Drag Force and Slip Correction of Aggregate Aerosols

The drag force on aggregate particles of uniform spheres was measured in a Millikan apparatus as a function of Knudsen number. Our experiment was designed to study the effect of particle orientation on the slip correction factor of nonspherical particles. The velocities of charged particles in a gravitational field with and without an applied electrical field were measured. An electrical field strength of 2000 V/cm was used to align doublet and triplet particles. Results showed that an aggregate particle moved in random orientation while in the gravitational field. The same particle moved with its polar axis parallel to the electric field (doublets) or with its plane of centers parallel to the electrical field (triangular triplets). Using a nonlinear regression method, both the dynamic shape factor and slip correction factor could be determined separately from the data. The dynamic shape factors at different orientations were in good agreement with those obtained previously in a sedimentation tank. The slip correction factor of singlet particles agreed with results previously obtained by Allen and Raabe for latex particles. Slip correction factors of doublets and triangular triplets can also be expressed in the Knudsen-Weber form: 1 + 2λ/d _a [1.142 + 0.558 exp(?0.999 d _a/2λ)]. The adjusted sphere diameter d _a was 1.21 d ₁ (primary diameter) for doublets moving parallel to the flow and 1.31 d ₁ for doublets randomly oriented. These results show that the slip correction factor of a nonspherical particle depends on the orientation and confirm the theory proposed by Dahneke.     

Drag of non-spherical solid particles of regular and irregular shape

The drag of a non-spherical particle was reviewed and investigated for a variety of shapes (regular and irregular) and particle Reynolds numbers (Re_p). Point-force models for the trajectory-averaged drag were discussed for both the Stokes regime (Re_p ? 1) and Newton regime (Re_p ? 1 and sub-critical with approximately constant drag coefficient) for a particular particle shape. While exact solutions were often available for the Stokes regime, the Newton regime depended on: aspect ratio for spheroidal particles, surface area ratio for other regularly-shaped particles, and min-med-max area for irregularly shaped particles. The combination of the Stokes and Newton regimes were well integrated using a general method by Ganser (developed for isometric shapes and disks). In particular, a modified Clift-Gauvin expression was developed for particles with approximately cylindrical cross-sections relative to the flow, e.g. rods, prolate spheroids, and oblate spheroids with near-unity aspect ratios. However, particles with non-circular cross-sections exhibited a weaker dependence on Reynolds number, which is attributed to the more rapid transition to flow separation and turbulent boundary layer conditions. Their drag coefficient behavior was better represented by a modified Dallavalle drag model, by again integrating the Stokes and Newton regimes. This paper first discusses spherical particle drag and classification of particle shapes, followed by the main body which discusses drag in Stokes and Newton regimes and then combines these results for the intermediate regimes.     

On the Effect of Particle Alignment in the DMA

The Differential Mobility Analyzer (DMA) is designed to measure particle mobility diameter, which for spherical particles is equal to particle volume equivalent diameter. In contrast, the mobility diameter of aspherical particles is a function of the particle shape and orientation. The magnitude of the DMA electric fields is such that it can cause aspherical particles to align preferentially in a specific orientation. The same electric field and the sheath flow rate ( q _sh ) define the particle mobility diameter. But, the fact that particle orientation depends on the electric field makes the dynamic shape factor and hence the mobility diameter depend on q _sh . Here, we describe an operating procedure that relies on a tandem DMA system, in which the second DMA is operated at a number of q _sh , to obtain information about particle shape by measuring the effect of particle alignment on the particle mobility diameter. We show how the relationship between the mobility diameter and q _sh can even be used to physically separate particles according to their shapes. In addition we explore the use of simultaneous measurements of particle alignment and particle vacuum aerodynamic diameters to gain further information on particle shape and account for particle alignment in the calculations of dynamic shape factor. We first test this approach on doublets and compact triplets of PSL spheres, for which the orientation dependent dynamic shape factors are known. We then investigate applications on a number of polydisperse particle systems of various shapes.     

Homogeneous drag models in gas–solid fluidization: Big data analytics and conventional correlation

The drag force model is vital for capturing gas–solid flow dynamics in many simulation approaches. Most of the homogeneous drag models in the literature are expressed as a function of phase fraction (ε) and particle Reynolds number (Re_s). In this work, we use a “big data” approach to analyze ~10⁸ data points for drag coefficient (F_d) for Geldart Group A particles at atmospheric pressure and find that the contribution of Re_s on F_d is much less than ε based on the Maximal information coefficient analysis. Thus, these drag models are separately reduced to machine learning and conventional expressions only related to ε. The reduced models achieve almost the same predictive performance as the originals in bubbling, turbulent, and jet fluidizations. Moreover, the reduced models provide better numerical stability for coarse grid simulations. These findings provide new insights into the drag coefficient for Geldart Group A particles under full fluidization conditions.     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Experimental estimation of the settling velocity and drag coefficient of the hollow cylindrical particles settling in non-Newtonian fluids in an annular channel

The drag coefficient data of particles settling in an annular channel is very much essential for designing different solid–fluid handling equipment, such as the fluidized bed. Experimental settling velocity, wall factor, and drag coefficient data of the hollow-cylinder particle are presented. Carboxymethyl cellulose solution has been used as the working fluid with a flow index of 0.64 ≤ n ≤ 0.91 and a consistency index of 0.31 ≤ K ≤ 1.81. The experimental results covered a wide diameter ratio range (0.14 ≤ d_eq/L ≤ 0.46), hollow cylinder inner to outer diameter ratio (0.2 ≤ d_i/d_o ≤0.8), and Reynolds number (0.05 ≤ Re ≤ 51 and 0.09 ≤ Re_∞ ≤ 55). d_eq, d_i, and d_o are the equivalent inner and outer diameters of the particle, L is the annular gap, and Re and Re_∞ are the Reynolds numbers in the presence and absence of the wall effect, respectively. The wall factor decreased, and the drag coefficient increased with d_eq/L and d_i/d_o ratios. The above parameters declined with the Reynolds number. The hollow cylinder experienced a lesser wall effect than the spherical particles settling in a non-annular channel. In some cases, the wall factor of the hollow cylinder is found to be equal to the spherical particles settling in an annular channel. The developed correlations have successfully predicted the drag coefficients of the hollow cylinder.     

The voidage function and effective drag force for fluidized beds

Here, an experimental investigation on the effective drag force in a conventional fluidized bed is presented. Two beds of different particle size distribution belonging to group B and group B/D powders were fluidized in air in a diameter column. The drag force on a particle has been calculated based on the measurement of particle velocity and concentration during pulse gas tests, using twin-plane electrical capacitance tomography. The validity of the voidage function “correction function”, (1−ε_s)ⁿ, for the reliable estimation of the effective drag force has been investigated. The parameter n shows substantial dependence on the relative particle Reynolds number , and the spatial variation of the effective static and hydrodynamic forces. It is also illustrated that, a simple correlation for the effective drag coefficient as function of the particle Reynolds number (Re_p), expressed implicitly in terms of the interstitial gas velocity, can serve in estimating the effective drag force in a real fluidization process. Analysis shows that, the calculated drag force is comparable to the particle weight, which enables a better understanding of the particle dynamics, and the degree of spatial segregation in a multi-sized particle bed mixture. The analogy presented in this paper could be extended to obtain a generalized correlation for the effective drag coefficient in a fluidized bed in terms of Re_p and the particle physical properties.     

Fluid‐particle drag in inertial polydisperse gas–solid suspensions

In this article, we extend the low Reynolds number fluid‐particle drag relation proposed by Yin and Sundaresan for polydisperse systems to include the effect of moderate fluid inertia. The proposed model captures the fluid‐particle drag results obtained from lattice‐Boltzmann simulations of bidisperse and ternary suspensions at particle mixture Reynolds numbers ranging from 0 ≤ Re_mix ≤ 40, over a particle volume fraction range of 0.2 ≤ ? ≤ 0.4, volume fraction ratios of 1 ≤ ?_i/?_j ≤ 3, and particle diameter ratios of 1 ≤ d_i/d_j ≤ 2.5. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

A Study of Density Effect and Droplet Deformation in the TSI Aerodynamic Particle Sizer

Effects of particle density and droplet deformation on the performance of a TSI aerodynamic particle sizer (APS) were studied using polystyrene latex (PSL), dioctyl phthalate (DOP), ammonium fluorescein (AF), fused aluminosilicate (FAP), and fused cerium oxide (FCO) monodisperse aerosols. Results indicated that, because of the sensitivity of the instrument, periodic cleaning of the APS inner nozzle is needed to maintain the consistency of its calibration curve. Density effects were experimentally confirmed with PSL, AF, FAP, and FCO aerosols of particle densities ranging from 1.05 to 4.33 g/cm³. Results, however, showed that this effect can only be experimentally detected for particles of density greater than 2 g/cm³ and aerodynamic diameter greater than 5 (μm. Effects of droplet deformation were studied with DOP.     

Performance of a TSI Aerodynamic Particle Sizer

Calibration curves of the aerodynamic particle sizer (APS) under different sets of operating conditions (i.e., pressure drop across the nozzle, flow rate, and ambient pressure) were obtained. Materials used included oleic acid (OA), dioctyl phthalate (DOP), polystyrene latex (PSL), and fused aluminosilicate particles (FAP). The effect of particle density on the calibration was not found to be significant among test aerosols (in the density range from 0.89 to 2.3 g/cm³). Calibration curves obtained at reduced ambient pressure were different from the manufacturer's curve, indicating that recalibration of the APS is required if other than standard operating conditions are used. However, all the curves can be consolidated into a unique curve that relates the Stokes number at the nozzle exit to the normalized particle velocity (particle velocity divided by gas velocity). Methods for calculating gas velocity, particle velocity, and other pertinent parameters for the APS were developed and the results are presented. Consequently, these parameters together with the unique curve can be used to generate calibration curves for any set of operating conditions without performing the experimental calibration in the laboratory. The geometric standard deviations of monodisperse aerosols measured by the APS are generally in good agreement (< 2%) with those determined by other methods, thus demonstrating the good resolution of the instrument.     

Effect of solid monodisperse particles on the pressure drop of fibrous filters

The increase in pressure drop across glass HEPA filters has been measured as a function of particle mass loading using polystyrene latex particles (PSL). PSL particles in several different sizes were generated as challenge aerosols. For each particle size distribution, the specific resistance (K₂) was calculated by measuring the mass of PSL particles loaded per unit area of filter and the pressure drop across the filters at a given filtration velocity. In all cases, the specific resistance of the filter cake increased as the aerodynamic mean particle diameter decreased at the same mass loading. This correlation equation was modified by using the lognormal conversion method suggested by Raabe [1971] for a polydisperse particle size distribution; then the modified equation was expressed as a function of geometric mean particle diameter and standard deviation which could be obtained by the measuring instruments (PDS 3603; TSI Inc.). The advantage of this approach over other methods is the use of a more convenient prediction of pressure drop, if we know the geometric mean particle diameter and standard deviation, which could be easily measured. The values of porosities, obtained from the pressure drop responses of loading in the filters using the Rundnick and First equation, were compared with other researches.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

Drag coefficient of Solid Recovered Fuels (SRF)

The numerical simulation of Solid Recovered Fuels (SRF) co-combustion in pulverised coal power plants requires a flexible particle model, which among other properties should be able to predict the aerodynamic behaviour of the irregular-shaped particles, especially their trajectories along the boiler axis. This will help to provide vital information on whether the SRF particles are entrained in the combustion gases or drop to the boiler bottom. One difficulty encountered in the process is the true value of the drag coefficient (C_D) of the coarse SRF particles. Most of the numerical simulation codes calculate the particle trajectories by integrating the force balance of the particles in which the C_D plays an important role. As a result, a true C_D of SRF will definitely lead to more realistic results.In this short communication, the authors have taken a practical approach in determining the C_D of the SRF. It was found that within the Newton’s law range the C_D of the SRF lies between 0.6 and 2.0 with a mean value of 1.5. The results were further validated by correlating the calculated lift velocities of SRF using different C_D values and that obtained through experiment.     

Experimental and Modeling Studies of the Stream-Wise Filter Vibration Effect on the Filtration Efficiency

The stream-wise vibration effect of a fibrous filter is studied experimentally and numerically for the purpose of evaluating filtration efficiency. The particle sizes range from 0.02 to 10 μ m and the face velocity ranges from 3 to 10 cm/s. The vibrational peak velocity also varied from 0 to 50 cm/s. The filtration efficiency for this wide size range is obtained by combining the individual test results for fine particles (0.02 to 0.5 μ m) and large particles (0.5 to 10.0 μ m). For the fine particle experiment, Arizona Road Dust (ARD) test particles are generated by an atomizer after an ultrasonic process and measured by a Scanning Mobility Particle Sizer (SMPS). For the large particle experiment, the test particles are generated by a fluidized bed and measured by an Aerodynamic Particle Sizer (APS). When the particles are generated by the atomizer after ultrasonicating, the majority of the particles are in nano scale without the agglomerates on the large particle surface, while particles generated by the fluidized bed are mostly in micro-scale because many nanoparticles are agglomerated on large particle surface. The filtration efficiency increases with the vibrational peak velocity in the impaction-dominant region (D _p > 0.1 μ m) and diffusion-dominant region (D _p < 0.1 μ m), due to the increased relative velocity between the particle and the filter fiber and the increased diffusion intensity from turbulence around the fiber, respectively. A model for the filter vibration effect is established with a modified Stokes number for the impaction-dominant region and an empirical analysis for the diffusion-dominant region.     

Terminal velocity of porous spheres

Terminal velocity of porous spheres was experimentally measured for a Reynolds number range of 0.2 to 120 for a normalized sphere radius, β = R/R of 15.6 to 33, where R and k are the sphere radius and permeability, respectively. The drag coefficient for 15 < β < 33 was found to be C_D = 24Ω/Re [1 + 0.1315 Re^{(0.82 - 0.05w)}] for 0.1 < Re ≤ 7 and C_D = 24Ω/Re [1 + 0.0853 Re^{(1.093 - 0.105w)}] for 7 < Re < 120 with w = log₁₀Re where Re is the sphere Reynolds number and Ω=2β² [1 - (tanh β/β)] / 2β² + 3[1 - tanh β/β)] At high Reynolds numbers, it was found that the porous sphere terminal velocity was less affected by the container walls than for the case of an impermeable sphere. However, at very low Reynolds numbers, the wall effects were found to be similar for both the permeable and the impermeable spheres.     

Synthesis and characterization of monodisperse uniformly shaped respirable aerosols

The top‐down, micromolding technique, referred to as Particle Replication in Nonwetting Templates (PRINT^®), affords a new opportunity for the generation of inhalation therapeutics. Powders were fabricated with predetermined particle size and shape; when dispersed with a collision jet nebulizer, these particles resulted in monodisperse aerosols with geometric standard deviations well below 1.2. Dynamic shape factors for this novel set of uniformly shaped particles were determined by correcting the drag of nonspherical particles in the ultra‐Stokesian flow conditions of the aerodynamic particle sizer (APS). This convenient approach for shape factor determination agreed well with current literature approaches and allowed for the correction of APS results for particles with known volumes. Determined shape factor values of PRINT geometries were used to estimate the theoretical median aerodynamic diameters of individual aerosols, which were then compared to actual inhalation powders. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3184–3194, 2013     

Fine particle turbulence modulation

Streamwise turbulence intensities of fine particulate suspensions were studied in a 26 mm N.B. horizontal pipe loop. Colloidal silica spheres were prepared in 10^?4M and 1M KNO₃ solutions to control the degree of aggregate formation in the suspension. Using an ultrasonic Doppler velocity profiling sensor, the turbulence intensities of the fine particle suspensions were compared with those of a particle‐free flow over a range of Reynolds numbers. At low electrolyte concentration, the silica particles remain dispersed, with the turbulence intensity of the suspension flow comparable with that of the particle‐free flow. At high electrolyte concentration, increased particle‐particle interaction leads to the formation of particle aggregates which support turbulence augmentation over a critical Reynolds number range. The range of Reynolds numbers over which this turbulence enhancement is observed is limited by both fluid dynamic effects at low Reynolds numbers (Re ≈ 5500) and aggregate breakup at high Reynolds numbers (Re ≈ 8000). © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A Method To Employ the Aerodynamic Particle Sizer Factory Calibration Under Different Operating Conditions

The dimensionless aerodynamic particle sizer (APS) response function (normalized particle velocity against particle Stokes number) first reported by Chen et al. (1985) is explored for much larger solid particles (diameters to 35 μm) over a similar range of instrument pressures (624–l740 mm Hg) and flow rates (4.2–6.0 L/min). An essentially unique response function is found for low and intermediate Stokes numbers under a variety of operating conditions, including the use of argon as the carrier gas. For large particles, however, non-Stokesian drag effects introduce systematic differences among calibration sets so that a unique response function no longer applies. The largest differences are observed between calibrations performed in air and argon, although even in this case the sizing error amounts to < 12% for a 20-μm polystyrene latex sphere. For intermediate Stokes numbers, a direct consequence of this work is that a reference calibration (channel number against Stokes number) can be used under different ambient conditions by setting the APS to operate at the same nozzle velocity as used in the reference calibration. With the single-velocity method, the factory-supplied calibration relating channel number to aerodynamic diameter can be used for air over a reasonable range of ambient temperatures and pressures. The same calibration can be used with an argon carrier gas provided that the aerodynamic diameters reported by the APS software are adjusted by the square root of the gas viscosity ratio. For the single-velocity mode of operation, a generalization of a correction proposed by Wang and John (1987, 1989) can be made and is shown to reduce by one half the sizing error introduced by non-Stokesian drag.     

Modeling of Particle Removal using Non-contact Brush Scrubbing in Post-CMP Cleaning Processes

Particle removal using non-contact brush scrubbing for post-CMP (Chemical Mechanical Planarization) cleaning is investigated analytically. The removal of Si O ₂ and A l ₂ O ₃ particles adhered onto Si O ₂ film coated on the wafer surface are considered. The cleaning fluid (H ₂ O/N H ₄ OH = 1:25 and 1:200) flowing between the brush and wafer surface is treated as a thin-film fluid flow. The flow field details and its effect on the drag force acting on the adhered particles are discussed. In addition to the drag force, the electrical double layer (EDL) and thermophoretic force effects on particle removal are also considered. It was found that the dominant force in achieving particle removal using a rolling mechanism is the drag force. The EDL and thermophoretic forces have an insignificant effect on particle removal. Based on the results from this study, particles of submicron size can be removed from a wafer surface using higher brush rotation speed and pure deionized (DI) water as the cleaning fluid.