Simulation of the Brownian coagulation of nanoparticles with initial bimodal size distribution via moment method

The Brownian coagulation of nanoparticles with initial bimodal size distribution, i.e., mode i and j, is numerically studied using the moment method. Evolutions of particle number concentration, geometric average diameter and geometric standard deviation are given in the free molecular regime, the continuum regime, the free molecular regime and transition regime, the free molecular regime and continuum regime, respectively. The results show that, both in the free molecular regime and the continuum regime, the number concentration of mode i and j decreases with increasing time. The evolutions of particle geometric average diameter with different initial size distribution are quite different. Both intra-modal and inter-modal coagulation finally make the polydispersed size distribution become monodispersed. As time goes by, the size distribution with initial bimodal turns to be unimodal and shifts to a larger particle size range. In the free molecular regime and transition regime, the inter-modal coagulation becomes dominant when the number concentrations of mode i and j are of the same order. The effects of the number concentration of mode i and mode j on the evolution of geometric average diameter of mode j are negligible, while the effects of the number concentration of mode j on the evolution of geometric average diameter of mode j is distinct. In the free molecular regime and continuum regime, the higher the initial number concentration of mode j, the more obvious the variation of the number concentration of mode i.     

Investigation on the granulation behavior of iron ore fine in a horizontal high-shear granulator

High-shear granulation is widely used in many particulate industries for its good capability to improve the size, strength and composition uniformity of powder substances. This work conducted an experimental study to investigate the granulation behavior of iron ore fine in a horizontal high-shear granulator, such as granules size distribution, granules growth rate, and permeability of the granules bed. The results show that the granule size and permeability of packed granules bed increase gradually with increasing the granulation time, and the growth of granules can be divided into three stages: the rapid growth stage, the slow growth stage and the relatively stable stage. Both the higher rotational speed and larger number of impellers increase the kinetic energy and collision frequency of the particles, which causes the increase of average granule size, growth rate and permeability of granules packed bed. On the other hand, the shear damage effect of the impellers on the granules is also enhanced with the increase of rotational speed and impeller number, resulting in significant granule size segregation.     

Variation of granule mass fraction with coordination number in wet granulation process

In granulation, fine particles combine to form a coarse granule in the form of a particle matrix partially or fully saturated with a binder liquid. The final product of granulation possesses a wide variety of granule size distributions with surface mean diameters which differ with operating conditions. The final granule size depends on the operating conditions, e.g. operating gas velocity, inlet air temperature, initial feed particle size, and viscosity of the binder. The objective of this paper is to find out the uniformity in the relation between the granule mass fraction in the final granule size distribution and the number of feed particles present in the granules. The total number of granules obtained depends on the experimental conditions but the granule mass fraction and the number of feed particles forming a single granule are independent of operating variables, feed material and method of granulation. The paper purports further to compare the uniform nature of mass fraction of the granules in final granule size distribution and the primary particles required to form that particular granule size irrespective of experimental conditions of granulation.     

Scale-up of a high shear wet granulation process using a nucleation regime map approach

Scale-up of the high shear wet granulation (HSWG) process is considered a challenge because HSWG is complex and influenced by numerous factors, including equipment, formulation, and process variables. For a system of microcrystalline cellulose and water, HSWG experiments at three scales (1, 2, and 4 L working vessel) were conducted with a granulator. Scale-up was implemented on the basis of a nucleation regime map approach. To keep dimensionless spray flux and drop penetration time constant, water addition time at three processing scales were 300, 442, and 700 s, respectively. The other process parameters were kept unchanged. Granule size distributions were plotted and compared, and scanning electron microscopy was used to analyze granule surface morphology. Physical characterization was undertaken using a modified SeDeM method. At nearly all scales, granule yield was greater than 85% and all the cosine values were larger than 0.89. At the same experiment points, granules at all scales had similar surface morphology and similar physical characteristics. The results demonstrate that a rational scaling-up of the HSWG process is feasible using a regime map approach.     

WET-GRANULATION RESEARCH WITH APPLICATION TO SCALE-UP

Summary Granulation is a unit operation by which larger granules are produced from fine, powdery particles to improve appearance, flow properties and mixedness, reduce dustiness and, in general, produce engineered particles with superior attributes. Agglomeration in wet granulation is achieved by introducing a “binder” fluid onto a shearing mass of fine powders. This paper gives a general overview of the process with emphasis on a simplified granulation model based on a dimensionless parameter containing inertia and viscous dissipation energies between colliding particles: the so-celled Stokes number. The model incorporates most common features of all granulation devices (mixers) used in the pharmaceutical industry.Also described in the paper is a computer simulation that captures the movement of flowing powder in an ideal mixer-granu/ator with constant shear rate. A fraction of the total number of particles is wet (covered by binder and there-fore “sticky”) while the rest of the particles are dry. The numerical simulation depicts two distinct regimes of agglomeration found in a typical granulator: granule growth and subsequent breakup. During granule growth-simulations, final granule size and shape distributions are obtained by analyzing the size and shape of formed granules using a pattern-recognition technique. A second kind of simulation, also using rapid granular flow modeling, follows the rotation and deformation of an “agglomerate” held together by a liquid binder. Results from these simulations yield critical values of the Stokes number. Below the cdticel value, the agglomerates are stable and only rotate in response to shear while above the critical value they break into two or more pieces. At the critical value, they attain a steady elongated shape. Using values of the critical Stokes number, the model predicts the size of formed granules.The existence of the critical state in which granules attain a characteristic elongated shape is used to measure shear forces in a granulator by employing calibrated “test” particles of known strength. This knowledge is employed in granulation scale-up to determine a kinematic rule that conserves stresses in the small and the large-scale machines. It was found that conserving the magnitude of internal stresses in the moving powder yields granules with similar attributes in granulators of different size.     

Experimental investigation of horizontal forced-vibration effect on air-water two-phase flow

In order to investigate the potential seismic vibrations effect on two-phase flow in an annular channel, experimental tests with air-water two-phase flow under horizontal vibrations were carried out. A low-speed eccentric-cam vibration module capable of operating at motor speed of 45–1200 rpm (f = 0.75–20 Hz) was attached to an annular channel, which was scaled down from a prototypic BWR fuel sub-channel with inner and outer diameters of 19.1 mm and 38.1 mm, respectively. The two-phase flow was operated in the ranges of 〈j_f〉 = 0.25–1.00 m/s and 〈j_g〉 = 0.03–1.46 m/s with 27 flow conditions, and the vibration amplitudes controlled by cam eccentricity (E) were designed for the range of 0.8–22.2 mm. Ring-type impedance void meters were utilized to detect the area-averaged time-averaged void fraction under stationary and vibration conditions. A systematic experimental database was built and analyzed with effective maps in terms of flow conditions (〈j_g〉-〈j_f〉) and vibration conditions (E-f and f-a), and the potential effects were expressed by regions on the maps. In the 〈j_g〉-〈j_f〉 maps, the void fraction was found to potentially decrease under vibrations in bubbly flow regime and relatively lower liquid flow conditions, which may be explained by the increase of distribution parameter. Whereas and the void fraction may increase at the region closed to bubbly-to-slug transition boundary under vibrations, which may be explained by the changes of drift velocity due to flow regime change from bubbly to slug flows. No significant change in void fraction was found in slug flow regime under the present test conditions.     

Micromechanics of coalescence in ductile fracture

Significant progress has been recently made in modelling the onset of void coalescence by internal necking in ductile materials. The aim of this paper is to develop a micro-mechanical framework for the whole coalescence regime, suitable for finite-element implementation. The model is defined by a set of constitutive equations including a closed form of the yield surface along with appropriate evolution laws for void shape and ligament size. Normality is still obeyed during coalescence. The derivation of the evolution laws is carefully guided by coalescence phenomenology inferred from micromechanical unit-cell calculations. The major implication of the model is that the stress carrying capacity of the elementary volume vanishes as a natural outcome of ligament size reduction. Moreover, the drop in the macroscopic stress accompanying coalescence can be quantified for many initial microstructures provided that the microstructure state is known at incipient coalescence. The second part of the paper addresses a more practical issue, that is the prediction of the acceleration rate δ in the Tvergaard-Needleman phenomenological approach to coalescence. For that purpose, a Gurson-like model including void shape effects is used. Results are presented and discussed in the limiting case of a non-hardening material for different initial microstructures and various stress states. Predicted values of δ are extremely sensitive to stress triaxiality and initial spacing ratio. The effect of initial porosity is significant at low triaxiality whereas the effect of initial void shape is emphasized at high triaxiality.     

Further investigations on the influence of scale-up of a high shear granulator on the granule properties

This study focuses on the characterisation of strength, density, and size of granules produced in various scales of a high shear granulator. Calcium carbonate （Durca165） was used as the feed powder and aqueous polyethylene glycol （PEG 4000） as the binder. The dried granules were analysed for their strength, density, size distribution, and wall make-up. Granules were produced in granulators with four scales, 1, 5, 50, and 250 L under three scale-up rules of constant tip speed, constant shear stress, and constant Froude number. The results show that regardless of equipment scale, increasing the impeller speed has a great effect on crushing strength and stress. The underlying cause is an increase in granule density due to more consolidation at higher impeller speeds. Wall make-up is significantly reduced to less than 5% as the scale is increased from 1 to 250 L. The results of this study corroborate our previous findings that the constant tip speed rule is the best criterion for scale-up of high shear granulators.     

Flow regime development analysis in adiabatic upward two-phase flow in a vertical annulus

In this work radial and axial flow regime development in adiabatic upward air-water two-phase flow in a vertical annulus has been investigated. Local flow regimes have been identified using conductivity probes and neural networks techniques. The inner and outer diameters of the annulus are 19.1 mm and 38.1 mm, respectively. The equivalent hydraulic diameter of the flow channel, D_H, is 19.0 mm and the total length is 4.37 m. The flow regime map includes 1080 local flow regimes identifications in 72 flow conditions within a range of 0.01 m/s < 〈j_g〉 < 30 m/s and 0.2 m/s < 〈j_f〉 < 3.5 m/s where 〈j_g〉 and 〈j_f〉 are, respectively, superficial gas and liquid velocities. The local flow regime has been classified into four categories: bubbly, cap-slug, churn-turbulent and annular flows. In order to study the radial and axial development of flow regime the measurements have been performed at five radial locations. The three axial positions correspond to z/D_H = 52, 149 and 230, where z represents the axial position. The flow regime indicator has been chosen as some statistical parameters of local bubble chord length distributions and self-organized neural networks have been used as mapping system. This information has been also used to compare the results given by the existing flow regime transition models. The local flow regime is characterized basically by the void fraction and bubble chord length. The radial development of flow regime shows partial and complete local flow regime combinations. The radial development is controlled by axial location and superficial liquid velocity. The radial flow regime transition is always initiated in the center of the flow channel and it is propagated towards the channel boundaries. The axial development of flow regime is observed in all the flow maps and it is governed by superficial liquid velocity and radial location. The prediction results of the models are compared for each flow regime transition.     

On the interaction of Taylor bubbles rising in two-phase co-current slug flow in vertical columns: turbulent wakes

An experimental study on the interaction between Taylor bubbles rising through a co-current flowing liquid in a vertical tube with 32 mm of internal diameter is reported. The flow pattern in the bubble's wake was turbulent and the flow regime in the liquid slug was either turbulent or laminar. When the flow regime in the liquid slug is turbulent (i) the minimum distance between bubbles above which there is no interaction is 5D-6D; (ii) the bubble's rising velocity is in excellent agreement with the Nicklin relation; (iii) the experimental values of the bubble length compare well with theoretical predictions (Barnea 1990); (iv) the distance between consecutive bubbles varied from 13D to 16D and is insensitive to the liquid Reynolds number. When the flow regime in the liquid slug is laminar (i) the wake length is about 5D-6D; (ii) the minimum distance between bubbles above which there is no interaction is higher than 25D; (iii) the bubble's rising velocity is significantly smaller than theoretical predictions. These results were explained in the light of the findings of Pinto et al. (1998) on coalescence of two Taylor bubbles rising through a co-current liquid. Received: 2 February 2000 / Accepted: 15 March 2001     

Boundary conditions for flows with chemical reactions occurring on a surface

With the use of a solution of a model Boltzmann equation for a binary mixture in the Knudsen layer, we obtain the boundary conditions for the equations of gas dynamics when the reactionl _iA_il _jA_j (l _i molecules of A_i change intol _j molecules of A_j, and vice versa) is occurring on a surface. The boundary condition that we obtain differs from those that are usually applicable by the presence of terms of the same order. This confirms the conclusion arrived at by the authors in [1], where it was shown that if the Knudsen layer is left out of account, which is precisely what is usually done, it is impossible to obtain correct boundary conditions.Moscow. Translated from Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–138, January–February, 1972.     

Soot Particle Size Distribution Functions in a Turbulent Non-Premixed Ethylene-Nitrogen Flame

A scanning mobility particle sizer with a nano differential mobility analyzer was used to measure nanoparticle size distribution functions in a turbulent non-premixed flame. The burner utilizes a premixed pilot flame which anchors a C₂H₄/N₂ (35/65) central jet with Re _D = 20,000. Nanoparticles in the flame were sampled through a N₂-filled tube with a 500- μm orifice. Previous studies have shown that insufficient dilution of the nanoparticles can lead to coagulation in the sampling line and skewed particle size distribution functions. A system of mass flow controllers and valves were used to vary the dilution ratio. Single-stage and two-stage dilution systems were investigated. A parametric study on the effect of the dilution ratio on the observed particle size distribution function indicates that particle coagulation in the sampling line can be eliminated using a two-stage dilution process. Carbonaceous nanoparticle (soot) concentration particle size distribution functions along the flame centerline at multiple heights in the flame are presented. The resulting distributions reveal a pattern of increasing mean particle diameters as the distance from the nozzle along the centerline increases.     

Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S _{i, j} dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S _{i, j} shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S _{i, j} shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S _{i, j} shocks and classical waves are described. As a physical application, all types of S _{i, j} shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

Product inhibition in packed-bed immobilized enzyme reactors for complex processes: Modelling of two-substrate processes

An analytical model was developed for describing the performance of packed-bed enzymic reactors operating with two cosubstrates, and when one of the reaction products is inhibitory to the enzyme. To this aim, the compartmental analysis technique was used. The relevant equations obtained were solved numerically, and the effect of the main operational parameters on the reactor characteristics were studied.Notation C _infa,i ^sup* local concentration of products in the pores of stage i - C _j,i concentration of substrate j in the pores of stage i - D _infa ^sup* internal (pore) diffusion coefficient for the reaction product a - D _j internal (pore) diffusion coefficient of substrate j - J _infa,i ^sup* net flux of product a, taking place from the pores of stage i into the corresponding bulk phase - J _j,i net flux of substrate j, taking place from the bulk phase of stage i into the corresponding pores - K _b inhibition constant - K _m,1, K _m,2 Michaelis constants for substrate 1 and 2, respectively - K _q inhibition constant - n total number of elementary stages in the reactor - Q volumetric flow rate throughout the reactor - R _j,i, R _infa,i ^sup* local reaction rates in pores of stage i, in terms of concentration of substrate j and product a respectively - S _infa,i ^sup* , S _infa,i-1 ^sup* bulk concentration of the reaction product a, in the stages i and i — 1, respectively - S _j,0 concentration of substrate j in the reactor feed - S _j,i-1, S _j,i concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectively - V total volume of the reactor - V _m maximal reaction rate in terms of volumetric units - y axial coordinate of the pores - y ₀ depth of the pores - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ^* dimensionless parameter, defined in Equation (22) - volumetric packing density of catalytic particles (dimensionless) - porosity of the catalytic particles (dimensionless) - V _infi ^sup* dimensionless concentration of reaction product in pores of stage i, defined in Equation (17) - j,i dimensionless concentration of substrate j in pores of stage i; defined in Equation (6) - _j,i-1, _j.i dimensionless concentration of substrate j in the bulk phase of stage i; defined in Equation (6) - dimensionless position along the pore; defined in Equation (6)     

Multi-level convolution filtering technique for digital laser-speckle-velocimetry

A new evaluation method for velocity measurements using digitized, single exposed speckle images is presented. The method is based on a convolution filtering technique used on different levels. Beginning with the computation of a small number of velocity vectors on the coarsest level, the solution is determined step by step on the finer levels, and the number of points is squared from one level to the next. On the coarsest level the vectors are computed with high accuracy, and good approximation is obtained through interpolation of the solution on the next, finer level. Preprocessing of the images considerably improves the accuracy and evaluation speed of the measurement. The computation of the displacement vectors on the finest level without interpolation shows that the number of erroneous vectors computed during the binarization of the images can be reduced by up to 70%. Using the convolution filtering technique on three levels allows for a further reduction of erroneous vectors by up to 40%. Use of smaller kernels and reduction of the kernel and the image area after every interpolation step reduces the computation time for a velocity vector field to 50% compared to the one-level algorithm.List of symbols A image area of the convolution - K kernel of the convolution - O overlapping area - Pc number of multiplications and additions necessary for the computation of a velocity vector - R correlation coefficient - S _c size of the field of the correlation coefficients - S _A size of the image area for the convolution - S _K size of the kernel - c column index - d displacement of the speckles - i, j index of the kernel - n _i number of points of gridi - r row index - t time of capture of the first speckle-image - t time difference between the capture of the two speckle-images - T temperature difference in the thermocapillary convection experiment - z distance of the gridpoints The authors acknowledge the support of the Deutsche Agentur für Raumfahrtangelegenheiten (DARA) under grant number 5o-QV 88981.     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

Effect of impeller design on homogeneity,size and strength of pharmaceutical granules produced by high-shear wet granulation

Small mixer impeller design is not tailored for granulation because impellers are intended for a wide range of processes. The aim of this research was to evaluate the performances of several impellers to provide guidance on the selection and design for the purposes of granulation. Lactose granules were produced using wet granulation with water as a binder. A Kenwood KM070 mixer was used as a standard apparatus and five impeller designs with different shapes and surface areas were used. The efficacy of granulate formation was measured by adding an optically sensitive tracer to determine variations in active ingredient content across random samples of granules from the same size classes. It was found that impeller design influenced the homogeneity of the granules and therefore can affect final product performance. The variation in active ingredient content across granules of differing size was also investigated. The results show that small granules were more potent than larger granules.