Population Balance Model Research Articles

Numerical simulation of hydrate particle behavior in vertical pipeline gas-liquid flow

During the extraction of deep terrestrial and deep-sea natural gas reservoirs, changes in temperature and pressure in the production well can lead to the formation of natural gas hydrates in the wellbore, blocking the flow channel and posing significant challenges to gas well production and management. To address this, we have established a mathematical model to describe the flow characteristics of hydrate multiphase flow systems in vertical pipelines. We employ Computational Fluid Dynamics simulations coupled with a Population Balance Model to predict the formation and breakage of hydrate particles, taking into account mechanisms such as particle collision, aggregation, and breakage. We have analyzed the impact of gas flow rate, slurry flow rate, and hydrate volume fraction on hydrate particle size in the pipeline. The results indicate that hydrate particles tend to aggregate at the bottom of the pipeline. The larger the hydrate slurry flow rate, the faster the hydrate particle size increases. As the gas phase flow rate increases, the increase in particle size slows down. High gas phase flow rates help form turbulence to suppress particle aggregation, velocity fluctuations contribute to particle aggregation growth, and stable velocities facilitate particle decomposition. The increase in hydrate volume fraction significantly increases the diameter of hydrates in the center of the pipeline. The research results can provide suggestions for wellbore blockage caused by hydrate particle aggregation on the wall and a decline in gas well production during actual natural gas well production.

Research on the gravity sedimentation law of watery crude oil in large oil tanks

ABSTRACT Based on field survey data, a numerical simulation method that combines an indoor settling experiment with the Mixture multiphase flow model and the Population Balance Model (PBM) was adopted in this article to study and analyze the sedimentation law of water-containing crude oil in a tank under different initial particle sizes and different continuous-phase viscosities. The results indicate that the maximum error in the comparison of the water content changes at different positions in the sedimentation cylinder after 1 h with experimental values is approximately 8.34%, confirming the precision of the developed settling model. When the initial particle size of liquid droplets in the tank increases from 8 μm to 64 μm, the maximum water content in the tank increases by 1.115%, and the magnitude of the increase in the average droplet size decreases by 90%. As the viscosity of the oil increases (from 35.26 mPa · s to 77.77 mPa · s), the maximum water content in the tank decreases by 1.331%, while the height of the stable water content region increases by 13.2%.

Assessment of different energy-based breakage distribution functions in population balance model of an industrial scale continuously operated wet stirred media mill

This study aimed at assessing various energy-based breakage distribution functions Bij within the context of a population balance model (PBM) of copper ore milling in a wet stirred media mill (WSMM). First, drop weight impact tests were performed to determine the size distribution parameters, tn. Second, various tn models called tn-fitted and tn-regenerated approaches were used to fit experimental tn, and their goodness-of-fits were compared. They were then used to construct the matrices of the cumulative breakage distributions, Bij, of the PBM, while a power-law form of the specific breakage rate function, Si, was assumed. By fitting the PBM to the product particle size distributions (PSDs) obtained under various rotor speeds, suspension volumetric flow rate, and solids content, we estimated the Si parameters. Results suggest that except the Napier-Munn model, all tn models fitted the experimental drop weight data reasonably well, which was reflected in the PBM fitting of the product PSDs. The PBM fits with Bij constructed with the tn-regenerated approach were more accurate than those with the tn-fitted approach. The PBM was validated with additional tests that were not considered in the parameter estimation. Overall, we have established that the choice of tn function and the methodology to determine Bij can affect the PBM predictions of the WSMM process significantly.

Analysis of Flow Pattern and Bubble Behavior in Slab Continuous Casting with Mold Electromagnetic Stirring

The electromagnetic stirring (EMS) and argon blowing are the widely recognized and extensively employed technologies to control the flow pattern in slab continuous casting, and their interaction is directly linked to the final product. To investigate the flow pattern, bubble characteristics under mold EMS and argon blowing, a mathematical model coupling the electromagnetic field, two‐phase flow is constructed. The population balance model (PBM) under the Eulerian frame is applied to describe the bubbles breakage and coalescence. The results show that the flow regime under the interaction of argon blowing and EMS transfers from bubble ascending flow to intermediate flow to horizontal recirculation flow as EMS current increases, and the large argon flow rate could suppress the formation of horizontal recirculation flow. Meanwhile, EMS can promote the coalescence of bubbles, with the increasing EMS current, the bubble numbers decrease first and then increase, and the bubble mean diameter increases first and then decreases. For obtaining the horizontal recirculation flow, the suitable argon flow rate and EMS current should be matched suitably.

Population Balance Modeling of Particle Size and Porosity in Fluidized Bed Spray Agglomeration

Population Balance Modeling of Particle Size and Porosity in Fluidized Bed Spray Agglomeration

Numerical investigation of dynamic gas–liquid separator by population balance model

Dynamic gas–liquid separator (DGLS) can efficiently separate gas and liquid phases and are widely used in aerospace, chemical, and petroleum engineering. The energy loss and separation efficiency within the DGLS are studied through the combination of numerical simulations and experiments. Three-dimensional transient Reynolds-Averaged Navier–Stokes equations were solved to analyze the fluid dynamics within the DGLS. The bubble aggregation and breakup in oil were simulated by using the population balance model. Experimental data were meticulously compared with numerical results to validate the accuracy and reliability of the numerical methods. The findings revealed a direct correlation between the inlet flow rate and various performance metrics of the DGLS. Specifically, as the inlet flow rate increased, the energy loss within the DGLS escalated, resulting in higher power consumption. The degassing rate of the DGLS exhibited a decreasing trend with increasing inlet flow rate, while the de-oiling rate showed an inverse relationship. The optimal performance of the separator was observed at an inlet flow rate of 140 m3·d−1, with ηg* and ηl* reaching 0.94 and 0.99, respectively. The relationship between the Qo and the η* and Po was fitted by a second-order polynomial. Moreover, the rotational speed of the DGLS demonstrated a positive correlation with energy consumption, accompanied by an increase in power output. However, the separation efficiency of the DGLS exhibited a non-linear relationship with rotational speed, peaking at a specific value before marginally declining. The optimization of degassing and dewatering rates occurred at a rotational speed of 2500 r·min−1. These findings underscore the importance of carefully adjusting operational parameters to achieve optimal performance and energy efficiency within DGLS.

Maximizing similarity: using correlation coefficients to calibrate kinetic parameters in population balance models

Crystallization plays a crucial role as a separation and purification technique, particularly in the chemical and pharmaceutical industries. By adjusting process parameters, the productivity, product quality, and efficiency of downstream processes can be improved. In complex processes, model-based design becomes invaluable. Population Balance Models (PBMs) have successfully aided the chemical industry in achieving more effective production processes for decades. These models can utilize various input data sources to identify the dominant mechanisms and calibrate model parameters. While inline particle monitoring tools serve as excellent qualitative descriptors, challenges arise from experimentation and data interpretation, hindering their direct application in the kinetic parameter estimation of PBMs. In this study, we present a novel approach that utilizes information from inline particle monitoring tools for the kinetic parameter estimation of PBMs, bypassing the associated obstacles. Our pioneering approach relies on offline product size data and the correlation-based utilization of inline particle monitoring information. The paper compares this novel strategy with two parameter estimation techniques: the classical method using solute concentration and product size data and a somewhat naïve approach, which assumes that the inline particle monitoring data can directly be compared with the simulations. The prediction capabilities are evaluated through two in-silico case studies. The results indicate that the precision and predictive capability of the correlation-based technique are comparable to the classical approach, both for noisy data and for a system undergoing significant agglomeration and deagglomeration. The use of Pearson’s correlation coefficient yields the best results in the novel cases. These findings in in-silico datasets provide a foundation and motivation for the practical application of this idea, unleashing the so-far hidden model development potential of such measurements.

Population balance modelling and reconstruction by quadrature method of moments for wet granulation

Population balance methods utilised in multiphase flow simulations mark a significant advancement in computational fluid dynamics. However, existing approaches exhibit shortcomings, such as being prone to inaccuracies or being computationally prohibitive. Addressing these challenges, a recent innovation in closure for the method of moments is the introduction of quadrature based moments methods (QBMM). Discretising a distribution by a number of discrete elements, QBMM facilitate efficient and accurate tracking of density distributions, particularly for particle size distributions (PSD). However, obtaining the full particle size distribution information using these methods requires reconstructing the distribution from a finite set of moments, which is not a trivial step.This study introduces a novel combination of the maximum entropy reconstruction (MER) and QBMM, establishing a robust and rapid framework for the time evolution and reconstruction of PSDs. As proof of concept for this framework, we focus on the direct quadrature method of moments (DQMOM) with spatially homogeneous and monovariate distributions. We show that coupling of MER with DQMOM has numerous advantages. To verify the framework, special cases of constant growth, aggregation, and breakage are considered for which analytical solutions can be found. Furthermore, we show the advantage of using DQMOM with volume-based over length-based distributions, and address numerical as well as theoretical issues.Application of the framework is successfully conducted on the evolution of the PSD from a twin-screw wet granulation dataset, considering all active primary physical mechanisms in a wet granulation process, namely growth, aggregation, and breakage. This showcases the consistency of the proposed framework and underscores its applicability to real-world scenarios.

Characteristics of gas–liquid two-phase flow in a stirred tank with double-layer punched impeller

This study investigates the gas–liquid two-phase flow characteristics in a stirred tank equipped with a double-layer punched impeller. Numerical simulations are conducted to analyze flow dynamics, gas holdup, bubble sizes, and distributions under various operational conditions. The results show a high degree of agreement between experimental and simulated power values and gas holdup distributions, validating the reliability of the computational fluid dynamics–population balance model coupling approach. The combination of the punched four-inclined-blade up-pumping turbine and the punched Rushton impeller exhibits excellent bubble dispersion characteristics, with overall small bubble sizes. Increasing the rotational speed can enhance turbulence within the flow field and accelerate the liquid phase velocity, which facilitates gas diffusion and improves gas–liquid mixing efficiency. Additionally, higher rotational speed further intensifies the shear effect of the punched impeller, resulting in a reduction in average bubble size.

Coupled Wall Boiling and Population Balance Model for High Void Fraction Flows Under Sub-Cooled Nucleate Boiling Regime: Model Development and Validation

Abstract The subcooled flow boiling of water in a vertical annulus channel is studied numerically at low-pressure conditions. The two-fluid model is developed with flow-regime dependent interfacial transfers for mass, momentum, and energy using the algebraic interfacial area density (AIAD) framework. A discrete population balance model is used to mechanistically determine the vapor bubble diameter in the flow channel by considering the bubble aggregation and breakup effects. Energy balance at the heated wall for the subcooled nucleate boiling is handled using a suitable wall boiling model. A coupling is achieved between the discrete population balance and the wall boiling model for the nucleation and the growth rate of the vapor bubbles along the heated wall. The developed model simulates the reference experimental cases of flow boiling in a vertical channel for various flow and thermal conditions. At low wall heat flux, the wall boiling generates vapor bubbles near the heated wall and within the bubbly flow regime. With an increase in the wall heat flux, the aggregation and evaporation cause the formation of larger bubbles, which progress toward the flow channel core region, a phase that is representative of the transitional flow regime. The model's capability to predict such flow regime transition is validated with the experimental results. The bubble aggregation is found to be dominant compared to the breakup, and thus, proper choice of the aggregation factor is important for the accurate prediction of vapor parameters for the subcooled flow boiling at low-pressure conditions.

Numerical simulation of hydrate flow in gas-dominated undulating pipes considering nucleation and deposition behaviors

The distribution and deposition of hydrates in deep-water gas transportation pipelines are crucial for safety. Current simulations of hydrate flow based on the population balance model have predominantly focused on water-dominant systems and neglect the nucleation and deposition of hydrates. By establishing a model for a hydrate flow in a gas-dominant system that considers the uneven distribution of the liquid film on the wall, hydrate nucleation, gas–liquid mass transfer, particle adhesion and aggregation, and dynamic deposition, we achieved simulation of the entire process of hydrate formation, aggregation, breakage, and deposition. Simulations in undulating pipelines were carried out to investigate the effects of gas velocity, liquid injection, pressure, and pipeline structure on distribution patterns. The results showed increasing the gas velocity enhanced the dispersion of particles and increasing the pressure increased the rate of aggregation. The formation of blocky aggregates posed significant risk in the rear section of the lower-bend pipe.

Numerical Study of Internal Flow Field in a Disc Stack Centrifuge Based on Mixture-PBM Model

Disc stack centrifuge belongs to one kind of sedimentation centrifuge, widely used in the environmental protection, pharmacy, and chemical industries, etc. The flow process inside the disc stack centrifuge seriously affects the separation efficiency. However, the flow process inside the disc stack centrifuge and its influence on the separation efficiency have not yet been detailed. We plan to study the flow process of oil and water phases inside the disc stack centrifuge and to explore the process of fragmentation and accumulation of water droplets. In this study, the Mixture-PBM (Population Balance Model) model is used to numerically simulate the two-phase flow of oil and water in the disc stack centrifuge and compare it with the tests. The research found that with the increase in rotational speed, the separation efficiency rises in both the test and numerical simulation results, and the difference between the test and simulation results is below 1%. The effect of ribs on the flow is considered, and the results show that the hysteresis of the liquid flow in the disc stack centrifuge is significantly reduced after considering the ribs, and the numerical simulation results can reach 98% of the theoretical results. Fluid entering the separation channel from the neutral pore creates a vortex, and as the dimensionless number λ increases, the degree of deviation of the fluid’s trajectory from the generatrix increases. The circumferential and radial velocities in the separation channel appear large in the center and small near the wall. The water content in the rising channel gradually decreases, and 90% of the water finishes settling in the distributor. The processing volume of the separation channel in each layer shows a small bottom layer, a large top layer, and a uniform law in the middle. The coalescence of water droplets occurs mainly in the separation channel, as found by analyzing the laws of the internal flow of the disc stack centrifuge, which provides the basis for improving the structure of the disc stack centrifuge, increasing the separation efficiency and reducing the floor space.

Solving crystallization/precipitation population balance models in CADET, Part II: Size-based Smoluchowski coagulation and fragmentation equations in batch and continuous modes

A particle size-based Smoluchowski coagulation and fragmentation equation was solved in the free and open source process modeling package CADET. The WFV and MCNP schemes were selected to discretize the internal particle size coordinate. Weights in these schemes were modified to preserve and conserve the zeroth and third moments for size-based equations. Modified propositions and proofs for the scheme are provided. Analytical Jacobians were derived and implemented to reduce the solver’s runtime. A two-dimensional Smoluchowski coagulation and fragmentation equation with axial position as external coordinate was formulated and discretized to support simulations of continuous particulate processes in dispersive plug flow reactors. Five 1D and four 2D test cases were used to validate the implementation and benchmark the solver’s performance. The runtime, L1 error norm, L1 error rate, particle size distribution moments up to sixth order and several scalar metrics were analyzed in detail.

Sensitivity analysis and uncertainty quantification based on the analytical solution for nanoparticle-stabilized foam flow in porous media

A model describing nanoparticle-stabilized foam displacement is proposed in this study. Based on literature experimental data, we include nanoparticle dependence in a simplified version of the Stochastic Bubble Population balance model in local equilibrium. To describe the gas–water fractional flow, we utilize quadratic Corey permeabilities. Our model consists of non-strictly hyperbolic conservation laws, and we investigate the existence of a global solution as a sequence of waves. This simpler model maintains the solution structure of classical Corey’s relative permeability model and allows us to obtain algebraic expressions to analyze the solution type and construct the water saturation profiles. We perform uncertainty quantification and sensitivity analysis of the model describing the foam injection assisted by nanoparticles, considering three relevant quantities of interest: breakthrough time, cumulative water production, and pressure drop. The analytical framework developed allows us to obtain these quantities. Our results show that the effect of nanoparticles is greater than the model’s uncertainty (for all quantities), suggesting that measuring it experimentally is statistically feasible. The presence of nanoparticles also significantly reduces the uncertainty propagation due to foam stabilization. From the sensitivity study, the endpoints of water and gas relative permeability are the most significant parameters for the breakthrough time. For the maximum pressure drop, the gas endpoint relative permeability and the concentration of nanoparticles stand out. Otherwise, water production is more sensitive to a foam-related parameter most of the time. We achieve convergence even using the classical Monte Carlo method, showing how analytical solutions drastically reduce computational costs.

Modeling Polymer Microencapsulation Processes Using CFD and Population Balance Models

Computational fluid dynamics (CFD) modeling has emerged as a valuable tool for investigating complex processes like microencapsulation. This paper aims to validate the ability of CFD simulations to predict particle size distribution in a polymer microencapsulation process. The CFD modeling approach employed a Eulerian multiphase framework, incorporating a discrete population balance model to track the evolution of the droplet population. A realizable k-ε turbulence model and a multiple reference frame strategy were utilized to capture the system’s flow dynamics. The results reveal that while the CFD simulations align well with experimental data at higher agitation speeds (>10,000 rpm), discrepancies arise at lower speeds (<7500 rpm), indicating a challenge in accurately capturing turbulent viscous regimes. Despite these challenges, the CFD model demonstrates robust predictive capabilities for droplet formation and distribution in microencapsulation processes, validated by error margins within the acceptable limits. The validated model can be used as a reliable tool to guide experimental efforts and optimize process parameters, contributing to an enhanced understanding and control of microencapsulation processes.

A two-stage mechanistic reduced-order model of pharmaceutical tablet dissolution: Population balance modeling and tablet wetting functions

We propose a two-stage reduced-order model (ROM) of pharmaceutical tablet dissolution that is comprised of (i) a mechanistic dissolution function of the active pharmaceutical ingredient (API) and (ii) a tablet wetting function. The former is derived from a population balance model, using a high-resolution finite volume algorithm for a given API crystal size distribution and dissolution rate coefficient. The latter is obtained from the mechanistic understanding of water penetration inside a porous tablet, and it estimates the rate at which the API is exposed to the buffer solution for a given formulation and the dimensions of the tablet, contact angle, and surface tension between the solid and liquid phases, liquid viscosity, and mean effective capillary radius of the pore solid structure. In turn, the two-stage model is mechanistic in nature and one-way coupled by means of convolution in time to capture the start time of the API dissolution process as water uptake, swelling, and disintegration take place. The two-stage model correlates dissolution profiles with critical process parameters (CPPs), critical material attributes (CMAs), and other crucial critical quality attributes (CQAs). We demonstrate the model’s versatility and effectiveness in predicting the dissolution profiles of diverse pharmaceutical formulations. Specifically, we formulate and fabricate acetaminophen and lomustine solid tablets using different API content and size distributions, characterize their dissolution behavior, and estimate capillary radius as a function of tablet porosity. The estimations generated by the proposed models consistently match the experimental data across all cases investigated in this study.

Investigation of Wall Boiling Closure, Momentum Closure and Population Balance Models for Refrigerant Gas–Liquid Subcooled Boiling Flow in a Vertical Pipe Using a Two-Fluid Eulerian CFD Model

The precise design of heat exchangers in automobile air conditioning systems for more sustainable electric vehicles requires an enhanced assessment of CFD mechanistic models for the subcooled boiling flow of pure eco-friendly refrigerant. Computational Multiphase Flow Dynamics (CMFDs) relies on two-phase closure models to accurately depict the complex physical phenomena involved in flow boiling. This paper thoroughly examines two-phase CMFD flow boiling, incorporating sensitivity analyses of critical parameters such as boiling closures, momentum closures, and population balance models. Three datasets from the DEBORA experiment, involving vertical pipes with subcooled boiling flow of refrigerant at three different pressures and varying levels of inlet liquid subcooling, are used for comparison with CFD simulations. This study integrates nucleate site density and bubble departure diameter models to enhance wall boiling model accuracy. It aims to investigate various interfacial forces and examines the S-Gamma and Adaptive Multiple Size-Group (A-MuSiG) size distribution methods for their roles in bubble break up and coalescence. These proposed approaches demonstrate their efficacy, contributing to a deeper understanding of flow boiling phenomena and the development of more accurate models. This investigation offers valuable insights into selecting the most appropriate sub-closure models for both boiling closure and momentum closure in simulating boiling flows.

Numerical simulation of sulfur particle agglomeration at bends of high sulfur natural gas gathering pipelines based on Euler–PBM coupling

Sulfur deposition can result in an increase in the wall thickness of high-sulfur natural gas gathering pipelines, leading to issues like unstable pipeline flow. It is crucial to reveal the aggregation of sulfur particles at key locations of high-sulfur natural gas gathering pipelines to predict the location and amount of sulfur deposition in the pipelines. In this paper, the Euler–PBM (Population balance model) coupling is used to establish a numerical simulation model of gas–solid two-phase pipe flow accompanied by sulfur particle agglomeration in the pipe bends, focusing on the influence of sulfur particle volume fraction, pipe inclination angle and inlet flow velocity on sulfur particles agglomeration behavior. The results show that the sulfur particles have a significant agglomeration effect at the bend of the collecting pipeline, and the agglomeration growth occurs to different degrees throughout the bend, and the main area of sulfur particles agglomeration is near the top wall of the pipeline, followed by other areas near the wall of the pipeline. When the inlet volume fraction of sulfur particles was increased from 0.05 to 0.25%, and the inclination angle of the pipe was increased from 30° to 90°, the distribution range of sulfur particle size after agglomeration became wider, and the maximum size of sulfur particles was 187.56 μm, and the effect of sulfur particle agglomeration was enhanced; the inlet flow rate was increased from 3.0 to 9.0 m/s, and the reduction range of sulfur particle size after agglomeration was 5.68–9.87 μm. The maximum particle size of sulfur particles also decreased, and the effect of sulfur particle agglomeration was weakened.

Column flotation hydrodynamics operating in the heterogeneous regime estimated by electrical resistance tomography and multi-phase CFD model

ABSTRACT This paper investigates the two-phase flow hydrodynamics of a column flotation operated from a homogeneous to a heterogeneous flow regimes using experimental and computational fluid dynamics (CFD) techniques. Experiments were conducted in a lab scale column flotation of 0.1 m internal diameter and 2.5 m length at various superficial gas velocities using a dual plane electrical resistance tomography (ERT). The mean gas holdup increased from 2 to 18% with the increase in superficial gas velocity from 0.6 to 7.2 cm/s. The gas holdup-based bubble swarm velocity method identified four distinct flow regimes, i.e. homogeneous bubbling regime, narrow discrete bubbling regime, a sustained helical flow regime and churn turbulent flow regime in the column. Further, transient simulations were conducted using a two-fluid model coupled with a population balance model to assess the flow behavior inside the column. Virtual mass, lift and drag forces were incorporated into the model to account for the interphase forces. Numerical simulations predicted mean gas holdup was matching with the experimental data at low gas velocities (<2.4 cm/s), and a marginal deviation was noted at higher gas velocities (>2.4 cm/s). The predicted radial gas holdup profiles were found to change from flatter to parabolic with the increase in gas velocities, i.e. similar to the experiments. The effect of particle size on the rate constant and recovery was estimated with Luttrell and Yoon’s performance model using the particle floatability input data and the two-phase experimental data. The developed CFD model can be useful to optimize the operating and design parameters for efficient operation of the column flotation process.

Numerical simulations of particle concentration and size effects in a slurry bubble column with a CFD‐PBM coupled model

AbstractThe combined effects of particle concentration (0–20 vol%) and particle size on the hydrodynamics of a slurry bubble column were studied by computational fluid dynamics coupled with population balance model (CFD‐PBM) in a wide range of superficial gas velocity (0.02–0.20 m/s). This CFD‐PBM coupled model included multiple effects of particles, namely increased slurry viscosity, reduced drag force, promoted bubble coalescence, and attenuated liquid turbulence, among which turbulence attenuation was crucial. To quantify liquid turbulence attenuation, a mechanistic model was established by equating the energy dissipated by particle motion to the attenuated liquid turbulence energy calculated from turbulent energy spectrum. The turbulent energy dissipation rate was reduced by over 60% at a solid concentration of 20%. Using this model, the effects of particle concentration and particle size on overall gas holdup were well predicted. This model advanced our understanding of how particles affect the gas holdup in a slurry bubble column.