Rosin-Rammler Distribution Function Research Articles

Numerical Study of the Drying Process of Different Sized Particles in an Industrial-Scale Spray Dryer

Development of high-performance spray dryers that are more energy efficient and are able to produce high-quality milk powders is very important for the future of the dairy powder industry. Understanding and optimization of the exiting dryers are also of great value. Computational fluid dynamics is a powerful tool to simulate and help understanding the characteristics of spray drying and to introduce potentially improved designs. The present study has concentrated on the multiphase flow in an industrial-scale spray dryer using the CFD package FLUENT. A Eulerian-Lagrangian approach is used in the simulations. A new drying model, REA model, for milk particles has been implemented for the first time in a CFD application. The numerical results match well with the plant data. It is argued that the “reflecting wall” boundary condition produced more physically correct results for normal dryer operation than the “escaping wall” boundary condition. The second one is an oversimplification. The influences of the particle size and particle size distribution, residence time, kinetic energy, and maximum temperature have been analyzed. The initial droplet size range was set to be from 100 to 500 µm with Rosin-Rammler distribution function. One significant result of this study is that rather dense particle clouds of medium-size particles (224–285 µm) are found near the side walls. They are transported upwards along the side wall (conical wall surface) and then are dispersed. It is found the particles with initial diameters of between 225 and 270 µm have the largest residence times.

Experimental study on drop formation in liquid–liquid fluidized bed

Drop formation in liquid–liquid fluidized bed was investigated experimentally. The normal water was injected via a fine-capillary spray nozzle into the co-flowing No. 25 transformer oil with jet directed upwards in a vertical fluidized bed. Experiments under a wide variety of conditions were conducted to investigate the instability dynamics of the jet, the size and size distribution of the drops. Details of drop formation, drop flow patterns and jet evolution were monitored in real-time by an ultra-high-speed digital CCD (charge couple device) camera. The Rosin–Rammler model was applied to characterize experimental drop size distributions. Final results demonstrate that drop formation in liquid–liquid system takes place on three absolutely different developing regimes: bubbling, laminar jetting and turbulent jetting, depending on the relative Reynolds number between the two phases. For different flow domains, dynamics of drop formation change significantly, involving mechanism of jet breakup, jet length pulsation, mean size and uniformity of the drops. The jet length fluctuates with time in variable and random amplitudes for a specified set of operated parameters. Good agreement is shown between the drop size and the Rosin–Rammler distribution function with the minimum correlation coefficient 0.9199. The mean drop diameter decreases all along with increasing jet flow rate. Especially after the relative Reynolds number exceeds a certain value about 3.5 × 10 4 , the jet disrupts intensely into multiple small drops with a diameter mainly ranging from 1.0 to 1.5 mm and a more and more uniform size distribution. The turbulent jetting regime of drop formation is the most preferable to the dynamic ice slurry making system.

An experimental study on the distribution of the drop size and velocity in asymmetric impinging jet sprays

Distributions of the drop size and velocity in an asymmetric impinging jet are investigated by injecting water and a sodium carbonate (Na2CO3) solution, which simulates the mixing process in impinging jet sprays of liquid oxidizer and liquid fuel for liquid propellants. The liquid sheet formed from the impinging jet is visualized and the drop size distributions are obtained by using image processing for the visualized images. The drop size distribution of the asymmetric impinging jets is fitted to the Rosin-Rammler distribution function. The obtained drop size distributions according to the azimuth angle in the impinging jet are compared with the theoretical predictions of previous research. The experimental results of the drop size distributions are located between the two curves obtained from the theoretical predictions by treating each jet in the asymmetric impinging jets as an independent wall-impinging jet. PIV images using a double-exposure method were processed to obtain the drop velocity vector in the impinging jets. Whether the drops shedding from the edge of the asymmetric impinging jets occurs radially or tangentially is also investigated from the PIV results.

Numerical and experimental study on hollow-cone fuel spray of highpressure swirl injector under high ambient pressure condition

A hybrid breakup model was proposed as a trustworthy prediction of hollow-cone fuel spray in the present study and the atomization process of the hollow-cone fuel spray of a high-pressure swirl injector in a Gasoline Direct Injection (GDI) engine under high ambient pressure conditions was studied by a new hybrid breakup model. The proposed hybrid breakup model is composed of the Linearized Instability Sheet Atomization (LISA) model as a primary breakup process. The Aerodynamically Progressed Taylor Analogy Breakup (APTAB) model, instead of the Taylor Analogy Breakup (TAB) model, was used as a secondary breakup process. The effects of the droplet deformation on a droplet aerodynamic external force are considered in the APTAB model. In addition, we replaced the x2 distribution function used in previous the APTAB model by the Rosin-Rammler distribution function to improve the prediction precision. The Laser Induced Exciplex Fluorescence (LIEF) technique and the Phase Doppler Anemometry (PDA) system were used to produce a set of experimental data for the model validation. The estimation of the prediction ability of the LISA+APTAB model was carried out, and spray characteristics, which are difficult to obtain by experimental method, were calculated and discussed. The suggested hybrid breakup model showed better prediction capability compared with the previous model (LISA+TAB model). From the calculated results, the effect of the ambient pressure on the SMD (Sauter Mean Diameter) and droplet velocity could be discussed quantitatively.

Vaporization Lengths of Hydrazine Fuels Burning with NTO

The performances of single hydrazines and hydrazine mixtures burning with NTO in a bipropellant rocket combustion chamber are compared. Systems using single hydrazines (N2H4, MMH, UDMH) and hydrazine mixtures (MMH=N2H4 and UDMH=N2H4) were simulated with total mass flow rates of 45:29 g=s and 89:25 g=s, respectively, in a chamber with diameter 40 mm. A one-dimensional mathematical model was developed, assuming thatcombustionandturbulentmixtureprocesses arecontrolledbydropletvaporization,andthatdropletdiameters follow the Rosin–Rammler distribution function with a finite number of diameters. The effects of drag, thermal expansion, transient vaporization, and heat transfer by convection and radiation to the droplets and to the chamber wallswereconsidered.Theinfluencesofchamberpressure,equivalenceratio,initialgastemperature,numberofsize classes, Sauter mean diameter, and parameters of the Rosin–Rammler distributions on vaporization lengths of dropletsareanalyzed.Profilesofnormalizeddiameters,temperatures,velocities,heatlosses,vaporizationrates,and other flow properties along the chamber are described. Nomenclature A = area or constant of Antoine’s equation B = constant of Antoine’s equation BM = transfer number C = constant of Antoine’s equation CD = drag coefficient Cp = specific heat at constant pressure D = diameter D32 = Sauter mean diameter F = drag f = mixture ratio h = specific enthalpy hfg = vaporization enthalpy ~ h � = convection coefficient for droplet heating

About mean diameter and size distributions of poly(lactide-co-glycolide) (PLG) microspheres

Despite the importance of microsphere size for controlled drug delivery, little work has been done to quantitatively predict the distribution of microspheres from manufacturing techniques. This work presents a quantitative study that describes the size distribution of poly(lactide-co-glycolide) (PLG) microspheres. A fluid mechanics based correlation for the mean microsphere diameter is formulated based on the theory of emulsification in turbulent flow under non-coalescing conditions. The correlation was constructed and validated with experimentally obtained mean microsphere diameters prepared at different stirring speeds. In addition, a Rosin Rammler distribution function was found to give an accurate representation of the microsphere distribution. The spread of the microsphere size distribution was found to decrease with stirring speed. With the validation of the mathematical correlation, it is now possible to have a good estimate of the average microsphere size prior to microsphere preparation. This is directly relevant to the pharmaceutical industry where microspheres of specified mean diameter and size distribution are desirable.

Continuous Grinding Kinetics of Ethenzamide Particles by Fluidized-Bed Jet-Milling

ABSTRACTContinuous grinding kinetics of Ethenzamide powder, as a model active pharmaceutical ingredient (API) was investigated by fluidized-bed jet-milling. Because the oversize fractions after the classification were well fitted by a modified Rosin-Rammler distribution function, an equation of grade efficiency curve was obtained, which was also characteristic of API. A continuous grinding model was developed on the basis of a batch model by using 1st Kapur function relating grinding rate, the grade efficiency curve, and the overall process flow model consisting of grinding, classification, and mixing zones. The residual ratio obtained was well fitted to the experimental results except for the particle size range smaller than 4 μm and larger than 100 μm. Furthermore, because the volume of the active grinding zone adopted as the fitting parameter was found to be 5 cm3 in all experiments and the value was considered to be appropriate dimensionally, this result supports the reliability of the model.

Conversibilidade entre distribuições probabilísticas usadas em modelos de hidrociclones

Apresenta-se uma análise conceitual sobre o relacionamento entre o parâmetro m, descritor da agudez da distribuição estatística de Rosin-Rammler, e o parâmetro alfa, seu correspondente na distribuição logística de Whiten. A curva de partição de sólidos, segundo uma função de distribuição de Rosin-Rammler, é adotada no modelo de hidrociclone de Plitt, enquanto que, alternativamente, a função de Whiten é adotada para a partição no modelo de Lynch e Rao. Os resultados mostram que os dois parâmetros podem ser relacionados através da seguinte equação de conversão, obtida por análise de regressão não-linear: alfa = 34,555 x exp [-3,9/ m0,5984]

On the diameter and size distributions of Bovine Serum Albumin (BSA)-based microspheres

A fluid mechanics-based correlation for the average size of Bovine Serum Albumin (BSA) microspheres, prepared using a water-in-oil emulsion technique, is presented. The correlation is formulated based on the theory of turbulent dispersion and a non-dimensional Weber number. Measured average diameters of the BSA microspheres prepared in olive oil at different stirring speeds are used to construct the correlation formula. The correlation gives good agreement with experimentally measured average diameters for a wide range of Weber numbers. This correlation is particularly useful to the pharmaceutical industry for predicting the size of encapsulated microspheres used in drug delivery prior to microsphere preparation. The effect of additives on microsphere size was also explored. The average diameter of the BSA microspheres was doubled by the addition of a bioadherent vitelline protein B solution. In addition, a Rosin-Rammler statistical distribution function is shown to accurately represent the microsphere size distribution obtained at different stirring speeds.

Use of a Fluidized Bed Hammer Mill for Size Reduction and Classification: Effects of Process Variables and Starting Materials on the Particle Size Distribution of Milled Lactose Batches

The process capability of a fluidized bed hammer mill was investigated with respect to four process variables, namely, rotational speeds of beater system and classifier wheel, airflow rates and length of grinding zones, as well as the particle size and flow property of the starting materials. The size distributions of all the milled lactose batches could be fitted to the Rosin Rammler distribution (RRD) function. The characteristic particle size (De) and uniform coefficient (n), which were derived from the RRD function, complemented the size at the 99th percentile of the cumulative undersize distribution (D99) to characterize the lactose batches. Lower De and D99 values indicate a finer powder while a higher n value indicates a narrower size distribution.The beater speed played a critical role. Increasing the beater speed from 12000 to 21000 rpm generally resulted in an increase in n and a decrease in D99 values due to the greater amount of milling energy supplied. The particle size and flow property of the starting material also played an important role at beater speed of 12000 rpm, where the lowest amount of milling energy was supplied. When a higher amount of milling energy was provided, the effect of particle size of the starting material was less significant. The other process variables exerted varying effects. Increasing the classifier wheel speed from 5000 to 15000 rpm decreased the De and D99 and increased the n values of the milled lactose batches, provided sufficient milling energy was supplied to the lactose particles. Changing airflow rates from 80 to 90 m3/h generally resulted in larger De and D99 values and lower n values as the higher airflow rate provided greater airflow‐induced kinetic energy that facilitated the passage of lactose through the classifier wheel. However, changing the long grinding zone to a short one did not significantly affect the De, D99 and n values of the milled lactose batches produced. Small lactose particles of narrow size distribution could be obtained using the fluidized bed hammer mill upon gaining a better understanding of the milling process.

Size Distribution Function of Droplet in Liquid-Liquid Mixing and Bubble in Gas-Liquid Mixing, Crystal in Crystallization and Crushed Product in Crushing.

The size distribution function of particles that present as droplets in liquid-liquid mixing, bubbles in gas-liquid mixing, crystals in crystallization, and crushed materials in crushing, is examined based on consideration of the balance of the external and the internal force/energy of the particle, and from the viewpoint of information entropy. Then, a new size distribution function (PSF), which is the product of the original size distribution function and the realizable probability function of the size of particles, is presented. The same formula style as that of newly presented PSF can be derived also by balancing the external energy calculated from the energy spectrum density distribution function (ESF) for the turbulent flow field and the internal force/energy of particles. Additionally, it is clarified that the size distribution of eddies in the turbulent flow can be deduced from the same viewpoint by considering that a large eddy is segmented by external force/energy and the style of the distribution is isomorphic with that of ESF. In addition, the usefulness of newly presented PSF is clarified by fitting it to the data generated by the traditional three common unimodal PSF; Rosin-Rammler, Normal and Log-Normal size distribution functions. The main advantages of the newly presented PSF are that it is defined based on a clear physical background and that, for all practical purposes, it can replace the three common PSF. Therefore, this newly presented PSF provides a means of presenting and comparing the different size distribution patterns in terms of a single mathematical expression.

Effect of GGBFS Particle-Size Distribution on Mortar Compressive Strength

This paper presents the results of an investigation into the influence of particle-size distribution (PSD) of ground granulated blast-furnace slag (GGBFS) on the compressive strength of slag mortar. The PSD of the GGBFS powder was defined by the parameters of the Rosin-Rammler distribution function, namely, position parameter, \ix\d0, and slope \in. These parameters describe the fineness and the size range characteristics of the PSD, respectively. Samples of GGBFS with different PSDs were prepared using a method by which a commercially available GGBFS sample was first separated and then recombined into predetermined proportions. The results showed that varying the PSD of the GGBFS by keeping \in constant and increasing \ix\d0 decreased the compressive strengths of mortar mixes. Varying \in at a constant \ix\d0 was found to produce only a small change in compressive strength. However, at 70% GGBFS replacement, and for both water/cement ratios considered, the mortar mix containing the GGBFS sample with the widest range of distribution showed the highest strength at 28 days of age.

Particle size distribution of GGBS and bleeding characteristics of slag cement mortars

This paper reports an investigation into the influence of particle size distribution (PSD) of ground granulated blast furnace slag (GGBS) on the bleeding characteristics of slag cement mortars. Samples of slag powder with different PSDs (as represented by the parameters of the Rosin-Rammler distribution function, namely, position parameter, x o, and slope, n, which describe the fineness and the size range of the PSD, respectively) were prepared using a method whereby a commercially available slag sample was first separated into different size fractions and later recombined in some pre-determined proportions. Bleeding rate and bleeding capacity of mortar mixes containing 0, 30, and 70% slag were investigated. The results showed that when varying the PSD of slag by altering n at constant x 0, no specific relationship between bleeding characteristics and psd could be established. However, the highest values of the bleeding rate and capacity were exhibited by the mixes in which the slag with the intermediate value of slope (i.e., having the medium range of PSD) was included. Furthermore, the results showed that for the slag samples with similar size range distribution (i.e., having a constant slope, n), the bleeding capacity increased with increases in x o, except the 30% slag mixes, which were made to 0.35 w/c.

A new dependent model for particle sizing with light extinction

The “dependent model” is one of the widely used algorithms in light extinction particle sizing. But how to determine the upper and lower limits of the integral to match the size range of measured particles is still a problem with this algorithm. By analysing the Rosin-Rammler (R-R) distribution function, the authors find that there is a relationship between integral range and the distribution parameter κ of the R-R function. The integral range depends entirely on the value of κ. The integral ranges corresponding to κ are presented in this paper. Based on these data, a self-adaptive program was written to process the measured data. Numerical simulation and experiments were performed to test this algorithm, and the results are presented in this paper.

MULTIPLE SCATTERING AND SIZE DISTRIBUTION EFFECTS ON THE PERFORMANCE OF A LASER DIFFRACTION PARTICLE SIZER

The effect of multiple scattering on the performance of the Malvern particle sizer was studied experimentally. Two particle samples with the same volume median diameter but with different size distributions were used in the experiments. One sample could be described by a log-normal distribution function, the other by a Rosin-Rammler distribution function. The particle samples were specifically designed to simulate drop size distributions in liquid sprays. The model-independent, log-normal, and Rosin-Rammler models were used in the data analysis, and the results were compared. It was found that even when the sample concentration was low and multiple scattering did not occur, the results obtained with the log-normal and Rosin-Rammler models could be misleading, particularly in the high end of the distribution (large particles). When the log-normal test sample was used, the data analysis with the model-independent and log-normal models gave good results, but the data analysis with the Rosin-Rammler model seriously underestimated the fraction of large particles. When the Rosin-Rammler test sample was used, the data analysis with the model-independent and Rosin-Rammler models gave good results, but the data analysis with the log-normal model seriously overestimated the fraction of large particles. When the sample concentration is high and multiple scattering occurs, data analysis with the model-independent model is not feasible. Correction equations are available in literature for data analysis with the log-normal and Rosin-Rammler models. It was found that when the correction equations were used, both the log-normal and Rosin-Rammler models provided the mean particle size reasonably well. However, there were considerable difficulties in the high end of the distribution. This was not surprising, since the experiments with a low particle concentration had already shown that the data analysis with the log-normal and Rosin-Rammler models can produce misleading results.

Description of normal, log—normal and Rosin—Rammler particle populations by a modified version of the beta distribution function

Data generated by the normal, log—normal and Rosin—Rammler distribution functions were normalized and fitted with a slightly modified version of the beta distribution function. As long as the frequency function had a zero or practically zero value at the two ends of a finite size range, the fitted curves were, for all practical purposes, indistinguishable from the normal and Rosin—Rammler distributions. The fit of the modified beta function to narrow log—normal distributions was also excellent but it declined significantly as the distribution spread increased. It appears, though, that for real particle populations, having a finite size range, and not necessarily a perfectly smooth size distribution, the modified beta function can replace all three functions, thus providing a way to present and compare the different size distribution patterns in terms of a single mathematical expression.

粉体のランダム充てんモデルによる充てん密度と粒度分布との関係

A previous study reported on the relationship between the packing density and the particle size ratio and/or the volume fraction in binary mixture as a part of the investigation for the effect of the particle size distribution on the packing density. In the present study a similar investigation was repeated for the particles which have continuous size distributions, the modified Rosin-Rammler ones. According to the computer simulation using a random packing model the packing density was related to the parameter m (distribution constant) of the modified Rosin-Rammler distribution function. Subsequently the relationship was introduced by using an analytical method based on the binary packing results, and the adequency of the analytical method was also confirmed. The study was further developed onto the packing density of the particle assembly composed of a mixture of two powders with different modified Rosin-Rammler size distributions to investigate the blending effects.The results are summarized as follows:(1) For the powder of modified Rosin-Rammler size distribution the packing density decreases with increase in m.(2) The packing densities of powders with any continuous size distributions can be obtained from the loose binary packing results by an improved analytical method.(3) For the loosely packed bed of two mixed powders with modified Rosin-Rammler size distributions, the packing density was calculated by the above analytical method. The densities change with the parameters, m1 and m2 (distribution constants), De2⁄De1 (ratio of absolute size constants) and X2 (volume fraction). These results give important information on practical use.

Distribution of particle sizes in food comminuted by human mastication

The masticatory performance of an individual may be quantified by describing the size distribution of particles of a comminuted test food as a function of the number of chewing strokes. A standardized sieving method and a standardized artificial test food (Optosil®) were used to obtain reproducible results. Some measurements were performed using peanuts. The distribution of particle sizes of the comminuted food was adequately described by a Rosin-Rammler distribution function for the seven participating individuals which characterizes the size distribution by the median particle size ( x 50) and the broadness of the distribution ( b). The broadness variable b hardly depends on the number of chewing strokes, and the differences in b between the individuals are small. Considerable differences in the median particle sizes at a given number of chewing strokes were observed between the subjects, reflecting differences in efficiency of comminution of a test food. For all subjects, the median particle size decreased as a function of the number of chewing strokes N according to the relation, x 50 = c × N − d . Variables c and d characterized the efficiency of comminution by the subjects.

The influence of electrostatic fields on the centrifugal atomisation of liquids

Using water as the working fluid, the atomisation process of the spinning disc atomiser in conjunction with an electrostatic field is investigated. An electric field can be set up at the disc edge by directly applying high voltage to the disc. The influence of the electric potential on the droplet diameter is observed to reduce mean droplet size under certain circumstances. A Malvern particle size analyser is used to monitor the spray droplet size distribution in real time. A Rosin Rammler distribution function provides a reasonably good fit of the droplet size distribution produced by the atomiser with and without applied potential. In the presence of the electric field the width of the distribution narrows down as the applied voltage increases.

浮選スラリー用オンストリーム粒度測定装置

An On-stream size measuring equipment has been developed for flotation pulp. It uses both vibrational output and light absorption. Sample flows into the vibration detection cell and next into the light absorption cell.The vibration pickup detects very weak signals that are produced by collisions of particles and the cell. Effective signals are separated through an acoustic filter from noises.The cross section area of light absorption cell is 1.3 cm2 and light pass length is 3 mm. Suitable pulp density for this cell is about 4 wt%.Several kinds of closed circuit tests have been done using SiO2 and complex sulphide ore particles. From these tests the following results were obtained.1) When n value of Rosin-Rammler distribution function of samples remained nearly constant, it is possible for this equipment to measure the size distributed samples as well as size sorted samples2) When the variation of pulp density is limited in narrow range, an on-stream size measurement is attainable by the vibrational output only.At the pilot plant tests, -200 mesh% of flotation pulp were measured by this equipment. The results of on-stream measured size, compared with screen analysis, were as follows.Number of samples 109-200mesh%(mean value) 73.91%Mean square root error 2.03%Correlation coefficient 0.923