Agglomeration and breakage of nanoparticles in stirred media mills—a comparison of different methods and models

The increasing industrial demand for nanoparticles challenges the application of stirred media mills to grind in the sub-micron size range. It was shown recently [Mende et al., 2003. Mechanical production and stabilization of submicron particles in stirred media mills. Powder Technology 132, 64-73] that the grinding behavior of particles in the sub-micron size range in stirred media mills and the minimum achievable particle size is strongly influenced by the suspension stability and thus the agglomeration behavior of the suspension. Therefore, an appropriate modeling of the process must include a superposition of the two opposing processes in the mill i.e., breakage and agglomeration which can be done by means of population balance models. Modeling must now include the influence of colloidal surface forces and hydrodynamic forces on particle aggregation and breakup. The superposition of the population balance models for agglomeration and grinding with the appropriate kernels leads to a system of partial differential equations, which can be solved in various ways numerically. Here a modified h-p Galerkin algorithm which is implemented in the commercially available software package PARSIVAL developed by CiT (CiT GmbH, Rastede, Germany) and the moment methodology according to [Diemer and Olsen, 2002a. A moment methodology for coagulation and breakage problems: Part I—analytical solution of the steady-state population balance. Chemical Engineering Science 57 (12), 2193-2209; Diemer and Olsen, 2002b. A moment methodology for coagulation and breakage problems: Part II—moment models and distribution reconstruction. Chemical Engineering Science 57 (12), 2211-2288] are used and compared to explicit data on alumina. This includes a comparison of the derived particle size distributions, moments and its accuracy depending on the starting particle size distribution and the used agglomeration and breakage kernels. Finally, the computational effort of both methods in comparison to the prior mentioned parameters is evaluated in terms of practical application.     

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.     

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.     

Direct measurement of droplet breakage in a pulsed disc and doughnut column

The lack of experimental data for the droplet breakup has been one of the limitations for the application of population balance model (PBM). In this work, a high‐speed camera was used to directly measure the droplet breakup frequency and daughter size distribution in a pulsed disc and doughnut column. It was found from the captured video that multiple breakup events were more frequently observed than binary breakup. The multiple breakup was treated as an original breakup and several intermediate breakups to characterize the process quantitatively. The effects of pulsation intensity, dispersed phase flow rates, and the spatial locations were investigated in detail. Empirical correlations were finally established for both the breakup frequency function and the daughter droplet size distribution function and fitted well with the experimental data. The correlation equations were then used in a simplified PBM to calculate the droplet number density, which further proved the feasibility of the correlations. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4188–4200, 2017     

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.     

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.     

A study of drop breakage in lean dispersions using the inverse-problem method

The inverse-problem approach has been applied to new experimental drop-size data in order to make direct estimations of breakage rates and fragment distributions in lean dispersions of moderately-viscous droplets. Evolving size distributions were measured over extended periods in three vessels of different scale with agitation conditions encompassing a range of average energy dissipation rates from 0.2 to 1.5 W/kg. The silicone oils used as the dispersed phase ranged in viscosity from 50 to 500 mPa s. Measured distributions were first tested for self-similarity which, if confirmed, allows the droplet breakage rates to be determined to within a constant factor. The novelty of the current work is centred upon the use of two dissimilar inversion methods (Sathyagal et al., 1996, Kostoglou and Karabelas, 2001) to provide both the daughter-drop size distributions and the final breakage rates across the full range of experimental conditions. The objective here was to ensure that the inversion process, which is known to be ill-conditioned, produced consistent results that are independent of the procedure employed. Both methods yielded near-identical results for the breakage kernel function with fragment numbers ranging from as many as six in the early stages of the agitation period to two for the smallest surviving parent drops. The inhomogeneous breakage kernels were found to be well-represented by the generalised functions proposed by Diemer and Olson (2002); the function parameters indicated a breakage process tending towards erosion. Droplet breakage rates showed a significant dependence upon vessel size and energy dissipation rate with the influence of dispersed-phase viscosity being particularly marked. The consistency of the results was tested by solving the population balance equations for pure breakage with the experimental breakage rate and kernel function being used to reconstruct the size distributions at chosen times. These realisations of the evolving distribution were found to be in close agreement with the experimental measurements.     

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.     

Experimental study on drop breakup time and breakup rate with drop swarms in a stirred tank

The direct experimental data for breakup parameters of drop breakup time, multiple breakage, and breakup rate are urgently required to understand drop breakup phenomena. In this regard, drop breakup experiments were carried out in a stirred tank using a high-speed online camera. The influences of the rotating speed, interfacial tension, and drop viscosity on the above breakup parameters were then quantitatively investigated. An mechanism correlation for the breakup time is proposed and is further verified by comparing with the results of Solsvik and Jakobsen (Chem Eng Sci, 2015;131:219-234). The percentage of multiple breakage comparing to binary breakup was statistically counted. The results indicated that the dimensionless drop diameter η = d/d_max can be adopted to characterize the proportion of binary breakup. Finally, the breakup rate was experimentally measured and the breakup probability was calculated using the inverse method.     

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.     

CFD simulation of bubble columns incorporating population balance modeling

A computational fluid dynamics (CFD)-code has been developed using finite volume method in Eulerian framework for the simulation of axisymmetric steady state flows in bubble columns. The population balance equation for bubble number density has been included in the CFD code. The fixed pivot method of Kumar and Ramkrishna [1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332] has been used to discretize the population balance equation. The turbulence in the liquid phase has been modeled by a k-ε model. The novel feature of the framework is that it includes the size-specific bubble velocities obtained by assuming mechanical equilibrium for each bubble and hence it is a generalized multi-fluid model. With appropriate closures for the drag and lift forces, it allows for different velocities for bubbles of different sizes and hence the proper spatial distributions of bubbles are predicted. Accordingly the proper distributions of gas hold-up, liquid circulation velocities and turbulence intensities in the column are predicted. A survey of the literature shows that the algebraic manipulations of either bubble coalescence or break-up rate were mainly guided by the need to obtain the equilibrium bubble size distributions in the column. The model of Prince and Blanch [1990. Bubble coalescence and break-up in air-sparged bubble columns. A.I.Ch.E. Journal 36, 1485-1499] is known to overpredict the bubble collision frequencies in bubble columns. It has been modified to incorporate the effect of gas phase dispersion number. The predictions of the model are in good agreement with the experimental data of Bhole et al. [2006. Laser Doppler anemometer measurements in bubble column: effect of sparger. Industrial & Engineering Chemistry Research 45, 9201-9207] obtained using Laser Doppler anemometry. Comparison of simulation results with the experimental measurements of Sanyal et al. [1999. Numerical simulation of gas-liquid dynamics in cylindrical bubble column reactors. Chemical Engineering Science 54, 5071-5083] and Olmos et al. [2001. Numerical simulation of multiphase flow in bubble column reactors: influence of bubble coalescence and breakup. Chemical Engineering Science 56, 6359-6365] also show a good agreement for liquid velocity and gas hold-up profiles.     

CFD simulation of gas-liquid contacting in tubular reactors

Gas-liquid contacting in tubular reactors was simulated using an Eulerian-Eulerian CFD approach in which accurate interphase momentum closure relations are incorporated, bubble-induced turbulence is accounted for, and population balance equations are used to describe bubble breakage and coalescence. The ability of two breakup kernels (Luo, H., Svendsen, H.F., 1996. Theoretical model for drop and bubble breakup in turbulent dispersions. A.I.Ch.E. Journal 42, 1225-1233; Lehr, F., Millies, M., Mewes, D., 2002. Bubble size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426-2443) and three coalescence kernels (Prince, M.J., Blanch, H.W., 1990. Bubble coalescence and breakup in air sparged bubble columns. A.I.Ch.E. Journal 36, 1485-1499; Luo, H., 1993. Coalescence, breakup and liquid recirculation in bubble column reactors. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim; Lehr, F., Millies, M., Mewes, D., 2002. Bubble size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426-2443) to accurately predict several flow parameters in pipe flow was tested.Good agreement between simulation and experimental results (radial profiles of gas holdup, turbulence intensity, and local Sauter bubble diameter) was achieved without the use of empirically derived relationships (such as Drift flux) by adjusting a single parameter which accounts for the deviation in the coalescence behaviour of tap water from that of pure water. The approach adopted in this investigation may thus be applicable to more complex hydrodynamic situations such as those encountered in mechanically agitated tanks and the need for extensive experimental testing may be replaced by single measurement of the effect interfacial properties have on coalescence rates.     

Theoretical model for multiple breakup of fluid particles in turbulent flow field

Generalized phenomenological model, based on the theories of probability and isotropic turbulence, is developed for multiple breakup of fluid particles in turbulent flow field. The approach uses a series of successive binary breakup events occur at a time scale comparable to the colliding eddy turnover time. It was found that the use of energy density, instead of energy, will increase the predicted binary breakup rate which is usually underestimated by the existing models in the literature. Generalization of the binary breakup model for multiple fragmentations is performed by defining a “remaining energy function” for the colliding eddy which means the contribution of original eddy to the later breakup events. For ternary breakage, the model shows a reasonably good agreement with the experimental data. The quaternary fragmentation frequency, however, is of negligible importance at lower energy dissipation rates but its contribution to breakage fraction at higher energy dissipation rates becomes considerable. The results also show that ternary and quaternary breakups have a considerable 90% contribution to the overall fragmentation, while pentenary and further fragmentations are of lower importance at low energy dissipation rates. At higher levels of energy dissipation rate, fragmentations up to six daughter particles contribute to more than 95% of the overall fragmentations. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4508–4525, 2016     

Applying nanofluid to a rotary disc contactor to investigate breakage phenomenon

Five different nanofluids have been prepared by use of two‐step procedure, in which SiO₂ nanoparticles with different volume fractions and hydrophobicities have been dispersed in butyl acetate as base fluid by applying ultrasonication method. Three different methods have been used to evaluate the nanofluids' stability: conventional sedimentation method, particle size analyser and UV‐Vis spectrophotometric. As a quite novel research, these nanofluids were applied to a Rotary Disc Contactor with 91 mm diameter and 12 rotating discs to investigate breakage probability and the first critical rotor speed. The effects of nanoparticles' volume fraction and hydrophobicity, as well as rotor speed, mother drop size and continuous phase height on the breakage phenomenon have been studied. © 2011 Canadian Society for Chemical Engineering     

A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions

Population balances for simultaneous coagulation and breakage (and their analogs, e.g., polymerization and depolymerization) are employed in describing many systems including aerosols, powders and polymers, and many unit operations including reactors, crystallizers, and size reduction/enlargement equipment. The birth term for the breakage process is usually formulated in terms of a distribution of breakage products known as the daughter distribution. There are many daughter distribution forms proposed in the literature in part because these distributions are notoriously difficult to determine experimentally. Here, a generalization of these forms is developed for multi-particle breakup which has the flexibility to represent a wide variety of distribution shapes. The simplicity of the generalized expression renders the population balance equations for simultaneous coagulation and breakage accessible to analytical attack, leading to an analytical expression for the fine end of the steady-state product size distribution. This expression has potential utility in both design and analysis of experiments aimed at measuring daughter distribution parameters.     

Characterization of liquid‐liquid dispersions with variable viscosity by coupled computational fluid dynamics and population balances

Sustaining stable liquid‐liquid dispersion with the desired drop size still relies on experimental correlations, which do not reflect our understanding of the underlying physics and have a limited prediction capability. The complex behavior of liquid‐liquid dispersions inside a stirred tank, which is equipped with a Rushton turbine, was characterized by a combination of computational fluid dynamics and population balance equations (PBE). PBE took into account both the drop coalescence and breakup. With the increasing drop viscosity, the resistance to drop breakage also increases, which was introduced by the local criteria for drop breakup in the form of the local critical Webber number (We_c). The dependency of We_c on the drop viscosity was derived from the experimental data available in the literature. Predictions of Sauter mean diameter agree well with the experimentally measured values allowing prediction of mean drop size as a function of variable viscosity, interfacial tension, and stirring speed. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2403–2414, 2015     

Break-up of droplets in Karr reciprocating plate extraction column

The breaking rate of individual drops was investigated in a Karr type reciprocating plate extraction column. The binary systems used were: water-1,2-dichloroethane, water-toluene and water-n-butanol. The breakage probability and the conditional probability of breaking-up into a given number of daughter droplets, as well as the drop size distribution of daughter droplets, were the measured characteristics. Relations between the breakage probabilities and the breakage frequency were derived.A mathematical model of the frequency of breaking-up into > and more droplets developed was based on the assumption that collisions with turbulent eddies of the Kolmogoroff region of universal equilibrium are responsible for the process. A probabilistic model of daughter drop size distribution was also derived in the form of a β-distribution. The models were compared with experimental results and a good agreement was obtained.     

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.     

Drop breakage model in static mixers at low and intermediate Reynolds number

Drop break-up process for the flow of liquid-liquid dispersion in a static mixer has been investigated. Two new theoretical models for the drop break-up at low and intermediate Reynolds number for variant viscosity ratio of the dispersed phase to the continuous phase have been developed assuming that the flow through the static mixer elements is analogous to the flow through porous media. This concept has recently been established by Morançais et al. (Chem. Eng. Commun. 179 (1999) 77) and Legrand et al. (Chem. Eng. Res. Des. 79 (2001) 949). The boundary-layer shear force concept has been applied to predict the drop break-up at low Reynolds number and at intermediate Reynolds number, the effect of inertia on the drop break-up has been considered. The predicted drop sizes are in reasonable agreement with experimental results.