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

Predicting the flocculation kinetics of fine particles in a turbulent flow using a Budyko-type model

Through an analogy with the Budyko curves in catchment hydrology science, this study defines the supply and demand limits for the flocculation system of cohesive particles (the supply limit can be represented by the potential floc size growth, whereas the demand limit can be represented by the steady-state floc size) and attempts to adopt Budyko-type models to estimate the temporal evolution of floc size during flocculation. Seventeen experimental datasets are collected to test the accuracy of the Budyko model with an average high correlation coefficient of 0.9784 and average low relative error and root mean squared error values of 0.1304 and 0.0605, respectively. Either the potential floc size growth or the steady-state floc size is a monotonic function of the flow shear rate, and a simple empiric power law function can be used to describe them. Other Budyko-type models are also found to show good prediction accuracies against the experimental datasets. This study indicates that Budyko-type models have potential as a good addition to existing flocculation models for predicting the temporal variation in the size population of flocs in a turbulent flow, provided that some coefficients have been calibrated by limited data points prior.

Particle Distribution and Heat Transfer of SiO2/Water Nanofluid in the Turbulent Tube Flow

In order to clarify the effect of particle coagulation on the heat transfer properties, the governing equations of nanofluid together with the equation for nanoparticles in the SiO₂/water nanofluid flowing through a turbulent tube are solved numerically in the range of Reynolds number 3000 ≤ Re ≤ 16,000 and particle volume fraction 0.005 ≤ φ ≤ 0.04. Some results are validated by comparing with the experimental results. The effect of particle convection, diffusion, and coagulation on the pressure drop ∆P, particle distribution, and heat transfer of nanofluid are analyzed. The main innovation is that it gives the effect of particle coagulation on the pressure drop, particle distribution, and heat transfer. The results showed that ∆P increases with the increase in Re and φ. When inlet velocity is small, the increase in ∆P caused by adding particles is relatively large, and ∆P increases most obviously compared with the case of pure water when the inlet velocity is 0.589 m/s and φ is 0.004. Particle number concentration M₀ decreases along the flow direction, and M₀ near the wall is decreased to the original 2% and decreased by about 90% in the central area. M₀ increases with increasing Re but with decreasing φ, and basically presents a uniform distribution in the core area of the tube. The geometric mean diameter of particle GMD increases with increasing φ, but with decreasing Re. GMD is the minimum in the inlet area, and gradually increases along the flow direction. The geometric standard deviation of particle diameter GSD increases sharply at the inlet and decreases in the inlet area, remains almost unchanged in the whole tube, and finally decreases rapidly again at the outlet. The effects of Re and φ on the variation in GSD along the flow direction are insignificant. The values of convective heat transfer coefficient h and Nusselt number Nu are larger for nanofluids than that for pure water. h and Nu increase with the increase in Re and φ. Interestingly, the variation in φ from 0.005 to 0.04 has little effect on h and Nu.

An Extended Entropic Model for Cohesive Sediment Flocculation in a Piecewise Varied Shear Environment

In this study, an extended model for describing the temporal evolution of a characteristic floc size of cohesive sediment particles when the flocculation system is subject to a piecewise varied turbulent shear rate was derived by the probability methods based on the Shannon entropy theory following Zhu (2018). This model only contained three important parameters: initial and steady-state values of floc size, and a parameter characterizing the maximum capacity for floc size increase (or decay), and it can be adopted to capture well a monotonic pattern in which floc size increases (or decays) with flocculation time. Comparison with 13 literature experimental data sets regarding floc size variation to a varied shear rate showed the validity of the entropic model with a high correlation coefficient and few errors. Furthermore, for the case of tapered shear flocculation, it was found that there was a power decay of the capacity parameter with the shear rate, which is similar to the dependence of the steady-state floc size on the shear rate. The entropic model was further parameterized by introducing these two empirical relations into it, and the finally obtained model was found to be more sensitive to two empirical coefficients that have been incorporated into the capacity parameter than those in the steady-state floc size. The proposed entropic model could have the potential, as an addition to existing flocculation models, to be coupled into present mature hydrodynamic models to model the cohesive sediment transport in estuarine and coastal regions.

Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors

Optimal oxygen supply is vitally important for the cultivation of aerobically growing cells, as it has a direct influence on cell growth and product formation. A process engineering parameter directly related to oxygen supply is the volumetric oxygen mass transfer coefficient kLa. It is the influences on kLa and computing time of different interfacial force and population balance models in stirred bioreactors that have been evaluated in this study. For this investigation, the OpenFOAM 7 open-source toolbox was utilized. Firstly, the Euler–Euler model with a constant bubble diameter was applied to a 2L scale bioreactor to statistically examine the influence of different interfacial models on the kLa value. It was shown that the kL model and the constant bubble diameter have the greatest influence on the calculated kLa value. To eliminate the problem of a constant bubble diameter and to take effects such as bubble breakup and coalescence into account, the Euler–Euler model was coupled with population balance models (PBM). For this purpose, four coalescence and five bubble breakup models were examined. Ultimately, it was established that, for all of the models tested, coupling computational fluid dynamics (CFD) with PBM resulted in better agreement with the experimental data than using the Euler–Euler model. However, it should be noted that the higher accuracy of the PBM coupled models requires twice the computation time.

A quasi-Monte Carlo based flocculation model for fine-grained cohesive sediments in aquatic environments

The quasi-Monte Carlo (QMC) method was enhanced to solve the population balance model (PBM) including aggregation and fragmentation processes for simulating the temporal evolutions of characteristic sizes and floc size distributions (FSDs) of cohesive sediments. Ideal cases with analytical solutions were firstly adopted to validate this QMC model to illustrate selected pure aggregation, pure fragmentation, and combined aggregation and fragmentation systems. Two available laboratory data sets, one with suspended kaolinite and the other with a mixture of kaolinite and montmorillonite, were further used to monitor the FSDs of cohesive sediments in controlled shear conditions. The model results show reasonable agreements with both analytical solutions and laboratory experiments. Moreover, different QMC schemes were tested and compared with the standard Monte Carlo scheme and a Latin Hypercube Sampling scheme to optimize the model performance. It shows that all QMC schemes perform better in both accuracy and time consumption than standard Monte Carlo scheme. In particular, compared with other schemes, the QMC scheme using Halton sequence requires the least particle numbers in the simulated system to reach reasonable accuracy. In the sensitivity tests, we also show that the fractal dimension and the fragmentation distribution function have large impacts on the predicted FSDs. This study indicates a great advance in employing QMC schemes to solve PBM for simulating the flocculation of cohesive sediments.