Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

Various particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.     

Modeling Emulsification in Static Mixers: Equilibrium Correlations versus Population Balance Equations

Two different modeling approaches are adopted to model turbulent breakage during continuous emulsification in static mixers. First, a correlation is developed to predict the droplet mean diameter. Second, a population balance equation (PBE) is applied to track the droplet size distribution (DSD) using two different breakage kernels. The performances of the two approaches are assessed against a large number of experimental data. The correlation is fast to develop and is found to be capable of predicting the mean diameter with an acceptable accuracy while the PBE‐based model gives an excellent prediction of the entire DSD.     

Numerical solution of the spatially distributed population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions

In liquid-liquid contacting equipment such as completely mixed and differential contactors, droplet population balance based modeling is now being used to describe the complex hydrodynamic behavior of the dispersed phase. For the hydrodynamics of these interacting dispersions this model accounts for droplet breakage, droplet coalescence, axial dispersion, exit and entry events. The resulting population balance equations are integro-partial differential equations (IPDE) that rarely have an analytical solution, especially when they show spatial dependency, and hence numerical solutions are sought in general. To do this, these IPDEs are projected onto a system of convective dominant partial differential equations by discretizing the droplet diameter (internal coordinate). This is accomplished by generalizing the fixed-pivot (GFP) technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996a) 1311) handling any two integral properties of the population number density for continuous flow systems by treating the inlet feed distribution as a source term. Moreover, the GFD technique has the advantage of being free of repeated or double integral evaluation resulting from the weighted residual approaches such as the Galerkin's method. This allows the time-dependent breakage and coalescence functions to be easily handled without appreciable increase in the computational time. The resulting system of PDEs is spatially discretized in conservative form using a simplified first order upwind scheme as well as first- and second-order non-oscillatory central differencing schemes. This spatial discretization avoids the characteristic decomposition of the convective flux based on the approximate Riemann solvers and the operator splitting technique required by classical upwind schemes. The time variable is discretized using an implicit strongly stable approach that is formulated by careful lagging of the non-linear parts of the convective and source terms. The algorithm is tested against analytical solutions of the simplified population balance equation for a differential liquid-liquid extraction column through four case studies. In all these case studies the discrete models converge successfully to the available analytical solutions and to solutions on relatively fine grids when the analytical solution is not available. Realization of the algorithm is accomplished by comparing its predictions to experimental steady-state hydrodynamic data of a laboratory scale rotating disc contactor of diameter. Practically, the combined algorithm is found fast enough for the computation of the transient and steady-state hydrodynamic behavior of the continuously and spatially distributed interacting liquid-liquid dispersions.     

Continued self-similar breakup of drops in viscous continuous phase in agitated vessels

Drop breakup in viscous liquids in agitated vessels occurs in elongational flow around impeller blade edges. The drop size distributions measured over extended periods for impellers of different sizes show that breakup process continues up to 15–20 h, before a steady state is reached. The size distributions evolve in a self-similar way till the steady state is reached. The scaled size distributions vary with impeller size and impeller speed, in contrast with the near universal scaling known for drop breakup in turbulent flows. The steady state size of the largest drop follows inverse scaling with impeller tip velocity. The breadth of the scaled size distributions also shows a monotonic relationship with impeller tip velocity only.     

Fluid flow and mass transfer in a counter-current gas-liquid inclined tubes photo-bioreactor

A novel type of photo-bioreactor is being studied. The main feature in this system is the inclined tube in which most of the photosynthesis occurs. Two phase gas-liquid counter-current flow takes place in this environment. The flow configuration in the range of interest has not been described previously. The trajectories of the liquid phase are the target of the present work. These trajectories are needed for a proper mathematical representation of the process. Experimental studies where carried out using two methods: studying the response of the system to a pulse disturbance, and tracking an optical tracer with a system developed in our laboratory. The influence of tube inclination and of gas flow rates was studied. A rough compartmental model is presented, which fits the transient experimental data.The mass transfer rate from the gas phase to the liquid was studied since it is needed to assess the capacity of providing enough CO₂ to match the light captured by photosynthesis. Both gas holdup and mass transfer rates are reported as a function of tube inclination and gas flow rates.     

Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.     

Modeling of Mixing and Reaction in Turbulent Multi‐Phase Flows with Distribution Functions

Numerical simulation of turbulent reacting or multiphase flows is gaining popularity as a tool for the analysis and optimization of many complex applications in process engineering. To make possible the accurate modeling of relevant reaction and transport processes, the respective distribution functions of mixture fraction or particle size must be considered in an adequate manner. In the present paper, novel approaches to make possible a more detailed yet efficient representation of distribution functions in turbulent, reacting multiphase flows are introduced. The application of the methods to the example of a system with mixing and reaction among three species is discussed.     

Scale‐up of Mini‐Plant Extractors on Energy Dissipation‐Based Population Balances

The new scale‐up concept for extraction columns relies on three identities being kept idem, as is the total specific flow rate, energy dissipation, and mean droplet residence time in a compartment. The droplet population balance‐based model allows maintaining hydrodynamic similarity in different geometries, as is in a mini‐ or a pilot plant. This leads to similar breakage and coalescence probabilities giving comparable droplet size distributions, thus mass transfer area and extraction efficiency. A new breakage frequency term has been developed relying on the energy dissipation rate and is thus independent from geometric constraints. The traditional scale‐up rules are based either on a constant tip velocity (≈ N) or on a constant energy input (≈ N³), whereas here it follows a constant energy dissipation (≈ N²). A step‐by‐step approach to the new procedure proved by case samples is given. Data from literature pilot experiments could be verified by computer simulations, without using adaptable parameters. All parameters in the correlations where derived in a lab‐scale apparatus and the coalescence parameters were obtained in the mini‐plant experiments. Derivation between simulated and experimental pilot data for stage numbers was less than 14 %, operating parameters (rotational speed N, throughput) were underestimated by 4 % leading to a slightly smaller HETS (Height Equivalent of Transfer Stages) value as measured, affecting the column height with less than 1 %.     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

Gas–liquid flows in medium and large vertical pipes

Gas–liquid bubbly flows with wide range of bubble sizes are commonly encountered in many industrial gas–liquid flow systems. To assess the performances of two population balance approaches – Average Bubble Number Density (ABND) and Inhomogeneous MUlti-SIze-Group (MUSIG) models – in tracking the changes of gas volume fraction and bubble size distribution under complex flow conditions, numerical studies have been performed to validate predictions from both models against experimental data of Lucas et al. (2005) and Prasser et al. (2007) measured in the Forschungszentrum Dresden-Rossendorf FZD facility. These experiments have been strategically chosen because of flow conditions yielding opposite trend of bubble size evolution, which provided the means of carrying out a thorough examination of existing bubble coalescence and break-up kernels. In general, predictions of both models were in good agreement with experimental data. The encouraging results demonstrated the capability of both models in capturing the dynamical changes of bubbles size due to bubble interactions and the transition from “wall peak” to “core peak” gas volume fraction profiles caused by the presence of small and large bubbles. Predictions of the inhomogeneous MUSIG model appeared marginally superior to those of ABND model. Nevertheless, through the comparison of axial gas volume fraction and Sauter mean bubble diameter profiles, ABND model may be considered an alternative approach for industrial applications of gas–liquid flow systems.     

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.     

具有高粘度分散相的油-水分散体系的液滴破碎与粒径分布研究

对湍流搅拌槽中原油-水分散体系的液滴破碎进行了研究。在不同的温度下原油表现出不同的流变学特征，对分散过程中的液滴破碎产生不同的影响。实验研究对比了不同温度下原油-水分散体系的液滴分布及最大稳定粒径，分析了触变性对破碎过程及最大稳定粒径的影响。经过模型计算与实验结果比较，发现以初始粘度计算的理论值预测最大稳定粒径更为合适。     

Investigations on hydrodynamics and mass transfer in gas–liquid stirred reactor using computational fluid dynamics

In this work, the hydrodynamics and mass transfer in a gas–liquid dual turbine stirred tank reactor are investigated using multiphase computational fluid dynamics coupled with population balance method (CFD–PBM). A steady state method of multiple frame of reference (MFR) approach is used to model the impeller and tank regions. The population balance for bubbles is considered using both homogeneous and inhomogeneous polydispersed flow (MUSIG) equations to account for bubble size distribution due to breakup and coalescence of bubbles. The gas–liquid mass transfer is implemented simultaneously along with the hydrodynamic simulation and the mass transfer coefficient is obtained theoretically using the equation based on the various approaches like penetration theory, slip velocity, eddy cell model and rigid based model. The CFD model predictions of local hydrodynamic parameters such as gas holdup, Sauter mean bubble diameter and interfacial area as well as averaged quantities of hydrodynamic and mass transfer parameters for different mass transfer theoretical models are compared with the reported experimental data of [Alves et al., 2002a] and [Alves et al., 2002b] . The predicted hydrodynamic and mass transfer parameters are in reasonable agreement with the experimental data.     

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.     

CFD simulation of dilute phase gas-solid riser reactors: Part I—a new solution method and flow model validation

A three-dimensional simulation of a dilute phase riser reactor (solid mass flux: ) is performed using a novel density based solution algorithm. The model equations consisting of continuity, momentum, energy and species balances for both phases, are formulated following the Eulerian-Eulerian approach. The kinetic theory of granular flow is applied. The gas phase turbulence is accounted for via a k-ε model. An extra transport equation describes the correlation between the gas and solid phase fluctuating motion. The solution algorithm allows a simultaneous integration of all the model equations in contrast to the sequential multi-loop solution in the conventional pressure based algorithms, used so far in riser simulations. The simulations show an unsteady behaviour of the flow, but a core-annulus flow pattern emerges on a time-averaged basis. The abrupt nature of the T type outlets causes a significant recirculation of gas and solid from the top of the riser. The flow near the outlets is highly non-symmetric and has a three-dimensional character. A significant decrease of the gas phase turbulence and particle granular temperature across the riser length is attributed to the presence of small particles, which is qualitatively consistent with the experimental data from literature.     

Implementation and analysis of numerical solution of the population balance equation in CFD packages

Simulation of polydisperse flows must include the effects of particle–particle interaction, as breakage and aggregation, coupling the population balance equation (PBE) with the multiphase modelling. In fact, the implementation of efficient and accurate new numerical techniques to solve the PBE is necessary. The direct quadrature method of moments, known as DQMOM, is a moment-based method that uses an optimal adaptive quadrature closure and came into view as a promising choice for this implementation. In the present work, DQMOM was implemented in two CFD packages: the commercial ANSYS CFX, through FORTRAN subroutines, and the open-source OpenFOAM, by directly coding the PBE solution. Transient zero-dimensional and steady one-dimensional simulations were performed in order to explore the PBE solution accuracy using several interpolation schemes. Simulation cases with dominant breakage, dominant aggregation and invariant solution (equivalent breakage and aggregation) were simulated and validated against an analytical solution. The solution of the population balance equation was then coupled to the two-fluid model, considering that all particles classes share the same velocity field. Momentum exchange terms were evaluated using the local instantaneous Sauter mean diameter of the size distribution function. The two-dimensional tests were performed in a backward facing step geometry where the vortex zones traps the particles and provides high rates of breakage and aggregation.     

Solution of the droplet breakage equation for interacting liquid-liquid dispersions: a conservative discretization approach

A conservative discretization approach for the population balance equation (PBE) with only droplet breakage describing the hydrodynamics of a continuously interacting liquid-liquid dispersion is presented. The approach is conservative in the sense that it conserves any two integral properties associated with the number droplet distribution and thus it is considered internally consistent. The discrete set of equations is laid down through applying the subdomain method where it is shown that this set of discrete equations is only internally consistent with respect to one integral property. The internal consistency is enforced by introducing a set of two auxiliary functions that are uniquely determined by matching the integral properties obtainable from the discrete set against those from the continuous PBE. However, it is shown that this conservation of integral properties is not exact for all the subdomains and hence it results in what we call the intrinsic discretization error (IDE). This IDE is not only associated with this approach, but also it is found inherently existing in the fixed-pivot (FP) technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996a) 1333). The derived equations of the IDE for the present discretization approach and the FP technique generalized to continuous flow systems show that the present approach enjoys a small value of the IDE. To validate the discretization approach, two analytical solutions for the continuous PBE are presented, where good agreement is found between the predicted and the analytical solutions. To assess the reliability of the present discretization approach two experimentally validated breakage frequency functions describing droplet breakage in a turbulent continuous phase as well as two daughter droplet distributions are considered. The convergence characteristics show that the present discretization approach has an identical convergence rate as that of the FP technique, and in some cases it is superior to it. This rate of convergence is found approximately proportional to the square of the inverse of the number of subdomains.