H. Lv, D. Li﹡, Simulations of the compressible dusty-gas flows by the hyperbolic quadrature method of moments, in preparation
Y. Wei, Z. Wang, D. Li﹡, Numerical study of two-phase jet by the Eulerian-Lagrangian method and Eulerian quadrature-based moments method, in preparation
Q. Cao, D. Li﹡, Investigation of the turbulence dispersion force in the Eulerian-Lagrangian method for poly-disperse bubbly flows, in preparation
D. Li﹡, Investigation of the boundedness of phase fraction by the flux-corrected transport on unstructured meshes, Computers & Fluids, submitted
The volume of fluid (VOF) method is promising for free-surface problems due to its simplicity. However, the solution of the phase volume fraction can be unbounded if a high order scheme is used, which may cause convergence problems. Physically, many transported variables are strictly bounded, such as the phase volume fraction (bounded between zero and one) and the density (larger than zero). The spirit of flux-correct transport (FCT) algorithm is to use high order scheme where possible, but do not insist on it in regions where the solution is overshot or undershot. In this work, we implemented the original FCT algorithm with Zalesak limiter to solve transport equations on unstructured meshes. It was used to solve the 1) Navier-Stokes equation with passive scalar transport, 2) one-dimensional Euler equation, 3) incompressible and 4) compressible VOF method. The algorithm is tested for 1) backward facing flow fields with passive scalar transport, 2) shock tube problem, 3) incompressible bubble dynamics and 4) compressible bubble dynamics. Simulations were carried out on structured meshes and unstructured meshes and the results were compared against analytical solutions or the data reported in the literature. Our results showed that the FCT algorithm can ensure the boundedness of the transported variables, no matter structured meshes or unstructured meshes were employed. On the other hand, the solutions predicted by the ordinary high order scheme oscillates. The predictions agree well with the analytical solutions or the results reported in the literature. Meanwhile, as an explicit solving method, we suggest to use a small time step to obtain better results.
Y. Li, D. Li﹡, Investigation of the force closure in the Eulerian-Eulerian method: a validation study of nine gas-liquid flow cases, Industrial & Engineering of Chemical Research, submitted
Gas-liquid flows can be simulated by the Eulerian-Eulerian (E-E) method. Whether to include a specific momentum interfacial exchange force model remains as a question with no answer. In this work, we employed different force combinations to simulate industrial bubbly flows. These test cases were selected from different industries including chemical, nuclear, bio-processing and metallurgical engineering. Simulations were launched by the OpenFOAM solver
reactingTwoPhaseEulerFoam, in which the E-E method was implemented with sophisticated numerical techniques to avoid numerical error. Predictions were compared against experimental data. It was found that the drag force and turbulent dispersion force play the most important role on the predictions and should be included for all simulations. The first one accounts for the two-way coupling while the second one accounts for the turbulence effect and ensures the E-E equations to be well-posed. The lift force and wall lubrication force should be included to address the phase fraction accumulation in the vicinity of the wall, especially for pipe flows with large aspect ratio. In other cases the lateral forces can be safely neglected. All the test case are open-sourced and are available as Mendeley Data for anyone to download as baseline test cases.
D. Li﹡, others, Comparison of Eulerian QBMM and classical Eulerian-Eulerian method for the simulation of poly-disperse bubbly flows, AIChE Journal
The spatial gas distribution of poly-disperse bubbly flows depends greatly on the bubble size. To reflect the resulting poly-celerity, more than two momentum balance equations (typically for the gas and liquid phases) have to be considered, as done in the multi-fluid approach. The inhomogeneous MUSIG model follows this approach, also combined with a population balance model. As an alternative, in a previous work, an Eulerian quadrature-based moments method (E-QBMM) was implemented in OpenFOAM; however, only the drag force was included. In this work, different non-drag forces (lift, wall lubrication and turbulent dispersion) are added to enable more complex test cases to be simulated. Simulation results obtained using E-QBMM are compared with the classical E-E method and validated against experimental data for different test cases. The results show that there is good agreement between E-QBMM and E-E methods for mono-disperse cases, but E-QBMM can better simulate the separation and segregation of small and large bubbles.
D. Li﹡, others, twoWayGPBEFoam: an open-source Eulerian QBMM solver for monokinetic bubbly flows, Computer Physics Communications, submitted
twoWayGPBEFoamis an open-source mescoscopic Eulerian QBMM solver for monokinetic bubbly flows. The solver is implemented within the OpenFOAM software framework. Unlike the existing macroscopic two-fluid model (TFM) solver
twoPhaseEulerFoam, it can predict the size segregation phenomenon and the size-conditional velocities of the disperse phase, although it will not be able to predict the particle trajectory crossing (PTC). On theoretical grounds, the evolution of the disperse phase in multiphase flows is dictated by the generalized population balance equation (GPBE), which can be transformed into moment transport equations and solved using the finite-volume method with higher-order realizable spatial-discretization schemes and time-integration schemes. In order to address the closure problem of the size-conditional spatial flux, the size-conditional velocities need to be modelled. In many previous works, these are assumed to be identical with the disperse phase velocity predicted by the TFM. In this work, the size-conditional velocities were modelled using the velocity polynomial approximation (VPA), for which the velocity polynomial coefficients (VPCs) can be obtained from the moments themselves. By carrying out several test cases with both one-way and two-way coupling, we show that the results predicted by our solver agree well with the analytical solutions and the existing experimental data.
D. Li, others, Compressibility induced bubble size variation in bubble column reactors: Simulations by the CFD–PBE, Chinese Journal of Chemical Engineering
Bubble column reactors can be simulated by the two fluidmodel (TFM) coupled with the population balance equation (PBE). For the large industrial bubble columns, the compressibility due to the pressure difference may introduce notable bubble size variation. In order to address the compressibility effect, the PBE should be reformulated and coupled with the compressible TFM. In this work, the PBE with a compressibility term was formulated from single bubble dynamics, the mean Sauter diameters predicted by the compressible TFM coupled with the PBE were compared with the analytical solutions obtained by the ideal gas law. Itwas proven that the mesoscale formulations presented in this work were physically consistent with the macroscale modeling. It can be used to simulate large industrial plants when the compressibility induced bubble size variation is important.
D. Li, others, Quadrature-based moment methods for the population balance equation: An algorithm review, Chinese Journal of Chemical Engineering
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
D. Li, others, Droplet Breakage and Coalescence in Liquid–Liquid Dispersions: Comparison of Different Kernels with EQMOM and QMOM, AIChE Journal
Droplet coalescence and breakage in turbulent liquid–liquid dispersions is simulated by using computational fluid dynamics (CFD) and population balance modeling. The multifractal (MF) formalism that takes into account internal intermittency was here used for the first time to describe breakage and coalescence in a surfactant-free dispersion. The log-normal Extended Quadrature Method of Moments (EQMOM) was for the first time coupled with a CFD multiphase solver. To assess the accuracy of the model, predictions are compared with experiments and other models (i.e., Coulalogou and Tavlarides kernels and Quadrature Method of Moments [QMOM]). EQMOM and QMOM resulted in similar predictions, but EQMOM provides a continuous reconstruction of the droplet-size distribution. Transient predictions obtained with the MF kernels result in a better agreement with the experiments
D. Li﹡, H. Christin, Simulation of bubbly flows with special numerical treatments of the semi-conservative and fully conservative two-fluid model, Chemical Engineering Science
Bubbly flows are found in a large number of chemical engineering applications. For the computational fluid dynamics (CFD) simulations of such multi-phase flows, both physical models and numerical treatment are crucial to obtain robust and accurate results. In this numerical study, we investigate the two-fluid model (TFM) under challenging conditions such as phase segregation and inversion. For the phase segregation, a singular problem arises in the phase momentum and the two-phase $k-\varepsilon$ equations when one phase fraction approaches zero. Another numerical issue is the accurate calculation of the drag coefficient, e.g., during the phase inversion. To address the singular problem, previous studies used a non-conservative formulation after dividing by the phase fraction; in our approach, we present a robust methodology for semi-conservative and fully conservative formulations. A special numerical treatment is introduced to the phase momentum equations and the turbulence equations, which avoids the singular problem in case of phase segregation. Concerning the drag force, two novel methods, the linear and the hyperbolic blending method, respectively, are presented to obtain accurate results. For testing the new numerical treatment, the analytical solution of a two-dimensional test case is first compared with the results predicted using a semi-conservative and a fully conservative formulation. The second test case investigated is a bubble column with different superficial velocities. The results from three-dimensional simulations using the novel formulations show good agreement with the literature data. Especially when phase segregation occurs, the semi-conservative and the fully conservative formulations using the two-phase $k-\varepsilon$ model formulation converge.
D. Li, others, Investigation of droplet breakup in liquid–liquid dispersions by CFD-PBM simulations: The influence of the surfactant type, Chinese Journal of Chemical Engineering
The accurate prediction of the droplet size distribution (DSD) in liquid–liquid turbulent dispersions is of fundamental importance in many industrial applications and it requires suitable kernels in the population balance model. When a surfactant is included in liquid–liquid dispersions, the droplet breakup behavior will change as an effect of the reduction of the interfacial tension. Moreover, also the dynamic interfacial tension may be different with respect to the static, due to the fact that the surfactant may be easily desorbed from the droplet surface, generating additional disruptive stresses. In this work, the performance of five breakup kernels from the literature is assessed, to investigate their ability to predict the time evolution of the DSD and of the mean Sauter diameter, when different surfactants are employed. Simulations are performed with the Quadrature Method of Moments for the solution of the population balance model coupled with the two-fluid model implemented in the compressibleTwoPhaseEulerFoam solver of the open-source computational fluid dynamics (CFD) code OpenFOAM v. 2.2.x. The time evolution of the mean Sauter diameter predicted by these kernels is validated against experimental data for six test cases referring to a stirred tank with different types of surfactants (Tween 20 and PVA 88%) at different concentrations operating under different stirrer rates. Our results show that for the dispersion containing Tween 20 additional stress is generated, the multifractal breakup kernel properly predicts the DSD evolution, whereas two other kernels predict too fast breakup of droplets covered by adsorbed PVA. Kernels derived originally for bubbles completely fail.
Z. Gao, D. Li, others, Simulation of droplet breakage in turbulent liquid-liquid dispersions with CFD-PBM: comparison of breakage kernels, Chemical Engineering Science
In this work droplet breakage in turbulent liquid-liquid dispersions is simulated by using computational fluid dynamics and population balance modelling. Model predictions are validated against experimental data for eight test cases, namely stirred tanks with different geometries and different continuous and disperse phases. The two first test cases correspond to two geometrically similar stirred tanks (one being the scale up of the other) working under the same power input per unit mass with water as continuous phase and a mixture of chlorobenzene and toluene as disperse phase. The last six test cases correspond to a slightly different geometry, working with water as continuous phase and different silicon oils as disperse phase, characterized by different viscosity values, stirred at different stirring rates. Simulations are performed with our implementation of the quadrature method of moments for the solution of the population balance model into the open-source computational fluid dynamics code OpenFOAM. Two different breakage kernels are considered in this work, based respectively on the classical homogeneous and multifractal turbulence theories. Analysis of the kernels and comparison with experiments reveal that the second kernel results is better agreement with experiments, a more accurate description of the underlying physics and presents the additional advantage of having no fitting constants.
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