A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions

A model for transitional breakage probability of droplets in agitated lean fiquid-liquid dispersions is proposed based on the mechanism of breakage of droplets due to their oscillations resulting from relative velocity fluctuations. A universal transitional breakage probability in terms of non-dimensionalized drop diameter is derived for all dispersed phases whose density and viscosity are almost the same as that of continuous phase. The maximum stable drop diameter d_s derived from the model, shows a dependence of N_We^?0.6. It is shown that a “power law” approximation Kvⁿ is valid for transitional breakage probability for d/d_s up to 2. The exponent 2.67, predicted by this model corresponds rather well with an estimate of 2, obtained from experimental observations. A functional relation for the rate constant K in terms of the parameters and physical properties of the system is derived. A universal non-dimensionalized equilibrium drop-size distribution for agitated lean liquid-liquid dispersions is derived by analytical solution of a population balance equation simplified by order of magnitude estimates. Interestingly enough, this analytical solution is the same as the Gaussian distribution suggested empirically by Chen and Middleman.     

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.     

Dynamic simulation of agitated liquid—liquid dispersions—II. Experimental determination of breakage and coalescence rates in a stirred tank

Using the theoretical procedure outlined in the first part of this work, breakage and coalescence rates were determined experimentally in a stirred tank. After reaching steady-state conditions, the intensity of agitation was suddenly changed and the variation in drop size distribution with time was monitored. The coalescence and breakage constants were evaluated by optimising the fit of the experimental results with the theoretical solution of the model equations. No a priori assumptions concerning the dependence of the interaction rates on drop size, system properties and operating conditions were made. Precise techniques for measuring the drop size distribution in turbulent dispersions were developed and tested. Empirical equations for dependence of breakage and coalescence constants on drop volume, holdup and system properties were derived.     

Drop breakage and coalescence in the toluene/water dispersions with dissolved surface active polymers PVA 88% and 98%

The influence of two PVAs of the same molecular weight (13,000–23,000) and different degree of hydrolysis equal to 88% and 98% on breakage and coalescence of toluene droplets in the liquid/liquid dispersion were considered for PVA concentration range 0.001–0.01%. Analysis of experimental results shows that drop coalescence is significant only in the system containing 0.001% PVA. Drop coalescence for polymer concentration range 0.002–0.01% is strongly limited. Therefore, population balance was solved using breakage and coalescence expressions with assumed partially mobile drop interfaces for the lowest PVA concentration. For c ≥ 0.002% drop size distributions were properly predicted using only breakage model. Multifractal formalism was used to take into account intermittent character of turbulence and explain drop behavior. Larger drop size reduction by PVA of lower degree of hydrolysis observed experimentally was confirmed by the model.     

湍流分散体系中液滴破碎频率模型的黏性修正

在分析研究分散相黏度对液滴变形和破碎影响的基础上,提出了一个改进的液滴破碎频率模型并拓展了液滴破碎判据标准.同时通过Monte Carlo模拟的随机方法,得到了湍流搅拌槽中液-液分散体系的液滴直径分布和Sauter平均直径d₃₂.通过与文献中关于d₃₂的实验结果比较发现,该模型预测的Sauter平均直径更接近实验值,对于黏性分散相改进的液滴破碎频率模型要优于Coulaloglou和Tavlarides提出的模型.计算结果表明对于黏性分散相液滴,其黏度限制了液滴变形,使得液滴破碎频率被大大减少, 液滴直径明显增加,液滴直径分布向右偏移.     

Drop break-up in turbulent pipe flow downstream of a restriction

This work addresses the drop fragmentation process induced by a cross-sectional restriction in a pipe. An experimental device of an upward co-current oil-in-water dispersed flow (viscosity ratio λ≈0.5) in a vertical column equipped with a concentric orifice has been designed. Drop break-up downstream of the restriction has been studied using a high-speed trajectography. The first objective of this work deals with a global analysis of the fragmentation process for a dilute dispersion. In this context, the operating parameters of the study are the orifice restriction ratio β, the flow Reynolds number, Re and the interfacial tension, σ. The break-up domain has been first mapped on a β(Re) graph and drop size distributions have been measured for different flow Reynolds numbers. It was observed that the mean drop diameter downstream of the restriction linearly increases as a function of the inverse of the square root of the pressure drop. This behaviour is in agreement with the observations previously made by Percy and Sleicher [A.I.Ch.E. Journal, 1983, 29(1), 161-164]. In addition, experiments based on the observation of single drop break-up downstream of the orifice have allowed the identification of different break-up mechanisms, and the determination of statistical quantities such as the break-up probability, the mean number of fragments and the daughter drop distribution. The drop break-up probability was found to be a monotonous increasing function of the Weber number based on the maximal pressure drop through the orifice. The mean number of fragments is also an increasing function of the Weber number and the reduced mean daughter drop diameter decreases as the Weber number increases. The daughter drop distributions are multimodal at low and moderate Weber numbers as a result of asymmetrical fragmentation processes. The statistical analysis of single drop break-up experiments was implemented in a simple global population balance model in order to predict the evolution of the size distribution across the restriction at different Reynolds numbers, in the limit of dilute dispersions.     

A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow

Several models for the daughter bubble/droplet size distribution are reviewed and a detailed discussion is given to get a better understanding of the daughter size distribution. A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow, based on an eddy-bubble/droplet collision method, is developed. It takes into account the energy distribution of turbulent eddies, effect of capillary pressure and surface energy increase during bubble/droplet breakup. An increase in the mother bubble/droplet size and energy dissipation rate increases the probability of unequal breakup. The model prediction is in good agreement with experimental results and the underlying physical situation.     

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)     

Breakage probability of irregularly shaped particles

The investigation of breakage probability by compression of single particles was carried out. The spherical glass particles and irregularly shaped particles of NaCl, sugar, basalt and marble were subjected to a breakage test. The breakage test includes the compression up to breakage of 100 particles to obtain the distribution of the breakage probability depending on the breakage force or compression work. The breakage test was conducted for five particle size fractions from each individual material, at two stressing rates. Thus obtained 50 breakage force distributions and corresponding 50 breakage work distributions were fitted with log-normal distribution function.Usually, the breakage probability distribution can be found by means of stress or energy approach. The first one uses the stress to calculate the breakage probability distribution. The second approach uses the mass-related work done to break the particle. We prefer to use the breakage force and energy as essential variables. The correlation between the force and energy at their breakage points is obtained by integrating the characteristic force–displacement curve, i.e. the constitutive function of elastic–plastic mechanical behavior of the particle. The irregularly shaped particle is approximated by comparatively “large” hemispherical asperities. In terms of elastic–plastic deformation of the contacting asperities with the plate, a transition from elastic to inelastic deformation behavior was considered. Thus, one may apply the model of soft contact behavior of comparatively stiff hemispheres. Based on this model a relationship between the breakage force distributions and corresponding energy distributions was analyzed. Every tested material exhibits a linear relationship between average breakage energy and average breakage force calculated for every size fraction.For future consideration both force and energy distributions were normalized by division by average force or energy, consequently. The relationship between the fit parameters of normalized energy distribution and corresponding fit parameters of normalized force distribution was established. The mean value and standard deviation of normalized force distribution can be found from mean value and standard deviation of normalized energy distribution by means of system of two linear equations. The coefficients of those linear equations remain the same for all of the above tested materials; particle size fractions and stressing rates. As a result the simple transformation algorithm of distributions is developed. According to this algorithm the force distribution can be transformed into energy distribution and vice versa.     

Crushing characteristics

A method of measuring the basic characteristics of comminution was developed. These characteristics are expressed by the major comminution functions: crushing probability function, energy function and breakage function. The crushing probability function is the strength distribution of particles of a given size. The energy function is the strength of the particles as a function of their sizes. And finally, the breakage function is the size distribution of the crushed material. The functions are defined mathematically. Several natural minerals were tested by drop tests in order to determine their individual comminution functions. From the tests, several crushing properties of the particulate materials can be derived. The comminution functions given in this paper would be the basic elements in developing mathematical models for various crushing and grinding processes.     

Break-up of a drop in a stirred tank

The drop break-up mechanism was studied in a stirred tank containing two immiscible liquids. The daughter drops formed by break-up of a single drop of known size were recorded photographically. From the experiments at constant agitator speed the following results were obtained. There is a critical drop size under which drops do not break up under given conditions. The break-up frequency increases approximately linearly with increase in drop volume. The number of daughter drops, v, is a random variable with a mean v > 2 which increases with the volume of the mother drop. The relative volume of a daughter drop has a β-distribution.     

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.     

Identical Feed Drops in a Flow Vessel May not Always Be Desirable

In this paper, the effect of feed size and daughter drop size distributions on the steady‐state drop size distribution in a continuous flow vessel is discussed. For identical sizes of feed drops, a new criterion has been proposed to ensure the complete breakage of the feed drops giving rise to smooth continuous drop size distributions. This has been predicted on the basis of two competing time scales, namely, breakage and residence times.     

液液静电雾化特性

对水静电雾化弥散于玉米油的液-液雾化过程进行了实验研究。通过拍照对雾滴形态进行了观察。观察表明,不同电压下液-液雾化会呈现出滴状和云状雾化两种较为典型的雾化形态,在两种形态下液滴具有不同的形貌和运动特点,本文给出了两种雾化形态的出现条件及特征描述。通过Winner99颗粒图像分析仪及雾滴尺寸的分布理论,对不同静电电压下雾滴直径的分布规律进行了定量分析。研究结果表明,液-液静电雾化中雾滴的直径分布服从Rosin-Rammler分布规律。随着电压的升高,雾滴直径分布趋向均匀,雾化细度得到改善。与在空气中雾化有所不同,液-液雾化中雾滴分布的概率密度曲线峰值两边呈现出显著的不对称性,小液滴数尺寸分布较窄而大雾滴数的尺寸分布较宽。随着电压的升高,大雾滴尺寸分布有所变窄,概率密度曲线趋近对称。     

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.     

Hydrate agglomeration modeling and pipeline hydrate slurry flow behavior simulation

Dynamic modeling and numerical simulation of hydrate slurry flow behavior are of great importance to offshore hydrate management.For this purpose, a dynamic model of hydrate agglomeration was proposed in this paper.Based on population balance equation, the frame of the dynamic model was established first, which took both hydrate agglomeration and hydrate breakage into consideration.Then, the calculating methods of four key parameters involved in the dynamic model were given according to hydrate agglomeration dynamics.The four key parameters are collision frequency, agglomeration efficiency, breakage frequency and the size distribution of sub particles resulting from particle breakage.After the whole dynamic model was built, it was combined with several traditional solid–liquid flow models and then together solved by the CFD software FLUENT 14.5.Finally, using this method, the influences of flow rate and hydrate volume fraction on hydrate particle size distribution, hydrate volume concentration distribution and pipeline pressure drop were simulated and analyzed.     

Modelling of droplet breakage probabilities in an oscillatory baffled reactor

The breakage of droplets dispersed in a continuous aqueous phase determines the performance of many mixing devices and reactors that rely on effective contact between two phases, e.g. emulsion mills, liquid-liquid extraction columns, stirred tank reactors and Oscillatory Baffled Reactors. Quantitative knowledge of the mechanisms involved in the breakage provides parameters for design and prediction. In the work presented here, oil was dispersed in water in a continuous OBR, and a High Speed Camera was used to record the events of breakage of individual oil droplets and probabilities of breakage were estimated. It was confirmed that breakage was more sensitive to changes in the amplitude of oscillation than in the frequency of oscillation. A novel integral model was developed based on an analysis of the total work effected on the deforming droplet in order to interpret the results. The quantitative results from direct observation were compared to the model predictions. The model with fitted parameters was finally extrapolated to smaller diameters, in an attempt to predict the critical drop diameter for breakage.     

Determination of breakage rates of oil droplets in a continuous oscillatory baffled tube

Following on our previous studies, the population balance model that was built on the earlier work from Jareš and Procházka [Break-up of droplets in Karr reciprocating plate extraction column. Chemical Engineering Science 42, 283-292] was modified to include the viscoelastic effect on droplet size distribution and to evaluate the breakage rates of oil-in-water dispersions in a continuous oscillatory baffled tube. In this work, experiments were performed showing that the breakage of droplets is the dominant mechanism in the system, and the physical properties of different oils had no significant influence on droplet size distributions. Under those conditions the model can be used to focus only on breakage rate constants, keeping the number of fitted parameters in the modelling process to a minimum. The droplet breakage results from this work suggest that the oscillation amplitude has more influence on the breakage rates than the oscillation frequency. This work is a major extension and includes droplet data from our previous studies so that the breakage rates can be compared; and the consistency of the rate constants is examined.     

Effects of Sand Fraction on Toluene‐diluted Heavy Oil in Water Emulsions in Turbulent Flow

The impact of the presence of sand on emulsification of toluene‐diluted heavy oil in simulated process water was systematically studied as a function of agitation time, in a stirred tank. Droplet size distributions were measured by light scattering technique. Optical microscopy and high‐speed video micrography were used to visually monitor agitation and emulsion stability. Results showed that the Sauter mean diameters of the droplets decreased with increasing sand content. Droplet breakage followed a first‐order kinetic model for all mixing speeds. Plots of droplet volume percent frequency versus droplet size followed lognormal distribution. The distribution span broadened into lower sizes with increased sand content. Emulsions were stable for over 48 h after formation.