Population Balance Theory Research Articles

Population Mass Balance Model for Precipitation with Turbidity Measurements.

The discretized population balance theory has been proven to be a useful method to simulate systems in which solid particles are present. In this work, we introduce a new approach to model precipitation reactions based on the temporal evolution of product concentration, from which particle size distribution, its dynamics, and the specific interfacial energies can be obtained. For a reference study, the previously investigated calcium oxalate precipitation was selected, where the reaction was followed via turbidity measurement. From the obtained particle size distribution, we can show that at low supersaturation, growth is the dominant process, while at higher supersaturation, nucleation is the dominant process. Moreover, the temporal change of the distribution curve has allowed us to split the precipitation into a nucleation, a growth-driven intermediate, and a saturation regime. Furthermore, the comparison between the experimental and calculated results has proved that the method is suitable for predicting particle size distributions and specific interfacial energies.

Study on microstructure evolution of waxy crude oil emulsions under dynamic cooling conditions.

A rheo-microscopy in situ synchronous measurement system was utilized to investigate the dynamic behavior of water droplets in W/O waxy crude oil emulsions subjected to dynamic cooling conditions, the microstructural evolution of water droplets aggregates can be categorized into three stages based on the various forms of wax crystals. The results show that under the joint action of wax crystals and water droplets, the water droplets aggregation trend and complexity in the system are negatively correlated with the changes of temperature and shear rate, and the water droplets movement behavior is positively correlated with the changes of temperature and shear rate. As the temperature decreases, the minimum edge distance of water droplets decreases by a maximum of 32.1%, the specific surface area (SA) decreases by a maximum of 12.0%, and the fractal dimension increases by a maximum of 11.7%. As the shear rate increases, the minimum edge distance of water droplets increases by up to 27.9%, the specific surface area (SA) increases by up to 10.1%, and the fractal dimension decreases by up to 8.5%. Additionally, an analysis is conducted on the collision aggregation behavior of water droplets in shear flow field based on population balance theory.

Gas–liquid–hydrate flow characteristics in vertical pipe considering bubble and particle coalescence and breakage

The gas–liquid flow behavior laden by hydrate particles is significant for the flow assurance of submarine natural gas hydrate production. An integrated multiphase flow model considering the coalescence and breakage of bubbles and hydrate particles was developed to investigate the gas–liquid–hydrate flow problem in a vertical pipe. The population balance theory was adopted to describe the evolution of the bubble and hydrate particle sizes in a flow field. The effects of particle behavior on flow pattern transition were analyzed. The results indicated that the phase distribution became inhomogeneous during the flow process, and bubbles tended to aggregate at the central region of the pipe. Most hydrate particles agglomerated in the liquid film region, and the particle size increased. The coalescence and breakage of bubbles induced a flow pattern transition from a bubbly to slug and churn flows. However, the agglomeration of hydrate particles increased the viscosity of the liquid phase, further reduced the turbulent intensity, and enhanced the stability of the bubbles. Thus, a higher gas superficial velocity was required for the formation of slug and churn flows. An annular flow was easier to form because the hydrate particles entrained in the liquid phase stabilized the liquid film. This study contributes to the flow assurance of submarine natural gas hydrate production and relevant multiphase flow modeling studies with hydrate particles.

Droplet Growth Model for Dropwise Condensation on Concave Hydrophobic Surfaces.

In this paper, mathematical models for predicting the droplet growth and droplet distribution of dropwise condensation on hydrophobic concave surface are developed and a theoretical analysis of the results of the model simulation is made. Under the assumptions of the Cassie–Baxter wetting mode and the consideration of noncondensable gases, the droplet growth model is not only established by heat transfer through a single droplet but also considered the thermal conductive resistance of the surface promoting layer. In addition, a droplet distribution model has been built based on the population balance theory. According to the calculation, the main thermal resistance in the droplet growth process is the conductive resistance inside the droplet. With the increase of contact angle, the above-mentioned thermal resistance increases; thus, the higher the hydrophobicity is, the slower the droplet growth and the less the droplet density are. Besides, the lower the temperature of the condensing surface is, the faster the droplets grow and the less the droplet density is. The models provide a mathematical tool for predicting the droplet radius at the initial stage of dewing on the concave surface and contribute to the design of functional surfaces in the field of water harvesting.

Consideration of the activity distribution using the population balance theory for designing a dual fluidized bed reactor-regenerator system. Application to the MTO process

The application of the population balance (PB) theory combined with a developed parallel compartment (PC) model is proposed for a rigorous consideration of the activity distribution function in a dual reactor-regenerator system with a continuous catalyst flow between the units. This original PB-PC model is used for the simulation of the fluidized bed reactor and regenerator in the methanol-to-olefins (MTO) process. The kinetic models for the main reaction, catalyst deactivation and reactivation were proposed from experimental data collected in an integral packed bed reactor with a SAPO-34 catalyst. The versatility of the model is proven by the individual design of each unit prior to the simulation of the dual reactor-regenerator system. The performance of the dual system is evaluated, mainly studying the steady-state activity distribution function and yields of products. The dynamics of MTO process is predicted, including the effect of temperature, space time and water content on the conversion and yields of products. The influence of the residence time in each unit on the MTO process is also simulated.

Numerical simulation of hydrate slurry flow characteristics in vertical pipes based on population balance theory

Solid fluidised mining uses mechanical means to break down hydrates in the seabed. Then the hydrates will be mixed with seawater and transported to secondary crushing and other pre-treatments. Finally, the hydrates will enter the vertical lift pipe and be sent to the platform or mining ship. Therefore, the study of the flow characteristics of hydrate slurry in vertical pipes is of great significance for hydrate mining. To study the effect of different working conditions on the flow characteristics of hydrate slurry in vertical pipes, the flow characteristics of hydrate slurry in vertical pipes were simulated using the population balance theory based on hydrate particle aggregation dynamics. This paper focuses on the simulation of the flow characteristics of hydrate slurry with different initial particle sizes, concentrations and velocities, and according to the simulation results, the flow characteristics of hydrate slurry such as pressure drop, concentration distribution, and hydrate particle size distribution are analysed. The results of this study can provide referenced for the flow characteristics of hydrate slurry in hydrate mining. [Received: October 6, 2018; Accepted: March 29, 2019]

Numerical simulation of hydrate slurry flow characteristics in vertical pipes based on population balance theory

Solid fluidised mining uses mechanical means to break down hydrates in the seabed. Then the hydrates will be mixed with seawater and transported to secondary crushing and other pre-treatments. Finally, the hydrates will enter the vertical lift pipe and be sent to the platform or mining ship. Therefore, the study of the flow characteristics of hydrate slurry in vertical pipes is of great significance for hydrate mining. To study the effect of different working conditions on the flow characteristics of hydrate slurry in vertical pipes, the flow characteristics of hydrate slurry in vertical pipes were simulated using the population balance theory based on hydrate particle aggregation dynamics. This paper focuses on the simulation of the flow characteristics of hydrate slurry with different initial particle sizes, concentrations and velocities, and according to the simulation results, the flow characteristics of hydrate slurry such as pressure drop, concentration distribution, and hydrate particle size distribution are analysed. The results of this study can provide referenced for the flow characteristics of hydrate slurry in hydrate mining. [Received: October 6, 2018; Accepted: March 29, 2019]

Study of Catalyst Coke Distribution Based on Population Balance Theory: Application to Methanol to Olefins Process

A coke distribution model of catalyst particles in three‐dimensional space was developed based on population balance theory, and an analytic expression of coke distribution for zero‐dimensional time‐independent problem was deduced. The expression shows that the coke distribution is determined by the average catalyst residence time, coke deposition (or burning) rate, and coke distribution of catalyst inflow. The coke distribution model was further applied to the methanol to olefins (MTO) process. The critical factors influencing coke distribution in MTO process, as well as the effect of coke distribution on product selectivity, were investigated. Three scales of MTO fluidized bed reactor–regenerator systems, i.e., pilot‐scale, demonstration‐scale and commercial‐scale with the reactor diameter of 0.261, 1.25, and 10.5 m, respectively, were simulated. The simulated results were in good agreement with the operation data. The model could be helpful in the operation optimization and reactor design. © 2018 American Institute of Chemical Engineers AIChE J, 65: 1149–1161, 2019

Dynamic scenario simulation of dropwise condensation on a superhydrophobic surface with droplet jumping

Dropwise condensation with coalescence-induced droplet jumping has been intensively studied. The coalescence dynamics between two droplets have been investigated, however, the complex interaction among the numerous droplets on the condensing surface remains less understood. This work conducts scenario simulation for the condensation process on structured hydrophobic surfaces. A number of droplets nucleate at prescribed nucleation sites on the surface, and then grow according to heat transfer equations and then coalesce with neighbors. The coalesced droplet is to jump or remain on the surface, depending on the absolute and also the relative size of the two coalescing neighbors. The gravity removal, as well as the sweeping downstream of the droplets, is also simulated. The simulation well reproduces the interaction scenario among the many condensing droplets. The simulated droplet size distribution is distinct from the widely used distribution that is based on the population balance theory, especially in that there are significant amounts of droplets have sizes beyond the critical jumping radius, which is highly consistent with experimental observations. The critical jumping radius is therefore not as important as expected. The optimization on the relative sizes and locations of the droplets could be the essential of further enhancement. The simulation method provides a powerful tool for large-scale condensation study.

Application of Population Balance Theory for Dynamic Modeling of Methane and Ethane Hydrate Formation Processes

Natural gas hydrates are considered as a promising fuel source in the relatively near future. Kinetic modeling of various steps of the natural gas hydrate formation process, such as dissolution, nucleation, and growth processes, has received numerous attentions. A novel mechanism is introduced for the entire nucleation and growth steps, and a proper mathematical model is presented to estimate the gas consumption rate in a constant temperature and pressure process. The proposed model covers the entire dissolution, nucleation, and growth stages. The combined Lax–Wendroff/Crank–Nicolson method is employed to solve the population balance equation for estimation of total surface area of evolved hydrate particles and corresponding particle size distributions. A special class of artificial neural network (known as the Regularization Network) is used to predict the solid–liquid equilibria. The proposed model is successfully validated using experimental data borrowed from the literature for both methane and ethane...

Numerical simulation of pipeline hydrate particle agglomeration based on population balance theory

In offshore operations, the agglomeration between hydrate particles is a significant reason that could lead to pipeline hydrate plugging. Dynamic modeling and numerical simulation of pipeline hydrate particle agglomeration are of great importance to offshore hydrate management. For this purpose, a dynamic model of hydrate agglomeration was proposed and then used to simulate pipeline hydrate particle agglomeration in this paper. The dynamic model was established based on population balance equation, which took both hydrate agglomeration and hydrate breakage into consideration. Collision frequency, agglomeration efficiency, breakage frequency and size distribution of the sub particles resulting from particle breakage are four key parameters that involved in the dynamic model. Combined with several traditional solid-liquid flow models, the dynamic model was solved by the CFD software FLUENT 14.5 to simulate the agglomeration process of hydrate particles in the pipeline at different conditions. The influences of flow rate and hydrate volume fraction on the agglomeration process were analyzed emphatically. The simulation results were also compared with the calculation results of hydrate particle growth model and hydrate rheological model. The conclusions of this paper can provide guidance for the development of deep water flow assurance.

Mathematical Modeling And Numerical Simulation of Dropwise Condensation on an Inclined Circular Tube

Dropwise condensation can improve heat transfer process and, consequently, leads to considerable reduction in size and weight of condensers as well as improvement in the dehumidification process in many applications, especially in civil transport aircraft. It can also be used as an efficient cooling tool for electronics and electrical systems in aircraft engineering and aerospace technology. In this paper, the stable dropwise condensation on an inclined tube is mathematically analyzed. To do this, the population of small droplets is estimated by population balance theory while an empirical correlation is used for large droplets. To calculate heat transfer across each droplet, sum of temperature drops due to droplet curvature, phase change at droplet-vapor interface, conduction through the droplet and promoter layer, are equated with surface subcooling. The total heat transfer is calculated with the given droplets population and heat transfer through single droplet. Subsequently, effects of various parameters, including surface subcooling, contact angle and contact angle hysteresis on the growth rate, maximum radius of droplet, droplets population, and total heat transfer rate, are investigated. Results show that growth rate and heat flux of small droplets are much higher than those of the larger ones; hence, surface with small droplets is preferred for dropwise condensation purposes. Droplets with low contact angle and contact angle hysteresis have higher heat transfer rates. Increasing the inclination of tube improves heat transfer process to such an extent that vertical tubes have higher heat transfer rate than the horizontal ones. This fact indicates that vertical tubes must be used for designing condensers with dropwise condensation, which is quite the opposite for condensers designed based on filmwise condensation.

The improved model of inter-particle breakage considering the transformation of particle shape for cone crusher

To describe the essence of the inter-particle breakage behavior of the granular materials in the cone crusher accurately, an improved model is established based on the inter-particle breakage theory and population balance theory. In this model, a relative particle size ratio is introduced for describing the breakage characteristics of particles with different size, and the transformation of particle shape is considered in the inter-particle breakage.According to the standards of BS 812 part 105, the cubic and flaky particles are selected as breakage materials of inter-particle breakage experiment. Based on the experimental data, the influence of compression ratio, particle size distribution ratio and relative particle size ratio on the inter-particle breakage is investigated, the particle shape transformation function of polynomial form is built, and then the improved inter-particle breakage model considering the transformation of particle shape is established. Thus, the dynamic process of inter-particles breakage with two kinds of particle shapes in the cone crusher is simulated, and the flakiness index of cone crusher product can be calculated theoretically.A full-scale test with ZS200MF cone crusher was conducted in Tangshan, China. The particle size distribution and particle shape component of the crushing product were investigated by simulation with the improved model. Through the comparison of the measurement and the simulation, the maximum error of cumulated weight distribution is 4.8%, and the maximum error of flakiness index is 4.6%, then the effectiveness and practicability of this model are proved. Thus, the improved model can predict the quality of crushing product by utilizing the granular property of feed, the geometry of crushing chamber and operational parameters of cone crusher, which provides a theoretical foundation for product quality improvement, high-energy crushing chamber design and operational parameters optimization of cone crusher.

Feasibility study of high-rate dissolved air flotation process for rapid wastewater treatment

One advantage of the flotation system is its high applicability as a compact facility. Recently, research for the introduction of additional high-rate dissolved air flotation (DAF) into conventional water treatment plants has been progressing in areas of remodeling water treatment plants, advanced treatment, and combined sewer overflow (CSO) treatment. For successful design and operation of high-rate DAF, laboratory experiments to verify model results and estimate standard values of major parameters by model simulation were performed in this study. With the model simulation having a population balance concept, an operational analysis of bubble-floc aggregates considering a hydraulic situation in the separation zone during high-rate DAF was conducted. Based on the model simulation results, a basic level for high-rate DAF design at a hydraulic loading rate of 30 m/h was determined. On the basis of population balance theory, it can describe the process of bubble-floc aggregate formation at a contact zone and calculate the rate of transfer into the separation zone.

The Effects of Nucleation on Soot Dynamic Evolution Based on the Method of Moments

The mathematical model of soot dynamical evolution process is built based on the discrete particles population balance theory, including particle nucleation, collision coagulation, oxidation and surface growth processes; a corresponding numerical model is established using Lagrange interpolation method of moments. Based on the perfectly-stirred reactor combustion model, coupled the numerical model with detailed chemical kinetic model. The computing platform of soot particle growth evolution is established. Meanwhile, three nucleation models are built for the soot nucleation in this paper. Two chemical reaction mechanisms are used to study different fuels combustion process. The related information of soot particle growth evolution is obtained, the effect of different nucleation model on other dynamical events such as coagulation, oxidation and surface growth processes is discussed.

Preparation and crystallization kinetics of micron-sized Mg(OH)2 in a mixed suspension mixed product removal crystallizer

Magnesium hydroxide is an important chemical, and is usually obtained from seawater or brine via precipitation process. The particle size distribution of magnesium hydroxide has great effects on the subsequent filtration and drying processes. In this paper, micron-sized magnesium hydroxide with high purity, large particle size and low water content in filter cake was synthesized via simple wet precipitation in a mixed suspension mixed product removal (MSMPR) crystallizer. The effects of reactant concentration, residence time and impurities on the properties of magnesium hydroxide were investigated by X-Ray diffraction (XRD), Scanning Electron Microscopy (SEM) and Malvern laser particle size analyzer. The results show that NaOH concentration and residence time have great effects on the water content and particle size of Mg(OH)2. The spherical Mg(OH)2 with uniform diameter of about 30 μm was obtained with purity higher than 99% and water content less than 31%. Furthermore, the crystallization kinetics based on the population balance theory was studied to provide the theoretical data for industrial enlargement, and the simulation coefficients (R2) based on ASL model and C-R model are 0.9962 and 0.9972, respectively, indicating that the crystal growth rate of magnesium hydroxide can be well simulated by the sizedependent growth models.

Precipitation kinetics of PMMA sub-micrometric particles with a supercritical assisted-atomization process

Sub-micrometric particles of PMMA were successfully prepared via a supercritical assisted-atomization (SAA) process using acetone as a solvent and supercritical carbon dioxide as a spraying medium. The effects of several key factors on the particle size were investigated. These factors included the concentration of polymer solution, temperature in saturator and volumetric flow rate ratio of carbon dioxide to polymer solution. The shape of the polymer's primary particles is spherical with the arithmetic mean size ranging from 82 nm to 176 nm and the mass-weighted mean size ranging from 127 nm to 300 nm. As evidenced from the experimental results, the lower concentrations of polymer solution, optimized volumetric flow rate ratios, and higher temperatures in saturator can effectively reduce the mean particle size. The precipitation kinetic parameters were determined from the particle size distributions with the aid of the population balance theory. This study found the primary nucleation to be dominant in the precipitation and diffusion may govern particle growth.

Numerical Analysis of Foam Motion in Porous Media Using a New Stochastic Bubble Population Model

We present a numerical analysis of the stochastic population balance (SPB) theory for foam motion in porous media. The theory condenses into a set of non-linear partial differential equations in the saturation, pressure, and bubble density. We solve the equations using the IMPES method and perform sensitivity and parametric analysis. Finally, we compare the saturation profiles obtained numerically with those obtained from CT scan foam experiments. The agreement between the theory and experiments confirms that the stochastic population balance model describes adequately foam dynamics in porous media.

Cone Crusher Optimization Based on Quality Control and Prediction

In view of the solution to the practical problems existing in cone crusher design, the quantity of rock materiel falling out of the crushing chamber during once eccentric rotation of the cone is analyzed, the productivity model for cone crusher with curvilinear chamber geometry is proposed; based on the population balance theory, the division of crushing chamber is analyzed, the product size distribution and corresponding calculation method are researched; combining the empirical model for predicting particle shape with the size distribution model, the flakiness prediction model based on size distribution is proposed. The cone crusher optimization model based on quality control and prediction is built. And the influences of structural and working parameters of cone crusher on productivity and product quality are analyzed. All these tasks are useful for future cone crusher optimization strategies.

Evaluation on simultaneous removal of particles and off-flavors using population balance for application of powdered activated carbon in dissolved air flotation process

Most of the water treatment plants applying the DAF process are faced with off-flavors control problems. For simultaneous control of particles of impurities and dissolved organics that cause pungent taste and odor in water, an effective method would be the simple application of powdered activated carbon (PAC) in the DAF process. A series of experiments were carried out to explore the feasibility for simultaneous removal of kaolin particles and organic compounds that produce off-flavors (2-MIB and geosmin). In addition, the flotation efficiency of kaolin and PAC particles adsorbing organics in the DAF process was evaluated by employing the population balance theory. The removal efficiency of 2-MIB and geosmin under the treatment condition with kaolin particles for simultaneous treatment was lower than that of the individual treatment. The decrease in the removal efficiency was probably caused by 2-MIB and geosmin remaining in the PAC particle in the treated water of DAF after bubble flotation. Simulation results obtained by the population balance model indicate, that the initial collision-attachment efficiency of PAC particles was lower than that of kaolin particles.