Binary Breakage Research Articles

Analysis of particle strain in stirred bioreactors by drop breakage investigations

Understanding of particle strain and drop breakage is relevant for various technical applications. To analyze it, single drop experiments in a breakage cell and evolving drop size distributions in an agitated system are studied. The mechanisms for particle strain and drop breakage are assumed to be comparable for the investigated turbulent flow regime. The agitation process is simulated using a population balance model. This model provides transient prediction capacities at different scales and can be used for scale-up/down projects. The number and the size distributions of daughter fragments for single drops have been studied. The results clearly support the assumption of binary breakage. The most common assumption of a Gaussian distribution for the daughter drop size distribution could not be supported. The evolution of a breakage-dominated toluene/water system was then simulated using different daughter drop size distributions from literature. The computational results were compared with experimental values. All simulations were able to predict the transient Sauter mean diameter excellently but varied strongly in the results on the shape of the distribution. In agreement with the experimental single drop results, the use of a bimodal or a very broad bell-shaped distribution of the daughter drops is proposed for the simulations. Although these results were obtained in a particular vessel for a specific phase system, it can be applied to simulate transient multiphase systems at different scales. We would expect that the general trends observed in this study are comparable to various applications in multiphase bioreactors.

Mathematical framework for higher order breakage scenarios

Droplet breakage has been widely studied due to its relevance to the gas purification industry. This phenomenon appears inside scrubbers on the mesh pad, due to the interaction of the droplets with the liquid drainage film or the walls of the equipment. The breakage process that has been most widely studied in the previous publications is binary breakup, mostly due to its simplicity and because it is the most common one in diluted systems. In this work, higher order breakups like ternary, quaternary, etc. have been considered. A higher order breakage model was developed based on an existing binary breakage model, and it was shown that this approach is suitable for higher order breakage computations. Despite the fact that this procedure is reasonably accurate for the first preliminary results, a more accurate mathematical model based on a detailed physical analysis of the problem in question should be developed.

Prediction of emulsion drop size distributions with population balance equation models of multiple drop breakage

Most population balance equation (PBE) models of emulsion drop breakage are based on the assumption of binary drop breakage. We previously developed such a PBE model for high-pressure homogenizers with a daughter drop distribution function exhibiting a maximum probability for two equal sized drops. In this paper, we present a PBE model accounting for multiple drop breakage and show that the model provides superior distribution predictions reflected by decrease in least-squares objective function for an oil-in-water emulsion processed in a pilot-scale high-pressure homogenizer. Following our previous work, two distinct rate functions for drop breakage, one due to turbulent eddies and another due to turbulent shear were used to reproduce the measured bimodal distributions. We found that multiple drop breakage was satisfactorily modeled with a uniform daughter drop distribution function if the assumed number of daughter drops formed was chosen to be sufficiently large. The PBE model with multiple drop breakage was shown to provide superior distribution predictions compared to the analogous binary breakage PBE model when adjustable model parameters were determined by nonlinear optimization. The multiple breakage PBE model was shown to be extensible to different emulsion formulations by using these base case model parameters to predict the effects of oil concentration, surfactant concentration, oil-to-surfactant ratio and emulsion premix distribution. Our experiments revealed that substantial breakage of the premix occurred during the first homogenization pass even under zero applied homogenization pressure operation, suggesting an unmodeled pressure independent breakage mechanism.

Comparison of discretization methods to solve a population balance model of activated sludge flocculation including aggregation and breakage

Population balance models (PBMs) can be used to describe the evolution with time of distributions of properties of individuals. In this study a PBM for activated sludge flocculation including aggregation and breakage processes was investigated. The PBM is an integro-differential equation and does not have an analytical solution. A possibe method of solving the equation at relatively low computational cost is to use discretization. Two different discretization techniques, the fixed pivot and the moving pivot, were compared using geometric grids of different coarseness. Simulations were performed for three different processes: pure aggregation, pure breakage and combined aggregation – breakage. The results for pure aggregation showed that the fixed pivot overpredicts the large particle sizes when using coarse grids since grid refinement results in a clear downward trend. The predictions of the moving pivot technique show even lower predictions for the large particle sizes, with a slight upward trend for finer grids. This suggests that these predictions are closer to the pseudo-analytical solution (i.e. at infinitely fine grid). For the pure breakage case it was found that the moving pivot predictions collapsed onto one curve. Since a binary breakage case was studied, a fixed pivot with a grid with geometric factor 2 also collapsed onto that curve. Grid refinement for the fixed pivot case resulted in overestimations. Similar conclusions could be drawn for the combined aggregation – breakage case. Overall, the moving pivot is found to be superior since it produces more accurate predictions, even for much coarser grids. Despite the computational burden, the latter implies a lower computational load.

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 T = 0.15 m 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.

Transient and asymptotic behaviour of the binary breakage problem

The general binary breakage problem with power-law breakage functions and two families of symmetric and asymmetric breakage kernels is studied in this work. A useful transformation leads to an equation that predicts self-similar solutions in its asymptotic limit and offers explicit knowledge of the mean size and particle density at each point in dimensionless time. A novel moving boundary algorithm in the transformed coordinate system is developed, allowing the accurate prediction of the full transient behaviour of the system from the initial condition up to the point where self-similarity is achieved, and beyond if necessary. The numerical algorithm is very rapid and its results are in excellent agreement with known analytical solutions. In the case of the symmetric breakage kernels only unimodal, self-similar number density functions are obtained asymptotically for all parameter values and independent of the initial conditions, while in the case of asymmetric breakage kernels, bimodality appears for high degrees of asymmetry and sharp breakage functions. For symmetric and discrete breakage kernels, self-similarity is not achieved. The solution exhibits sustained oscillations with amplitude that depends on the initial condition and the sharpness of the breakage mechanism, while the period is always fixed and equal to ln 2 with respect to dimensionless time.

Comparison and pitfalls of different discretised solution methods for population balance models: a simulation study

One way of solving population balance equations in a time efficient way, necessary for using the model in non-linear parameter estimation, process control or for combination with computational fluid dynamics (CFD), is by means of discretisation of the population property of interest. Several methods are available in literature, but which one should be used in which case? Do they all perform equally well or do some of them have unexpected pitfalls? In this study, three discretisation methods (the Hounslow method, the fixed pivot and the moving pivot) were compared for three different processes (pure aggregation, pure binary breakage into equally sized daughters and combined aggregation/breakage) and six different initial conditions (two monodispersions, one uniform polydispersion and three partially uniform polydispersions). Their performances were compared in terms of accuracy and calculation speed and their pitfalls addressed and explained where possible. Overall, it was concluded that the moving pivot with a geometric factor of 2 performs good in all studied cases (not for finer or coarser grids). The eventual choice of discretisation scheme is highly dependent on the particular goal of the study and is a compromise between accuracy and calculation speed.

Simulation of coalescence and breakage: an assessment of two stochastic methods suitable for simulating liquid–liquid extraction

Industrial problems such as liquid–liquid extraction often require the use of population balances, so a thorough understanding of the advantages and disadvantages of all the available solution techniques is essential. The mathematical literature on stochastic solution methods for the population balance equation is often quite fragmentary and tends to focus mainly on existence proofs rather than examining the applicability of a particular algorithm to a practical problem of interest to researchers or industrialists. In this paper, a practical study is made of two stochastic solution methods for the population balance equation, simulating coalescence and binary breakage. The first algorithm studied is the existing direct simulation algorithm (DSA), proposed in (Eibeck and Wagner, Stoch. Anal. Appl. 18(6) (2000) 921). The second is an extension of the Mass Flow Algorithm (MFA), which was proposed in (Eibeck and Wagner, Ann. Appl. Probab. 11(4) (2001) 1137) for coagulation only and in (Jourdain, Markov Processes and Related Fields 9(1) (2003) 103) for discrete coagulation-fragmentation. MFA (mass flow algorithm) is extended to include breakage in the continuous case, and a binary search method of distribution generation is introduced, leading to improved efficiency. Numerical investigation of the performance of the two algorithms is carried out by applying them both to a test case, for which an analytical solution is calculated. For both algorithms, convergence of the predicted moments to the analytical solution goes as the inverse of the number of stochastic particles, N , except for the zeroth moment predicted by MFA. This exhibits large fluctuations, due to the presence of very small particles, and converges approximately as N - 1 / 3 . The new algorithm, MFA, exhibits significant variance reduction—and therefore improved simulation efficiency—for the prediction of higher moments, but for our test case the zeroth moment (the total number of particles) is predicted with better efficiency by DSA. In many breakage models for liquid–liquid systems however, the introduction of a minimum particle size reduces the advantage held by DSA for predicting the zeroth moment. Depending on the minimum particle size, MFA can perform comparably with DSA for predicting the zeroth moment. Using models for coalescence and breakage from literature, DSA is successfully applied to the case of a laboratory scale rotating disc contactor.

Kinetics of fluidised bed melt granulationIV. Selecting the breakage model

The main purpose of this paper is to develop and verify a suitable breakage model to describe the breakage kinetics in fluidised bed melt granulation (FBMG), by means of population balance modelling. A two internal coordinate population balance equation (PBE) [AIChE J. 47 (2001) 1984] that describes the granule size and tracer mass is employed in the modelling process. In addition, a new analytical PBE and discretized population balance equation was developed for the tracer mass distribution for the growth and nucleation process. Three physically plausible breakage models (fragmentation, attrition and random binary breakage) were also implemented in a discretized population balance model (PBM) to describe the breakage process in FBMG. Three phases of the modelling procedures were performed. In Phase 0, the equipartition of kinetic energy (EKE) kernel is fitted to the mass-based granule size distributions to extract an aggregation rate constant to represent the net growth process. In Phase I, a fitting exercise to the different breakage models was performed, and the best fit model is that by random binary breakage with a size-independent selection rate constant. In Phase II, a refined two-parameter breakage model was developed to account for granule breakage by a combination of random binary breakage and attrition processes. A new error-weighted integral fit technique was also developed to extract the aggregation and breakage rate constant simultaneously. For this purpose, we have conducted experiments with and without the presence of tracer granules. Each of these models is quite effective at describing the granule size distribution to which they are fitted, although the Phase 0 model gives a poor description of the numbers of granules. These same models are then used to predict the results of the tracer experiments. From this, we are able to show that the Phase 2 model is the most effective.

Kinetics of fluidised bed melt granulation: IV. Selecting the breakage model

The main purpose of this paper is to develop and verify a suitable breakage model to describe the breakage kinetics in fluidised bed melt granulation (FBMG), by means of population balance modelling. A two internal coordinate population balance equation (PBE) [AIChE J. 47 (2001) 1984] that describes the granule size and tracer mass is employed in the modelling process. In addition, a new analytical PBE and discretized population balance equation was developed for the tracer mass distribution for the growth and nucleation process. Three physically plausible breakage models (fragmentation, attrition and random binary breakage) were also implemented in a discretized population balance model (PBM) to describe the breakage process in FBMG. Three phases of the modelling procedures were performed. In Phase 0, the equipartition of kinetic energy (EKE) kernel is fitted to the mass-based granule size distributions to extract an aggregation rate constant to represent the net growth process. In Phase I, a fitting exercise to the different breakage models was performed, and the best fit model is that by random binary breakage with a size-independent selection rate constant. In Phase II, a refined two-parameter breakage model was developed to account for granule breakage by a combination of random binary breakage and attrition processes. A new error-weighted integral fit technique was also developed to extract the aggregation and breakage rate constant simultaneously. For this purpose, we have conducted experiments with and without the presence of tracer granules. Each of these models is quite effective at describing the granule size distribution to which they are fitted, although the Phase 0 model gives a poor description of the numbers of granules. These same models are then used to predict the results of the tracer experiments. From this, we are able to show that the Phase 2 model is the most effective.

Characterization of asphaltenes aggregation and fragmentation in a shear field

AbstractAggregation and breakup of asphaltenes flocs in a Couette flow device are studied for various values of shear rate (G), volume fraction of particles (φ), and solvent composition (ratio of toluene to n‐heptane in the solution Xs). The number average asphaltene aggregate size initially increases before achieving a maximum size and then decreases until reaching a steady‐state value that reflects a balance between aggregation and fragmentation. Increasing the applied shear rate leads to higher aggregation and fragmentation rates, reducing the steady‐state average aggregate size and the characteristic relaxation time required to achieve a steady state. For a fixed value of G, when either φ is increased or Xs is decreased, the steady‐state floc size increases. The kinetics of the aggregation‐fragmentation process and the attainment of steady state by asphaltenes aggregates were modeled using a population balance approach. The model describes the time evolution of the average floc size using fitted parameters (the breakup coefficient for shear fragmentation, b and the porosity of the aggregates ϵ). Binary breakage was included in calculations and collision efficiency was assumed to be equal to one.

Flocculation of biological cells: Experiment vs. theory

AbstractFlocculation of biological cells is important in the biotechnology industry, as it could lead to improved efficiencies for bioreactor harvesting operations such as microfiltration. Experimental studies for flocculation of yeast and CHO cells using cationic polyelectrolytes suggest the existence of a steady‐state, self‐similar floc size distribution. The experimentally determined floc size distributions were modeled using a population balance approach. For flocculated yeast suspensions, the variation of the floc volume fraction with dimensionless particle diameter is predicted by the population balance model assuming a binary breakage distribution function. However, the variation of floc number fraction with dimensionless particle diameter is better predicted assuming a log normal fragment distribution function probably due to the presence of submicron‐sized yeast cell debris. For CHO cell flocs, the floc volume and number fractions are predicted using a log normal fragment distribution function. CHO cells are far more fragile than yeast cells. Thus, individual CHO cells in a CHO cell floc can lyse leading to the formation of a number of small particles.

Dynamics Of Dispersions With MultipleBreakage And Volume Scattering

In some previous papers we have developed a model for the dynamics of liquid-liquid dispersions including some new features, among which the socalled volume scattering phenomenon, as a third mechanism of evolution, besides coalescence and breakage. So far we confined to binary breakage. The present paper analyses the more general case in which more than two droplets may be produced as the result of breakage or volume scattering. The main point is the introduction of a probability function for each breakage mode: from this point of view the paper completes our approach to the description of the dynamics of droplets dispersions which are spatially homogeneous.

Evolution of the Gouge Particle Size Distribution: A Markov Model

—The hydrologic and mechanical properties of faults are determined by their internal structures and zoning due to fracturing, grain breakage and diagenesis. The proposed model in this study is a first step in the development of a more complete model of fault internal structure. It describes the evolution of the gouge particle size distribution (PSD) with shear induced grain breakage within a small, assumed uniform volume element. A Markov formalism is applied for binary breakage within each time step over the many time steps constituting the evolution. The particle size, strain- and stress-dependent breaking probability is constructed based on physical arguments and data on natural and simulated gouge PSDs. The model is calibrated by using the results of on Ottawa sand PSD experiments. A perfect fit is obtained between experimental and numerically simulated PSDs for a range of normal effective stress (NES). Results of the numerical simulations capture the bimodal form of gouge PSD and also clearly define two different breakage mechanisms. The gouge development behavior makes a dramatic transition as the normal effective stress exceeds the grain crushing stress of the gouge mineral.¶For a more complete intrafault structure analysis, the interplay of diagenesis and the mechanics of grain breakage can be integrated into a single mechano-chemical model of the type presented here. The model proposed here has great potential for predicting the complex roles of faults as seals or fluid conduits.

Laminar and turbulent shear‐induced flocculation of fractal aggregates

AbstractA population balance model is used to model simultaneous coagulation and fragmentation in turbulent shear. The fractal‐like aggregate structure is quantified using a mass fractal dimension, Df = 2.05, derived from light‐scattering measurements and is incorporated into the model using appropriate kinetic expressions. In addition, fluid viscous effects on aggregate collisions are taken into account using existing theories on viscous retardation. Flocculation experiments of a polystyrene particle/Al(OH)3/water system in a stirred tank are compared to the model results. Good agreement is found for both average‐size evolution and steady‐state size distribution using only one fitting parameter and assuming binary breakage. The average polystyrene‐Al(OH)3 aggregate size initially increases before reaching a constant steady‐state value during coagulation–fragmentation in a stirred tank. Increasing the applied shear rate increases the coagulation and fragmentation rates, decreasing the steady‐state average aggregate size, and the time lag before steady state. The model developed in this work is applied to laminar shear data from the literature, showing excellent agreement.

On the steady-state size distribution of dispersions in breakage processes

Breakage processes are considered in the absence of agglomeration or coagulation. A new method is proposed, based on a population balance type of formulation, applicable to systems (such as dispersions) that may be characterized by a maximum stable particle size. In this method, considerable simplification is achieved by means of a transformation that effectively eliminates the breakage frequency, thus allowing the convenient computation of steady state through solution of an integral equation. To compute the steady state, apart from the maximum size and the breakage kernel, only an estimate of the initial distribution is required. Two functional forms of binary breakage kernels which can represent a large variety of possible breakage mechanisms are proposed (by an appropriate selection of parameter values). For the sake of completeness, analytical solutions are also presented for several, relatively simple kernels. Finally, a study is made to assess the influence of initial conditions on the steady-state size distribution, which is helpful in tackling the inverse problem of determining the breakage kernel using limited experimental data.

Application of the master equation to coalescence and dispersion phenomena

In the present work the master equation is applied to the phenomena of coalescence and dispersion. Specifically, expressions are derived to model binary coalescence and breakage with a particular emphasis on the results which are unique to the stochastic treatment, namely the covariances and correlation functions. A probabilistic analysis is also introduced in an attempt to rationally derive the daughter size distribution from knowledge about breakage probabilities of individual droplets or particles. This analysis is valid for binary or higher order breakage and any combination thereof. One possible assignment of the probability for binary breakage is proposed; it is based on the excess surface energy of a dispersed phase droplet. Finally, the general contributions of stochastic models are discussed.