2001. Science Foundation of China (11472260).

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Research Article Stochastic Simulation of Soot Formation Evolution in Counterflow Diffusion Flames

Xiao Jiang,1 Kun Zhou ,1 Ming Xiao ,1 Ke Sun,1 and Yu Wang2

1 e State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan, China 2School of Automotive Engineering, Wuhan University of Technology, Wuhan, China

Correspondence should be addressed to Kun Zhou; [email protected]

Received 4 December 2017; Accepted 3 April 2018; Published 9 May 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Xiao Jiang et al. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Soot generally refers to carbonaceous particles formed during incomplete combustion of hydrocarbon fuels. A typical simulation of soot formation and evolution contains two parts: gas chemical kinetics, which models the chemical reaction from hydrocarbon fuels to soot precursors, that is, polycyclic aromatic hydrocarbons or PAHs, and soot dynamics, which models the soot formation from PAHs and evolution due to gas-soot and soot-soot interactions. In this study, two detailed gas kinetic mechanisms (ABF and KM2) have been compared during the simulation (using the solver Chemkin II) of ethylene combustion in counterflow diffusion flames. Subsequently, the operator splitting Monte Carlo method is used to simulate the soot dynamics. Both the simulated data from the two mechanisms for gas and soot particles are compared with experimental data available in the literature. It is found that both mechanisms predict similar profiles for the gas temperature and velocity, agreeing well with measurements. However, KM2 mechanism provides much closer prediction compared to measurements for soot gas precursors. Furthermore, KM2 also shows much better predictions for soot number density and volume fraction than ABF. .e effect of nozzle exit velocity on soot dynamics has also been investigated. Higher nozzle exit velocity renders shorter residence time for soot particles, which reduces the soot number density and volume fraction accordingly.

1. Introduction Diffusion flames, where fuel and air are introduced separately into the combustion chamber and form an ig- Around 80%–85% world energy comes from combustion of nitable mixture by diffusion, are widely used in practical fossil fuel [1]. .e formation and evolution of soot (i.e., carbon combustion systems for safety reasons. Counterflow diffusion particles resulting from incomplete combustion of hydro- flames are frequently used in experimental and theoretical carbons) is an important and constantly studied field in research because they represent essentially a one-dimensional combustion due to its practical significance in the production structure of diffusion flame, which provides valuable data of technical carbon (such as filler in rubber, component of with respect to optimizing combustion processes and ex- printing paints), as well as in the combustion efficiency and perimental data for validation of flame modelling. human health [2, 3]. Understanding the mechanism of soot Generally, there are three categories of methods to simulate formation is a long-standing challenge in combustion re- the general aerosol dynamics (including soot), that is, the direct search. Quantitative knowledge of soot formation has been discretization method (e.g., section method [5]), method of largely derived from three types of work [4]: measurement of moments (interpolative [6], quadrature/direct quadrature soot volume fraction, number density, and particle size dis- [7–9], Taylor expansion [10, 11], etc.), and stochastic method tributions (PSDs); development of detailed chemical mecha- (also called the Monte Carlo method) [12]. nisms for the formation of polycyclic aromatic hydrocarbons; Recently, the authors [13] developed an efficient oper- and development of soot population dynamics models to ator splitting Monte Carlo method for the simulation of describe the evolution of the particle ensemble. general aerosol dynamics. .e Monte Carlo method has 2 Journal of Nanotechnology

2.2. Soot Dynamics. .e particle size distribution, n, of soot Oxidizer Nozzle particle satisfies the following general dynamic equation [20]: zn → zn zn zn + · n u � · D n +� � +� � +� � . Flame ∇ ∇ ∇ zt zt nucl zt growth zt coag Soot zone Stagnation (1) plane .e right most three terms refer to the nucleation, growth, and coagulation, respectively. .e diffusion term can be neglected owing to large Schmidt number for soot particles [20]. .e velocity on the left hand side is the gas Nozzle Gaseous fuel velocity corrected by the thermophoretic effect for particles [21]. .e convection term can be implicitly solved with the ∗ Figure 1: Configuration of counterflow diffusion flame. introduction of the Lagrangian time t [22, 23]. → x ∗ → dx′ t ( x ) � � → → . (2) been coupled with the chemical kinetics solver Chemkin II 0 u x ′ � to simulate soot formation and evolution in a counterflow diffusion flame [14]. .is work is to further explore the .en, the general dynamic equation is simplified as (the ∗ simulation framework of coupling the kinetics solver with star notion in the Lagrangian time t has been omitted) the stochastic method and to provide more detailed simu- zn zn zn zn lation results on the soot dynamics in counterflow diffusion �� +� � +� � . (3) zt zt zt zt flames, so as to investigate the effects of different popular nucl growth coag chemical kinetic mechanisms and nozzle exit velocity on the soot formation and evolution. .is transformation converts the Eulerian point of view in (1) to the Lagrangian point of view in (3). Nucleation is the process of a large number of gas 2. Methodology molecules forming a stable nucleus. .e nucleation term is modelled as .e simulation of soot formation and evolution in diffusion �� flames are accomplished in two steps, that is, gaseous zn 4πk � � � P B N d2 T0.5x2. (4) chemical kinetics simulation to determine the concentration zt vdw i m AVO PAH of gaseous soot precursors (i.e., polycyclic aromatic hy- nucl PAH 0 drocarbons, PAHs) and stochastic simulation for soot In the KM2 mechanism, there are 8 PAHs products. Any particle dynamics. two PAHs molecules may nucleate to form a nascent soot particle, which results in 36 different nucleation processes. .e above nucleation model is derived from a simple 2.1. Gaseous Chemical Kinetics. Gaseous chemical kinetics particle-particle collision model. A collision coefficient is are handled by the open source software Chemkin II [15]. In assigned to every PAH, which is [0.006, 0.01, 0.01, 0.01, the current setting of counterflow diffusion flame (Figure 1), 0.011, 0.011, 0.014, and 0.02] for the 8 PAHs (ordered from fuel gas (C2H4) and oxidizer eject from two opposing the smallest to the biggest, see the legend in Figure 2). .e nozzles, respectively. When fuel and oxidizer mix in the pair collision coefficient is chosen as the smaller one between middle, a stable sheet of flame forms. Along the nozzle axial two collision PAHs. direction, the model equations (derived from mass, mo- Coagulation is the process that two particles collide and mentum, and energy conservations) can be reduced to one coalesce into a bigger particle. .e coagulation term is dimension, and gas velocity, temperature, and reactant modelled by the well-known Smoluchowski’s equation product concentration can be obtained through solving the 1D model equations with the Newton iteration method [16]. zn 1 v In Chemkin II, the chemical and physical properties of � � � � β(v, v)n(v)n(v − v) dv zt coag 2 0 materials are stored in a library file, which can be adapted (5) according to a user’s need. A user provides the kinetic ∞ mechanism so as to determine the reaction route. − � β(v, v)n(v)n(v) dv. 0 Here, two kinetic mechanisms (i.e., ABF and KM2) are used to investigate the effects of the different mechanisms. Surface growth includes surface chemical reactions and .e ABF mechanism [17] contains 101 species and 543 physical condensation. .e surface chemical reaction in- reactions, leading to the maximum product pyrene (C16H10). cludes oxidation and the hydrogen-abstraction/acetylene- .e KM2 mechanism [18, 19] contains 202 species and addition (HACA) mechanism [24]. Oxidation is modelled 1351 reactions, leading to the maximum product coronene by the reaction of soot particles with OH and O2 molecules. (C24H12). .e HACA process can be described as Journal of Nanotechnology 3

2.5 2500

2 2000

–5 10 ×

1.5 T (K) 1500

1 1000 Temperature Temperature

PAHs concentration concentration PAHs 0.5 500

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Distance from fuel nozzle (cm) Distance from fuel nozzle (cm)

A4 BAPYR ABF mechanism PYC2H-1 BEPYREN KM2 mechanism PYC2H-2 BGHIPER B. C. Choi PYC2H-4 CORONEN Figure 3: Temperature proﬁle. Circles are measurement data Figure 2: Simulated proﬁles of large PAHs by KM2 mechanism. from [28].

∞ zn flame of ethylene-oxygen in the canonical counterflow � � � k C αx � m ΔS N . (6) zt s g s i i i configuration is investigated. On the fuel side, the gas is growth i�1 a mixture of 75% C2H4 and 25% Ar. On the oxidizer side, it is Condensation modelling is similar to nucleation (4), a mixture of 22% O2 and 78% Ar. .e nozzle gas exit ve- except that one of the two PAHs concentrations is replaced locities are 13.16 and 16.12 cm/s on the fuel side and oxidizer by the soot particle concentration. side, respectively. .e corresponding strain rates are 9.4 and 11.51 #/s. .ese settings are the same as those in the experiment [28]. Figure 3 shows the temperature profiles 2.3. Stochastic Simulation of Soot Dynamics. .e stochastic obtained from the two mechanisms. .e measurement data simulation adopts the operator splitting Monte Carlo method from [28] is also included in the plot. Both mechanisms developed recently by the authors [13]. .is method is nu- predict almost the same temperature profile, very close to the merically highly efficient and quite flexible in accommodating measurement data. .e peak position of the simulated various models for the general aerosol dynamics. Details can temperature profile differs from the measurement only be found in the original paper. a little bit. Both mechanisms predict the temperature profile quite satisfactorily. 3. Results and Discussion Soot particles are formed from precursor PAH molecules. In the ABF mechanism, it contains four PAHs, 3.1. Verification and Validation. In principle, this simulation denoted as A1 to A4, corresponding to benzene C6H6, work used two methods: Chemkin for gas chemical kinetics naphthalene C10H8, phenanthrene C14H10, and pyrene and stochastic simulation tool for soot dynamics. Chemkin C16H10, respectively. In the KM2 mechanism, much larger is the de facto standard for the simulation of chemical ki- PAH molecules are included, up to coronene (C24H12). .e netics. A converged numerical solution is generally believed precursors mostly determine the soot particle number to be true for the model equations, and it requires no density through the nucleation process, that is, conversion of verification. Since the simulation results greatly depend on PAH gas to particles. Figure 4 compares the profiles of mole the kinetic mechanism used, most of the validation work fractions of PAHs from A1 to A4 from simulations with the aims to make accurate prediction as possible for quantities two mechanisms along with the measurement [29]. .e available in experimental measurements. KM2 mechanism gives better prediction than the ABF .e stochastic method used here for soot dynamics was mechanism for all PAHs, when compared with the exper- developed by the authors, which has been thoroughly ver- iment data. .e ABF mechanism is found to underpredict ified against classical testing cases [13], and has been suc- A4 considerably. It is worth noting that the comparison with cessfully used to simulate soot dynamics in combustion experiment data for KM2 modelling is not completely flames [14, 25] and general aerosol in turbulent flows satisfactory either. .ere are large uncertainties in com- [26, 27]. bustion kinetic models [30], which may come from the In order to compare simulation results from the two extrapolation of knowledge of smaller species reaction, kinetic mechanisms, that is, ABF and KM2, a nonpremixed missing reaction pathways, uncertainty in the reaction 4 Journal of Nanotechnology

10–3 10–3

10–4

10–4 10–5

10–6 10–5 A2 mole fractionA2 mole A1 mole fractionA1 mole

10–7

10–6 10–8 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Distance from fuel nozzle (cm) Distance from fuel nozzle (cm)

ABF mechanism ABF mechanism KM2 mechanism KM2 mechanism S. Senkan S. Senkan

(a) (b) 10–3 10–4

10–4 10–5

10–5 10–6 10–6 A3 mole fraction A3 mole A4 mole fractionA4 mole 10–7 10–7

10–8 10–8 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Distance from fuel nozzle (cm) Distance from fuel nozzle (cm)

ABF mechanism ABF mechanism KM2 mechanism KM2 mechanism S. Senkan S. Senkan

Figure 4: Profiles of A1–A4 mole fraction. Circles are measurement data from [29]. (a) A1, (b) A2, (c) A3, (d) A4. coefficient, and others. Figure 2 shows the mole fraction of much faster on the fuel side than on the oxidizer side. .at is larger PAHs, which is not available in the ABF mechanism. because nucleation and condensation renders the volume Coronene concentration is much higher than that of A4. fraction to increase quickly towards the peak. .e decrease .is fact has a very large impact on the soot volume faction. of volume fraction off the peak on the oxidizer side is mostly It is shown in Figure 5 that the ABF mechanism under- due to gas transport (convection and diffusion), which is predicts the soot volume fraction considerably, while the a process much slower than the soot dynamics. result from KM2 compares rather well with the measurement data. Neglect of the larger PAHs in the ABF mechanism is the underlying reason for such underprediction. 3.2. Effects of Nozzle Exit Velocity. In the last subsection, it is Overall, the KM2 mechanism gives much better predictions shown that the KM2 mechanism gives much better pre- than the ABF mechanism, although with higher numerical dictions than the ABF mechanism. So from now on, only the cost due to much higher number of species in the simulation. simulation results from the KM2 mechanism are presented. In Figure 5, it is observed, both in measurement data and Figure 6 shows the profiles of soot volume fraction under simulation, that the volume fraction drops from the peak various nozzle exit velocity, from 20 to 40 cm/s (the same on Journal of Nanotechnology 5

2.5 ×10–6 6

2 5 6 – 10 × (cm)

1.5 30 4 D

3 1

Soot volume fraction Soot volume 0.5

Average particle size Average 1

0 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Distance from fuel nozzle (cm) Distance from fuel nozzle (cm) ABF mechanism 20 cm/s KM2 mechanism 30 cm/s B. C. Choi 40 cm/s Figure 5: Proﬁle of soot volume fraction. Circles are measurement Figure 7: Proﬁles of average soot particle size under various exit data from [28]. velocities.

2.5 Figure 7 shows the average soot size, which is calculated thorough the third and zeroth order of moments (the moment is deﬁned with respect to the soot diameter) of the soot particle 2 6

– size distribution. When the nozzle exit velocity is smaller, soot 10

× particles are larger on average. .is comes from the same fact 1.5 of shorter residence time for higher exit velocity. When the nozzle exit velocity increases from 20 to 40 cm/s, the peak average soot diameter decreases from 5.1 to 2.5 nm, around 1 a factor of 2, which corresponds to a factor of 8 on the decrease of volume fraction. From Figure 6, it can be noted that the magnitude of volume fraction for the case of 20 cm/s is nearly Soot volume fraction Soot volume 0.5 10 times higher than that of 40 cm/s; hence, it can be concluded that the nozzle exit velocity has a strong effect on the particle 0 growth process, while it has only a relatively mild effect on the 0.4 0.5 0.6 0.7 0.8 0.9 nucleation process. Distance from fuel nozzle (cm) Soot particle size distribution is the most informative 20 cm/s statistical description on the collective behavior of soot 30 cm/s particles. Figure 8 shows the size distribution at various height 40 cm/s locations from the fuel nozzle. Since the flame is stable, the Figure 6: Profiles of soot volume fraction under various exit distribution at various heights reflects the evolution of the velocities. distribution along the axial direction. From the temperature profile in Figure 3, it is known that the flame sheet is at the height H � 0.82 cm. Figure 8 shows the distribution at three both sides). It is clear that higher exit velocity renders much locations, H � 0.775, 0.75, and 0.675 cm. Further away lower soot volume fraction, due to the reduction of residence from the flame sheet, nascent soot particles grow bigger due to time of soot particle, which causes soot to have less time to surface growth and coagulation. A noticeable trough in the form and grow. On the other hand, higher nozzle exit ve- distribution is formed away from the flame sheet, when locity means higher strain rate, and it is found [31] that particles of intermediate size are greatly consumed due to increasing the strain rate could reduce concentrations of coagulation while nascent nano-size particles are not able to PAHs, which are the precursors to soot formation. When the grow quickly enough to make up the depletion. Such dis- nozzle exit velocity doubles from 20 to 40 cm/s, the mag- tribution has been observed very often in experiments. nitude of the corresponding volume fraction decreases sharply, by a factor nearly 10. .e combined effect of less 4. Conclusions newly nucleated particles and less growth time makes the nozzle exit velocity a crucial factor in determining the soot Soot formation and evolution in counterflow diffusion flames volume fraction. have been investigated by coupling the chemical kinetics 6 Journal of Nanotechnology

H = 0.775 cm H = 0.75 cm 102 102

100 100

10–2 10–2 Normalized number density number Normalized density number Normalized 10–4 10–4

2 4 6 8 10 30 50 2 4 6 8 10 30 50 D D Particle diameter p (nm) Particle diameter p (nm) 20 cm/s 20 cm/s 30 cm/s 30 cm/s 40 cm/s 40 cm/s (a) (b) H = 0.7 cm H = 0.675 cm 102 102

100 100

10–2 10–2 Normalized number density number Normalized Normalized number density number Normalized 10–4 10–4

2 4 6 8 10 30 50 2 4 6 8 10 30 50 D D Particle diameter p (nm) Particle diameter p (nm) 20 cm/s 20 cm/s 30 cm/s 30 cm/s 40 cm/s 40 cm/s (c) (d)

Figure 8: Normalized number density distribution under various exit velocities at different heights from the fuel nozzle H. (a) H � 0.775 cm, (b) H � 0.75 cm, (c) H � 0.7 cm, (d) H � 0.675 cm. solver Chemkin II with the operator splitting Monte Carlo However, the KM2 mechanism provides much closer method, which is an efficient stochastic method for simulating prediction for soot gas precursors, that is, PAHs, when aerosol dynamics. compared with the measurements. In the ABF mechanism, In this study, two detailed gas kinetic mechanisms (ABF the maximum PAH molecule is pyrene, while in the KM2 [17] and KM2 [18, 19]) have been compared during the mechanism, the maximum PAH molecule is coronene. On simulation (using the solver Chemkin II) of ethylene one hand, coronene is found to have much higher propen- combustion in counterflow diffusion flames. Subsequently, sity to nucleate than pyrene. On the other hand, coronene the operator splitting Monte Carlo method is used to concentration observed in KM2 is found to be much higher simulate the soot dynamics. Both the simulated data from than the pyrene concentration (which is the PAH of highest the two mechanisms for gas and soot particles are com- nucleation propensity in ABF). When compared with mea- pared with the experimental data available in the literature. surements on the soot number density and volume fraction, It is found that both mechanisms predict similar profiles for KM2 also shows much better predictions than ABF. the gas temperature and velocity, agreeing well with the .e effect of nozzle exit velocity on soot dynamics has measurements. also been investigated. Higher nozzle exit velocity renders Journal of Nanotechnology 7 shorter residence time for soot particles to form and grow, [6] M. Frenklach, “Method of moments with interpolative clo- which reduces the soot number density and volume fraction sure,” Chemical Engineering Science, vol. 57, no. 12, accordingly. However, the effect of nozzle velocity on the pp. 2229–2239, 2002. number density is far more mild than that on the volume [7] R. McGraw, “Description of aerosol dynamics by the quad- fraction, which means nucleation is slightly affected by the rature method of moments,” Aerosol Science and Technology, nozzle velocity, while surface growth is greatly affected by vol. 27, no. 2, pp. 255–265, 1997. [8] D. L. Marchisio, R. D. Vigil, and R. O. Fox, “Quadrature that. .e residence time is a crucial factor to determine the method of moments for agrregation-breakage processes,” soot particle size. Journal of Colloid and Interface Science, vol. 258, no. 2, pp. 322–334, 2003. Nomenclature [9] M. Z. Yu, J. Z. Lin, and T. L. 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A Fortran Program for Modeling Steady Laminar One- Dimensional Premixed Flame, Sandia National Laboratories, Livermore, CA, USA, 1985. Conflicts of Interest [16] J. F. Grcar, R. J. Kee, M. D. Smooke, and J. A. Miller, “A hybrid .e authors declare that they have no conflicts of interest. Newton/time-integration procedure for the solution of steady, laminar, one-dimensional, premixed flames,” Symposium (International) on Combustion, vol. 21, no. 1, pp. 1773–1782, Acknowledgments 1988. [17] J. Appel, H. Bockhorn, and M. Frenklach, “Kinetic modeling .e financial support from the National Natural Science of soot formation with detailed chemistry and physics: Foundation of China (no. 11402179, 11572274, 11602179) is laminar premixed flames of C2 hydrocarbons,” Combustion greatly acknowledged. and Flame, vol. 121, no. 1-2, pp. 122–136, 2000. [18] Y. Wang, A. Raj, and S. H. 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Research Article Design and Numerical Study of Micropump Based on Induced Electroosmotic Flow

Kai Zhang , Lengjun Jiang, Zhihan Gao, Changxiu Zhai, Weiwei Yan, and Shuxing Wu

Institute of Fluid Engineering of China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Kai Zhang; [email protected]

Received 1 February 2018; Accepted 5 April 2018; Published 9 May 2018

Academic Editor: Xiaoke Ku

Copyright © 2018 Kai Zhang et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Induced charge electroosmotic flow is a new electric driving mode. Based on the Navier–Stokes equations and the Poisson– Nernst–Planck (PNP) ion transport equations, the finite volume method is adopted to calculate the equations and boundary conditions of the induced charge electroosmotic flow. In this paper, the formula of the induced zeta potential of the polarized solid surface is proposed, and a UDF program suitable for the simulation of the induced charge electroosmotic is prepared according to this theory. At the same time, on the basis of this theory, a cross micropump driven by induced charge electroosmotic flow is designed, and the voltage, electric potential, charge density, and streamline of the induced electroosmotic micropump are obtained. Studies have shown that when the cross-shaped micropump is energized, in the center of the induction electrode near the formation of a dense electric double layer, there exist four symmetrical vortices at the four corners, and they push the solution towards both outlets; it can be found that the average velocity of the solution in the cross-flow microfluidic pump is nonlinear with the applied electric field, which maybe helpful for the practical application of induced electroosmotic flow in the field of micropump.

1. Introduction a large number of scholars such as Lin Bingcheng, Qin Jianhua, and so on, all of which contributed to the development of *e huge technological advances have made people’s de- microfluidic chip technology in China and even in the world. mand for technology products continue to move toward the *e internationally renowned magazine Lab on a Chip even direction of portability, miniaturization, and intelligence. published a special album titled “Focus on China” on the 10th With the improvement of living standards, more and more anniversary of its founding, which is to affirm the important attention is being paid to health problems and the accuracy contribution made by Chinese scholars to the research of of the results, and the monitoring methods have also been set microfluidic technology. at higher standards. At this moment, the microfluidic chip is *e delivery mixing, reaction, separation, and control a good choice for further development and popularization of of microfluidics are key components of microfluidics real-time diagnostic technology. system. Because of its small characteristic scale, most of the *e so-called microfluidics chip refers to a chemical or fluid flowing in the microchannels is laminar. Particles, biological lab built on a chip that is only a few square centi- droplets, or bubbles generally within the microchannels meters. It integrates the processes of biological and chemical belong to the field of low Reynolds number flow theory reactions and separation and detection into microchannels, [1–3]. Due to the sharp decrease of the volume to surface while using the design of microchannel networks for micro- area ratio, the study found that the laws and phenomena of fluidic control and transport and ultimately enables various fluid movement at the microscale are different from the functions in chemical or biological laboratories, that is, lab on macro environment; the continuity equation in the three a chip. Much research has been made under the leadership of equations of hydrodynamics may no longer be suitable for 2 Journal of Nanotechnology use in microfluidics [4]. With the reduction of the char- speed under the same voltage, so we design a micropump acteristic scale of the study of fluid motion, a new flow effect [9–11] based on it, which can be applied to the driving of was found. *e coupling of the electric field force, the flow microfluidic chip. field, the temperature field and the ion motion within the microchannel, the electroosmotic flow, electrophoresis, induced electroosmotic flow, and other electrical phe- 2. Theoretical Analysis nomena can be used to achieve microfluidic (microparticle) 2.1. Governing Equations and Boundary Conditions. In transport and control [5, 6]. Electrokinetic phenomena in this study, the theoretical model is based on the Navier– microfluidics are caused by the interaction between the Stokes equation [12] of viscous fluid flow, combined with applied electric field and the diffusion layer in the electric the Poisson–Nernst–Planck (PNP) ion transport equation. double layer. More electrokinetic phenomena studied at *e model can successfully predict new phenomena when the present include electrophoresis, dielectric electrophoresis, applied voltage is too small to disrupt the salt concentration. electroosmotic flow, and induced electroosmotic flow. As the flow is considered steady and incompressible, the Electrokinetic phenomena can be divided into linear governing equations are shown below: (electrophoresis and electroosmotic flow) and nonlinear −1 ⇀ ( ) (dielectric electrophoresis, electrophoresis, and induced Pe ∇ · ∓n+∇φ − ∇n ± � + v · ∇n ± � 0, 1 electroosmotic flow) electrokinetic phenomena according to whether the zeta potential in the electrokinetic phe- ∇ ·(Pe⇀vq) � ∇ ·(c∇φ + ∇q), (2) nomenon changes with an applied electric field. In this paper, the phenomenon of induced electroosmotic flow is ∇ ·(Pe⇀vc) � ∇ ·(q∇φ + ∇q), (3) used. Induced electroosmosis (ICEO) is a phenomenon driven 2λ2∇2φ � −q, (4) by electrostatic forces under applied electric field and is 0 a variant of the electroosmotic phenomenon [7]. *e phe- ⇀ nomenon of induced electroosmotic flow mainly depends on ∇ · v � 0, (5) the interaction of polarizable solids with an applied electric field to generate an electromotive phenomenon. *e induced Re(⇀v · ∇⇀v) � −∇p + ∇2⇀v − ∇2φ∇φ, potential on the polarizable surface is critical to the induced (6) charge electroosmotic flow. *e magnitude of its zeta po- 2 2 2 2 tential is related to the applied electric field. *e earliest where λ0 � εfkT/(2z e apn∞), Pe � Uap/D, Re � uap/v � induced electroosmotic flow was discovered by Romans et al. PeSc. *e other variables are the characteristic speed u, the at the end of the 20th century. Subsequently, in 2004, Bazant characteristic length ap, the kinematic viscosity v, the di- and Squires perfected the relevant theory and formally electric constant ε, the valence of ions z, the absolute proposed the concept of inducing electroosmotic flow. And temperature T, the ion concentration n∞, the diffusion the study of the mixing [8] and transporting of the fluid in coefficient D, and Boltzmann’s constant k. the simple microchannel is accomplished by using this Equations (2)–(4) are solved to obtain ion concentration theory. By 2005, Levitan used experimental methods to and density distribution, and then (5) and (6) are solved to confirm the correctness of the basic model of induced get the information of flow field. *e zeta potential in the electroosmotic flow. electroosmotic flow is induced by an applied electric field, *e induced charge electroosmotic flow (ICEOF) has and the magnitude of the potential depends on the applied been studied and applied to the microfluidic systems ex- electric field. According to the relevant theoretical study, it is tensively in the last two decades. *e phenomenon is used found that the induced tangent slip velocity of the electric by Wu and Li to realize the function of fluid mixing and double layer on the polarizable solid surface in the elec- flow regulation in microfluidic chips; Zhao and Bau used troosmotic flow is induced electroosmotic flow to enhance chaotic flow to r improve the mixing efficiency of microfluidics; Yariv, Bau, εε0 2 uICEO � E , (7) and Li et al. gave attention and conducted preliminary μ studies on inducing particle-wall effect in electroosmotic where ε is the dielectric constant, ε is the dielectric constant flow; Peng then experimentally found that the higher the 0 of vacuum, r is the radius, μ is the dynamic viscosity, and E is zeta potential of the electrical double layer around the the applied electric field strength. surface of the polarizable solid, the more particles ag- glomerated; demonstrating the feasibility of using micronanoparticle manipulation to induce electroosmosis. Harbin Institute of Technology, Peng and Jia innovated the 2.2. Zeta Potential Verification. Induced charge electroos- use of ITO conductive glass as the electrode, based on the mosis flow (ICEOF) phenomenon, which is caused by the principle of induced electroosmotic flow and imple- interaction between the applied electric field and the electric mentation of micronanoparticle manipulation. double layer formed on the polarizable surface, and zeta Compared with the classical electroosmotic flow, the potential changes on the polarizable solid are shown in induced electroosmotic flow can obtain a higher driving Figure 1. Journal of Nanotechnology 3

0.5

0 0246

Zeta potential (V) Zeta potential β –0.5

–1 β Figure 3: Mesh division of cross micropump. Theoretical data Simulation data independency verification, the cross geometry model of Figure 1: Zeta potential on the polarizable solid surface under the micropump is shown in Figure 3, and the total number of E electric filed of 0. grids finally confirmed is 20,000. *e model’s boundary conditions are set as follows: Outlet 1 (1) Boundary conditions of the surface potential Inlet and outlet: φinlet−1 � φa, φinlet−2 � 0 V; φoutlet−1 � φoutlet−2 � 0 V; *e surface of polarizes solid: zφ/z⇀n � −(zc/z⇀n)/q. 200 μm (2) Boundary conditions of ion concentration Inlet and outlet: c � 2; Side wall: c � 2; Polarizable metal ⇀ ⇀ Ø =100 μm *e surface of polarized solid: zc/zn � −q(zφ/zn). (3) Boundary conditions of ion density Inlet 1 Inlet 2 Inlet and outlet: q � 0; Side wall: q � 0; *e surface of polarized solid: zq/z⇀n � −c(zφ/z⇀n). 2000 μm In this simulation, the flow field, the applied electric field, 200 μ m and the zeta potential control equation of the wall surface of the polarizable obstacle are shown in (4). *e water used in Pt electrode Pt electrode the solution medium is related to the physical parameter: −1 −1 −3 −1 −1 εr � 80, ε0 � 8.85e − 12 C · V m , μ � 1.003e kg · m s , ρ � 998.2 kg/m3. Outlet 2

Figure 2: *e geometry of micropump. 3.2. Results and Discussions. First of all, the electric field of the cross channel is analyzed. In the simulation, an additional electric field is added to the two inlets to generate Under the two-dimensional uniform electric field, the an electric field from the positive electrode to the negative analytical formula of zeta potential at ideal polarizable cy- electrode in the solution medium in the microchannel, as lindrical surface is shown below: shown in Figure 4. At the same time, under the action of an

ζ � 2E0a cos β. (8) applied electric ﬁeld, the centrally located polarizable electrode is polarized, and the opposite ion in the adsorption solution forms a close-packed charge layer on the 3. Numerical Simulation surface, eventually producing an electric double layer near the surface. *e potential is the zeta potential, and the 3.1. Model and Boundary Conditions. As shown in Figure 2 charge density around the polarizable solid is shown in of the cross-shaped induced electroosmotic micropump, Figure 5. In the program, the negative terminal defaults to a cylindrical polarizable solid is embedded in the middle of zero, so the potential and charge in the positive direction the cross-shaped channel. *e distance between the ener- will be more dense, but after the power is applied, an gized electrodes is L � 2000 μm, the width of the micro- electric ﬁeld will be generated between the positive and channel is W � 200 μm, and the diameter of a circular negative electrodes. polarizable solid is ϕ � 100 μm. Using the Gambit software *erefore, when the center of the polarizable solid to mesh the 2D micropump model and pass the grid surface produces an electric double layer under the action 4 Journal of Nanotechnology

fluid can not flow out from the outlet but do swirling 20 movement in volatile solids around. As the applied electric field increases, the shape of the vortex around the polarizable solid can also be found from Figure 6 above. When the voltage at the inlet is φa � 10 V, the four vortices are basically at four corners and dis-

20 tributed evenly. With the increase of voltage, the four 40 30 vortices around the polarizable solid gradually move toward the exit channel. When the voltage at the inlet is

17 φa � 300 V, it can be clearly seen that the four vortices basically entered the interior of the exit channel. At the same time, the distance between the two vortices of polarizable solids increases with increasing voltage. *e above results show that the greater the voltage, the more easily the fluid flows into the outlet channel and also can result in a more efficient driving effect. In general, the performance of a micropump is mainly 20 measured by its microfluidic driving ability, which can be compared with the fluid velocity at the exit. *is paper Figure 4: Voltage diagram in the cross channel. mainly simulates cross induced electroosmotic micro- pumps with a two-dimensional structure. *erefore, it is necessary to study the speed of its outlet. According to the simulation results, under the ideal conditions, micropump at the upper and lower exit has the same speed. *erefore,

0.1 the speed of one of the outlets will be studied separately in this paper. Figure 7 shows the velocity proﬁle at one outlet, where the vertical axis is the exit speed v (mm/s) and the 0.15 abscissa is the distance between the solution and the exit 0.2 distance l (mm). As can be seen from the ﬁgure, the velocity

0.05 at the outlet is parabolic, and the larger the voltage is, the greater the velocity is, and the driving eﬀect of the 0.2 0.25 micropump is better. When the voltage at the inlet is 100 V, 0 the maximum ﬂuid velocity at the outlet of the electrical microchannel reaches 10 mm/s, and as the voltage in-

0.1 creases, the drive speed increases faster and faster. *is shows that the use of the micropump can produce a good driving effect. Figure 8 shows the relation between the average velocity of single outlet and applied electric field strength, where the ordinate is the average speed at a single exit v (mm/s) and the abscissa is the voltage at the power source Figure 5: Charge density diagram the in cross channel. U (V). It can be seen from the figure that the average speed at a single outlet is a quadratic nonlinear relationship with the power supply voltage, and when the power supply of an applied electric field, the ions in the solution are voltage is higher, the average speed of the micropump attracted by the electric double layer, and finally the liquid increases faster. When the voltage is greater than 100 V, is driven to form an induced electroosmotic flow. Figure 6 the average speed of the microchannel outlet at this time is shows the micropump flow diagram of induced electro- already close to 10 mm/s. At this point, we linearly fit the osmotic flow in the cross channel under different electric numerical simulation results to get the cross structure of field intensities obtained from simulation. What can be the micropump single-exit average velocity and applied seen from the diagram is that some of the fluids will flow electric field curve: y � 0.001x2 − 0.081x + 2.0618. along the polarizable solid surface from the left and right inlet to the outlet. Fluid at a distance farther away from the 4. Conclusions polarizable solids does not enter the exit channel but instead creates vortices around the polarizable solids. *is is In summary, the mechanism of induced electroosmotic mainly due to the fact that the ion concentration in the flow is studied in depth. *e analytical solution of induced diffusion layer in the electrical double layer is smaller in zeta potential at polarizable solid surface is proposed by distance from the polarizable solid and less in drag force on analyzing the governing equations of induced electroos- the fluid driven by the external electric field, so that the motic flow. Based on this theory, a UDF program suitable Journal of Nanotechnology 5

(a) (b)

(c) (d) Figure 6: Streamline diagram of cross channel under diﬀerent electric ﬁeld intensities. (a) φa � 10 V, (b) φa � 100 V, (c) φa � 200 V, and (d) φa � 300 V.

120 80

) 70

100 –1 s · 60

) 80 –1 50 2 s

· y = 0.001x – 0.0506x 60 40 30

40 speed (mm Average

Velocity (mm Velocity 20

20 10 0 0 0 100 200 300 400 –0.1 –0.05 0 0.05 0.1 Voltage (V) Distance (mm) Figure 8: Relation between average velocity of single outlet and 10 V 200 V applied electric field strength. 100 V 300 V Figure 7: Velocity profiles at individual outlets at different the cross induced charge electroosmosis micropump has voltages. a nonlinear relationship with the applied electric field, which is more powerful than that of the traditional elec- for induced charge electroosmotic flow simulation is de- troosmotic pump. veloped. At the same time, the cross micropump driven by induced charge electroosmotic flow was designed, and the Conflicts of Interest voltage, potential, charge density, and flow field of the induced micropump were obtained. *e results show that *e authors declare that they have no conflicts of interest. 6 Journal of Nanotechnology

Acknowledgments *is study was supported by the National Natural Science Foundation of China (11472260). References [1] J. Z. Lin, Y. L. Wang, P. J. Zhang, and X. K. Ku, “Mixing and orientation behaviors of cylindrical particles in a mixing layer of an Oldroyd-B fluid,” Chemical Engineering Science, vol. 176, pp. 270–284, 2018. [2] J. Lin, X. Pan, Z. Yin, and X. Ku, “Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation,” Applied Mathematics and Mechan- ics, vol. 37, no. 10, pp. 1275–1288, 2016. [3] M. Z. Yu, J. Z. Lin, and T. Chan, “Effect of precursor loading on non-spherical TiO2 nanoparticle synthesis in a diffusion flame reactor,” Chemical Engineering Science, vol. 63, no. 9, pp. 2317–2329, 2008. [4] M. Z. Yu, Y. Y. Liu, J. Z. Lin, and M. Seipenbusch, “Gener- alized TEMOM scheme for solving the population balance equation,” Aerosol Science and Technology, vol. 49, no. 11, pp. 1021–1036, 2015. [5] K. Ward and Z. Hugh Fan, “Mixing in microfluidic devices and enhancement methods,” Journal of Micromechanics and Microengineering, vol. 25, no. 9, p. 094001, 2015. [6] K. Masilamani, S. Ganguly, C. Feichtinger, D. Bartuschat, and U. Rüde,“Effects of surface roughness and electrokinetic heterogeneity on electroosmotic flow in microchannel,” Fluid Dynamics Research, vol. 47, no. 3, p. 035505, 2015. [7] C. Canpolat, “Induced-charge electro-osmotic flow around cylinders with various orientations,” Proceedings of the In- stitution of Mechanical Engineers, Part C: Journal of Me- chanical Engineering Science, vol. 231, no. 21, pp. 4507–4066, 2017. [8] C. Wang, Y. Song, X. Pan, and R. Dongqing Li, “A novel microfluidic valve controlled by induced charge electroosmotic flow,” Journal of Micromechanics and Micro- engineering, vol. 26, no. 7, p. 075002, 2016. [9] K. Bengtsson and N. D. Robinson, “A large-area, all-plastic, flexible electroosmotic pump,” Microfluidics and Nano- fluidics, vol. 21, p. 178, 2017. [10] A. K. R. Lai, C. C. Chang, and C. Y. Wang, “Optimizing electroosmotic pumping rates in a rectangular channel with vertical gratings,” Physics of Fluids, vol. 29, no. 8, p. 082002, 2017. [11] M. Gao and L. Gui, “A handy liquid metal based electroosmotic flow pump,” Lab on a Chip, vol. 14, no. 11, pp. 1866– 1872, 2014. [12] M. Z. Yu, J. Z. Lin, and T. Chan, “Numerical simulation of nanoparticle synthesis in diffusion flame reactor,” Powder Technology, vol. 181, no. 1, pp. 9–20, 2008. Hindawi Journal of Nanotechnology Volume 2018, Article ID 8672106, 10 pages https://doi.org/10.1155/2018/8672106

Research Article Simulation of Motion of Long Flexible Fibers with Different Linear Densities in Jet Flow

Peifeng Lin ,1 Wenqian Xu,1 Yuzhen Jin,1 and Zefei Zhu 2

1Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou 310008, China 2School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310008, China

Correspondence should be addressed to Zefei Zhu; [email protected]

Received 1 February 2018; Accepted 12 March 2018; Published 2 May 2018

Academic Editor: Mingzhou Yu

Copyright © 2018 Peifeng Lin et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Air-jet loom is a textile machine designed to drive the long fiber using a combination flow of high-pressure air from a main nozzle and a series of assistant nozzles. To make the suggestion of how to make the fiber fly with high efficiency and stability in the jet flow, in which vortices also have great influence on fiber movement, the large eddy simulation method was employed to obtain the transient flow field of turbulent jet, and a bead-rod chain fiber model was used to predict long flexible fiber motion. ,e fluctuation and velocity of fibers with different linear densities in jet flow were studied numerically. ,e results show that the fluctuation amplitude of a fiber with a linear density of 0.5 ×10−5 kg·m−1 is two times larger than that of a fiber with a linear density of 2.0 ×10−5 kg·m−1. ,e distance of the first assistant nozzle from the main nozzle should be less than 120 mm to avoid collision between the fiber and the loom. ,e efficient length of the main nozzle to carry the fiber flying steadily forward is about 100–110 mm. For fibers with a linear density of 0.5 ×10−5 kg·m−1, it is suggested that the distance of the first assistant nozzle from the main nozzle is about 110 mm. With the increase of fiber linear density, the distance could be appropriately increased to 140 mm. ,e simulation results provide an optimization option for the air-jet loom to improve the energy efficiency by reasonably arranging the first assistant nozzle.

1. Introduction Matsuoka [2] proposed a method for simulating the dynamic behavior of rigid and flexible fibers in a flow field. ,e fiber is Carried flight of a long, flexible fiber in a jet flow field is regarded as made up of spheres that are lined up and bound to a fundamental science problem for air-jet loom [1], which is each neighbor. Each pair of bonded spheres can stretch, bend, designed to drive the yarn using a combination flow of high- and twist by changing the bond distance, bond angle, and pressure air from a main nozzle and a series of assistant torsion angle between spheres, respectively. For the rigid nozzles. Different from other pipeline pneumatic conveying fiber, the computed period of rotation and distribution of systems, the jet flow of the main nozzle will rapidly de- orientation angle agree with those calculated using Jeffery’s celerate in a free area and cannot guarantee a sustained high- equation with an equivalent ellipsoidal aspect ratio. For the speed flight of the fiber. ,erefore, assistant nozzles must be flexible fiber, the period of rotation decreases rapidly with the added to the system. On the other hand, the presence of the growth of bending deformation of the fiber and rotation orbits reed groove requires that the fluctuation of the fiber must be deviate from a circular one for the rigid fiber. De Meule- less than a certain range. How to make the fiber fly forward meester et al. [3] tried to study the dynamic properties of fiber rapidly and steadily is always a core issue of fiber flight and is movement and developed a one-dimensional mathematical jointly determined by two aspects: the fiber model and the model, in which the behavior of the fiber is described by flow simulation. Newton’s second law. Lindstro¨m and Uesaka [4] proposed In general, a long fiber can be modeled as a rigid or flexible a model for flexible fibers in viscous fluid flow; namely, the fiber for different dynamic research purposes. Yamamoto and fibers are modeled as chains of fiber segments which interact 2 Journal of Nanotechnology with the fluid through viscous and dynamic drag forces. y Fiber segments, from the same or different fibers, interact with each other through normal, frictional, and lubrication forces. ,e simulations using the proposed model successfully reproduced the different regimes of motion for thread-like particles that range from rigid fiber motion D to complicated orbiting behavior including coiling and self-entanglement. x Vahidkhah and Abdollahi [5] used the lattice Boltzmann method (LBM) to solve the Newtonian flow field and the immersed boundary method (IBM) to simulate the deformation of the flexible fiber interacting with the flow. ,e variations of the fiber length during the simulation time for Figure 1: Schematic diagram of vortices in jet flow. different values of stretching constant are studied. Kabanemi and He´tu [6] carried out a direct simulation study to analyze body, and the bending deformation is more obvious than that the effect of fiber rigidity on fiber motion in simple shear flow. of unconstrained flexible body. Yang et al. [13] studied the ,e fiber is modeled as a series of rigid spheres connected by two-way coupling turbulent model and rheological properties stiff springs, which is similar to that used by Yamamoto and for fiber suspension in the contraction based on the RANS Matsuoka [2]. ,e model correctly predicts the orbit period of simulation. fiber rotation, as well as the trend of critical flow strength, Pei and Yu [14] studied the motional characteristics of versus fiber aspect ratio, during which the breakage occurs in the flexible fibers in the airflow inside the Murata vortex simple shear flow. Meirson and Hrymak [7] extracted rota- spinning (MVS) nozzle. A two-dimensional fluid structure tional friction coefficients from Jeffery’s model, created a interaction (FSI) model combined with the fiber wall contact general case long flexible fiber orientation model, and applied is introduced to simulate a single fiber moving in the airflow it in a simple shear flow. Nan et al. [8] presented a linear inside the MVS nozzle. ,e model is solved using a finite viscoelastic sphere-chain model based on the discrete element element code ADINA. Based on their simulation results, the method to quantify the material damping of deformed flexible formation principle and the influence of some nozzle pa- fibers. A correlation is formulated to quantify the relationship rameters on the tensile property of the MVS fiber were between the damping coefficient of the local bond and that of discussed. the flexible fiber. Meulemeester et al. [9] developed a three- More and more investigators put their interest on the dimensional mathematical model of the yarn. ,e three- air-jet loom [15–18] and other fluid machineries [19, 20], dimensional model for the weft insertion on air-jet looms but due to the computation cost gap between scientific has been successfully tested. researches and engineering needs, only a few of them On the other side, as we have mentioned, fiber flying in considered the influence of turbulent fluctuation [21]. ,e fluid is also affected by turbulent flow characteristics. Kim jet flow caused by the main nozzle is a typical free shear et al. [10] analyzed the flow in an air-jet loom by using turbulence, and there are strong vortices which play a time-accurate characteristic-based upwind flux-difference an important role on the momentum and energy transport splitting compressible Navier–Stokes method. ,e unsteady of flow field [22, 23]. ,e vortices also have great influence pressure and Mach number behavior along the center line of on the fiber movement. Large eddy simulation (LES) is the main nozzle were analyzed. Andric´ et al. [11] analyzed the a believed mathematical model for turbulence vortex dynamics of individual flexible fibers in a turbulent flow, the simulation, by directly calculating the large-scale flow direct numerical simulation of the incompressible Navier– motion. Stokes equations is used to describe the fluid flow in a plane In this paper, we simplified the two-phase flow system channel, and a one-way coupling is considered between the and were able to employ the LES method to simulate the fibers and the fluid phase. ,ey found that the fiber motion is development of a vortex in jet flow and the Lagrangian bead- primarily governed by velocity correlations of the flow fluc- rod model to give the time evolution of a long flexible fiber tuations. In addition, they reported that there is a clear ten- distribution with different linear densities. ,en, the fiber dency of the thread-like fibers to evolve into complex fluctuation and the velocity were discussed to make the geometrical configurations in a turbulent flow field, and the suggestion of how to make the fiber fly with high efficiency fiber inertia has a significant impact on reorientation time- and stability in an air-jet loom. scales of fibers suspended in a turbulent flow field. Jin et al. [12] simulated the turbulent flow by solving the Reynolds- averaged Navier–Stokes (RANS) equations and con- 2. Mathematical Formulation ducted the three-dimensional numerical simulation of the movement of the flexible body. ,e numerical results 2.1. Fluid Flow. ,e 2D jet flow is shown in Figure 1, in which show that an unconstrained flexible body would turn over x and y are the streamwise and cross-stream directions, re- forward along the airflow’s diffusion direction, while spectively. ,e width of the nozzle D is 3.5 mm, the fluid a constrained flexible body in the flow field will make velocity at the nozzle U0 is 240 m/s, and the flow Reynolds a periodic rotation motion along the axis of the flexible number Re � 5.68 × 104. Journal of Nanotechnology 3

,e LES equations governing the jet flow obtained by i i + 1 filtering the Navier–Stokes equations are as follows [24]: zρ zρu � + n � 0, zt zxn (1) N zρu � zρu u � −zp z zσ m + m n � + mn , �μ − τmn � 1 zt zxn zxm zxn zxn Figure 2: Schematic diagram of the flexible fiber model. where ρ is the fluid density, um is the filtered velocity, p is the filtered pressure, index m, n is taken as 1, 2 and refers to the x, y, and μ is the dynamic viscosity. ⎧⎪ 24 ⎪ Re ≤ 1, ,e SGS (subgrid stress) tensor τmn and Smargorinsky– ⎪ ⎪ Re Lilly model [24] which are based on the mixing length hy- ⎪ ⎨⎪ pothesis are used to calculate the SGS stress. C � 24 4 ( ) d ⎪ + √�� , 5 ⎪ 3 1 < Re ≤ 1000 ⎪ Re Re ⎪ ⎪ 2.2. Flexible Fiber Model. A single flexible long fiber is ⎩⎪ modeled as a bead-rod chain, which is similar to that used by β Re > 1000, Guo et al. [25]. ,e fiber model is composed of N beads, where Re � 2r ρ|V − V | is the Reynolds number of N s v qi 0i which are connected by − 1 massless rods (Figure 2). Only equivalent spherical and β is a constant between 0.4 and 0.45. the beads are affected by forces, and the rods maintain the According to Newton’s second law, the equations of motion configuration of the fiber. Using the model, the chain is for the bead i that constitute the fiber are as follows: allowed to be stretched by changing the distance of adjacent d2r rods. m i � F, i dt2 If the distance between adjacent beads is not equal to the (6) equilibrium distance, the stretching restoring force Fni exerted on the bead i will be F � Fni + Fdi, πR2E where m is the mass of the bead i and r is the position vector F � − ΔL, (2) i i ni L of the bead i. where R is the bead radius, E is the elastic modulus, L is the equilibrium distance, and ΔL is the distance variation, which 3. Results and Discussion is equal to the transient distance of each two adjacent beads 3.1. Computation Conditions. ,e computational domain subtracted by the equilibrium distance. covers x × y � 21D × 40D, the nozzle width D � 3.5 mm, When immersed in the unsteady flow field which is fully developed boundary conditions at the outlet, and static calculated in the last section, the fiber is subjected to hy- surrounding environment conditions are assumed. ,e local drodynamic forces, which are also changed with time. In this mesh is shown in Figure 3. Fiber motion is solved in one-way paper, only drag force is considered in order to decrease the coupling between the fiber and flow; fiber-wall and fiber- computational cost. Other hydrodynamic forces such as fiber interactions are neglected. Basset history term, additional mass, slip-rotational lift force, and fluid inertia are negligible. For bead i, the drag force Fdi is contributed by fiber sections (i − 1, i) and 3.2. Method Verification (i, i + 1). It can be calculated as follows: 3.2.1. Jet Flow Field Verification. According to the experi- 1 i i ( ) mental results of [26, 27], the streamwise velocity satisfies Fdi � �Fdi−1, i + Fdi, i+1�, 3 2 the self-preservation distribution at x ≥ 8D and the profile is expressed as a Gaussian curve. ,e self-preservation profiles where Fdi−1, i and Fdi, i+1 are the drag forces acting on bead i, which are devoted to the fiber section (i − 1, i) and (i, i + 1), of the mean velocity are shown in Figure 4, where the data are normalized by the centerline mean velocity U and the respectively. ,e drag Fdi−1, i acting on the fiber section m (i − 1, i) can be expressed as follows: half-width r0.5, which is the distance between the position where the streamwise velocity u/Um � 0.5 and the center 2 � � i πrv � � line of the jet. ,e figure shows that the air flow which is F � C ρ�V − V ��V − V �, (4) di−1,i 2 d qi di qi di mentioned in Section 2.1 has been successfully simulated. where Vqi and Vdi are the fluid and fiber velocities at the mass center of the fiber section (i − 1, i), and Cd is the drag 3.2.2. Flexible Fiber Model Verification. As the limitation of coefficient associated with Reynolds number, which can be the visible area and the resolution of the high-speed camera, represented as follows: an experiment about the motion of microscale fixed flexible 4 Journal of Nanotechnology

0.006

0.004

0.002

y 0

Figure 5: Test equipment of the experiment. –0.002

,ere are some less significant differences between the –0.004 simulation models used in this paper and the actual fiber properties: the bending modulus of the fiber, which is much lower than the elasticity modulus, is not taken into account; –0.006 0 0.002 0.004 0.006 0.008 0.01 0.012 the hydrodynamic forces such as Basset force and slip- x rotational lift force, which are relative small compared to the drag force, are neglected as well. Despite all the sim- Figure 3: Mesh at the nozzle exit. plifications, the experimental and simulation results are still in relatively good agreement (Figure 6). ,erefore, the following simulation of the flight fiber is reasonable and 1.0 credible.

3.3. Discussions and Analysis. When a micro- or nanoscale free fiber is being carried by the high-velocity airflow after m 0.5 being injected into the jet flow field, as described above, the u / U flexible fiber model assumes that the fiber mass is concentrated at a series of beads, and the intervals between every two beads can vary as a result of the force-displacement balance on the beads. Subsequently, the transient equivalent 0.0 radius of the fiber varies too; they can even be reduced from –3 –2 –10 1 2 3 y/r microscale to nanoscale. Fiber linear density is another 0.5 important characteristic and has a significant effect on 8D 16D calculating the force acted on the beads which would de- 12D Gaussian termine the fiber motion in the jet flow. So the fiber motion −5 −5 Figure with different linear densities of ρL � 0.5 × 10 , 1.0 ×10 , 4: Self-preservation profiles of the mean streamwise ve- −5 −5 −1 locity by the LES method. 1.5 ×10 , and 2.0 ×10 kg·m are separately studied. ,e initial velocity of the fiber is νin � 70 m/s. ,e initial length of static free fiber is Lf � 40 mm. ,e initial bead fiber was carried out to verify the long flexible fiber model. radius is given as R � 100 μm. ,e two ends of the fibers are ,e fixed fiber length Lf � 100 mm, and the linear density marked as end A and B, respectively. At the initial stage, end −5 −1 (mass per unit length) ρL � 0.2 × 10 kg·m . As shown in A is at the position x � 40 mm and end B is at x � 0 mm Figure 5, the noncontact test bed consists of an air jet loom, (Figure 7). ,e motional characteristic of the fiber is stable at high-speed camera, and professional image analysis software the initial stage. ,e farther from the nozzle, the lower the air Image Pro-Plus. ,e fiber is fixed at the nozzle so its motion velocity. So, the velocity of end B will be faster than that of state can be recorded by the high-speed camera after each end A after a period of time. As a result, fiber bending parameter adjustment of the loom. deformation is observed. ,e transient fiber shape comparison of numerical simulation results and that recorded by the high-speed camera are shown in Figure 6. It can be seen that the It 3.3.1. Effect of Fiber Linear Density on Fiber Fluctuation. Due can be seen that the free end of the fiber fluctuates signif- to different densities, the time of fiber flying across the icantly all the time, but the fluctuation of the fiber segment simulation domain is different. ,e fiber movement is an- close to the main nozzle is not obvious. fluctuates signifi- alyzed until the end A of fiber arrives at the calculating cantly all the time, but the fluctuation of the fiber segment boundary. close to the main nozzle is not obvious. ,is is due to the ,e motion and bending deformation of fibers with turbulence characteristics of the jet flow, fiber flexibility, and different linear densities over time are shown in Figure 8. As the shape characteristics of the fiber. the jet flow goes on, the vortex keeps rolling-up, transporting Journal of Nanotechnology 5

0.03

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Figure 6: Result comparison of ﬁber between numerical simulation and high-speed photography. 6 Journal of Nanotechnology

B A 0 B A

y/D B A

–5

–10 0 5 10152025303540 x/D

t = 0s t = 0.000650s t = 0.001200s

Figure 7: Motion of free ﬁber in jet ﬂow.

10 10

5 5

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–10 –10 0 5 10152025303540 0 5 10152025303540 x/D x/D

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5 5

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t = 0s t = 0s t = 0.000675s t = 0.000675s t = 0.001250s t = 0.001275s (c) (d)

Figure −5 −1 −5 −1 8: Motional characteristic of the fiber with different linear densities: (a) ρL � 0.5 × 10 kg·m , (b) ρL � 1.0 × 10 kg·m , −5 −1 −5 −1 (c) ρL � 1.5 × 10 kg·m , and (d) ρL � 2.0 × 10 kg·m . Journal of Nanotechnology 7 and mixing with each other (Figure 1), and the fluctuation 20 of vortex driven on fiber becomes more and more strong. By comparing Figures 8(a)–8(d), it can be seen that the fiber −5 −1 15 fluctuation relating with linear density ρL � 0.5 × 10 kg·m is more obvious than that of fiber with linear density ρ � 2.0 × 10−5 kg·m−1, which shows that the smaller linear 10

L (m/s) y density causes the more unstable fiber motion. ,is is be- v cause the fiber transverse acceleration generates from the 5 velocity difference between transverse velocity of flow field Velocity Velocity and that of fiber. Moreover, the smaller linear density results in the larger transverse acceleration and the larger transverse 0 acceleration lead to the more unstable fiber motion. ,e transverse velocity of end A of fibers with different –5 linear densities is shown in Figure 9. ,e fiber transverse 0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 velocity is 0 m/s at the initial time, and it becomes larger with Time t (s) the increasing of running time. ,e end transverse velocity 0.5e – 5 kg/m 1.5e – 5 kg/m −5 −1 e e of fiber with linear density ρL � 2.0 × 10 kg·m increases 1.0 – 5 kg/m 2.0 – 5 kg/m to about 2 m/s after 0.0012 s. At the same time, the end Figure 9: Transverse velocity distribution of end A of fibers with transverse velocity of fiber with linear density ρL � 0.5 × 10−5 kg·m−1 increases to about 18 m/s. ,us, the smaller fiber different linear densities. linear density leads to the larger transverse velocity. In other words, the fiber transverse fluctuation becomes more obvious. In order to learn more about the influence of air velocity on fiber fluctuations, several equidistant points are selected on the fiber (Figure 10). To represent the amplitude of fiber B y fluctuation, the fluctuation standard deviation of fiber is Δ calculated using (7) when the end A of fiber arrives at 40 mm, A xA 80 mm, 100 mm, 120 mm, and 140 mm (i.e., xA � 40 mm, 80 mm, 100 mm, 120 mm, and 140 mm) from the main nozzle: �� 2 � Δy Figure σ � , (7) 10: Illustration of free fiber in jet flow. n where Δy is the vertical distance from the point to the With the increase of fiber linear density, the distance could center line and n is the sum of the selected points in the be appropriately increased. fiber (n � 41). ,e fluctuation standard deviation is listed in Table 1. As shown in Table 1, the fiber is affected by the airflow 3.3.2. Effect of Fiber Linear Density on Fiber Velocity. ,e and the fiber fluctuation standard deviation increases x-velocity (vA) and distance from the nozzle (xA) of end A with gradually in the motion process with time. When the fiber different linear densities and time are shown in Figure 11. with different linear densities arrives at the same location in ,e corresponding flow axial velocity (at the positions where jet flow, the smaller fiber linear density causes the greater the end A is) is given in Figure 12. Within the running time fluctuation standard deviation. ,e fluctuation amplitude of 0–0.0002 s, the fiber velocity increases obviously. ,en, −5 −1 of fibers with ρL � 0.5 × 10 kg·m is two times larger than the fiber is in a state of slow acceleration during the time of −5 −1 that of fibers with ρL � 2.0 × 10 kg·m . ,at is to say, the 0.0002–0.0004 s. At the next stage, there is no longer a sig- fiber with smaller linear density will fluctuate more obvi- nificant change on fiber velocity. After about 0.0008 s, the ously as it flies across the computation domain. As the cross fiber is in a state of deceleration. It can be explained as section of the reed groove in the loom is about follows. ,e fiber velocity is relatively low at the initial 5 mm × 5 mm, so when the fluctuating amplitude of the moment. When the end A velocity is around 70 m/s, the air fiber end is larger than 2.5 mm, that is, after the head of velocity of flow field is around 160 m/s (Figure 12). ,ere is −5 −1 fiber with ρL � 0.5 × 10 kg·m arrived at xA � 120 mm a large velocity difference between the flow field and fiber. −5 −1 and the head of fiber with ρL � 1.0 × 10 kg·m arrived at ,e fiber is in rapid acceleration because of the large drag xA � 140 mm, the fiber will much more likely impact with force as listed previously. It means the fiber gets a lot of the groove if there is no another assistant jet flow. So it is energy from the flow in the period of 0–0.0002 s, and the end strongly recommended that the distance of the first as- A flies about 15 mm forward in this ultra-short time. ,en, sistant nozzle from the main nozzle is less than 120 mm for the velocity of flow field, where the fiber is, decreases rapidly, the fiber whose linear density is less than 0.5 ×10−5 kg·m−1. which we can see from Figure 12. During the time of 8 Journal of Nanotechnology

Table 1: Standard deviation of fiber fluctuations with different fiber linear densities at different positions.

Linear density Amplitude of ﬁber ﬂuctuation, σ (mm) –1 ρL (kg·m ) xA � 40 mm xA � 80 mm xA � 100 mm xA � 120 mm xA � 140 mm 0.5 ×10−5 0 0.75 1.62 3.34 5.03 1.0 ×10−5 0 0.40 1.01 2.04 3.20 1.5 ×10−5 0 0.29 0.75 1.54 2.39 2.0 ×10−5 0 0.22 0.61 1.13 1.81

100 150 120 140 95 130 120 110 90 110 (mm) (m/s) A A 100 85 x 100 90 80 80 Velocity v Velocity 70 Position, 90 75 60 v (m/s) Velocity 50 80 70 40 0.0000 0.0005 0.0010 0.0015 Time t (s) 70 v x v x 0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 A A A A t 0.5e – 5 kg/m 2.0e – 5 kg/m Time (s) e v v v v 1.0 – 5 kg/m Te distance of A B A B 1.5e – 5 kg/m v = 70e3 mm/s 0.5e – 5 kg/m 1.5e – 5 kg/m 1.0e – 5 kg/m 2.0e – 5 kg/m Figure 11: vA and xA curves with different linear densities and time. Figure 13: X-velocity of ends A and B of fiber with different linear densities. 250

200 Figure 11, and the velocity of airflow is lower than that of the fiber. ,e fiber cannot get energy from the flow any more for 150 all the fibers simulated here. In other words, the efficient (m/s) f length of the main nozzle to carry the fiber flying rapidly 100 forward is about 100–110 mm. So the addition of an assistant nozzle is suggested in this place. 50 Velocity v Velocity Besides, the difference of velocity distribution is obvious 0 with different fiber linear densities. According to Newton’s second law, the lower linear density leads to greater fiber –50 acceleration. So when the velocity of airflow is greater than 0.0000 0.0005 0.0010 0.0015 that of the fiber, the greater fiber acceleration causes the Time t (s) greater velocity. But, when the velocity of airflow is lower than that of the fiber, the velocity of fiber with a lower 0.5e – 5 kg/m 1.5e – 5 kg/m e e density falls quickly. It means that the fiber with lower linear 1.0 – 5 kg/m 2.0 – 5 kg/m density is more sensitive to the flow, and the flow velocity Figure 12: Airflow velocity at the position of fiber end A. change along the fiber will easily cause a nonuniform velocity distribution at different segments of the fiber. ,e x-velocity of ends A and B of the fiber with different 0.0002–0.0005 s, the velocity of airflow is larger than the fiber, linear densities is shown in Figure 13. ,e velocity difference but the velocity difference decreases. So the fiber is in a state between end A and B of the fiber with liner density ρL � of slow acceleration. As the velocity difference between the 0.5 × 10−5 kg·m−1 is about 35 m/s when the end A arrives at flow field and fiber keeps decreasing in 0.0005–0.0008 s, the calculating boundary, and the fiber with linear density −5 −1 the influence of flow field on fiber velocity becomes un- ρL � 2.0 × 10 kg·m is only about 10 m/s. ,e non- apparent and the fiber remains at the same velocity mag- uniform velocity distribution leads to obvious bending nitude especially for the fibers with a relatively large linear deformation of the fiber (Figure 8(a)), which is undesirable density. At t � 0.0008 s, the fiber end A arrives around for pneumatic conveying especially jet loom. In order to 100 mm away from the main nozzle, which we can see from guarantee the stability of movement of the fiber with lower Journal of Nanotechnology 9

density, a more stable velocity distribution of flow is needed. assistant nozzle is 110 mm for the fiber with ρL � On the other hand, the decrease of the distance between first 0.5 × 10−5 kg·m−1. When the fiber linear density in- the assistant nozzle and the main nozzle can reduce the axial creases, the distance could be appropriately increased velocity change and improve the stability of flow [28]. So, in to 140 mm. order to make the flight more stable, for the fiber with lower linear density, the distance of first assistant nozzle from the Conflicts of Interest main nozzle should be appropriately smaller. But, it is obviously worse for saving energy. Actually, in industrial ,e authors declare that they have no conflicts of interest. production, people will try their best to add the spacing to make full use of the high-speed air jet flow. Acknowledgments ,e discussions above are summarized as follows. First, as we have discussed before, for fiber with linear density ,is work was supported by the National Natural Science −5 −1 Foundation of China (Grant nos. 51576180 and 51676173), ρL � 0.5 × 10 kg·m , if there is an assistant nozzle within 120 mm from the main nozzle, it will be highly likely for the Fluid Engineering Innovation Team of Zhejiang Sci-Tech fiber to impact with the groove. Second, the efficient length University (ZSTU) of China (Grant no. 11132932611309), of the main nozzle to carry the fiber forward is less than National Scholarship Foundation of China, and 521 Talents 110 mm. ,ird, to make full use of the high-speed air jet Fostering Program Funding of Zhejiang Sci-Tech University flow, the distance between the main nozzle and the first of China. assistant nozzle cannot be very small, although the smaller nozzle spacing could lead to a more stable fiber flight. References Considering all the factors, there is only a narrow range of suggestion distance to set the first assistant nozzle. For fiber [1] K. Mitugu, “Air jet loom,” US Patent 4,673,005, 1987. � . × −5 · −1 [2] S. Yamamoto and T. Matsuoka, “A method for dynamic with a linear density ρL 0 5 10 kg m , it is 110 mm, simulation of rigid and flexible fibers in a flow field,” Journal and when the fiber linear density increases, the distance of Chemical Physics, vol. 98, no. 1, pp. 644–650, 1993. could be appropriately increased to 140 mm. [3] S. De Meulemeester, J. Githaiga, L. Van Langenhove, D. V. Hung, and P. Puissant, “Simulation of the dynamic yarn behavior on air jet looms,” Textile Research Journal, vol. 75, 4. Conclusions no. 10, pp. 724–730, 2005. [4] S. B. Lindströmand T. Uesaka, “Simulation of the motion of To make the suggestion of how to make the fiber fly with flexible fibers in viscous fluid flow,” Physics of Fluids, vol. 19, high efficiency and stability in a jet flow, we employed the no. 11, p. 113307, 2007. LES method to simulate the development of vortices in the [5] K. Vahidkhah and V. Abdollahi, “Numerical simulation of jet flow and Lagrangian bead-rod model to give the time a flexible fiber deformation in a viscous flow by the immersed evolution of long flexible fiber distribution with different boundary-lattice Boltzmann method,” Communications in linear densities. ,e fluctuation and velocity of fiber in jet Nonlinear Science & Numerical Simulation, vol. 17, no. 3, flow were then studied numerically, and the simulation pp. 1475–1484, 2012. results can provide an optimization option for the air-jet [6] K. K. Kabanemi and J. Hétu,“Effects of bending and torsion loom to improve the energy efficiency by reasonably rigidity on deformation and breakage of flexible fibers: a direct simulation study,” Journal of Chemical Physics, vol. 136, no. 7, arranging the first assistant nozzle. ,e results are as follows. p. 74903, 2012. (1) As the primary vortex rolls up, transports, and mixes [7] G. Meirson and A. N. Hrymak, “Two-dimensional long- with each other, the fiber fluctuation becomes flexible fiber simulation in simple shear flow,” Polymer stronger and the motion becomes more unstable as Composites, vol. 37, no. 8, pp. 2425–2433, 2015. the linear density decreases. [8] W. Nan, Y. Wang, and H. Tang, “A viscoelastic model for flexible fibers with material damping,” Powder Technology, (2) ,e fluctuation amplitude of a fiber with ρL � 0.5 × vol. 276, pp. 175–182, 2015. 10−5 kg·m−1 is two times larger than that of a fiber [9] S. D. Meulemeester, P. Puissant, and L. V. Langenhove, −5 −1 with ρL � 2.0 × 10 kg·m . 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Nilsson, “A study of a flexible fiber model and its behavior (4) ,e efficient length of the main nozzle to carry the in DNS of turbulent channel flow,” Acta Mechanica, vol. 224, fiber flying steadily forward is about 100–110 mm. So no. 10, pp. 2359–2374, 2013. [12] Y. Jin, J. Li, L. Zhu, J. Du, Y. Jin, and P. Lin, “,ree- an assistant nozzle should be added in this place. dimensional numerical simulation of the movement of the (5) To save energy, according to (2) and (3), the sug- flexible body under different constraints,” Journal of >ermal gested distance between the main nozzle and the first Science, vol. 23, no. 6, pp. 593–599, 2014. 10 Journal of Nanotechnology

[13] W. Yang, K. Zhou, Z. Zhao, and Z.-H. Wan, “Study on the two-way coupling turbulent model and rheological properties for fiber suspension in the contraction,” Journal of Non- Newtonian Fluid Mechanics, vol. 246, pp. 1–9, 2017. [14] Z. Pei and C. Yu, “Numerical study on the effect of nozzle pressure and yarn delivery speed on the fiber motion in the nozzle of Murata vortex spinning,” Journal of Fluids and Structures, vol. 27, no. 1, pp. 121–133, 2011. [15] M. Ishida and A. Okajima, “Flow characteristics of the main nozzle in an air-jet loom part I: measuring flow in the main nozzle,” Textile Research Journal, vol. 64, no. 1, pp. 10–20, 1994. [16] D. D. Liu, Z. H. Feng, B. H. Tan, and Y. P. Tang, “Numerical simulation and analysis for the flow field of the main nozzle in an air-jet loom based on Fluent,” Applied Mechanics and Materials, vol. 105, pp. 172–175, 2012. [17] J. M. Lu, “Discussion on weft insertion processing parameters of air-jet loom,” Applied Mechanics & Materials, vol. 556–562, pp. 1005–1008, 2014. [18] L. Chen, Z.-H. Feng, T.-Z. Dong, W.-H. Wang, and S. Liu, “Numerical simulation of the internal flow field of a new main nozzle in an air-jet loom based on Fluent,” Textile Research Journal, vol. 85, no. 15, pp. 1590–1601, 2015. [19] X. Liu and J. Lu, “Unsteady flow simulations in a three-lobe positive displacement blower,” Chinese Journal of Mechanical Engineering, vol. 27, no. 3, pp. 575–583, 2014. [20] J. Zhao, S. Zhou, X. Lu, and D. Gao, “Numerical simulation and experimental study of heat-fluid-solid coupling of double flapper-nozzle servo valve,” Chinese Journal of Mechanical Engineering, vol. 28, no. 5, pp. 1030–1038, 2015. [21] H. Sun, R. Xiao, F. Wang, Y. Xiao, and W. Liu, “Analysis of the pump-turbine s characteristics using the detached eddy simulation method,” Chinese Journal of Mechanical Engi- neering, vol. 28, no. 1, pp. 115–122, 2014. [22] J. Lin, X. Shi, and Z. Yu, “,e motion of fibers in an evolving mixing layer,” International Journal of Multiphase Flow, vol. 29, no. 8, pp. 1355–1372, 2003. [23] L. Jianzhong, S. Xing, and Y. Zhenjiang, “Effects of the aspect ratio on the sedimentation of a fiber in Newtonian fluids,” Journal of Aerosol Science, vol. 34, no. 7, pp. 909–921, 2003. [24] J. H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics, Springer, Berlin, Germany, 2002. [25] H.-F. Guo, B.-G. Xu, C.-W. Yu, and S.-Y. Li, “Simulating the motion of a flexible fiber in 3D tangentially injected swirling airflow in a straight pipe-Effects of some parameters,” In- ternational Journal of Heat and Mass Transfer, vol. 54, no. 21, pp. 4570–4579, 2011. [26] W. Forstall and A. H. Shapiro, “Momentum and mass transfer in coaxial gas jets,” Journal of Applied Mechanics-transactions of the ASME, vol. 17, pp. 399–408, 1950. [27] P. Lin, D. Wu, M. Yu et al., “Large eddy simulation of the particle coagulation in high concentration particle-laden planar jet flow,” Chinese Journal of Mechanical Engineering, vol. 24, no. 6, pp. 947–956, 2011. [28] Y. Jin, Research on the Characteristics of Yarn-Air Flow Field Stimulated by Series Nozzles, Zhejiang University, Hangzhou, China, 2010. Hindawi Journal of Nanotechnology Volume 2018, Article ID 6548689, 6 pages https://doi.org/10.1155/2018/6548689

Research Article Simulation and Visualization of Flows Laden with Cylindrical Nanoparticles in a Mixing Layer

Wenqian Lin 1 and Peijie Zhang2

1College of Computer Science and Technology, Zhejiang University, Hangzhou 310027, China 2Department of Mechanics, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Wenqian Lin; [email protected]

Received 31 January 2018; Accepted 27 February 2018; Published 5 April 2018

Academic Editor: Mingzhou Yu

Copyright © 2018 Wenqian Lin and Peijie Zhang. /is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. /e motion of cylindrical particles in a mixing layer is studied using the pseudospectral method and discrete particle model. /e effect of the Stokes number and particle aspect ratio on the mixing and orientation distribution of cylindrical particles is analyzed. /e results show that the rollup of mixing layer drives the particles to the edge of the vortex by centrifugal force. /e cylindrical particles with the small Stokes number almost follow fluid streamlines and are mixed thoroughly, while those with the large Stokes number, centrifugalized and accumulated at the edge of the vortex, are poorly mixed. /e mixing degree of particles becomes worse as the particle aspect ratio increases. /e cylindrical particles would change their orientation under two torques and rotate around their axis of revolution aligned to the vorticity direction when the shear rate is low, while aligning on the flow-gradient plane beyond a critical shear rate value. More particles are oriented with the flow direction, and this phenomenon becomes more obvious with the decrease of the Stokes number and particle aspect ratio.

1. Introduction transferred them onto mica substrates; they showed that enhancing the DTMA (+) in the mixed PSS-DTMA system Mixing of nanoparticles in a flow was found in many ap- would increase the hydrophobic property of the complexes. plications and has been a subject of interest during the recent Finally, they inferred that the polyelectrolyte-surfactant twenty years. /e mixing of nanoparticles affects the template can offer a potential of designing structures of property of the flow and the behaviors of the final pro- polyelectrolyte-nanoparticle materials. Liggieri et al. [4] duction. /erefore, understanding deeply such a flow is utilized different techniques to measure the dilational vis- important to design a novel technology for nanoparticle coelasticity in a wide frequency range and used these data to manipulation. provide qualitative and quantitative information about Some efforts have been put into the numerical simula- structural behaviors of complex mixed layers. Xie et al. [5] tion and visualization of nanoparticles. Singh et al. [1] de- simulated numerically the impact coherent structure on the posited In2O3 and In2O3 : Ag nanoparticle layers and Brownian coagulation of particles in a mixing layer, and it is indicated the presence of Ag2O and Ag in air- and vacuum- found that the number density of nanoparticles decreases annealed samples, respectively. Aruna et al. [2] reported the gradually as the flow evolves, while the particle average modifications in CO sensing of SnOx nanoparticle layers by volume increases. /e impact of fluid advection on particle using Pd nanoparticles; they found that the homogeneously coagulation is small in the regions far away from the eddy mixed nanoparticle layers show capability between CO and structure. /e particle coagulation in the eddy core has ethanol as a manifestation of the dual conductance response. a wave-like distribution. Yazhgur et al. [6] studied the ad- Hsiao et al. [3] used mixed PSS-DTMA Langmuir layers to sorption films of silica nanoparticles modified by a cationic incorporate with silver precursors from the subphase and surfactant at the air-water interface. /ey divided the 2 Journal of Nanotechnology whole surfactant concentration range into four regions y y p characterized by different surface rheological behaviors. Kim ϕ θ et al. [7] studied the optical, electrical, and morphological U 1 x O x behaviors of the Ga2O3 NP/SWNT layers by enhancing the U thickness of SWNTs. Murfield and Garrick [8] performed 2 z numerical simulations of nanoparticle nucleation in tur- Figure bulent wakes and showed that nucleation initially occurs in 1: Mixing layer flow and cylindrical particle. the shear layers where molecular transport dominates and across the span of the wake. Fager and Garrick [9] presented ignored; thus, the governing equations are the following the results of direct numerical simulation of zinc nano- continuity and momentum equations: particle nucleation in a turbulent round jet; they indicated that particle nucleation occurs along the outer region of the ∇ · u � 0, (1) jet, and the regions over which nucleation occurs increase significantly after collapse of the jet potential core. Guzman zu |u|2 ∇2u + ∇�p + � � u × ω + , (2) et al. [10] carried out measurements of the dilational vis- zt 2 Re coelastic modulus against the frequency by the oscillatory barrier method at different degrees of compression of the where u and p are the fluid velocity and pressure, re- monolayer in order to deepen the understanding of the spectively; ω is the vorticity; and Re is the Reynolds number. impacts of nanoparticles on the interfacial properties of biosystems. Orsi et al. [11] studied the interfacial dynamics of a 2D self-organized mixed layer consisted of silica 2.2. Equation for Cylindrical Particles. /e size of a non- nanoparticles interacting with phospholipid monolayers at Brownian and rigid cylindrical particle is smaller than the the air-water interface and found a dynamical transition characteristic scale of the flow field. /e motion of particles from the Brownian diffusion to an arrested state. Garrick is caused by force F and torque T exerted by the fluid: [12] showed that, in the proximal region of the jet, con- du l densation is the dominant mechanism; however, once the jet p mp � F � � f(s) ds, potential core collapses and turbulent mixing begins, co- dt −l agulation is the dominant mechanism. Kerli and Alver [13] (3) investigated the mixture of ZnO and NiO effect on the solar dΩ l cell and observed that the solar cells made with ZnO have the I · + Ω × I · Ω � T � � sp × f(s) ds, dt −l highest performance with the efficiency of 0.542%. /e researches available in the literature are mainly where mp, up, and Ω are the particle mass, velocity, and concerned with spherical particles as shown above. How- angular velocity, respectively; I is the particle moment of ever, many particles in the practical usage are nonspherical inertia; f(s) is the force distribution along the particle axis; s among which the cylinder particle is a typical example. is the distance from any point to the particle center; l is the Compared with the spherical particle, the mixing of cylin- half length of the particle; and p is the orientation vector of drical particles in a flow is more complicated because the the particle. change of particle orientation is coupled with the translation Equation (3) can be nondimensionalized with the scale motion. Understanding the property of cylindrical particles of the flow field as in the mixing layer is a great importance in the optimization 1 dup 1 of production processes. As far as we know, the motion and � � f(s) ds, dt St −1 (4) mixing behaviors of cylindrical nanoparticles in the mixing layer have not been reported in the literatures yet. /erefore, the objective of the present paper is to explore the impact of / dΩ 3 4St −1 2 the Stokes number and particle aspect ratio on the mixing x � � � T + Ω Ω , property and orientation distributions of cylindrical parti- dt aSt Reρ∗ x y z cles in a mixing layer. −1/2 dΩy 3 4St � � � Ty − ΩzΩx, (5) 2. Flow and Basic Equations dt aSt Reρ∗ / 2.1. Mixing Layer Flow and Fluid Equation. A mixing layer dΩ 2a 4St −1 2 z � � � T , consisting of two streams and a cylindrical particle are dt St Reρ∗ z shown in Figure 1 where velocities U1 and U2 are different. In the mixing layer, the hydrodynamic instabilities will where a is the particle aspect ratio, St is the Stokes number 2 happen, which leads to the flow to roll up and form coherent (St � ρpr U0/2μθ0, in which ρp is the particle density, r is the vortex and even pairing of the vortices. particle radius, U0 � U1 − U2, and θ0 is the initial mo- Dilute cylindrical particle suspension is studied here, mentum thickness of the mixing layer), and ρ∗ is the that is, the effect of the cylindrical particle on the fluid can be particle-to-fluid density ratio. Journal of Nanotechnology 3

(a) (b) (c) (d)

Figure 2: Rollup of the mixing layer.

(a) (b) (c) (d)

Figure 3: Vortex pairing.

3. Numerical Simulation 28 for the vortex rollup, and both are doubled for the vortex pairing. 3.1. Initial Condition ofthe Flow. For the temporally evolving mixing layer, the initial condition of the flow consists of two parts: 3.3. Computational Approach. Equations (1) and (2) are (1) /e velocity profile with hyperbolic-tangent form of numerically solved using the pseudospectral method. /e the base flow: Adams–Bashforth scheme is used for the nonlinear term, and the implicit Crank–Nicolson scheme is used for other terms. /e time step is set to be 0.03. /e collection points in U � 0.5 tanh(y). (6) the streamwise and transverse directions are taken to be 128 and 256, respectively. Equations (4) and (5) are solved with the fourth-order (2) /e lower wavenumber disturbances with stream Runge–Kutta method for 400 cylindrical particles. /ese function: particles are distributed and oriented homogeneously in the upper layer of the flow as shown in Figure 1. /e time step of the integration is small enough to guarantee a stable tra- (x, y) � A Ɽ�(y)eiax � + A Ɽ�(y)ei(a/2)x �, ( ) ψ 1 φ1 2 φ2 7 jectory of the particles. where ψ is the stream function; A1 and A2 are the amplitude of the fundamental and its subharmonic modes, respectively, 4. Results and Discussions and we set A1 � 0.1 and A2 � 0 and A1 � 0.1 and A2 � 0.06 for the process of vortex rollup and vortex pairing; Ɽ means 4.1. Visualization of Vortex Rolling Up and Vortex the real part for a complex; φ1(y) and φ2(y) are the Pairing. /e Kelvin–Helmholtz instability makes the mix- eigenfunctions for the fundamental and subharmonic ing layer roll up and forms the large-scale vortex as shown in modes, respectively; and α is the fundamental wavenumber Figure 2, where the center of the vortex is called the center and is set to be 0.4446 [14]. point and the middle point of two center points is called the saddle point. /e flow can be divided into the inhomogeneous region and the homogeneous region. /e 3.2. Boundary Condition. /e periodic boundary conditions streamlines connecting two saddle points are defined as the in the streamwise (x) and transverse (y) directions are demarcation lines; the region encompassed by the stream- imposed by introducing an image flow far enough from the lines, that is, vortex core, is considered as the in- mixing layer center because all the perturbations vanish as homogeneous region, and the other region is called the y → ∞. /erefore, the Fourier spectral method can be homogeneous region. exerted in both directions, and fast Fourier transformation /e process of vortex pairing is shown in Figure 3 where can be used. /e periodic L1 and L2 in the streamwise and we can see that two vortices rotate each other and merge into transverse directions are, respectively, taken to be 2π/r and one vortex finally. 4 Journal of Nanotechnology

(a) (b)

Figure 4: Particle distribution for diﬀerent Stokes numbers (t � 100, Re � 200, and a � 15): (a) St � 0.02 and (b) St � 1.0.

4.2. Particle Distribution. /e mixing of cylindrical parti- 1.25 cles can be described by the particle spatial distributions. /e rollup of the mixing layer drives the particles to the 1.20 edge of the vortex by the inertial centrifugal force, and at the same time it brings about an asymmetric shear rate 1.15 distribution around the particle and generates a normal stress imbalance on the particle surface. /e cylindrical n 1.10 particle migration and mixing are dependent on the above M two factors. 1.05 /e mixing degree of the particles is negatively corre- lated with the inhomogeneous degree of particle distribu- 1.00 tion. /erefore, the inhomogeneous degree of particle distribution Mn is used in order to quantitatively describe 0.95 the mixing behavior: 0 102030405060708090100 �� t 2 2 �N �x (t) − x � +�y (t) − y � i�1 pi c pi c St = 0.02 Mn(t) � ��, (8) N 2 2 St = 0.1 �i� �xpi(0) − xc � +�ypi(0) − yc � 1 St = 1.0 (x , y ) (x , y ) where pi pi and c c are the coordinates of the center Figure 5: Change of inhomogeneous degree with time at diﬀerent of the ith particle and vortex center, respectively. A larger St (Re � 200 and a � 15). value of Mn is corresponding to a poorer mixing.

1.25 4.2.1. Eﬀect of the Stokes Number on the Mixing Degree. Figure 4 shows the particle distribution at t � 100 for St � 0.02 and 1. 1.20 Stokes number (St) is directly related to the function of particle inertia. In the process of rollup of the mixing layer, 1.15 the cylindrical particles for St � 0.02 almost follow ﬂuid

n 1.10

streamlines and distribute more homogeneously and are M mixed thoroughly. On the contrary, the particles for St � 1.0 are centrifugalized from the vortex core to the edge by inertia 1.05 and are poorly mixed. Figure 5 shows the change of inhomogeneous degree 1.00 with time at diﬀerent Stokes numbers. We can see well 0.95 mixed for St � 0.02. /e mixing degree becomes worse with 0 10 20 30 40 50 60 70 80 90 100 the increase of the Stokes number because more particles t are centrifugalized to the edge of the vortex when St is a =10 larger. a =15 a =20

4.2.2. Effect of the Particle Aspect Ratio on the Mixing Figure 6: Change of inhomogeneous degree with time at different Degree. Figure 6 shows the change of inhomogeneous de- a (Re � 200 and St � 0.1). gree with time for different particle aspect ratios. /e mixing degree becomes worse as the particle aspect ratio increases because the angular acceleration of the particles is inversely are smaller for particles with larger particle aspect ratios so proportional to the particle aspect ratio. Particle angular that the centrifugal force drives the particles towards the acceleration and translational motion induced by the fluid edge of the vortex. Journal of Nanotechnology 5

4.3. Orientation Distribution of the Cylindrical Particle. /e 0.6 cylindrical particles will change the orientation angles φ and θ (shown in Figure 1) under two torques. One torque makes the particle rotate around the vorticity axis, and another 0.4 torque leads to the particle spin around the ﬂow direction. For describing the orientation distribution, the orientation

distribution function ψ is used: ψ 0.2 N ψ θ − θ � � i+1−i, (9) i+1 i N where θ is shown in Figure 1, N is the total particle number, 0.0 and Ni+1−i is the number of particles with the angles ranging from i to i + 1. 0 10 20 30 40 50 60 70 80 90 θ (°) 4.3.1. Effect of the Stokes Number on the Particle St = 0.02 Orientation. Figure 7 shows the orientation distribution St = 0.1 function of particles for different St. We can see that, with St = 1.0 the decrease of the Stokes number, more particles are oriented with the flow direction. /is may be attributed Figure 7: Orientation distribution of cylindrical particles for that the angular acceleration of the particles is inversely different St (Re � 200 and a � 15). proportional to the Stokes number, and the angular acceleration induced by the fluid stress is larger at smaller 0.6 Stokes number so that the particles change their orientation more rapidly.

0.4 4.3.2. Effect of the Particle Aspect Ratio on the Particle Orientation. /e orientation distribution functions of cylindrical particles for different particle aspect ratios are ψ 0.2 shown in Figure 8. /e particle would rotate around its axis of revolution aligned to the vorticity direction when the shear rate is low, while aligning on the flow-gradient plane beyond a critical shear rate value. From the figure, we can see 0.0 that more particles are oriented on the flow-gradient plane (θ � 0°) and more particles are oriented towards the flow 0 102030405060708090 direction with the decrease of the particle aspect ratio. θ (°)

a =10 5. Conclusion a =15 a =20 /e effect of the Stokes number and particle aspect ratio on the mixing and orientation distribution of cylindrical par- Figure 8: Orientation distribution of cylindrical particles for ticles in a mixing layer is numerically studied. /e mixing of different a (Re � 200 and St � 0.1). cylindrical particles can be described by the particle spatial distributions. /e rollup of the mixing layer drives the particles to the edge of the vortex by the inertial centrifugal while aligning on the flow-gradient plane beyond a critical force and at the same time brings about an asymmetric shear shear rate value. More particles are oriented with the flow rate distribution around the particle. /e cylindrical parti- direction, and this phenomenon becomes more obvious with cles with the small Stokes number almost follow fluid the decrease of the Stokes number and particle aspect ratio. streamlines and are mixed thoroughly. On the contrary, the particles with the large Stokes number are centrifugalized to the edge of the vortex and are poorly mixed. /e mixing Data Availability degree of particles becomes worse as the particle aspect ratio increases. Particle angular acceleration and translational /e data used to support the findings of this study are motion are smaller for that particle with a larger particle available from the corresponding author upon request. aspect ratio so that the centrifugal force drives the particles towards the edge of the vortex. /e cylindrical particles will Conflicts of Interest change the orientation angle under the two torques with which the particles would rotate around its axis of revolution /e authors declare that there are no conflicts of interest aligned to the vorticity direction when the shear rate is low, regarding the publication of this paper. 6 Journal of Nanotechnology

Acknowledgments /e authors would like to thank the Major Program of National Natural Science Foundation of China (Grant no. 11631216). References [1] V. N. Singh, B. R. Mehta, R. K. Joshi, and F. E. Kruis, “Effect of silver addition on the ethanol-sensing properties of indium oxide nanoparticle layers: optical absorption study,” Journal of Nanomaterials, vol. 2007, Article ID 28031, 5 pages, 2007. [2] I. Aruna, F. E. Kruis, S. Kundu, M. Muhler, R. /eissmann, and M. Spasova, “CO ppb sensors based on monodispersed SnOx : Pd mixed nanoparticle layers: insight into dual conductance response,” Journal of Applied Physics, vol. 105, no. 6, p. 064312, 2009. [3] F. W. Hsiao, Y. L. Lee, and C. H. Chang, “Formation and characterization of mixed polyelectrolyte-surfactant Lang- muir layer templates for silver nanoparticle growth,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 351, no. 1–3, pp. 18–25, 2009. [4] L. Liggieri, E. Santini, E. Guzman, A. Maestro, and F. Ravera, “Wide-frequency dilational rheology investigation of mixed silica nanoparticle-CTAB interfacial layers,” Soft Matter, vol. 7, no. 17, pp. 7699–7709, 2011. [5] M. L. Xie, M. Z. Yu, and L. P. Wang, “A TEMOM model to simulate nanoparticle growth in the temporal mixing layer due to Brownian coagulation,” Journal of Aerosol Science, vol. 54, pp. 32–48, 2012. [6] P. A. Yazhgur, B. A. Noskov, L. Liggieri et al., “Dynamic properties of mixed nanoparticle/surfactant adsorption layers,” Soft Matter, vol. 9, no. 12, pp. 3305–3314, 2013. [7] K. H. Kim, H. M. An, H. D. Kim, and T. G. Kim, “Trans- parent conductive oxide films mixed with gallium oxide nanoparticle/single-walled carbon nanotube layer for deep ultraviolet light-emitting diodes,” Nanoscale Research Let- ters, vol. 8, no. 1, pp. 507–518, 2013. [8] N. J. Murfield and S. C. Garrick, “Large eddy simulation and direct numerical simulation of homogeneous nucleation in turbulent wakes,” Journal of Aerosol Science, vol. 60, pp. 21–33, 2013. [9] A. J. Fager and S. C. Garrick, “Metal particle nucleation in turbulent jets visualization of the large and small scales,” Journal of Visualization, vol. 16, no. 4, pp. 297–302, 2013. [10] E. Guzman, E. Santini, M. Ferrari, L. Liggieri, and F. Ravera, “Interfacial properties of mixed DPPC-hydrophobic fumed silica nanoparticle layers,” Journal of Physical Chemistry C, vol. 119, no. 36, pp. 21024–21034, 2015. [11] D. Orsi, E. Guzman, L. Liggieri et al., “2D dynamical arrest transition in a mixed nanoparticle-phospholipid layer studied in real and momentum spaces,” Scientific Reports, vol. 5, no. 1, p. 17930, 2015. [12] S. C. Garrick, “Growth mechanisms of nanostructured titania in turbulent reacting flows,” Journal of Nanotechnology, vol. 2015, Article ID 642014, 10 pages, 2015. [13] S. Kerli and U. Alver, “Preparation and characterization of ZnO/NiO nanocomposite particles for solar cell applications,” Journal of Nanotechnology, vol. 2016, Article ID 4028062, 5 pages, 2016. [14] A. Michalke, “On the inviscid instability of the hyperbolic- tangent velocity profile,” Journal of Fluid Mechanics, vol. 19, pp. 543–556, 1964. Hindawi Journal of Nanotechnology Volume 2018, Article ID 6513634, 7 pages https://doi.org/10.1155/2018/6513634

Research Article Modeling of Scattering Cross Section for Mineral Aerosol with a Gaussian Beam

Wenbin Zheng 1 and Hong Tang 2,3

1College of Software Engineering, Chengdu University of Information Technology, Chengdu 610225, China 2College of Engineering, Sichuan Normal University, Chengdu 610068, China 3College of Metrology and Measurement Engineering, China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Hong Tang; [email protected]

Received 31 December 2017; Accepted 28 January 2018; Published 28 March 2018

Academic Editor: Xiaoke Ku

Copyright © 2018 Wenbin Zheng and Hong Tang. -is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the generalized Lorenz Mie theory (GLMT), the scattering cross section of mineral aerosol within the Gaussian beam is investigated, and an appropriate modeling of the scattering cross sections for the real mineral aerosols including the feldspar, quartz, and red clay is proposed. In this modeling, the spheroid shape is applied to represent the real nonspherical mineral aerosol, and these nonspherical particles are randomly distributed within the Gaussian beam region. Meanwhile, the Monte Carlo statistical estimate method is used to determine the distributed positions of these random nonspherical particles. Moreover, a method for the nonspherical particles is proposed to represent the scattering cross section of the real mineral aerosols. In addition, the T matrix method is also used to calculate the scattering cross sections of the spheroid particles in order to compare the scattering properties between the plane wave and the Gaussian wave. Simulation results indicate that fairly reasonable results of the scattering cross sections for the mineral aerosols can be obtained with this proposed method, and it can provide a reliable and eﬃcient approach to reproduce the scattering cross sections of the real randomly distributed mineral aerosols illuminated by the Gaussian beam.

1. Introduction Gouesbet is a generalization of the Lorenz Mie theory for an arbitrary incident-shaped beam such as the Gaussian -e problem of light scattering by particles has been an beam (laser in the fundamental mode TEM00) and the important topic of research interest in the wide areas of light sheet [10–12]. applications [1–5]. Since the light scattering has been In this paper, the scattering cross sections for the established, some researches have studied the electromag- practical nonspherical mineral aerosols are investigated, and netic light scattering of particles for the plane wave case [1]. modeling of the scattering cross sections for the feldspar, And some common theories and methods have been utilized quartz, and red clay is conducted. Actually, the nonspherical to analyze this problem. When the plane light wave is in- calculations and measurements show significant differences cident into the particle, the classical Mie theory, the discrete from the sphere particles [13–16]. Here, we choose the dipole approximation (DDA), the T matrix method, and the spheroid to represent the nonspherical mineral aerosol in finite difference time domain (FDTD) method can be used order to study the scattering cross sections of nonspherical for sphere and nonsphere particles [6–9]. Nevertheless, none particles within the Gaussian beam incidence. of those methods can be applied to analyze and calculate the scattering of particles for the nonplane waves such as the 2. Theoretical Method and Calculations Gaussian beam incidence or a top-hat beam. In recent years, with the development of laser tech- In the GLMT framework, the incident Gaussian beam field nique and the expansion of its application areas, the including the electric and the magnetic fields can be de- laser has been used for the measuring of particle sizing scribed by the Bromwich Scalar Potentials (BSP) in the and other particle prosperities. It is well known that spherical coordinate system (r, θ, φ) [10–12]. -e field the generalized Lorenz Mie theory (GLMT) proposed by components are then found to be 2 Journal of Nanotechnology

x,y

w0 O z

Figure 1: Geometry of coordinate of the incident Gaussian beam.

Table 1: Scattering cross section of sphere particle with the diﬀerent position (r � 0.1, λ � 0.5, m � 1.33). Position (0, 0, 0) (0, 1, 0) (0, 2, 10.88) (0, 0, 10.88) (0.1, 0.1, 0.1) (0.1, 0.1, 1)

Csca 0.601405 E−14 0.828364 E−15 0.206664 E−15 0.153682 E−14 0.578179 E−14 0.564966 E−14

Table 2: Scattering and absorbing cross section of sphere particle with the diﬀerent location. (r � 0.1, λ � 0.5, m � 1.33 + 0.1i). Position (0, 0, 0) (0, 1, 0) (0, 2, 10.88) (0, 0, 10.88) (0.1, 0.1, 0.1) (0.1, 0.1, 1)

Csca 0.579785 E−14 0.798824 E−15 0.199261 E−15 0.148166 E−14 0.557396 E−14 0.544659 E−14 Cabs 0.105321 E−13 0.147600 E−14 0.365064 E−15 0.270385 E−14 0.101278 E−13 0.989770 E−14 (r � 0.1, λ � 0.5, m � 1.33 + 0.1i).

2 where gm and gm are the generalized functions of i z U 2 n,TM n,TE E � TM + k U , a b r zr2 TM GLMT, n and n are the scattering coefficient of Mie theory, 2 and E and H are the magnetic and electric energy. i 1 z UTM iωμ zUTE Eθ � − , According to the GLMT, the particle is randomly located r zrzθ r sin θ zφ in the Gaussian beam, and the scattering properties are 1 z2U iωμ zU also determined by the location information in the beam. Ei � TM + TE, φ r sin θ zrzφ r zθ In Figure 1, the beam propagates alone the z axis from negative (1) z to positive z, and the electric field component is essentially z2U Hi � TE + k2U , vibrating in the x axis. -e coordinate origin o is the beam r zr2 TE waist center, and its waist radius is w0. -e unit of the incident 2 i iωε zUTM 1 z UTE wavelength λ and the particle radius r, as well as the beam Hθ � + , r sin θ zφ r zrzθ waist radius w0, is μm [17]. For a measured particle system, the particles can be 1 z2U iωε zU Hi � TE − TM, distributed anywhere within the Gaussian beam [18]. Here, φ r sin θ zrzφ r zθ we define√� the particles in the semicircular region, that is, 0 ≤ z ≤ 3πw2/λ, x2 + y2 ≤ (2w )2. where UTM and UTE are the transverse magnetic√�� (TM) and 0 0 transverse electric (TE) BSP, respectively; i � −1; Ei and Table 1 shows the scattering cross section of the sphere Hi are called electric field and magnetic introduction field, particle with different location in the Gaussian beam. -e respectively; (r, θ, φ) is the spherical coordinate system; k is particle radius r is 0.1, the incident wavelength λ is 0.5, the the wave number; ω is the angular frequency of the elec- waist radius w0 is 1, and the relative complex reflective index tromagnetic wave; and μ and ε are the permeability and the of particle m is 1.33. -e six positions in the Cartesian co- permittivity of the medium, respectively. ordinate are (0, 0, 0), (0, 1, 0), (0, 2, 10.88), (0, 0, 10.88), (0.1, 0.1, -e scattering cross section and extinction cross section 0.1), and (0.1, 0.1, 1). It is very obvious that the scattering cross of particle are evaluated by section of sphere particle is also different when the position of ∞ n particle is different. In Table 2, the scattering cross section and λ2 2n + 1 (n +|m|)! Csca � ∑ ∑ absorbing cross section of the sphere particle are calculated in π n�1 m�−n n(n + 1) (n −|m|)! the Gaussian beam. -e relative complex reflective index of � �� 2�� m �2 � 2�� m �2 particle m is 1.33 + 0.1i, the particle radius r is 0.1, the incident × ��an��gn,TM� + �bn��gn,TE� �, wavelength λ is 0.5, and the waist radius w0 is 1. We can see π 2π 1 i s∗ s i∗ i s∗ s i∗ 2 that the absorbing cross section of sphere particle is still Cext � � � Re�EφHθ + EφHθ − EθHφ − EθHφ �r 0 0 2 different with the different position in the Gaussian beam. In order to investigate the general average scattering × sin θdθdφ, cross sections of nonspherical mineral particles, three (2) aerosols, that is, the feldspar, quartz, and red clay, are Journal of Nanotechnology 3

×10−10 ×10−10 1.8 1.6

1.6 1.4 1.4 1.2 1.2 1 1 sca sca 0.8 C 0.8 C 0.6 0.6 0.4 0.4

0.2 0.2

0 0 0 12345678910 0 12345678910 Equal-surface diameter Equal-surface diameter

w0 = 1 w0 = 1 T matrix T matrix w0 = 10 w0 = 10

Figure 2: Scattering cross sections of quartz particles (λ � 0.5, (a) m � 1.54). ×10−11 1.8

1.6 ×10−10 1.8 1.4

1.6 1.2

1.4 1 abs

1.2 C 0.8

1 0.6 sca C 0.8 0.4

0.6 0.2

0.4 0 0 12345678910 0.2 Equal-surface diameter 0 0 2 4 6 8 10 12 14 16 18 w0 = 1 T matrix Diameter of rotation axis w0 = 10 Aspect ratio = 1/3, w0 = 1 T matrix (b) Figure 4: Scattering and absorbing cross sections of feldspar Figure 3: Scattering cross sections of quartz particles (aspect particles (λ � 0.5, m � 1.5 + 0.001i). ratio � 1/3, λ � 0.5, m � 1.54). studied, and the spheroid model is used to represent the Figure 2 gives the scattering cross sections of spheroid real nonspherical mineral particle. Since the scattering quartz particles with diﬀerent sizes. For a spheroid, there are prosperities of particles within the Gaussian beam are two parameters representing its shape, that is, the aspect ratio related with the position information, the Monte Carlo and the radius of rotation axis. Here, we use the equal-surface statistical estimate method is used to determine the dis- area sphere to represent the spheroid particle, and the equal- tributed positions of these random nonspherical particles surface diameter can be calculated by the spheroid particle. In [19, 20]. After making the average of the position information, Figure 2, w0 is equal to 1 and 10, respectively, and the Tmatrix we can obtain the general location for the random particles, method is also used to obtain the scattering cross sections of and then the average scattering cross section of particles is also spheroid particles for the plane wave as a comparison [21]. calculated with the GLMT framework. -e relative complex reﬂective index of quartz particle m is 4 Journal of Nanotechnology

×10−10 ×10−10 1.6 1.5

1.4

1.2 1 1 sca

sca 0.8 C C 0.6 0.5 0.4

0.2

0 0 0 12345678 0 12345678910 Diameter of rotation axis Equal-surface diameter

1 Aspect ratio = 2, w0 = 1 w0 = T matrix T matrix w0 = 10 (a) (a) ×10−11 1.8 ×10−11 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1

abs 1 C 0.8 abs

C 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 12345678 0 Diameter of rotation axis 0 12345678910 Equal-surface diameter Aspect ratio = 2, w0 = 1 T matrix w0 = 1 T matrix (b) w0 = 10 Figure 5: Scattering and absorbing cross sections of feldspar (b) particles (aspect ratio � 2, λ � 0.5, m � 1.5 + 0.001i). Figure 6: Scattering and absorbing cross sections of red clay particles (λ � 0.5, m � 1.7 + 0.001i). 1.54, and the incident wavelength λ is 0.5. We can see that serious differences are occurred between the Gaussian beam and the plane wave, and the differences are decreasing with absorbing cross sections of feldspar particles with aspect larger w0. -at is because the Gaussian beam gradually be- ratio � 2. For the feldspar particles, the imaginary part of comes the plane wave when w0 is infinite. Figure 3 describes the complex reflective index is not zero, and then the the scattering cross sections of spheroid quartz particles with absorbing cross sections of feldspar particles can be cal- aspect ratio � 1/3 and w0 � 1. -e parameter in the horizontal culated. With the increasing equal-surface diameter or the axis is the diameter of rotation axis, and the T matrix method diameter of rotation axis, the differences between the is also used to obtain the scattering cross sections of spheroid Gaussian beam and plane wave are enlarged. particles for the plane wave as a comparison. Figure 6 gives the scattering and absorbing cross sections Figure 4 gives the scattering and absorbing cross sec- of red clay particles. Figure 7 is the scattering and absorbing tions of feldspar particles. Figure 5 is the scattering and cross sections of red clay particles with aspect ratio � 1/2. For Journal of Nanotechnology 5

×10−10 ×10−10 1.5 1 0.9 0.8 0.7 1 0.6 sca sca 0.5 C C 0.4 0.5 0.3 0.2 0.1 0 0 0 5 10 15 0 12345678910 Diameter of rotation axis Equal-surface diameter

Aspect ratio = 1/2, w0 = 1 w0 = 1 T matrix T matrix w0 = 10 (a) (a) ×10−11 1.8 ×10−11 8 1.6 7 1.4

1.2 6

1 5 abs

C 0.8

abs 4 C 0.6 3 0.4 2 0.2 1 0 0 51015 0 Diameter of rotation axis 0 12345678910 Equal-surface diameter Aspect ratio = 1/2, w0 = 1 T matrix w0 = 1 T matrix (b) w0 = 10 Figure 7: Scattering and absorbing cross sections of red clay (b) particles (aspect ratio � 1/2, λ � 0.5, m � 1.7 + 0.001i). Figure 8: Scattering and absorbing cross sections of spheroid particles (λ � 0.5, m � 1.5 + 0.01i). the red clay particles, the real part of the complex reflective index is larger than that of the feldspar particles and the quartz particles. -e differences are relatively small between 3. Conclusions the Gaussian beam and plane wave with w0 � 10, compared with the differences of scattering and absorbing cross sec- In this paper, the scattering cross sections of nonspherical tions of the feldspar particles and the quartz particles. mineral particles are investigated with in the Gaussian beam Figures 8 and 9 show the scattering and absorbing cross based on the GLMT. In the framework of GLMT, the general sections of spheroid particles with 1.5 + 0.01i. In Figure 9, the location information is statistic by the Monte Carlo statis- incident wavelength is 1.2. According to this simulation, tical estimate method, and the scattering cross sections there are still differences between the Gaussian beam and of spheroid particles including the feldspar, quartz, and plane wave, and the differences are smaller with larger in- red clay are calculated. Actually, the spheroid shape can cident wavelength. represent the nonspherical feldspar, quartz, and red clay 6 Journal of Nanotechnology

×10−10 incident wavelength and the complex reﬂective index of 1.2 mineral particles also have eﬀects on the scattering cross sections of nonspherical mineral particles.

1 Conflicts of Interest 0.8 -e authors declare that they have no conﬂicts of interest.

sca 0.6 C Acknowledgments 0.4 -is research was supported by the Zhejiang Province Natural Science Funds (LY15A020003), the Major Project of 0.2 Education Department in Sichuan (18ZA0409), the Applied Basic Research Project of Sichuan Province, and the Open 0 project Fund of Laboratory of Meteorological Information 0 12345678910 Sharing and Data Mining (QGX18009). -e authors are Equal-surface diameter grateful to Gerard´ Gouesbet from INSA de Rouen, France, for useful discussions on GLMT and some help. w0 = 1 T matrix w 0 = 10 References (a) [1] J. A. Lock and G. Gouesbet, “Rigorous justiﬁcation of the ×10−11 localized approximation to the beam-shape coeﬃcients in 6 generalized Lorenz–Mie theory. I. On-axis beams,” Journal of the Optical Society of America A, vol. 11, no. 9, pp. 2503–2513, 1994. 5 [2] K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow 4 grains,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, no. 10, pp. 1581–1595, 2011. [3] Z. M. Li, J. Shen, X. M. Sun, and Y. J. Wang, “Nanoparticle size

abs 3

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Research Article Numerical Research on Convective Heat Transfer and Resistance Characteristics of Turbulent Duct Flow Containing Nanorod-Based Nanofluids

Fangyang Yuan ,1,2 Jianzhong Lin ,3 and Jianfeng Yu1

1School of Mechanical Engineering, Jiangnan University, Wuxi, China 2Jiangsu Key Laboratory of Green Process Equipment, Changzhou University, Changzhou, China 3School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China

Correspondence should be addressed to Jianzhong Lin; [email protected]

Received 25 October 2017; Accepted 21 January 2018; Published 6 March 2018

Academic Editor: Carlos R. Cabrera

Copyright © 2018 Fangyang Yuan et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A coupled numerical model for nanorod-based suspension flow is constructed, and the convective heat transfer and resistance characteristics of the nanofluid duct flow are investigated. *e numerical results are verified by experimental results and theoretical models. Most of nanorods are located randomly in the bulk fluid, while particles near the wall aligned with the flow direction. Friction factor of nanofluids with nanorods increases with higher particle volume concentration or aspect ratio, but the increment reduces when the Reynolds number gets larger. *e relative Nusselt number is obtained to characterize the intensity of convective heat transfer. *e results show that the Nusselt number of nanofluids increases when the particle volume concentration or aspect ratio becomes larger. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial applications although it will cause a slight increase of resistance.

1. Introduction after adding nanoparticles into base fluid [1, 6]. *e rheology of nanofluids also leads to the complexity on characterizing Nanofluids, which are fluids with nanosized particles sus- physical properties of bulk flow [8, 9]. pended, are considered to be next-generation mediums for Although many researchers have reviewed lots of measured advanced heat transfer or cooling technology [1]. Since Choi experimental data and demonstrated that particle shape plays put forward the concept firstly [2], the study of nanofluids has an important role on convective heat transfer characteristic of been intensively performed and reported in worldwide. Ab- nanofluids [10, 11], few models can be found in literatures normal heat transfer characteristics were reported by exper- proposed for nanofluid flow containing nonspherical nano- iments including the effective heat conductivity, natural and particles. Nanorods, as well as nanotubes, nanofibers, and forced convection heat transfer, and boiling heat transfer of nanobelts are used more and more for nanofluid preparation nanofluids [3–5]. Factors that can influence thermophysical because of larger aspect ratio and surface area compared with properties have been discussed in many literatures, such as spherical nanoparticles. Such kind of one-dimensional nano- nanoparticle size and shape, material of particle and base fluid, materials are available for heat transfer application with the particle loading, temperature, and so on, but quantitative development of nanoscale synthesis technology [12]. Xie et al. understanding is still a lack on the topic [1, 3]. Another hot [13] first found the thermal conductivity of nanofluids with topic is the flow resistance characteristic of nanofluids [6, 7]. elongated nanoparticles is superior to that of spherical *e pumping power needed for the bulk fluid did not increase nanoparticles. Murshed et al. [14] measured enhanced thermal dramatically as the particle loading grows in experiments [5], conductivity of water-based nanofluid containing TiO2 whereas the viscosity of nanofluids shows a marked increase nanorods with the aspect ratio of 4; they showed that the results 2 Journal of Nanotechnology give 12% higher than that predicted by classical Hamilton– As summarized above, high potential of rod-like nano- Crosser model. *en, two models derived considering in- particles in enhancing thermal conductivity of nanofluids has got terfacial layer existence between particle and base fluid for the more attention by researchers. Related reports focus mostly on effective thermal conductivities of nanofluids with spherical viscosity and thermal conductivity properties, while progress and cylindrical nanoparticles [15]. Yang and Han [16] in- made on convective heat transfer characteristics is still highly vestigated the thermal conductivity of nanofluids dispersed by needed. In addition, lack of numerical modelling on nanofluid Bi2Te3 nanorod in perfluoro-n-hexane and oil. *ey found the flow brings an obstacle to industrial applications. *e objective thermal conductivities of nanofluids decrease with increasing of this work is to develop a coupled model for nanorod-based temperature due to nanorod aspect ratio. *e results measured suspension flow and investigate the convective heat transfer and by Zhang et al. [17, 18] employing carbon nanotubes (CNTs) resistance characteristics of the nanofluid duct flow numerically. agreed well with the predictions of the unit-cell model pro- *e orientation and movement of nanorod are considered, and posed by Yamada and Ota [19] by considering shape factor for key factors that can influence the friction factor and Nusselt cylindrical particles. By using the excluded volume concept, number of the fluid flow are discussed in sequence. Koo et al. [20] constructed a model which can reproduce the nonlinear increase of the thermal conductivity for carbon nanotube and nanofiber suspensions. Timofeeva et al. [21] 2. Model for Nanorod-Based Nanofluids studied the particle shape effect on thermophysical properties 2.1. Governing Equations. *e duct flow for nanofluid is of alumina-EG/H2O suspensions and concluded that elongated considered as impressible, and the governing equations particles and agglomerates resulted in higher viscosity at the include the continuity equation, modified Navier–Stokes same volume fraction due to structural limitation of rotational equation with the additional term of nanorods, and the enhancements in effective thermal conductivities diminished energy conservation equation [28, 30, 31]: by interfacial effects. Molecular dynamics (MD) simulation zu based on the model developed by Ghosh et al. [22] have shown i � 0, (1) that cylindrical nanoparticles pick up thermal energy much zxi faster than spherical nanoparticles during a collision with 2 Dui 1 zp μ z ui a block-shaped heat source [23]. Comparative experimental � − + 2 Dt ρnf zxi ρnf z x results reported by Yu et al. [10] indicated that convective heat j (2) transfer characteristic of nanofluids containing nonspherical μa z 1 nanoparticles is affected significantly by the shear-induced + �aijklεkl − �Iijakl �εkl�, ρnf zxj 3 alignment and orientational motion of particles. By consid- DT z2T ering the thickness and thermal conductivity of the interfacial � C + C , nf T � 2 (3) nanolayer, Jiang et al. [24] proposed a model which provides Dxj zxj good predictions for the effective thermal conductivity of CNT- based nanofluids. Other existing models are summarized by where ui, p, ρnf, and T are the nanofluid velocity, pressure, Yang et al. [11], and three similar theoretical models are density, and temperature, respectively; μ is the dynamic proposed for nanofluids with finite cylindrical particles by viscosity of the pure fluid; μa is the apparent viscosity; akl and anisotropy analysis [25–27]. All of the three models give aijkl are the mean second- and fourth-order tensors of minimal relative errors to experimental results compared to particle orientation, respectively; εij � (zui/zxj + zuj/zxi)/2 other models. Lin et al. [7, 28] took numerical simulation on is the rate-of-strain tensor; and Cnf is the thermal diffusivity 2 the phaolefins-based nanofluids fluid flow containing cylin- coefficient of the nanofluid, CT � Cμk /εPrT (k is the tur- drical Al2O3 nanoparticles in laminar and turbulent pipes, bulent kinetic energy, ε is the turbulent dissipation rate, respectively. *e results show that the friction factor of flow Cμ � 0.09, and turbulent Prandtl number, PrT � 0.9) is the decreases when the Reynolds number and particle aspect ratio eddy thermal diffusivity coefficient. become larger. *e Nusselt number of convective heat transfer Substituting instantaneous velocity, pressure, tempera- increases with the increase of Reynolds number, particle aspect ture, rate-of-strain tensor, and tensors of particle orientation ratio, and volume concentration. By comparing the defined which consist of mean and fluctuation part in (1–3) and performance evaluation criterion (PEC), nanofluids containing averaging, we have rod-like nanoparticles with large aspect ratio and a suitable zU particle volume concentration are more effective for convective i � 0, (4) heat transfer process at higher Reynolds number. A coupled zxi numerical model is constructed by Yuan et al. [29] to simulate 2 DUi 1 zP μ z Ui zu′iuj′ convective heat transfer and resistance characteristics of � − + 2 − Dt ρnf zxi ρnf z xj zxj TiO2/water nanofluids with cylindrical particles in laminar (5) channel flow. *e results show that the Nusselt number of μa z 1 + �aijklεkl − �Iijakl �εkl�, nanofluid flow related to the Reynolds number, axial length, ρnf zxj 3 Prandtl number, and particle volume concentration and a fitted DT z2T formula is proposed to predict the Nusselt number of nanofluid � C + C , nf T � 2 (6) flow containing cylindrical nanoparticles. Dt zxj Journal of Nanotechnology 3

2 where Ui, P, and T are the mean nanofluid velocity, pressure, where eddy viscosity μT � 0.09ρnf k /ε. For solving (5), the and temperature, respectively, u′i is the fluctuation flow k-equation and ε-equation for turbulent flow are given by velocity, and εkl is the mean rate-of-strain tensor. zk zUi z μT zk In (5), the nanofluid density is ρnf Uj � −ρnf u′iuj′ − ρnf ε + ��μa + � �, zxj zxj zxj σk zxj ρ �1 − ϕ �ρ + ϕ ρ , (7) nf e f e p (14) 2 where the subscripts “f ” and “p” stand for pure fluid and zε ε zUi ε nanoparticles, respectively, and ϕ is the effective particle ρnf Uj �−C1 ρnf u′iuj′ − C2ρnf e zxj k zxj k volume concentration, which is used for replacing the (15) nominal particle volume fraction ϕ, in consideration of the z μT zε + ��μa + � �, particle aggregation: zxj σε zxj � s �3−fi , (8) ϕe ϕ /se where C1 �1.44, C2 �1.92, σk � 1.0, and σε � 1.3. where s and se are the effective size of aggregates and primary particle size, respectively, and fi is the fractal index and varies 2.2. Nanoparticle Orientation Distribution. *e mean sec- from 1.5 to 2.45 for those of rod-like nanoparticles. Yu et al. ond- and fourth-order tensors of particle orientation in (5) [10] gives the value as: s/se � 1.48 and fi � 1.95 according to describe the orientation distribution of nanoparticle pop- experimental data. ulation in flow field; they can be calculated by Advani and Batchelor’s theory was extended by Mackaplow and Tucker [36] as Shaqfeh [32] to account for two-body interactions and gave the apparent viscosity in (5): aij � � pipjψ(p) dp, 3 (16) 1 3 1 ln(2r) + 0.640 1.659 0.1515 nl μa � πnl μ� � + �+ �, a � � p p p p ψ(p)dp, 6 ln(2r) ln(2r) − 1.5 ln(2r)2 ln(2r)3 ijkl i j k l (9) where pi is a unit vector parallel to the particle’s axis and where n and l are the number density and half length of ψ(p) is the probability density function for particle orien- nanorods, respectively, μ is the dynamic viscosity of the pure tation at any position with p being the orientation vector. fluid, and r is the particle aspect ratio, which is the ratio of The governing equation of ψ(p) is the modified Fokker– length of diameter. Planck equation considering the rotary effect caused by *e thermal diffusivity coefficient in (6) is [33, 34] Brownian movement [37, 38]: 2 k zψ zψ z ψ z�ψp_j � nf + u � D , ( ) Cnf � , (10) j rB 2 − 17 �C � zt zxj zpj zpj ρ p nf where the heat capacitance of the nanofluid is given by where DrB is the Brownian rotary diffusion coefficient, z/zpj is the gradient operator projected onto the surface of the unit �ρCp � �1 − ϕe ��ρCp � + ϕe�ρCp � (11) nf f p sphere, and p_j is the particle angular velocity. Cintra and Tucker [39] expressed p_ as In (10), the classical Hamilton–Crosser model is i employed for predicting thermal conductivity of nanorod- DrI zψ p_i � −ωijpj + λεijpj − λεklpkplpi − , (18) based nanofluid as follows: ψ zpi

kp + kf(K − 1) +(K − 1)ϕe�kp − kf � where ω � (zu /zx − zu /zx )/2 is the vorticity tensor, k � k ⎣⎡⎢ ⎦⎤⎥, ( ) ij j i i j nf f 12 λ � (r2 − 1)/(r2 + 1). *e last term in (18) is introduced to kp + kf(K − 1) − ϕe�ks − kf � model the behavior at higher concentrations, in which DrI is in which kf and kp are the thermal conductivity of pure fluid a rotary diffusion coefficient resulting from particle-particle and nanoparticles, respectively, and K is the shape factor interactions. Folgar and Tucker�� [40] suggested that DrI is given by K � 3/c, where c is the particle sphericity, defined as isotropic and given by CI 2εijεji, where the interaction the ratio of the surface area of a sphere with the same volume coefficient CI is fixed to 0.01. as that of the particle and the surface area of the particle. *e For cylindrical particles, the Brownian rotary diffusion model predicts well the thermal conductivity of nanofluid coefficient in (17) is with ZnO nanoparticles with experimental data by Ferrouillat kbT et al. [35] in rod-like shape. DrB � ��. 2 2 (19) crL + c *e Reynolds stress tensor −ρu′iuj′ in (5) is rS zU zUi j 2 With aspect ratio r, the rotational friction coefficients −ρnf u′iuj′ � 2μT�+ � − ρnf kδij, (13) zxj zxi 3 around long axis and short axis are [41, 42]. 4 Journal of Nanotechnology

3.84πμL31 + δ � to obtain the distribution of ϕ in the flow, it is necessary to c � rL , rL r2 solve the general dynamics equation for nanorods: ( ) Dn(v) z zn(v) 20 − D � 0. ( ) 3 tB 26 πμL Dt zxj zxj c � , rS 3 ln r + δ � rS Based on the Reynolds average, the above equation is with transformed to 0.677 0.183 Dn(v) z zn(v) zn′(v)u′ δ � − , j (27) rL r r2 − DtB + � 0, Dt zxj zxj zxj (21) 0.917 0.05 where n(v) is the mean particle volume distribution function δ � −0.662 + − . rS r r2 based on the volume and the last term on the left-hand side represents the change in n(v) resulting from turbulent _ Substituting instantaneous ψ and pj which consist of diffusion and is usually assumed to be [45]: mean and fluctuation part in (17) and averaging, we have the zn(v) mean equation of probability density functions for the n′(v)uj′ � −εi , (28) particle orientation: zxj zψ zψ z2ψ zψ zψ + u D p + K p where εi is the eddy diffusion coefficient which is a function j − rB 2 − ωji i εji i zt zxj zpj zpj zpj of position and εi � ]t/Sct, where vt and Sct are the turbulent (22) viscosity and the Schmidt number, respectively. zψ z2ψ z2ψ DtB in (26) is the Brownian translational diffusion co- K p p p K p p D � , − εkl k l j − εklψ k l − rI 2 αψp 2 efficient for rod-like nanoparticles: zpj zpj zpj k T D � ��b , � . ( k2] )1/2 � . ( ])1/2 tB (29) where αψx 1 3 5 /3ε and αψp 0 7 4ε/15 are c2 + c2 the dispersion coefficients of linear and angular displace- tL tS ] ment in which is the kinetic viscosity of the pure fluid [43], and the translational friction coefficients parallel and per- which appear based on the dispersion mechanism: pendicular to the long axis are [41, 42] zψ −ψ′uj′ � αψx , 2πμL zxj c � , tL ln r + δ ( ) tL 23 (30) zψ L _ 4πμ −ψ′pj′ � αψp . crS � , zpj ln r + δtS Integral on the angle after multiplying component of with unit vector pi, (22) is transformed to the equation about aij: 0.980 0.133 δ � −0.207 + − , tL r r2 Daij � −�ω a − a ω � + K�ε a + a ε − 2ε a � (31) Dt ik kj ik kj ik kj ik kj kl ijkl 0.185 0.233 δ � 0.839 + + . rS r r2 + 2�αψp + DrI + DrB��δij − αaij �. Substituting (29) in (27), we have averaged general (24) dynamics equation for nanoparticles. With the moment transformation which involves multiplying the equation by *e mean fourth-order particle orientation tensor needs to k be closured by model. Considering the shearing-stretch feature v and then integrating over the entire size regime, the of duct flow, the orthotropic closure model [44] is fit for (24): equation for rod-like nanoparticles finally becomes det�a � ij zm zm z ⎢⎡ k T zm ⎥⎤ aijkl � �δijδkl + δikδjl + δilδjk� k k ⎢⎝⎜⎛��b ⎠⎟⎞ k⎥ 6 + uj − ⎣⎢ + εi ⎦⎥ � 0, (32) ( ) zt zxj zxj 2 2 zxj 25 ctL + ctS 1 + �a a + a a + a a �, 3 ij kl ik jl il jk where the zero moment and the first-order moment in which det(a ) is the determinant of second-order tensor a . M � �∞ n(v) � N, ij 2 0 0 dv ∞ (33) M1 � � vn(v)dv � V, 2.3. General Dynamics Equation for Rod-Like Nanoparticles. *e 0 particle volume concentration ϕ in (7), (11), and (12) is describe the total particle number density and volume of considered distribution inhomogeneity in the flow. In order nanoparticles at a given point. Journal of Nanotechnology 5

r 0.35 z 0.30

0.25

0.20 f 0.15 L D 0.10

0.05 Figure 1: Schematic of a nanofluid pipe flow. 0.00 0 2000 4000 6000 8000 10000 12000 14000 16000 Re Table 1: *ermophysical properties of materials. Numerical results H O ZnO Experimental results *ermophysical properties 2 (base fluid) (nanorods) Blasius equation ρ (kg·m−3) 998.2 5606 −1 −1 Figure 2: Comparison of friction factor of nanofluids. Cp (J·kg ·K ) 4182 5200 k (W·m−1·K−1) 0.62 60

Δp f � , ( ) 2 34 3. Numerical Method ρnf (L/D)Uav�2 � � hD k (zT�zr)| zT� Schematic of a turbulent duct flow containing nanorod- − nf r�±D/2 � Nu � � � − � , (35) based nanofluid is shown in Figure 1. *e length and di- knf knf zr r�±D/2 ameter of the channel are L and D, respectively. where ∆p is the pressure drop, U is the average velocity of *e boundary conditions are given as follows: inlet: av the fluid in the flow direction, and h is the heat transfer U � U , U � 0, T � T , m � m ; outlet: zU /zx � 0, U � 0, x 0 y in 1 10 x y coefficient. zT/zx � 0, zm /zx � 0; and wall: U � U � 0, m �0, 0 x y 1 Figure 2 shows the comparison of friction factor of zm /zx � 0. Notably, numerical simulations performed here 0 nanofluids varying with the Reynolds number. *e solid are forced convection process with uniform wall tempera- points in figure are measured by Ferrouillat et al. [35], and ture, other than those with uniform wall heat flux. the solid curve is the Blasius equation which is *e ratio of channel length to height is L/D � 125. *e f � 0.316Re−0.25 for turbulent duct flow. *e numerical re- nanofluid is a mixture of water and rigid ZnO nanorods. sults are drawn by hollow points and are obtained by setting *ermophysical properties of water and ZnO are presented the nanorod aspect ratio as 8, because the aspect ratio of in Table 1. *e temperatures of inlet nanofluid and wall are nanoparticles in reference experiments are about 6∼10. From T � 293 K and T � 323 K, respectively. Reynolds number in w the figure, the numerical results show agreement with the of the fluid flow is defined as Re � u D/] ranging from 2500 0 nf experimental results and the curve of Blasius equation. In to 15,000. *e nominal particle volume fraction ϕ for turbulence regime, the friction factor of nanofluid flow nanofluids is 0.4%, 0.93%, and 1.3%, and aspect ratio r is 8, 12 drops slowly when the Reynolds number grows. and 16 for comparing with experimental results in Fer- Comparison of calculated Nusselt number of nanofluids rouillat et al. [35]. with Dittus–Boelter relation is shown in Figure 3. *e *e SIMPLEC algorithm is employed for solving cou- classical Dittus–Boelter relation is commonly used for pled continuity equation and moment equation, where the predicting the Nusselt number in fully developed turbulent convection item and diffusion item are discreted using duct flow, which is Nu � 0.023Re0.8Pr0.4. *e difference of QUICK scheme and second-order center difference, re- Nusselt number predicted by current model and D-B respectively. No slip condition is set for wall boundary lation is within about 10%, and the difference declines in condition. high Nu region.

4. Results and Discussion 4.2. Orientation Distribution of Nanorods. *e orientation of 4.1. Calculation Verification. To validate the model and code nanorods in fluid makes an effect on nanofluid fluid flow. for nanofluid flow, classical models and experimental results *e orientation distribution of nanorods population can be are used for comparing the numerical results of friction described by the mean second-order orientation tensor of factor and the Nusselt number of duct fluid flow in Figures 2 particle. *e components of aij in z direction are shown in and 3. *e definition of f and Nu is as follows: Figure 4, in which the volume concentration and aspect ratio 6 Journal of Nanotechnology

120 0.35

0.30 100 0.25 80 0.20

60 ij a 0.15

40 0.10

0.05 20 0.00

Nusselt number calculated by current model current by calculated number Nusselt 0 0 20 40 60 80 100 120 Nusselt number predicted by Dittus-Boelter relation 0.0000 0.0005 0.0010 0.0015 0.0020 − 0.0020 − 0.0015 − 0.0010 − 0.0005 Numerical results 10% z Dittus-Boelter relation −10% a11 a23 a a Figure 12 33 3: Comparison of calculated Nusselt number of nanoﬂuids a a =1/π with Dittus–Boelter relation. 13 ij a22 aij =0

Figure are set as ϕ � 1.4% and r � 16, respectively. It can be seen 4: Components of aij in z direction (ϕ �1.4%, r � 16). from the figure that the distributions of components of aii coincide with each other, and the median distribution of a ii 0.06 which along the axial direction is 1/π. *e value of aii near the wall is larger that that of central area, which means that the nanorods align with the flow direction due to the existence of wall. *e orientation state of nanorods can hardly 0.05 keep in y-z component because a23 is always zero in z direction. But a13 is not zero near the wall, which illustrates the wall effect on orientation state of nanorods. f 0.04

4.3. Friction Factors. *e friction factor and pressure drop of 0.03 the duct flow are of importance to the practical applications. Figure 5 shows the friction factors of nanofluids at different volume concentrations as a function of the Reynolds 0.02 number. *e aspect ratio of nanorods in experiments and 0 2000 4000 6000 8000 10000 12000 14000 16000 simulations are set as r � 8. All the friction factors of Re nanofluid flow are higher than those of base fluid and de- ϕ =0 ϕ = 0.93% crease when the Reynolds number grows higher. *e friction ϕ = 0.4% ϕ = 1.3% factor increases slightly with the increase of particle volume concentration, and the difference diminishes when the Figure 5: Friction factor as a function of the Reynolds number by Reynolds number becomes larger. different volume concentrations (r � 8). In Figure 6, the friction factor as a function of the Reynolds number by different aspect ratio is given, and the volume concentration ϕ is 0.93%. As can be seen in figure, Nusselt number of nanofluids to that of base fluid, that is, the shape of nanoparticles takes effect on friction factor of Nunf/Nuf. Figures 7 and 8 show the calculated and exper- nanofluid fluid flow. Nanorods with larger aspect ratio bring imental results of Nunf/Nuf as a function of the Reynolds more flow resistance when the Reynolds number is the same. number with different particle aspect ratios or different When the Reynolds number gets larger, the friction factor of volume concentrations. *e solid points give the measured nanofluid approaches to that of base fluid, which means the Nu with particle volume concentration as 0.93% and reveal additional pumping power for nanofluid is not significant in that the Nusselt number of nanofluid is smaller than that of high Reynolds number region. base fluid when Re is less than 7000. *e overall experimental data indicate that Nunf/Nuf grows as the Reynolds number gets larger except the point when Re is about 2100. 4.4. Convective Heat Transfer. *e Nusselt number describes *e calculated Nunf/Nuf by model in this paper is larger than the ratio of convective heat transfer across the wall. *e that of experiments when the volume concentration is set as relative Nusselt number is defined here as the ratio of the 0.93%. In addition, all the numerical results are greater Journal of Nanotechnology 7

0.060 1.10

0.055

0.050 1.05

0.045 ) f

f 0.040 1.00 (Nu nf

0.035 Nu

0.030 0.95

0.025

0.020 0.90 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 Re Re ϕ=0 r=12 r =8 r =16 r=8 r=16 r =12 Experimental results

Figure Figure 6: Friction factor as a function of the Reynolds number by 8: Nunf/Nuf as a function of the Reynolds number by diﬀerent aspect ratios (ϕ � 0.93%). diﬀerent aspect ratios (ϕ � 0.93%).

1.10 5. Conclusions 1.08 *e convective heat transfer and resistance characteristics of 1.06 the nanofluid duct flow are investigated based on a coupled numerical model for nanorod-based suspension flow. *e 1.04 numerical results are verified by experimental results and

) 1.02 f theoretical models. Most of nanorods are located randomly 1.00

(Nu in the bulk fluid, while particles near the wall aligned with nf 0.98 the flow direction. Friction factor of nanofluids with Nu nanorods increases with higher particle volume concen- 0.96 tration or aspect ratio, but the increment reduces when the 0.94 Reynolds number getting larger. *e relative Nusselt 0.92 number is obtained to characterize the intensity of con- 0.90 vective heat transfer. *e results show that the Nusselt 0 2000 4000 6000 8000 10000 12000 14000 16000 number of nanofluids increases when the particle volume Re concentration or aspect ratio becomes larger. Compared to ϕ = 0.4% ϕ = 1.3% increasing the aspect ratio of nanorods, increasing the ϕ = 0.93% Experimental results, ϕ = 0.93% particle volume concentration would be more effective on enhancing the convective heat transfer intensity in industrial Figure 7: Nunf/Nuf as a function of the Reynolds number by applications although it will cause a slight increase of different volume concentrations (r � 8). resistance. than 1, which means the convective heat transfer of nano- Conflicts of Interest fluid is better than base fluid at the same Reynolds number. *e authors declare that they have no conflicts of interest. *e Nunf/Nuf is monotone increasing as a function of the Reynolds number, no matter the particle volume concentration or aspect ratio varies in figures. In Figure 7, the Acknowledgments relative Nusselt number increases as the particle volume concentration gets larger. Similarly, in Figure 8, the relative *is work is supported by the National Natural Science Nusselt number increases as the particle aspect ratio gets Foundation of China (no. 11632016) and Jiangsu Key Lab- larger. Both the particle loading and particle shape influence oratory of Green Process Equipment (GPE201705). the convective heat transfer characteristic obviously. Compared to increasing the aspect ratio of nanorods, increasing the particle volume concentration would be more References effective on enhancing the convective heat transfer intensity [1] S. S. Murshed and P. Estellé, “State of the art review on in industrial applications although it will cause a slight viscosity of nanofluids,” Renewable and Sustainable Energy increase of resistance. Reviews, vol. 76, pp. 1134–1152, 2017. 8 Journal of Nanotechnology

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Research Article The Asymptotic Behavior of Particle Size Distribution Undergoing Brownian Coagulation Based on the Spline-Based Method and TEMOM Model

Qing He 1 and Mingliang Xie 2

1Key Laboratory of Distributed Energy Systems of Guangdong Province, School of Chemical Engineering and Energy Technology, Dongguan University of Technology, Dongguan 523808, China 2State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Qing He; [email protected]

Received 12 December 2017; Accepted 16 January 2018; Published 1 March 2018

Academic Editor: Yu Feng

Copyright © 2018 Qing He and Mingliang Xie. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the particle size distribution is reconstructed using 3nite moments based on a converted spline-based method, in which the number of linear system of equations to be solved reduced from 4m × 4m to (m + 3) × (m + 3) for (m + 1) nodes by using cubic spline compared to the original method. ,e results are veri3ed by comparing with the reference 3rstly. ,en coupling with the Taylor-series expansion moment method, the evolution of particle size distribution undergoing Brownian coagulation and its asymptotic behavior are investigated.

1. Introduction Using a given number of moments to reconstruct the PSD is known as the 3nite-moment problem or inverse Particle size distribution (PSD) is one of the most important problem in mathematical analysis [7]. Generally, this properties of aerosol particles, including transport, sedi- problem is distinguished between the three types for the mentation, and so on [1]. It is also of utmost interest in many monovariate case: the HausdorB moment problem with the industrial applications, such as powder preparation and PSD supported on the closed interval [a, b], where [a, b] are particle synthesis [2, 3]. ,e evolution of the PSD un- the lower and upper limits of the domain of PSD; the Stieltjes dergoing diBerent dynamic processes is usually described in moment problem with the PSD supported on [0, +∞); and the framework of population balance equations (PBEs) the Hamburger moment problem with the PSD supported mathematically [4], which have a strong nonlinear structure on (−∞, +∞) [8]. Until now, there exist several frequently in most cases and cannot be solved analytically. With a high- used reconstruction methods in the literature mainly for the computational eDciency, the moment-based method has HausdorB moment problem, including but not limited to become a powerful tool for investigating aerosol micro- parameter-3tting method, Kernel density function-based physical processes, in which cases some statistical charac- method, maximum entropy method, and spline-based teristics of the PSD, namely, the moments of the PSD, are method. ,e parameter-3tting method is to assume the obtained [5]. However, the detailed information about the PSD as a simple function (i.e., log-normal distribution or target PSD is out of reach. ,eoretically, the PSD is equal to gamma distribution), where the parameters in the function the moments of all orders. ,e proof about the uniqueness of are determined by the given low-order moments [7]. It is the reconstruction in the case all moments are known is given fastest and easiest method but with drawbacks that need with an appropriate condition that the range of the PSD is in a priori knowledge about the solution and limited to simple a 3nite interval [6]. But in practice, only a 3nite number of shapes, even though a weighted sum of diBerent simple moments are obtained. functions can be used [9]. ,e kernel density function-based 2 Journal of Nanotechnology method is a positivity-preserving representation and can be In this paper, we use a converted ansatz for s(l)(x) to regarded as the development of parameter-3tting method, reduce the number of the linear system. For cubic spline, the which approximates the PSD by a superposition of weighted second derivative in each node is set as Li, then s″(x) can be kernel density functions [10]. ,is method gives rise to an ill- written in the following form using linear interpolation: posed problem for determining weights, and a large number ″ x − xi+1 x − xi of available moments are needed to ensure accuracy. Based si (x) � Li + Li+1 , xi − xi+ xi+ − xi on the maximization of the Shannon entropy or the min- 1 1 (1) imization of the relative entropy from information theory, x ∈ 􏼂xi, xi+1 􏼃, (i � 1, 2, ... , m). the maximum entropy method is a notable method which needs relatively less knowledge of the prior distribution or ,en, s(x) and their 3rst derivatives can be gotten the number of moments compared to the previous two through integrating: methods [8, 11]. With the advantage of no priori assump- L x − x 􏼁2 L x − x 􏼁2 tions on the shape of the PSD as well as that the needed ′ i i+1 i+1 i si (x) � + + Ci1, number of moments only depends on that of interpolation 2 xi − xi+1 2 xi+1 − xi nodes, the spline-based method proposed by John et al. [7] (2) L x x 3 L x x 3 has attracted some researchers’ attention, such as the in- i − i+1 􏼁 i+1 − i 􏼁 si(x) � + + Ci1x + Ci2, vestigation on particle aggregation and droplet coalescence 6 xi − xi+1 6 xi+1 − xi [12, 13]. And an adaptive spline-based algorithm with C C a wider application for nonsmooth and multimodal distri- where i1 and i2 are integral constants. With the continuity of � ... butions was developed later [6]. More relevant research the spline and their 3rst derivatives at xi (i 2, 3, , m), we can get about the comparisons between these methods can be found i 􏼐Δxj−1 + Δxj 􏼑 in the literature [14, 15]. Ci1 � C11 + ∑ Lj , In this paper, we will use the spline-based method to j�2 2 reconstruct the PSD coupling with PBEs describing Brow- (3) nian coagulation in the free molecule regime and continuum i 􏼐Δxj−1 + Δxj 􏼑 􏼐xj−1 + xj + xj+1􏼑 regime. Compared to the original method, the number of Ci � C − ∑ Lj , 2 12 2 3 linear system of equations to be solved is signi3cantly re- j�2 duced through substituting the continuous conditions. ,e C correctness of this new treatment is veri3ed by comparing in which Δxi is the length of the ith subinterval and 11 and C with the reference results in [7]. ,en with the moments 12 are related to the left boundary conditions. ,us, the obtained by the Taylor-series expansion moment method sum of the number of moments and boundary conditions (TEMOM) [16], the evolution of PSD due to Brownian needed to solve the equations is m + 3. coagulation and its asymptotic behavior are investigated. Usually, we consider that the value of PSD out of the support [a, b] is small enough and can be set as zero:

s1x1 􏼁 � 0, 2. Theory and Modeling (4) smxm+1 􏼁 � 0, 2.1. Modeling of Spline-Based Method. In the original method, the support of the target PSD [a, b] is divided into m and the 3rst derivatives are denoted as � ··· � subintervals: a x1 < x2 < < xm+1 b. In each subinterval, s′ x 􏼁 � q , the PSD is approximated by a spline (piecewise polynomial) 1 1 1 (5) (l) sm′ xm+ 􏼁 � q , si (x) of degree l; thus, there exist (l + 1)m unknowns. For 1 2 cubic spline (l � 3), the splines s(l)(x) and their 3rst and where q and q are zero for smooth boundary conditions. second derivatives are continuous at each node x (i � 2, 1 2 i ,en, (3) can be simpli3ed by substituting the left boundary 3,..., m), which give 3(m − 1) conditions. With the smooth conditions: boundary conditions, which means s(l)(x) and their 3rst and j�i second derivatives are null at nodes x1 and xm+1, there still Ci1 � q1 + ∑ Ljdj, require m − 3 additional conditions, which have to be j�1 (6) supplemented by the known moments. ,en, a 4m × 4m ill- j�i conditioned linear system is obtained. In order to improve Ci2 � −q1x1 − ∑ Ljdjej. the accuracy of calculation, the interval should be set as j�1 small as possible, which is controlled by tolred. For example, the last (or the 3rst) subinterval is divided into n Together with the right boundary conditions, we can get the following formula: smaller subintervals: xm � xm1 < xm2 < ··· < xmn � xm+1; if i�m+ (l) (l) 1 the ratio of 2-norm of s (xmn) to the maximum of s (x) is q1 − q2 + ∑ Lidi � 0, less than tolred, the last node is reset as xm+1 � (xm + xm i�1 (7) +1)/2. Furthermore, tolneg and tolsing are introduced to i�m+1 guarantee that the value of s(l)(x) is nonnegative. More de- q1x1 − q2xm+1 + ∑ Lidiei � 0, tailed procedure is shown in [7]. i�1 Journal of Nanotechnology 3

in which di and ei are given as follows (i � 2, 3,..., m): I − 3x I + 3x2 I − x3 I G � − i4 i+1 i3 i+1 i2 i+1 i1 Δx + Δx i x d � i−1 i, 6Δ i i 2 xk+2 xk+2 xk+1 xk+1 Δx1 n+1 − i n+1 − i ( ) d � , + di − diei , 13 1 2 k + 2 k + 1 Δx d � m, 2 3 m+1 Ii4 − 3xiIi3 + 3xi Ii2 − xi Ii1 2 ( ) Hi � . x + x + x 8 6Δx e � i−1 i i+1, i i 3 2x + x Now together with (7), a (m + 3) × (m + 3) linear system e � 1 2, for L , q , and q is obtained. Next, we will discuss the 1 3 i 1 2 number of moments that should be supplemented (note xm + 2xm+1 em+1 � . that, in this paper, all cases calculated with q1 and q2 are 3 zero): ,e kth order moment Mk of the PSD is de3ned as (1) If q and q are unknowns, (m + 1) moments are follows: 1 2 ∞ needed to solve these equations. k Mk � ｚ x f(x)dx. (9) 0 (2) If q1 and q2 are zero or any other constants given, s″(x) ,us, the kth order moment of s(x) is (m − 1) moments are needed; if the value of m at boundary is given (such as smooth conditions xi+1 k in [7], namely, L1 � Lm+1 �0), (m − 3) moments ｚ x ∑ si(x)dx xi i�1 are needed. And in this case, the order of the coeDcient matrix is (m + 1) × (m + 1) or (m − 1) × m −Li 2 3 (m − 1). � ∑ 􏼠 􏼐Ii4 − 3xi+1Ii3 + 3xi+1Ii2 − xi+1Ii1􏼑 i�1 6Δxi (3) If q1 and q2 obey some relationships, for example, � � L q1 (s1(x2) − s1(x1))/Δx1, q2 (sm(xm+1) − sm(xm))/ i+1 2 3 � � + Ci1Ii2 + 􏼐Ii4 − 3xiIi3 + 3xi Ii2 − xi Ii1􏼑 + Ci2Ii1􏼡, Δxm, then L1 −L2/2 and Lm+1 −Lm/2 can be de- 6Δxi rived and (m − 1) moments are needed. (10) For quadratic spline, we can also get a (m + 1) × (m + 1)

in which Ii are linear system in the same way by denoting that xk+1 − xk+1 I � i+1 i , i1 x − x x − x k + 1 s′ (x) � l i+1 + l i , i ix − x i+1x − x xk+2 − xk+2 i i+1 i+1 i (14) I � i+1 i , i2 k + 2 x ∈ [xi, xi + 1], (i � 1, 2, ... , m), (11) xk+3 − xk+3 I � i+1 i , where li is the 3rst derivative in each node. ,e corre- i3 k + 3 sponding s(x) and linear system are as follows: xk+4 − xk+4 i+1 i 2 2 Ii4 � . k + 4 li x − xi+1 􏼁 li+1 x − xi 􏼁 si(x) � + + ci1, 2 xi − xi+1 2 xi+1 − xi In order to represent Li explicitly, (10) is arranged as follows: j�m+1 m xi+1 k+2 k+2 k+1 k+1 k xm+1 − x1 xm+1 − x1 ∑ ljdj � 0, ｚ x ∑ si(x)dx � q1 − q1x1 j�1 xi i�1 k + 2 k + 1 m m m x i+1 k + ∑ GiLi + HiLi+1􏼁, ｚ x ∑ si(x)dx � ∑gili + hili+1 􏼁, i�1 xi i�1 i�1 (12) (15) where where 4 Journal of Nanotechnology

3000 3000

2500 2500

2000 2000

1500 1500 s ( x ) s ( x )

1000 1000

500 500

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (×10−3) x (×10−3)

m = 6 m = 8 m = 6 m = 8 m = 7 Reference [7] m = 7 Reference [7] (a) (b)

Figure 1: Reconstruction of the PSD for diBerent moments (m) about Example 2.1 in [7] by using (a) quadrature spline and (b) cubic spline.

k+ k+ 1/6 1/2 I − 2x I + x2 I x 1 − x 1 in which B1 � (3/4π) (6kbT/ρp) and B2 � 2kbT/3μ, where g � − i3 i+1 i2 i+1 i1 + d n+1 i , i 2Δx i k + 1 kb is the Boltzmann constant, T is the temperature, ρp is the i particle density, and μ is the gas viscosity. 2 With the de3nition of the kth order moment, Mk, (17) is Ii − 2xiIi + x Ii h � 3 2 i 1, transformed to a series of original diBerential equations by i 2Δx (16) i multiplying both sides with vk and then integrating over all j�i particle sizes: ∞ ∞ dMk 1 k k k ci � ∑ ljdj. 1 � ｚｚ 􏽨υ + υ1 􏼁 − υ − υ1􏽩 j�1 dt 2 0 0 (19)

× β υ, υ1 􏼁n(υ, t)nυ1, t 􏼁dυdυ1. 2.2. Modeling of PBE and TEMOM. ,e population balance Using the Taylor-series expansion technology to ap- equation describing irreversible Brownian coagulation proximate the collision frequency function and fractional with continuous monovariable can be written as follows moments, the moment equations are closed without any other [17]: arti3cial assumption [16, 18]. In the original TEMOM model, zn(υ, t) 1 v the 3rst three moments can be obtained easily using the � ｚ β υ, υ − υ1 􏼁nυ1, t 􏼁nυ − υ1, t 􏼁dυ1 fourth-order Runge-Kutta method with M1 remaining con- zt 2 0 (17) stant due to the mass conservation requirement. ,e corre- ∞ sponding higher and fractional moments are as follows [19]: − ｚ β υ1, υ 􏼁n(υ, t)nυ1, t 􏼁dυ1, k 0 M k(k − 1)M − 1 􏼁 M � 1 􏼢1 + C 􏼣, (20) k Mk−1 where n(υ, t) is the number density function of the 0 2 particles with volume from υ to υ + dυ at time t and where M � M M /M2 is a dimensionless moment. Obvi- ( , ) C 0 2 1 β υ1 υ is the collision frequency function between ously, the reconstruction depends heavily on the reliability particles with volume υ and υ1. In the free molecule and of known moments. Based on the log-normal size distri- continuum regime, β(υ, υ1) are represented separately as bution assumption, the maximum relative error for Mk of 1/2 this model is discussed by Xie [19], and the results dem- 1 1 1/3 1/3 2 βFMυ, υ1 􏼁 � B1􏼠+ 􏼡 􏼐υ + υ1 􏼑 , onstrate that the error of Mk for k ≤ 2 with a small standard υ υ1 deviation is acceptable. Furthermore, theoretical analysis of ( ) 18 the PBE is feasible because of the relative simple form of this 1 1 , 􏼁 � B 􏼠+ 􏼡􏼐1/3 + 1/3 􏼑, model [20, 21], and the explicit asymptotic solutions are as βCR υ υ1 2 1/3 1/3 υ υ1 υ υ1 follows: Journal of Nanotechnology 5

0.9 1 0.8 0.7 0.8 0.6

0.6 0.5 ψ ( η ) ψ ( η ) 0.4 0.4 0.3 0.2 0.2 0.1 0 0 10−4 10−3 10−2 10−1 100 101 102 10−4 10−3 10−2 10−1 100 101 102  

m = 6 m = 8 m = 6 m = 8 m = 7 Reference [22] m = 7 Reference [23] (a) (b)

Figure 2: Reconstruction of ψ(η) for diBerent moments (m) by using cubic spline in the (a) free molecule regime and (b) continuum regime.

， M ， ⟶ 0.313309932 × B−6/5M−1/5t−6/5, reconstruction about Example 2.1 in [7] by using quad- 0 FM 1 1 rature spline and cubic spline proposed in this paper. And ， M ， ⟶ 7.022205880 × B6/5M11/5t6/5, the parameters tolred, tolneg, and tolsing are set as the same 2 FM 1 1 of those in the literature to maintain consistency. It should ， (21) be noted that the tolerance values have an inOuence on the ， 81 −1 −1 M ， ⟶ B t , results [7, 14]. ,e great agreements with the references 0 CR 169 2 verify the validity of this new converted method. How- ，， 338 2 ever, an underlying Oaw is that only the continuity of s(x) M2， ⟶ B2M1t, CR 81 is necessary in practice. Moreover, the sensitivity of tolsing to solution may increase when the number of the linear and MC tends to a constant 2.200126847 or 2, respectively. system sharply decreases. It can also be seen that some Using the similarity transformation η � v/(M1/M0), the PSD inOexion points appear with m increasing. ,is may be can be arranged as follows: caused by the increasing condition number of the linear M2 system. 0 ( ) n(υ, t) � ψ(η). 22 ,e reconstruction of ψ(η) in the free molecule regime M1 and continuum regime for diBerent moments (m) by cubic According to the theory of self-preserving, ψ(η) does not spline is shown in Figure 2, where the references are from Lai change with time at a large t [1], and its moments only et al. [22] and Friedlander and Wang [23]. ,e initial interval depend on k and MC: is set as [1e − 5, 10], and the spacing of adjacent nodes are ∞ equidistant logarithmically. Both the left and right bound- k k(k − 1)MC − 1 􏼁 mk � ｚ ψ(η)η dη � 1 + . (23) aries are adjusted according to the comparison results of 0 2 2 ||s(ηmn)|| /max(s(η)) and tolred, with tolred � 1e − 2 for m � 6 ,en, ψ(η) can be approximated by s(η) using the spline- and 1e − 4 for m � 7 or 8, respectively. Besides, based method, and the asymptotic behavior of n(υ, t) is also tolred � −1e − 2 and the initial value of tolsing are set as known together with (21) and (22). 1e − 36. In addition to the 3rst three integral moments m0, m1, and m2, the fractional moments m1/3, m2/3, m4/3, and 3. Results and Discussions m5/3 are chosen to reconstruct ψ(η), for the reason that these fractional moments with volume-based variable are pro- One diDculty of the inverse problem is the ill-conditioned portional to the integral moments with length-based vari- coeDcient matrix of the linear system. Another is that the able. It can be seen that the results for m � 7 show relatively value of s(x) is nonnegative. By using the pseudoinverse small diBerences compared to the references. Generally, the routine, a least-squares solution of the linear system is diBerences may be caused by two parts: one is the error of obtained, in which the singular values smaller than tolsing TEMOM model and the other is the error of spline-based are set as zero (see Remark 4.2 in John et al. [7]). method. Scaling the moments Mk and time t by ∗ (k−1) k 5/6 1/6 Moreover, the parameter α is introduced to avoid large Mk � MkM00 /M10, τ � tB1M00 M10 , or τ � tB2M00, the diBerence in the order of magnitude. In this paper, we will evolution of dimensionless n(υ, t) for m � 7 at long time is follow this treatment. Figure 1 shows the results of the presented in Figure 3 with the initial conditions given as 6 Journal of Nanotechnology

×10−5 ×10−5

7 6

6 5 5 4 4 n ( υ ) n ( υ ) 3 3 2 2

1 1

0 0 10–1 100 101 102 103 104 10−1 100 101 102 103 104 υ υ

τ = 20 τ = 40 τ = 20 τ = 40 τ = 30 τ = 50 τ = 30 τ = 50 (a) (b)

Figure 3: ,e evolution of dimensionless n(υ, t) with time for m � 7 in the (a) free molecule regime and (b) continuum regime.

∗ ∗ ∗ M00 � 1, M10 � 1, and M20 � 4/3. Obviously, the particle Oame reactor,” Chemical Engineering Science, vol. 63, number decreases and the average volume increases with pp. 2317–2329, 2008. time advancing due to coagulation. [4] S. K. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, Oxford University Press, London, UK, 2nd edition, 2000. 4. Conclusion [5] S. E. Pratsinis, “Simultaneous nucleation, condensation, and coagulation in aerosol reactor,” Journal of Colloid and In- By establishing the ansatz s(x) on the basis of the continuity terface Science, vol. 124, no. 2, pp. 416–427, 1988. of second derivation, the number of linear ill-conditioned [6] L. De Souza, G. Janiga, V. John, and D. ,évenin, “Re- system can be reduced signi3cantly from 4m × 4m to construction of a distribution from a 3nite number of mo- (m + 3) × (m + 3) for (m + 1) nodes by using cubic spline, ments with an adaptive spline-based algorithm,” Chemical although only the continuity of s(x) is necessary in practice. Engineering Science, vol. 65, no. 9, pp. 2741–2750, 2010. ,en, coupling with the asymptotic solutions of TEMOM [7] V. John, I. Angelov, A. A. Oncül,and¨ D. ,évenin,“Tech- [20] and the theory of self-preserving, the evolution of the niques for the reconstruction of a distribution from a 3nite PSD due to Brownian coagulation in the free molecule number of its moments,” Chemical Engineering Science, regime and continuum regime and its asymptotic behavior vol. 62, no. 11, pp. 2890–2904, 2007. are obtained easily. [8] R. V. Abramov, “An improved algorithm for the multidimensional moment-constrained maximum entropy problem,” Journal of Computational Physics, vol. 226, no. 1, pp. 621–644, Conflicts of Interest 2007. [9] E. Aamir, Z. K. Nagy, C. D. Rielly, T. Kleinert, and B. Judat, ,e authors declare that they have no conOicts of interest. “Combined quadrature method of moments and method of characteristics approach for eDcient solution of population Acknowledgments balance models for dynamic modeling and crystal size distribution control of crystallization processes,” Industrial & ,is work was supported by the National Natural Science Engineering Chemistry, Research, vol. 48, no. 18, pp. 8575–8584, Foundation of China with Grant nos. 50806023 and 11572138. 2009. [10] G. A. Athanassoulis and P. N. Gavriliadis, “,e truncated References HausdorB moment problem solved by using kernel density functions,” Probabilistic Engineering Mechanics, vol. 17, no. 3, [1] J. Z. Lin, P. F. Lin, and H. J. Chen, “Research on the transport pp. 273–291, 2002. and deposition of nanoparticles in a rotating curved pipe,” [11] R. Abramov, “A practical computational framework for the Physics of Fluids, vol. 21, no. 12, p. 122001, 2009. multidimensional moment-constrained maximum entropy [2] M. Z. Yu, J. Z. Lin, and T. Chan, “Numerical simulation of principle,” Journal of Computational Physics, vol. 211, no. 1, nanoparticle synthesis in diBusion Oame reactor,” Powder pp. 198–209, 2006. Technology, vol. 181, no. 1, pp. 9–20, 2008. [12] W. Hackbusch, V. John, A. Khachatryan, and C. Suciu, “A [3] M. Z. Yu, J. Z. Lin, and T. Chan, “EBect of precursor loading numerical method for the simulation of an aggregation- on non-spherical TiO2 nanoparticle synthesis in a diBusion driven population balance system,” International Journal Journal of Nanotechnology 7

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