CFD simulation of gas–liquid flow in a high-pressure bubble column with a modified population balance model

In this study, based on the Luo bubble coalescence model, a model correction factor C_e for pressures according to the literature experimental results was introduced in the bubble coalescence efficiency term. Then, a coupled modified population balance model (PBM) with computational fluid dynamics (CFD) was used to simulate a high-pressure bubble column. The simulation results with and without C_e were compared with the experimental data. The modified CFD-PBM coupled model was used to investigate its applicability to broader experimental conditions. These results showed that the modified CFD-PBM coupled model can predict the hydrodynamic behaviors under various operating conditions.     

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.     

Population balance modelling for bubbly flows with heat and mass transfer

Population balance equations combined with a three-dimensional two-fluid model are employed to predict bubbly flows with the presence of heat and mass transfer processes. Subcooled boiling flow belongs to this specific category of bubbly flows is considered. The MUSIG (MUltiple-SIze-Group) model implemented in CFX4.4 is further developed to account for the wall nucleation and condensation in the subcooled boiling regime. Comparison of model predictions against local measurements near the test channel exit is made for the radial distribution of the bubble Sauter diameter, void fraction, interfacial area concentration and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Additional comparison was also performed against existing boiling model in CFX4.4 and the modified model developed in our previous work (Int. J. Heat Mass Transfer 45 (2002) 1197). Good agreement is better achieved with the local radial bubble Sauter diameter, void fraction, interfacial area concentration and liquid velocity profiles against measurements using the newly formulated MUSIG boiling model over the simpler boiling models. However, significant weakness of the model is still evidenced in the prediction of the vapour velocity. Work is in progress to circumvent the deficiency of the model by the consideration of additional momentum equations or an algebraic slip model to account for bubble separation.     

Population balance models for the thermal degradation of PMMA

A widely accepted view of the thermal degradation of polymers such as PMMA is that an initiation reaction produces radical fragments that undergo rapid depropagation and are also converted back to molecules by a termination reaction. This mechanism is applied to a population of linear molecules and radicals and the evolution of the population is modelled by appropriate discrete sets of ordinary differential equations. In particular, end-chain and random initiation reactions with first-order termination are analysed and compared with experimental data. We find on comparison with TG data for PMMA that the initiation reaction is important in dictating the qualitative behaviour of the overall rate of thermal degradation. Furthermore, the behaviour of degradation rate with initial degree of polymerisation is also investigated and interpreted within the framework of the model.     

Modeling and simulation of mixing in water-in-oil emulsion flow through a valve-like element using a population balance model

Emulsion flows are very common in natural processes as well as in several engineering areas, such as in the process of desalting crude oil that occurs in refineries. This kind of flow is described as a polydispersed multiphase flow. In this work, we evaluated the behavior of water-in-oil emulsion flowing through a duct with an element used to mimic the effect of a globe valve. An Eulerian multi-fluid approach was employed by solving the population balance equation coupled with computational fluid dynamics. Coalescence and breakage models recently developed were extended to this inhomogeneous model. A bivariate population balance problem was also solved to demonstrate the mixing caused by the valve-like element. The simulated results showed good agreement with the available experimental data for the Sauter and DeBroukere mean diameters.     

Population balance modeling. Promise for the future

Population balance modeling has received an unprecedented amount of attention during the past few years from both academic and industrial quarters because of its applicability to a wide variety of particulate processes. In this article, a fresh look is taken of the basic issues of the application of population balances towards strengthening the approach as well as widening the scope of their applications with regard to formulation, computational methods for solution, inverse problems, control of particle populations and stochastic modeling.     

PDF method for population balance in turbulent reactive flow

Turbulent reactive flows with particle formation, such as soot formation and precipitation, are characterized by complex interactions between turbulence, scalar transport, particle formation and particle transport and inter-particle events such as coagulation. The effect of formation, growth and coagulation on the particle size distribution (PSD) must be modelled by the population balance equation (PBE). While the PBE has been studied extensively in homogeneous systems and, recently, in simple flows, its coupling with turbulent reactive flows poses a wealth of new questions. Processes such as nucleation, growth and coagulation are described by kinetic laws that link them to the local concentrations of the reactive scalars, which are random in a turbulent flow. This accounts for additional mechanisms that induce randomness and fluctuations to the particle concentration and PSD. Furthermore, conventional RANS closure of the coagulation term PDE (which describes the evolution of the PSD) leads to unknown correlations. In this work a new pdf approach is developed, based on the transport of the joint pdf of reactive scalars and particle number densities at different sizes, which overcomes the additional closure problems. It is also shown how the pdf method can be solved numerically via Monte-Carlo methods, and this is demonstrated via two applications in a partially stirred reactor: precipitation via nucleation-growth and coagulation. In each case the pdf method is compared with models that neglect correlations at various levels, and it is demonstrated that the interaction of turbulence with particle formation mechanisms accounts for significant deviations in the PSD.     

Population balance modelling for a flow induced phase inversion based granulation in a two-dimensional rotating agglomerator

A novel two-dimensional rotating agglomerator was developed to carry out the flow induced phase inversion (FIPI) based granulation. The process in this agglomerator shows that a continuous paste flow (mixed with liquid binder and primary particles) is extruded into the interstice of two relatively rotating disks, as the paste becomes solidified due to the loss of heat to the disks, it is then broken into granules by the shearing force imposed by the rotating disk. Experimental measurements have shown that the size of these granules is enlarged along the positive radial direction of the disks. It is also found that these granules contain approximately the same quantity of binder in terms of its volume fraction. The paper thus proposes a population balance (PB) model to describe the growth of the granules by considering a size independent agglomeration kernel. The PB simulated results are found to be well capable of describing the change of the particle size distribution (PSD) of the granules in the radial direction. This study also proposes a velocity profile for the paste flow and attempts to establish a quantitative relationship between the granulation rate and the deformation rate as this would help us understand the mechanism of the agglomeration. It is hoped that this study would be used to improve the design of the agglomerator and to assure the control of the process and the granular product quality.     

A safe and efficient process for the preparation of difluoromethane in continuous flow

Difluoromethane is typically produced vialiquid-phase fluorination as performed in a batch reactor. However,this process suffers from some problems, e.g., severe corrosion of the reactor, high safety risk, and the regeneration of the catalyst. In this paper, a flow process as performed in the tubular reactor was designed. The optimum conditions for continuous synthesis of difluoromethane were obtained as follows: the reaction temperature was 100℃, the molar ratio of dichloromethane to hydrogen fluoride was 1.6:1 and the reaction time was 300 s. The operation of the cyclic process was stable for 24 h with the conversion per pass of hydrogen fluoride up to 16.2%.The unreacted raw materials were easily reused. The deactivation of the common catalyst, antimony pentachloride, was investigated by catalyst concentration curve and XPS analysis. The approach proposed in this work is proven to be safe, efficient and low amount of catalyst.     

Population balance model versus lumped model for emulsion polymerisation: Semi-batch and continuous operation

In this article, a systematic comparison is made of a detailed population balance model of the emulsion polymerisation process that accounts for the dynamic evolution of the entire particle size distribution with a simple model of the process that accounts only for the dynamic evolution of the average particle size and the total particles. Both models account for the underlying mechanisms of the process to the level admissible within their respective frameworks. The predictions of the two models are compared under both batch and continuous operation. The aim is to elucidate the degree of disparity of model predictions for batch operation, and to elucidate the ability of the simple lumped models to predict oscillatory dynamics for continuous operation. The focus is on the comparison of the predictions of important control variables such as solids content (conversion), total particles and average particle size, but also those of important particle phenomena of nucleation, growth and coagulation. It is found that with respect to these lumped control variables, the simple model performs well in matching the detailed model, and hence will be a very valuable tool for the purpose of on-line feedback control. However, the detailed models will be important for distributed control variables such as the entire particle size distribution.     

Population balance modelling of protein precipitates exhibiting turbulent flow through a circular pipe

The breakage of liquid-liquid, solid-liquid and solid-gas dispersions occurs in many industrial processes during the transport of particulate materials. In this work, breakage of whey protein precipitates passing through a capillary pipe is examined and an experimentally derived breakage frequency is applied to construct a suitable population balance model to characterize the breakage process. It has been shown that the breakage frequency of precipitate particles is highly dependent on their shear history and on the turbulent energy dissipation rate in the pipe. The population balance equation (PBE) uses a volume density based discrete method which is adapted from mass density based discretization. In addition to comparing the model with experimental data, predicted results at different velocities are presented. It was found that the population balance breakage model provides satisfactory results in terms of predicting particle size distributions for such processes.     

A generalized population balance model for the prediction of particle size distribution in suspension polymerization reactors

In the present study, a comprehensive population balance model is developed to predict the dynamic evolution of the particle size distribution in high hold-up (e.g., 40%) non-reactive liquid-liquid dispersions and reactive liquid(solid)-liquid suspension polymerization systems. Semiempirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersed monomer droplets in terms of the type and concentration of suspending agent, quality of agitation, and evolution of the physical, thermodynamic and transport properties of the polymerization system. The fixed pivot (FPT) numerical method is applied for solving the population balance equation. The predictive capabilities of the present model are demonstrated by a direct comparison of model predictions with experimental data on average mean diameter and droplet/particle size distributions for both non-reactive liquid-liquid dispersions and the free-radical suspension polymerization of styrene and VCM monomers.     

Multi-scale product property model of polypropylene produced in a FBR: From chemical process engineering to product engineering

A multi-scale product model has been built to characterize the polypropylene (PP) formation dynamics in a catalytic FBR. For the first time, the gas–solid flow field, the morphological and molecular properties of particles, as well as their dynamics can be simultaneously obtained by solving the unique model that couples a CFD model, a population balance model (PBM) and moment equations. The quantitative relationships between the operating conditions and the multi-scale particle properties have been further established. The results demonstrate that the product model can be used to guide a multi-scale generalization of the polymer product from chemical process to product engineering.     

Theoretical prediction of flow regime transition in bubble columns by the population balance model

Theoretical prediction of flow regime transition in bubble columns was studied based on the bubble size distribution by the population balance model (PBM). Models for bubble coalescence and breakup due to different mechanisms, including coalescence due to turbulent eddies, coalescence due to different bubble rise velocities, coalescence due to bubble wake entrainment, breakup due to eddy collision and breakup due to large bubble instability, were proposed. Simulation results showed that at relatively low superficial gas velocities, bubble coalescence and breakup were relatively weak and the bubble size was small and had a narrow distribution; with an increase in the superficial gas velocity, large bubbles began to form due to bubble coalescence, resulting in a much wider bubble size distribution. The regime transition was predicted to occur when the volume fraction of small bubbles sharply decreased. The predicted transition superficial gas velocity was about 4 cm/s for the air-water system, in accordance with the values obtained from experimental approaches.     

Population balance modeling of non-linear effects in milling processes

Specific breakage rate (selection) and breakage distribution functions are used to describe the birth and death terms in population balance models (PBMs) for milling processes. Traditional PBMs for milling processes are inherently linear because the breakage rate is assumed first-order. The specific breakage rate is independent of the population density while it depends on particle size and possibly on time. Even though the linear theory has been applied with some success to the modeling, optimization, and design of various mills in the last 50 years, many researchers have indicated its restrictions and subjected it to serious criticism. In this paper, we first categorize the experimentally observed deviations from the linear theory and suggest the multi-particle interactions as the origin of these deviations. To account for the peculiar non-linear effects, a phenomenological theory has been proposed via multiplicative decomposition of the specific breakage into a size-dependent apparent breakage rate and a population density dependent functional. The proposed theory recovers the first-order breakage kinetics in the limit, yet it is sufficiently general to explain all experimentally observed deviations. Numerical simulations of a batch milling process have demonstrated the potential of the non-linear theory.     

Population balance modeling of bubble size distributions in a direct-contact evaporator using a sparger model

The study of bubble size distributions in direct-contact evaporators was addressed both theoretically and experimentally. Recently developed models for calculating bubble coalescence and breakage frequencies in isothermal bubble columns were adapted to the population balance equation using the bubble mass as the internal coordinate which was discretized using an expansion of the number density function by impulse functions. A sparger model was developed based on experimental data for a non-coalescing system and using bubble formation models for isothermal and non-isothermal conditions. Bubble size distributions in a direct-contact evaporator operating in the quasi-steady-state regime for four different gas superficial velocities, including the homogeneous and heterogeneous regimes, together with the sparger model, were used for estimating the three empirical parameters from the population balance model, which were observed to be functions of the gas superficial velocity. In all cases considered, the population balance model fitted the experimental data rather well and the regressed parameters exhibit the physically expected behavior with changes in the gas superficial velocity.     

基于相群平衡模型的浮选气泡聚并模拟

气泡尺寸分布直接影响气浮分离效率,而聚并是导致气浮池内气泡尺寸变化的主要因素。首先用实验方法测量气浮接触区气泡尺寸分布,然后用计算流体力学方法对气泡/水两相流动及气泡聚并进行模拟,最后通过对实验和数值模拟结果进行对比建立基于相群平衡模型的浮选气泡聚并行为的模拟方法,分别运用Luo、Free molecular和Turbulent聚并模型对气浮接触区气泡聚并行为进行模拟。结果表明:Turbulent聚并模型计算所得气泡尺寸分布与实验值最接近,适合模拟接触区气泡聚并;气泡平均直径随高度升高先变大后保持不变,气泡聚并主要发生在接触区中下部;气泡的加入增强了接触区流动混乱程度,上部产生对称涡流,中下部呈由边壁向中心的水平流动。     

Population balance modelling of droplet coalescence and break-up in an oscillatory baffled reactor

This paper presents new population balance analysis to describe simultaneous coalescence and break-up in the formation of methylmethacrylate droplets in a batch oscillatory baffled reactor. It is concluded that the droplet data can very well be described by a model in which coalescence is taken to be shear induced, selection for break-up proceeds at a rate proportional to droplet volume and approximately four equally sized break-up fragments are produced per break-up event. It is shown that the experimental droplet size distribution data are self-similar in form and exhibit asymptotic behaviour characteristics also seen in the model. The coalescence rate is found to vary as the square of the oscillation frequency and the selection rate to vary with the oscillation frequency to the power five. As a result the asymptotic mean droplet volume is inversely proportional to the oscillation frequency.