Experimental studies and population balance equation models for breakage prediction of emulsion drop size distributions

A population balance equation (PBE) model for pure drop breakage processes was developed from homogenization experiments and used to investigate model extensibility over a range of emulsion formulation and homogenizer operating variables. Adjustable parameters in the mechanistic breakage functions were estimated from measured drop volume distributions by constrained nonlinear least-squares optimization. Satisfactory prediction of measured bimodal distributions was achieved by the incorporation of two different breakage functions that accounted for large drop breakage due to turbulent shear and for small drop breakage due to collisions between drops and turbulent eddies. Model extensibility to different emulsion compositions and homogenizer pressures was investigated by comparing model predictions generated with the base case parameters to drop volume distributions measured under different conditions. The PBE model satisfactorily accounted for changes in the dispersed phase volume fraction and the interfacial tension with the base case parameters. By contrast, significantly improved predictions for the continuous phase viscosity or multiple formulation variables were obtained through re-estimation of the model parameters using multiple data sets in which the associated variables were systematically varied. The model was not able to satisfactorily predict drop volume distributions resulting from homogenizer pressure changes, perhaps due to the assumption of a constant pressure throughout the homogenizer. We conclude that PBE models of drop breakage can be used to reasonably predict the effects of emulsion formulation variables on drop volume distributions and have the potential for guiding experimental efforts aimed at the design of novel emulsified products.     

Self-similar inverse population balance modeling for turbulently prepared batch emulsions: Sensitivity to measurement errors

We investigate the sensitivity of an inverse population balance equation (PBE) modeling technique for extracting single particle functions from transient size distribution measurements. A dynamic PBE model of a turbulently agitated batch emulsification vessel is used to generate volume size distribution data under the assumption of negligible drop coalescence. The distribution data are subjected to various types of error consistent with available measurement technologies and then introduced as input data to the inverse PBE modeling algorithm, which includes validation of the self-similar assumption. The errors considered include measurement noise, data skewed towards smaller or larger drops, skewed data due to the presence of large dust peaks, and reduced resolution caused by data binning. For each case, the computed functions for the drop breakage rate and the distribution of daughter drops are compared to the actual functions to assess the impact of input data errors on the effectiveness of the inverse PBE modeling approach. The type of measurement errors considered generally lead to underprediction of the breakage rate and, consequently, to overprediction of the number of large drops. Because the estimated and actual breakage rates tend to converge at small drop sizes, the inverse algorithm generates accurate predictions of the drop size distribution at sufficiently long batch times when small drops dominate. Implications for our future work on PBE modeling of drop size distributions in pharmaceutical emulsions prepared with high pressure homogenization are discussed.     

振动筛板槽中液滴分散动力学的研究(Ⅰ)实验研究及理论分析

本文通过实验观察发现,液滴的破碎只有在液滴与振动筛板孔口发生碰撞时才发生.基于这个实验现象,建立了振动筛板槽内液滴破碎的新模型.把筛板孔口附近的剪应力作为破碎力,破碎速率可表示为:G(d)=C_12Af/H(?)~n_4(d/d_h)~n_5[1-(d_(cr)/d)~(2n_1+1)]~0.5n(d)液滴的凝聚可以按气体分子碰撞过程来处理.凝聚速率可以用下式表示:ω(d_1,d_2)=C_Ⅱ(d_1+d_2)~(7/3)∈~（1/3)[β_d∈~(2/3)d_1+d_2/σ(d_1+d_2）~（1/3）]~n_6n(d_1)n(d_2)     

Drop break-up in intermittent turbulence: Maximum stable and transient sizes of drops

A multifractal method describing the fine-scale structure of turbulence, including its intermittency, is applied to derive the drop breakage functions for drops whose diameter falls within the inertial subrange of turbulence, including effects of the viscosity of the dispersed phase. The model predicts well the transient drop size distributions of dispersions undergoing breakage at long stirring times. Concepts of quasi-stable and asymptotically stable drop sizes are presented and discussed in relation to the exponent on the Weber number. The functions for drop break-up in the viscous subrange are proposed and discussed.     

Drop size distribution and holdup in a rotating impeller extraction column

A stagewise hydrodynamic model, applying drop population balance equations derived from models for breakage and coalescence of drops in a countercurrent liquid-liquid extraction system, was developed to predict the drop size distribution and the holdup of the dispersed phase in a rotating impeller extraction column. The drop size distributions were obtained by taking the photographs of the dispersions at the same locations through the rectangular shaped glass box filled with distilled water. The experimental variables were the impeller speed and flow rates of the continuous and dispersed phases. The solutions of the model equations were obtained by performing the computer simulation and the optimum parameter values were determined. The results predicted by the model were in good agreement with the experimental results obtained from the present rotating impeller extraction column.     

Modelling of high viscosity oil drop breakage process in intermittent turbulence

A multifractal model of the fine-scale structure of turbulence is applied to describe breakage of viscous drops of immiscible liquid immersed in a fully developed turbulent flow. A population of drops whose diameter falls within the inertial subrange of turbulence is considered here. The population balance equation is used to predict the drop size distributions. Calculations are performed for binary and multiple breakage. Several daughter distribution functions are applied and the results of their application are compared with experimental data. Experimental investigations of drop breakup were carried out in a flat bottom stirred tank having the diameter of and equipped with Rushton type agitator and four baffles. Silicone oils with viscosity of 10, 100, 500 and 1000 m Pa s were dispersed in the aqueous continuous phase. Measurements were performed using high resolution digital camera. Experimental results as well as numerical simulations show that after the initial period of multiple breakage, the strongly asymmetric type of binary breakage dominates.     

Description of interaction processes in agitated liquid-liquid dispersions

Phenomenological models are proposed to describe drop breakup and coalescence in a turbulently agitated liquid-liquid dispersion. Based on these models, breakage and coalescence rate functions are developed and used to solve the general population balance equation describing drop interactions in a continuous flow vessel. Parameters of the models are evaluated by comparison with experimental data on drop size distributions and mixing frequencies obtained in a continuous flow vessel over a range of operating conditions. The favorable agreement between experimental observation and the model are encouraging that the model is suitable for predicting dispersion properties such as drop size distributions, interfacial areas and mixing frequencies.     

An analysis of ball-and-race milling part II. The babcock E 1.7 mill

A mathematical model was developed for a ball-and-race mill based on specific rates of breakage and primary fragment distributions. The model includes internal classification of particles falling back into the race and external classification due to the built-on classifier. It was demonstrated that the normalized primary fragment distribution produced in a pilot-scale Babcock E-type mill of 17 in. race diameter was the same as in the Hardgrove laboratory test mill and that the specific rates of breakage varied with particle size in the same manner. Steady-state continuous tests on the pilot-scale mill showed that breakage rates depended on the rate of feed, since the mill pulled less power at low feed rates. This effect plus the residence time effect gave coarser product size distributions at low and high feed rates than at a medium feed rate. Model simulations based on parameters measured in the Hardgrove mill correctly predicted the product size distribution from the E-type mill.     

The Interaction of solute transfer,contaminants and dro break-up in rotating disc contactors: Part II. The coupling of the mass transfer and breakage processes via interfacial tension

In order to investigate the influence of solute transfer and of surface active agents on the drop breakage process in liquih liquid extraction columns, their effect on the interfacial tension has to be studied in detail. The difficulty encountered is that the interfacial tension during solute transfer continuously changes and that no simple apparatus is commercially available which can measure these varying interfacial tension values. An attempt has been made here to theoretically predict them. The equations developed to predict the interfacial tension variation can be combined with a model for the breakage process and hence drop size distributions can be calculated from stage to stage. Applying a new combined film mass transfer coefficient model which takes into account the effect of contaminants, single drop extraction performance has been calculated for simplified conditions of constant bulk concentration in the continuous phase. Calculated efficiencies have been compared with experimental data and a good simulation of contaminant effects and dependency on drop size has been found. The calculations were restricted to low dispersed phase hold-up values, so that coalescence effects could be ignored. This work provides the required support for a procedure to be applied to counter-current flow extraction columns.     

MAXIMIZING SELECTIVITY OF LIQUID-LIQUID REACTION SYSTEMS. CONTROL OF THE DISPERSION PROCESS

Currently available information on droplet coalescence and break-up rates in turbulent flows in mixing vessels can be used to control drop sizes in dispersed phase equipment. The effect of drop size distributions on the selectivity and productivity in multi-reaction systems is examined in this paper.

The reaction system features the primary desired product (C) as resulting from reaction (in the bulk phase) between a reactant (A) in the drop phase and a second reactant (B) in the bulk phase. An adverse reaction is also envisaged which consumes (C) by further reaction with (B) to form a waste product. While small drops promote conversion because of large interfacial area, larger drops promote selectivity because of the facility of the product to re-enter the drop phase avoiding further reaction (to form waste) in the bulk phase. The effect of the bivariate distribution of drop size and reactant (A) concentration in the feed to a continuous stirred tank reactor on the selectivity and productivity of (C) is investigated within the framework of film theory while neglecting drop dynamics such as coalescence and break-up.

The results show the selectivity can be substantially improved by controlling drop size and distribution of the reactants among the differently sized droplets. Contrary to conventional wisdom which emphasizes creation of interfacial area by promoting very small droplets, it emerges that optimal distributions of drop size and reactant concentration which maximize productivity of the desired product exist. The practical implications are discussed.     

Dispersion of high-viscosity liquid-liquid systems by flow through SMX static mixer elements

The pressure drop and the dispersed phase drop size distribution have been measured for flow through SMX static mixer elements, in columns of diameter 41.18 and 15.75 mm, for a continuous phase of aqueous corn syrup and a dispersed phase of silicone oil. For single-phase flow the pressure drops were consistent with known literature correlations. In the presence of the dispersed phase the pressure drops were increased about 20% above the expected single-phase values, showing more short-term fluctuations but with no significant effect of the flow fraction of the dispersed phase. Droplet size distributions were measured by the computer-aided analysis of images from a digital camera. For shorter lengths of packing the distributions showed a significant “tail” at the large-diameter end, but as the packing length was increased the tail decreased or became non-existent. The mean drop sizes have been compared with a new model based on drop formation at equivalent point sources within the packing.     

DYNAMIC BEHAVIOUR OF LIQUID DISPERSION IN A VIBRATING PLATE TANK

In this paper a new model of the drop breakage in a vibrating plate tank has been developed by a me-chanism based on the shearing stress in the neighborhood of the holes in the plates as a breakage force,andthe coalescence of drops may be dealt with by the process of gas molecule collision.Based on the theoreticalanalyses,simulation of drop breakage and coalescence which are random processes has been carried out byMonte-Carlo simulation technique.The parameters of breakage and coalescence rate have been obtained.     

The interaction of solute transfer,contaminants and drop break-up in rotating disc contactors: Part I. Correlation of drop breakage probabilities

The breakage process of single drops in RDC liquid-liquid extraction columns has been investigated. The breakage probability and daughter drop size distribution were the measured characteristics. Binary systems, non-equilibrated ternary systems with mass transfer in both directions (c → d and d → c) and systems with surface active agent added were used in the experiments. A model of the breakage probability was developed based on a modified Weber number, taking into account the applied shearing stress and the resisting interfacial tension force. It is shown that breakage probability can be estimated if interfacial tension is known as a function of interfacial conditions.     

A model for continuous grinding in a laboratory hammer mill

The equations of fully mixed, steady continuous grinding with rapid ideal classification through a discharge screen have been modified to allow for non-rapid removal of material less than the screen size. Values of specific rates of breakage, daughter fragment distributions and the variation of specific rates with mill hold-up determined in prior batch tests on a hammer mill were used in the equations, to predict hold-up versus feed rate and product size distribution versus feed rate. The predictions were in reasonable agreement with the experimental values from continuous grinding. There is a maximum production rate which can obtained, and energy utilization is inefficient at low production rates.     

Creeping flow mass transfer in a cloud of size distributed round drops at high Schmidt numbers

For large Schmidt numbers, analytical expressions of the continuous phase mass transfer coefficient for an active test drop or bubble located in a random fixed array of inactive falling drops or bubbles having an arbitrary size distribution have been obtained for the two limiting cases of rapidly circulating drops and almost solid-like drops in slow creeping motion. The theoretical basis for using a statistically expected velocity field around the test drop in the random cloud to calculate the statistically expected concentration field has been examined. Replacing the size distributed cloud of inactive drops by a uniform cloud of drops of size equal to the Sauter mean radius $\text{b}$₃₂ of the size distributed system causes a minor increase in the predicted mass transfer coefficient for the active test drop unless it is much larger than $\text{b}$₃₂. Theoretical calculations show a similar behavior if one deals with an active cloud and utilizes mass transfer coefficient expressions from an inactive cloud. The conclusion of Tan, Prasher and Guin that mass transfer coefficient in non-uniform active packings obtained experimentally is adequately described by correlations for uniform packings if $\text{b}$₃₂ is used is supported by the theoretical calculations for the size distributions used in the experiments of Tan et al. The predicted mass transfer coefficient in an inactive uniform cloud are somewhat lower than those predicted by the Waslo—Gal-Or model. There is reasonable agreement between the predicted values for a uniform inactive cloud of drops and available experimental data on active clouds of drops and bubbles.     

油水两相分散流液滴粒径预测模型

油水两相分散流的液滴粒径及其分布在很大程度上影响管路压降等流动参数,研究液滴粒径预测模型对揭示油水两相流的流动特性具有重要意义。通过研究湍流脉动动能与乳状液的界面能之间的平衡、管流径向速度脉动与摩擦速度之间的关系以及泵的剪切作用,建立了油水两相管流中分散相液滴粒径预测模型;在水平管道上对油水两相分散流的液滴特性进行了实验研究,采用高速摄像和显微镜拍摄获得液滴数据,探索含油率、流量和温度等因素对粒径的影响。预测模型计算结果与不同流量、温度和含油率条件下的实验数据吻合较好。根据预测模型计算了有泵和无泵情况下分散流液滴粒径,发现泵的剪切和扰动作用使得分散液滴具有更小的粒径,泵对液滴粒径及其分布起到了显著作用。     

A stochastic model for breakage frequency of viscous drops in turbulent dispersions

In industrial liquid-liquid mass-transfer equipment, many a times the dispersed phases involved are highly viscous. The viscosity of dispersed drops influences the rate processes, especially their breakage rate. A new stochastic model for predicting the breakage frequency of viscous drops in a turbulent dispersion applying the random behavior of the turbulent fluctuations, has been proposed. It has been assumed that the correlation time of turbulent fluctuations across the viscous drops is so small compared to the time scale of drop deformation, that the turbulent fluctuation can be considered as a white-noise process.     

Studies of drop break-up in liquid-liquid systems in a rotating disc contactor. Part II: Effects of mass transfer and scale-up

The number fraction of drops of a given size which break up at rotor level in a rotating disc contactor has been observed during mass transfer in either direction to or from solvent or aqueous drops. Critical rotor speeds for a given drop size undergoing mass transfer can be used to find an effective interfacial tension. Using this interfacial tension value, the break-up fractions are correlated within experimental uncertainties in the same manner as for no mass transfer. Drop break-up fractions depend on column size and relevant empirical correlations of the data are presented. The results may be used to estimate the effect of mass transfer on drop size distributions in an RDC.     

Development of a multi-scale simulation method for design of novel multiphase reactors

In this paper a multi-scale simulation method for modelling dispersions in a novel multiphase reactor is presented. This novel reactor is a continuous reactor which consists of repeated identical small mixing elements. The reactor is excellent for studying the effect of turbulence on drop size distributions since turbulence is continuously produced and dissipated along the reactor. Furthermore the energy dissipation within each element is very homogeneous. In addition it allows optical access at all positions along the reactor.Simulations were performed for a wide range of turbulence intensities for different dispersed phase hold-up. Each simulation was validated with measurements of the size distribution along the reactor. Good quantitative agreement was obtained at low hold-up in terms of prediction of the breakage rates and prediction of the size distributions. At higher hold-up the model gave reasonable predictions at low turbulence intensity however too large drops were predicted at high turbulence intensity. This can be a result of turbulence modulation and shows that reliable turbulence models for multiphase flows are necessary in this simulation method. The results show that physical models describing breakup and coalescence combined with CFD provide a good tool for efficient development and optimisation of novel multiphase reactors.     

Breakage and coalescence of drops in a batch stirred vessel—I Comparison of continuous and discrete models

Conditions of analogy between a continuous and a discrete models describing the processes of breakage and coalescence of drops in a batch stirred tank are defined. The discrete approach may simplify the numerical solution of the problem but an error is introduced into the resulting drop size distribution. Thus the question arises of minimizing this error for a selected number of drop size classes and of estimating its magnitude. The error is a function of the number of classes as well as of the width of the class intervals. An optimum width can be approximated using the principle of equal residuals. The residuals—parts of the area under the continuous distribution curve uncovered by the discrete distribution—can also be used to find the upper and lower limits of the error estimate.