Direct Quadrature Method Of Moments Research Articles

Application of the Direct Quadrature Method of Moments for the modelling of the enzymatic hydrolysis of cellulose: II. Case of insoluble substrate

In the second part of this study, the Population Balance Model developed for the enzymatic hydrolysis of cellulose polymer chains is extended to particulate cellulose substrates. In this case, the particle fragmentation process is complex. This phenomenon is induced by the hydrodynamic shear stress when the particle cohesion is lower than the hydrodynamic force. The particle cohesion depends on the mechanical properties of the substrate particles which are affected by the Endoglucanase (EG) activity that hydrolyses accessible glycosidic bonds all along the reaction. These two effects are taken into account in the model. The Exoglucanase (CBH) activity is assimilated to an erosion process as for soluble substrates. The β-glucosidase activity as well as the inhibition effect are also considered in the global population balance-based model. The numerical resolution of the Population Balance Equation (PBE) is based on the Direct Quadrature Method of Moments (DQMOM) coupled with the Maximum Entropy (ME) reconstruction technique. Numerical results in different cases show coherent prediction of the Particle Size Distribution (PSD) evolution during the hydrolysis reaction as well as the kinetics of simple sugars release. The modelling procedure is supported by experimental data which reveal the importance of the particles cohesion and its evolution during the hydrolysis reaction.

Application of the Direct Quadrature Method of Moments for the modelling of the enzymatic hydrolysis of cellulose: I. Case of soluble substrate

The modelling of the enzymatic hydrolysis of cellulosic polymers is investigated through a population balance approach. Both Endoglucanase (EG) and Exoglucanase (CBH) activities are taken into account. EG achieves random attacks along cellulosic chains and cleaves the β-glycosidic bonds whereas CBH produces cellobiose molecules by chain-end scission mechanism. The EG activity is modelled as a pure breakage while the CBH activity is assimilated to an erosion process with a specific product (cellobiose). In the two cases, the inhibition of the cellulases activity by the end-product is incorporated. The population balance equation (PBE) accounting for breakage processes is solved using the Direct Quadrature Method of Moments (DQMOM) coupled to a distribution reconstruction technique based on the Maximum Entropy (ME) principle in order to track the time evolution of the chain length distribution (CLD) during the hydrolysis reaction. The β-glucosidase activity transforming the produced cellobiose into glucose is modelled as a Michaelis–Menten type kinetic with a competitive inhibition effect and solved simultaneously with the PBE. The numerical results show the time-evolution of the CLD during the hydrolysis reaction as well as the rate of conversion of the substrate into simple sugars. These results are in concordance with those predicted analytically. The synergistic action of the EG and CBH is highlighted and discussed and the inhibition effect is investigated. The approach is promising by its accuracy and fastness for the analysis of dynamic experimental data of the enzymatic hydrolysis reaction.

방위각 정보만을 이용한 비선형 표적추적필터

Dept. of Aviation Maintenance, Far East University, Abstract : The optimal estimation of a bearing only target tracking problem be achieved through the solution of the Fokker-Planck equation and the Bayesian update. Recently, a nonlinear filtering algorithm using a direct quadrature method of moments in which the associated Fokker-Planck equation can be propagated efficiently and accurately was proposed. Although this approach has demonstrated its promising in the field of nonlinear filtering in several examples, the “degeneracy” phenomenon, similar to that which exists in a typical particle filter, occasionally appears because only the weights are updated in the modified Bayesian rule in this algorithm. Therefore, in this paper to enhance the performance, a more stable measurement update process based upon the update equation in the Extended Kalman filters and a more accurate initialization and re-sampling strategy for weight and abscissas are proposed. Simulations are used to show the effectiveness of the proposed filter and the obtained results are promisingKey Words : Nonlinear Estimation, Kalman Filter, Fokker-Planck Equation, Bearing Only Tracking

Discussion on DQMOM to solve a bivariate population balance equation applied to a grinding process

A bivariate population balance equation applied to a grinding process is implemented in a model (PBM). The particles are simultaneously characterized by their size and their mechanical strength, expressed here by the minimum energy needed to break them. PBM is solved by the Direct Quadrature Method of Moments (DQMOM). The mixed moments of the distribution are expressed by the quadrature form of the population density defined for one order (N) and incorporating the weights and the abscissas defined for the two properties. The effect of the quadrature order (N=2,3,4) and the selected set of the 3N moments needed to solve the system on the accuracy of the results is discussed. For a given order of the quadrature, the selected set of the initial mixed moments slightly affects first the weights and abscissas derived from the initial particle distribution. The set of moments also affects the precision of the moments calculated versus time but only those having high orders in relation with the respective range of the solid properties considered. Problems of convergence and significant differences in the predicted mixed moments are also observed when the order of the quadrature is equal to 2. However, the changes of a bivariate distribution versus time applied to a grinding process are well predicted using the DQMOM approach, choosing a number of nodes equal to 3, associated with a smart selection of the moment set, incorporating all the moments of interest.

A time-dependent Eulerian model of droplet diffusion in turbulent flow

Industrial sprays are common in many operations where liquid application to surfaces is required. Maximization of the spray process efficacy depends on accurate simulation of spray characteristics and surrounding flow conditions. Traditionally, droplet modeling has been performed using a Lagrangian approach which requires tracking a sufficient number of droplets to form statistical estimates of droplet deposition. Alternatively for this study, an Eulerian-based flow and droplet transport model using the Direct Quadrature Method of Moments (DQMOM) is applied to spray modeling and is coupled with the Reynolds Averaged Navier Stokes (RANS) equations and Shear Stress Transport (SST) turbulence model. A new turbulent diffusion model is developed for droplet transport which accounts for inertia and time-limit effects and covers a wide range of operational parameters associated with aerial spraying. The model is based on a full parametric study using Lagrangian particle tracking in a uniform, homogeneous and isotropic turbulent flow field and is subsequently implemented within the Eulerian DQMOM framework and compared to: 1) Lagrangian tracking predictions of a log-normal particle size distribution injected into a homogeneous, isotropic turbulent flow field, and 2) full-scale experimental wind tunnel measurements in the anisotropic, non-homogeneous wake of a hollow cone hydraulic nozzle.

Hydrodynamics and particle mixing/segregation measurements in an industrial gas phase olefin polymerization reactor using image processing technique and CFD-PBM model

Particle size distribution (PSD) has a significant impact on the performance of fluidized bed reactors due to uneven distribution in the segregation and mixing phenomena. This paper develops a new method of digital image processing that investigates the hydrodynamics of an industrial gas phase olefin polymerization reactor and studies the fluidization structure of a wide range of particle size distribution in an industrial gas phase polymerization reactor by means of a CFD-PBM coupled model, where the direct quadrature method of moments (DQMOM) was implemented to solve the population balance model. It was shown that the applied parameter assumptions and closure laws were appropriately chosen to satisfactorily predict the available operational data in terms of pressure drop and bed height. The transient CFD-PBM/DQMOM coupled model and image analysis technique are then implemented extensively to analyze bubble fluidization structure and segregation phenomena at different velocities. The particle segregation indicates that the small bubbles present in the bed are unable to induce vigorous mixing at low superficial gas velocity while particle mixing improves at a velocity above the minimum fluidization velocity. Further, the predicted results show higher axial segregation phenomena when compared to the radial direction.

A CFD–PBM coupled model of hydrodynamics and mixing/segregation in an industrial gas-phase polymerization reactor

The particle size distribution (PSD) has a significant influence on the performance of fluidized bed reactors, as uneven distribution usually results from segregation and mixing tendencies. The objective of this paper is to study the segregation of wide range of particle size distribution in an industrial gas phase polymerization reactor by means of a CFD–PBM coupled model, where the direct quadrature method of moments (DQMOM) was implemented to solve the population balance model. It was shown that the model is able to satisfactorily predict the available operational data in terms of pressure drop and bed height. Model sensitivities of discretization scheme, maximum solid packing and fluidization/de-fluidization were also studied. The transient CFD–PBM/DQMOM coupled model is then utilized extensively to analyze minimum fluidization velocity, fluidization behavior and segregation phenomena at different velocities. The results suggested that third-order MUSCL discretization scheme, maximum solid packing value which is 0.01 higher than specific solid volume fraction and also fluidization process were mathematically and physically consistent with real observation. In addition, the segregation is strongly affected at minimum fluidization velocity range of particles. The PSD becomes well-mixed at high gas velocity while the quasi-layer inversion was predicted in low gas velocity.

A class of exact solutions for population balances with arbitrary internal coordinates

We develop a novel transformation that maps the linear, nonhomogeneous, multidimensional population balance equation (PBE) into an advection equation that is readily solved using the method of characteristics. The PBEs targeted by this transformation exclude aggregation, breakage, and time dependent birth and death rates. In addition, internal coordinates are assumed to grow independently of each other. The ensuing general formulation is then used to recover closed‐form analytical solutions for problems with monosurface and bulk‐diffusion growth‐rates as well as Gaussian‐type nucleation. For completeness, we derive the multidimensional Green's functions for our approach. This is followed by a brief discussion on how the proposed framework may be used for code verification of moment methods such as the quadrature method of moments and the direct quadrature method of moments. Finally, a sample Mathematica code is provided to derive analytical solutions for the single‐internal‐coordinate case given user‐specified growth, birth, and death rates. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1691–1698, 2015

Efficient DQMOM for Multivariate Population Balance Equations and Application to Virus Replication in Cell Cultures

Population balances play an important role in process and bioprocess engineering. They represent PDEs which are often multidimensional. The numerical solution of these is quite challenging in particular for high dimensional problems. For this reason the distribution dynamics are usually represented by the respective moments. As the corresponding dynamic moment equations can only be computed in a closed form under strict assumptions, approximate moment methods have to be applied. However, existing techniques can not be implemented efficiently for high dimensional problems. In this manuscript an efficient implementation of the Direct Quadrature Method of Moments (DQMOM) is derived using monomial cubature rules. Furthermore, application is demonstrated for a five dimensional PBE which is based on a single cell model for viral replication in cell cultures. The algorithm is used to analyze the effects of different model assumptions on the overall dynamics.

Numerical solution of mixed continuous–discrete population balance models for depolymerization of branched polymers

A simulation technique to describe the depolymerization of branched polymers via bivariate population balance modeling was developed. The polymers were characterized by two internal coordinates: the number of monomer units and branching bonds. Three commonly used mechanisms for depolymerization (random chain, end chain, and random debranching scission) were applied and formulated such that only physically possible polymers were created. The mechanisms and the population balance equation were formulated in a mixed continuous–discrete manner. The population balance equation was solved using the Direct Quadrature Method of Moments (DQMOM). With this algorithm, the time evolution of the distribution with respect to the internal coordinate was computed. In addition, the algorithm was validated through comparison with Monte Carlo simulations. Notably, the accuracy of the mixed continuous–discrete formulation was significantly higher that of the continuous formulation. However, DQMOM was found to be unsuitable for describing the temporal evolution of the distribution for random scission.

Multi-environment probability density function approach for turbulent CH4/H2 flames under the MILD combustion condition

The multi-environment probability density function approach has been applied to simulate turbulent CH4/H2 flames under moderate or intense low-oxygen dilution (MILD) conditions. In order to circumvent excessive computational burden, the direct quadrature method of moments (DQMOM) has been adopted as an alternative approach to solve the transported probability density function (PDF) equation. The multi-environment PDF approach has the form of a conventional Eulerian scheme and retains the desirable property of a particle-based method. In this study, the joint-composition PDF is approximated using the two-environment PDF, expressed via the combination of weights and abscissas on the composition and physical space. Micromixing is represented by the IEM model, and the chemical mechanism is based on detailed chemistry. In terms of the unconditional means and conditional statistics for temperature and mass fractions of species and pollutants, the predicted profiles are in reasonable agreement with experimental data. Numerical results clearly indicate that the two-environment PDF transport model has the capability to realistically predict the effects of oxygen mass fraction on the flame lift-off, auto-ignition, flame structure, and NOx formation characteristics in turbulent CH4/H2 jet flames issuing from a jet in hot coflow (JHC). It was also found that the two-environment PDF approach reproduces the sensitivity of the CO and CO2 peak levels versus the O2 concentration variation, as well as the peak levels of NO conditional fluctuations under the MILD combustion condition. Even if considerable discrepancies in the unconditional means exist mainly due to inconsistent treatment of weight combination and shortcomings of singularity treatment procedure, the present three-environment PDF model well captures the measured high-level conditional temperature and CO fluctuations on the fuel-lean side where the local extinction occurs via mixing of the shrouded cold air and the vitiated flame field.

CFD-PBM modeling polydisperse polymerization FBRs with simultaneous particle growth and aggregation: The effect of the method of moments

Real solid systems in polymerization fluidized bed reactors (FBRs) are characterized by wide particle size distributions that change continuously due to particle micro-behavior, growth and aggregation, etc. Simulations of such gas–solid flow hydrodynamics require the solution of a coupled computational fluid dynamics (CFD)-population balance equation (PBE) model, i.e. the CFD-PBM. Therefore, the analysis of the existing numerical methods for solving the PBE is important for the ability of the numerical prediction of the coupled model. Three representatively numerical moment-based methods, namely the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM) and the fixed pivot quadrature method of moments (FPQMOM), were used to solve the PBE for evaluating the effect of numerical method. Comparative results demonstrated the suitability of the FPQMOM for modeling polymerization FBRs with simultaneous polymerization particle growth and aggregation.

Computational Modeling of Biomass Thermochemical Conversion in Fluidized Beds: Particle Density Variation and Size Distribution

The design and scale-up of fluidized-bed reactors is an important step to commercialize viable conversion pathways (such as fast pyrolysis) for biomass into hydrocarbon intermediates and fuels that lead to “drop-in” replacements for jet fuel, diesel, gasoline, and other petroleum-based products. Detailed information about the particle size distribution (PSD) and particle density evolution throughout the fluidized-bed reactor can play a critical role in determining in situ catalyst selectivity, intermediate components, and reactor performance. This work presents an Euler–Euler computational fluid dynamics (CFD) model applied to biomass thermochemical conversion for use in fluidized-bed reactor simulations. The complex chemical and physical processes of particle devolatilization and their interaction with the reacting gas environment are described within a multifluid framework based on the kinetic theory of granular flows. The direct quadrature method of moments is used to describe the biomass PSD. Continuo...

Model-based analysis of the impact of the distributor on the hydrodynamic performance of industrial polydisperse gas phase fluidized bed polymerization reactors

A two dimensionally Eulerian–Eulerian multiphase flow model coupled with a population balance modeling (CFD–PBM) simulation was implemented to investigate the fluidization structure in an industrial scale gas phase polymerization reactor (FBR). Direct quadrature method of moments (DQMOM) was employed in this model to solve the PBM. Two cases including perforated distributor and complete sparger have been applied to examine the flow structure through the bed. A simulation of the reactor with perforated distributor was performed first to validate and evaluate the impact of distributor's characteristics on the fluidization behaviors. The predicted results were in good agreement with the industrial data in terms of pressure drop and bed height. The results showed that different heterogeneous flow patterns were created in a perforated distributor, due to more kinetic energy and jet formation above the distributor. A dead zone is expected to be formed near the corners of the perforated distributor. In addition, the cluster formation is expected to be decreased in comparison with the complete sparger plate distributor. Furthermore, the results predicted bigger bubble diameter in the case of the perforated distributor by using an image processing technique. The information obtained from this study could be important to assure efficient industrial operations of FBRs.

Efficient simulation of gas–liquid pipe flows using a generalized population balance equation coupled with the algebraic slip model

The inhomogeneous generalized population balance equation, which is discretized with the direct quadrature method of moment (DQMOM), is solved to predict the bubble size distribution (BSD) in a vertical pipe flow. The proposed model is compared with a more classical approach where bubbles are characterized with a constant mean size. The turbulent two-phase flow field, which is modeled using a Reynolds-Averaged Navier–Stokes equation approach, is assumed to be in local equilibrium, thus the relative gas and liquid (slip) velocities can be calculated with the algebraic slip model, thereby accounting for the drag, lift, and lubrication forces. The complex relationship between the bubble size distribution and the resulting forces is described accurately by the DQMOM. Each quadrature node and weight represents a class of bubbles with characteristic size and number density, which change dynamically in time and space to preserve the first moments of the BSD. The predictions obtained are validated against previously published experimental data, thereby demonstrating the advantages of this approach for large-scale systems as well as suggesting future extensions to long piping systems and more complex geometries.

Reaction precipitation of amorphous calcium phosphate: Population balance modelling and kinetics

Precipitation of amorphous calcium phosphate being a precursor of hydroxy-apatite is investigated using a Y-mixer-tubular reactor device and a population balance model including nucleation, growth and agglomeration of particles. The set of moment equations generated by the population balance equation is solved using the direct quadrature method of moments. The population density function is approximated by gamma distribution with the Laguerre polynomial correction terms. The model parameters are estimated by fitting the population density function to experimental particle size distribution data using the least squares method. Identification of the kinetic parameters and the population density function proves to be satisfactory. The results of sensitivity analysis obtained by simulation with the estimated parameters show similar tendencies as those of experiments. The results demonstrate well that nano-scaled amorphous calcium phosphate can be obtained applying high temperature and high calcium ion concentrations.

Large eddy simulation of polydisperse particles in turbulent coaxial jets using the direct quadrature method of moments

Modeling coal particles in turbulent coal flames is a challenging but nonetheless critical step for predicting flame characteristics. Traditional approaches such as Reynolds-Averaged Navier-Stokes (RANS) coupled with Lagrangian solvers are unable to correctly predict particle concentrations and velocities in complex systems. This study shows how Large Eddy Simulations (LES) coupled with an Eulerian solver can address those issues. Previous Direct Numerical Simulations and Stokes number analysis suggest that LES has the capabilities to resolve all the turbulent length scales affecting coal particles in a coal flame and could therefore lead to more accurate simulation. The effects of fluid aerodynamics on the particle motion in the near field region of non-reacting coaxial particle-laden jets were simulated using LES. The Direct Quadrature Method of Moments (DQMOM) was used to track the particle phase in an Eulerian framework. Simulation results were compared to experimental data and accurately modeled a wide range of particle behaviors, such as particle jet waviness, spreading, break up, particle clustering and segregation, in different conditions. Simulations also predicted the mean axial velocity along the centerline for both the gas phase and the solid phase with a maximum error of 12% relative to experimental data. This study therefore provides a solid validation of the LES with DQMOM approach to model particles in turbulent flows and justifies its use for coal flame simulations.

A Detailed Validation Study of Multi-Environment Eulerian Probability Density Function Transport Method for Modeling Turbulent Nonpremixed Combustion

The current work involves the validation of presumed shape multi-environment Eulerian probability density function (PDF) transport method (MEPDF) using direct quadrature method of moments (DQMOM)-interaction by exchange with mean (IEM) approach for modeling turbulence chemistry interactions in nonpremixed combustion problems. The joint composition PDF is represented as a collection of finite number of Delta functions. The PDF shape is resolved by solving the governing transport equations for probability of occurrence of each environment and probability-weighted mass fraction of species and enthalpy in Eulerian frame for each environment. A generic implementation of the MEPDF approach is carried out for an arbitrary number of environments. In the current work, the MEPDF approach is used for a series of problems to validate each component of MEPDF approach in an isolated manner as well as their combined effect. First of all, a nonreactive turbulent mixing problem with two different Reynolds numbers (Re = 7000 and 11,900) is used for validation of the mixing and correction terms appear in the MEPDF approach. The second problem studied is a diffusion flame with infinitely fast chemistry having an analytical solution. The reaction component is validated by considering a 1D premixed laminar flame. In order to validate the combined effect of mixing and turbulence chemistry interactions, two different turbulent nonpremixed problems using global one-step chemistry are used. The first reactive problem used is H2 combustion (DLR Flame H3), while the second reactive validation case is a pilot-stabilized CH4 flame. The current predictions for all validation problems are compared with experimental data or published results. The study is further extended by modeling a turbulent nonpremixed H2 combustion using finite-rate chemistry effects and radiative heat transfer. The current model predictions for different flame lengths as well as minor species are compared with experimental data. The current model gave excellent predictions of minor species like OH. The differences in the current predictions with experimental data are discussed.

Simulation of the compressible flow with mass transfer of semi-continuous mixtures using the direct quadrature method of moments

We developed a method for the solution of the compressible flow with mass transfer of semi-continuous mixtures and it is based on the quadrature method of moments (QMoM) for continuous thermodynamics. The method extends the adaptive characterization of the continuous mixture to field problems and solves the mass transport equation for the continuous component. The method is referred as direct QMoM, or DQMoM, for continuous thermodynamics. It was implemented in the OpenFOAM® as a compressible ideal gas flow solver. The DQMoM was applied to the mixing flow of two gas streams with different compositions of a mixture with 57 hydrocarbons diluted in nitrogen. We showed that 4 adaptive components reproduced the mixture properties within 3% accuracy. Furthermore, the DQMoM CFD solution was approximately two times faster than the solution using 58 discrete components.

Comminution process modeling based on the monovariate and bivariate direct quadrature method of moments

Population balance equations (PBE) applied to comminution processes are commonly based on selection and breakage functions, which allow the description of changes of particle‐size distributions vs. time. However, other properties, such as the particle strength, also influence the grinding kinetics. A bivariate PBE was developed and resolved by the direct quadrature method of moments. This equation includes both the particle size and strength, the latter of which is defined as the minimum energy required for breakage. The monovariate case was first validated by comparing the predicted moments with those calculated from the size distributions given by an analytical solution of the PBE derived for specific selection and breakage functions. The bivariate model was then compared with a discretized model to evaluate its validity. Finally, the benefit of the bivariate model was proven by analyzing the sensitivity of some parameters and comparing the results of the monovariate and bivariate cases. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1621–1631, 2014