On the Comparison between Population Balance Models for CFD Simulation of Bubble Columns

CFD simulations of bubble columns have received much attention, and several multiphase models have been developed, tested, and validated through comparison with experimental data. It is well-known that bubble coalescence and breakup can lead to significant variations in the bubble size distribution and that, to model the evolution of the dispersed gas phase, the population balance equation has to be solved. In this work, a classes method (CM) and a method of moments (MOM) are investigated and compared. The MOM represents an attractive alternative in which, instead of tracking the entire bubble distribution, only the lower-order moments of the distribution are tracked. The above two approaches have been implemented in the commercial CFD code FLUENT, version 6.0, in conjunction with the Eulerian multiphase model

CFD Analysis of Premixed Methane Chlorination Reactors with Detailed Chemistry

With the implementation of efficient algorithms for the accurate calculation of reaction source terms, computational fluid dynamics (CFD) is now a powerful tool for the simulation and design of chemical reactors with complex kinetic schemes. The example studied in this work is the methane chlorination reaction for which the detailed chemistry scheme has 152 reactions and 38 species. The adiabatic, jet-stirred chlorination reactor used for the CFD simulations is an insulated right cylinder with a coaxial premixed feed stream at one end. In order for this reactor to remain lit, recirculation of hot products is crucial, and hence, reactor stability is sensitive to both macroscale and microscale mixing. By neglecting density variations, a Lagrangian composition probability density function (PDF) code with a novel chemistry tabulation algorithm (in-situ adaptive tabulation or ISAT) for handling complex reactions is used to simulate the species concentrations and temperature field inside of the reactor. In addition, a reduced mechanism with 21 reactions and 15 species is tested for accuracy against the detailed chemistry scheme, a simplified CSTR model is used to illustrate the shortcomings of zero-dimensional models, and a pair-wise mixing stirred reactor (PMSR) model is used to show the stabilizing effect of micromixing on reactor stability. The CFD simulations are generally in good agreement with results from pilot-scale reactors for the outlet temperature and major species

Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) nd quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3

Numerical Description of Dilute Particle-Laden FLows by a Quadrature-Based Moment Method

The numerical simulation of gas-particle flows is divided into two families of methods. In Euler-Lagrange methods individual particle trajectories are computed, whereas in Euler-Euler methods particles are characterized by statistical descriptors. Lagrangian methods are very precise but their computational cost increases with instationarity and particle volume fraction. In Eulerian methods (also called moment methods) the particle-phase computational cost is comparable to that of the fluid phase but requires strong simplificaions. Existing Eulerian models consider unimodal or close-to-equilibrium particle velocity distributions and then fail when the actual distribution is far from equilibrium. Quadrature-based Eulerian methods introduce a new reconstruction of the velocity distribution, written as a sum of delta functions in phase space constrained to give the right values for selected low-order moments. Two of the quadrature-based Eulerian methods, differing by the reconstruction algorithm, are the focus of this work. Computational results for two academic cases (crossing jets, Taylor-Green flow) are compared to those of a Lagrangian method (considered as the reference solution) and of an existing second-order moment method. With the quadrature-based Eulerian methods, significant qualitative improvement is noticed compared to the second-order moment method in the two test cases

Conditional-moment Closure with Differential Diffusion for Soot Evolution in Fire

The conditional-moment closure (CMC) equation for the evolution of a large Lewis number scalar, soot, is derived starting from the joint probability density function (pdf) equation for the gas-phase mixture fraction, ξ g , and the soot mass fraction, Y s . Unlike previous approaches starting with the joint pdf, the residual terms that result from the typical closure models were retained. A new formulation of the one-dimensional turbulence (ODT) model suitable for spatially evolving flows with buoyant acceleration and radiative transport in participating media was employed to carry out simulations of a prototypical ethene fire. The resulting ODT evolution of ξ g and Y s was used to assess the significance of various terms in the CMC equation including the residual correlations. The terms involving differential diffusion are found to be important along with the soot source terms and the large-scale evolution of both ξ g and Y s . Of particular importance in the regions in mixture-fraction space around the soot production and consumption is a residual term, not previously identified, related to the correlation between the differential diffusion and Y s . This term results in a diffusion-like behavior of Y s in the mixture fraction coordinate that has an apparent Lewis number near unity. In scenarios where the large Lewis number component is a non-negligible component of the mixture fraction (i.e., large soot loading), it is found easier to employ a mixture fraction neglecting this component. Such a mixture-fraction variable has a chemical source term, but this appears easier to model than the differential diffusion and dissipation terms that result when the large Lewis number component is retained in the mixture-fraction definition

A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence (CIT). In the present work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian probability-density-function formalism. This framework includes the Eulerian approach, and hence can be useful also for the development of two-fluid models. A rigorous formalism and a general model have been put forward focusing, in particular, on the two ingredients that are key in moderately dense flows, namely, two-way coupling in the carrier phase, and the decomposition of the particle-phase velocity into its spatially correlated and uncorrelated components. Specifically, this last contribution allows us to identify in the stochastic model the contributions due to the correlated fluctuating energy and to the granular temperature of the particle phase, which determine the time scale for particle–particle collisions. The model is then validated and assessed against direct-numerical-simulation data for homogeneous configurations of increasing difficulty: (i) homogeneous isotropic turbulence, (ii) decaying and shear turbulence and (iii) CIT

A multi-Gaussian quadrature method of moments for simulating high Stokes number turbulent two-phase flows

With the great increase in computational resources, Large Eddy Simulation (LES) of industrial configurations is now an efficient and tractable tool. Numerous applications involve a liquid or solid disperse phase carried by a gaseous flow field (eg, fuel injection in automotive or aeronautical engines, fluidized beds, and alumina particles in rocket boosters). To simulate this kind of flow, one may resort to a Number Density Function (NDF), which satisfies a kinetic equation

A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers

With the large increase in available computational resources, large-eddy simulation (LES) of industrial configurations has become an efficient and tractable alternative to traditional multiphase turbulence models. Many applications involve a liquid or solid disperse phase carried by a gas phase (eg, fuel injection in automotive or aeronautical engines, fluidized beds, and alumina particles in rocket boosters)