A new thermodynamic function for phase‐splitting at constant temperature,moles, and volume

We introduce a new thermodynamic function for phase‐split computations at constant temperature, moles, and volume. The new volume function F_i introduced in this work is a natural choice under these conditions. Phase equilibrium conditions in terms of the volume functions are derived using the Helmholtz free energy. We present a numerical algorithm to investigate two‐phase equilibrium based on the fixed point iteration and Newton method. We demonstrate usefulness and powerful features of the new thermodynamic function for a number of examples in two‐phase equilibrium calculations. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

等温反应湍流流动标量关联量输运的直接模拟

用谱方法对等温反应槽道湍流流动中标量关联量输运进行了三维直接模拟。所得到的无反应情况下的单标量时均值和脉动值与文献中差分法的直接模拟（DNS）数据一致。瞬态模拟结果显示有反应情况下标量脉动量也有条带结构。用直接模拟数据库对标量关联量输运方程各项的贡献进行了先验性统计,发现产生项和耗散项的贡献最大,扩散项和反应项的贡献很小,然而化学反应对各项的大小和分布规律影响很大。对标量关联量方程RANS模拟各项的封闭模型进行了后验性检验。与DNS的统计值相比,除了近壁区之外,在流场的大部分区域,模拟值与精确的统计值基本一致。     

Bifurcation analysis of index infinity DAE parabolic models describing reactors and reacting flows

We show that most steady‐state models of chemical reactors and reacting flows in which convection effects are dominant and diffusion/conduction is neglected in the flow direction but included in the transverse directions, may change from parabolic type with a unique solution to index infinity differential‐algebraic equation (DAE) type with an infinite number of steady‐state solutions depending on the values of the reaction parameters. When a model is of index infinity, standard numerical methods may find only one of the solutions corresponding to latest possible ignition. We present complete bifurcation analysis of these models, a method for finding all solutions, determine the stability and, for some simpler cases, the domain of initial conditions attracted to these states. We also demonstrate that the various steady‐state solutions of the DAE systems are best found by integrating the transient hyperbolic versions of the models with appropriately selected capacitance terms and initial conditions. © 2016 American Institute of Chemical Engineers AIChE J, 63: 295–305, 2017     

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

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.     

A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows

This paper investigates the first part of a two-stage methodology for the detailed fully coupled modelling of nanoparticle formation in turbulent reacting flows. We use a projected fields (PF) method to approximate the joint composition probability density function (PDF) transport equation that describes the evolution of the nanoparticles. The method combines detailed chemistry and the method of moments with interpolative closure (MoMIC) population balance model in a commercial computational fluid dynamics (CFD) code. We show details of the implementation and present an extensive set of numerical experiments and validation. We consider the example of the chloride process for the industrial synthesis of titania. We show good agreement with experimental data and present fully coupled detailed chemistry CFD simulations of nanoparticle formation in a representative ‘slot’ reactor geometry. The simulations show that inception occurs in a mixing zone near the reactor inlets. Most of the nanoparticle mass is due to surface growth downstream of the mixing zone with a narrower size distribution occurring in the regions of higher surface growth. The predicted temperature and particle properties are compared to a perfect mixing case. The implications for the second part of the methodology, where it is proposed to post-process the data using a more detailed particle model, are discussed critically.     

Comparison of the stochastic fields method and DQMoM-IEM as turbulent reaction closures

This paper compares two mean reaction rate closures for turbulent reacting flow: the Stochastic Fields (SF) method and the Direct Quadrature Method of Moments using the Interaction by Exchange with the Mean micromixing model (DQMoM-IEM). The methods have many common features and have received significant attention in recent literature, yet have not been systematically compared. We present both methods in the same mathematical framework and compare their numerical performance. In addition, we introduce antithetic sampling as a variance reduction technique to increase the efficiency of the SF algorithm. We extend the methodology to take advantage of this development and show details of the implementation of each method in a commercial computational fluid dynamics code. We present a systematic investigation and consider both axisymmetric and 3D formulations of a problem known from the literature. DQMoM-IEM showed excellent agreement with experimental and transported probability density function data. SF gave reasonable agreement, but retained a minor grid-dependence not seen with DQMoM-IEM and did not fully resolve the sub-grid segregation of the species. The antithetic sampling was demonstrated to significantly increase the efficiency of the axisymmetric SF cases.     

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

The method of moments with interpolative closure (MOMIC) for soot formation and growth provides a detailed modeling framework maintaining a good balance in generality, accuracy, robustness, and computational efficiency. This study presents several computational issues in the development and implementation of the MOMIC-based soot modeling for direct numerical simulations (DNS). The issues of concern include a wide dynamic range of numbers, choice of normalization, high effective Schmidt number of soot particles, and realizability of the soot particle size distribution function (PSDF). These problems are not unique to DNS, but they are often exacerbated by the high-order numerical schemes used in DNS. Four specific issues are discussed in this article: the treatment of soot diffusion, choice of interpolation scheme for MOMIC, an approach to deal with strongly oxidizing environments, and realizability of the PSDF. General, robust, and stable approaches are sought to address these issues, minimizing the use of ad hoc treatments such as clipping. The solutions proposed and demonstrated here are being applied to generate new physical insight into complex turbulence-chemistry-soot-radiation interactions in turbulent reacting flows using DNS.

Copyright 2014 American Association for Aerosol Research     

MODELING OF THE REACTANT CONVERSION RATE IN A TURBULENT SHEAR FLOW

Results are presented of direct numerical simulations (DNS) of a spatially developing shear flow under the influence of an infinitely fast chemical reactions of the type A + B → Products. The simulation results are used to construct the compositional structure of the scalar field in a statistical manner. The results of this statistical analysis indicate that the use of a Beta density for the probability density function (PDF) of an appropriate Shvab-Zeldovich mixture fraction provides a very good estimate of the limiting bounds of the reactant conversion rate within the shear layer. This provides a strong justification for the implementation of this density in practical modeling of non-homogeneous turbulent reacting flows. However, the validity of the model cannot be generalized for predictions of higher order statistical quantities. A closed form analytical expression is presented for predicting the maximum rate of reactant conversion in non-homogeneous reacting turbulence.     

An improved collision damping time for MP-PIC calculations of dense particle flows with applications to polydisperse sedimenting beds and colliding particle jets

This paper describes several improvements to a numerical model introduced by O’Rourke et al. (2009) for collisional exchange and damping in dense particle flows. O’Rourke et al. (2009) use a Bhatnagar, Gross, and Krook (BGK) approximation to the collision terms in a particle distribution function transport equation to model the effects of particle collisions on damping fluctuating particle velocities and, in gas/liquid/solid beds, fluctuating temperatures and compositions of liquid films on particle surfaces. In this paper we focus on particle flows in which the particles have no liquid films and report on an improved expression we have developed for the collision damping time of particle velocity fluctuations used in the BGK approximation. The improved expression includes the effects on the collision damping time of the particle material coefficient of restitution and of non-equilibrium particle velocity distributions. The collision model improvements are incorporated into the general-purpose computational-particle fluid dynamics (CPFD) numerical methodology for dense particle flows. Three computational examples show the benefits of using the new collision time in calculations of particle separation in polydisperse dense particle flows and calculations of colliding particle jets.     

Numerical study on the turbulent reacting flow in the vicinity of the injector of an LDPE tubular reactor

Three-dimensional (3-D), transient numerical simulations of the turbulent reacting flow in the vicinity of the initiator injection point of a low-density polyethylene (LDPE) tubular reactor using a large eddy simulation (LES) approach combined with a filtered density function (FDF) technique are presented. The numerical approach allows for detailed predictions of the turbulent flow field and the associated (passive and reactive) scalar mixing. The aim is to study the influence of the injector geometry and initiator injection temperature on the LDPE process in terms of product quality (average polymer chain length, and polydispersity) and process efficiency (such as initiator consumption).     

Spin dynamics of an ultra-small nanoscale molecular magnet

We present mathematical transformations which allow us to calculate the spin dynamics of an ultra-small nanoscale molecular magnet consisting of a dimer system of classical (high) Heisenberg spins. We derive exact analytic expressions (in integral form) for the time-dependent spin autocorrelation function and several other quantities. The properties of the time-dependent spin autocorrelation function in terms of various coupling parameters and temperature are discussed in detail. The author wants to thank Dr. Gary Erickson for proof-reading the final draft of the paper.     

Numerical simulation of inviscid flows with hydrogen combustion behind shock waves and in detonation waves

A new numerical algorithm for simulation of nonequilibrium chemically reacting flows of an inviscid multicomponent gas is described. Application of this algorithm to the numerical solution of several problems of air-hydrogen mixture combustion in oblique detonation waves is demonstrated.Zhukovskii Central Aerodynamic Institute, Zhukovskii 140160. Translated from Fizika Goreniya i Vzryva, Vol. 31, No. 3, pp. 118–133, May–June, 1995.     

Numerical prediction of three-dimensional fiber orientation in Hele-Shaw flows

A numerical technique is developed to determine the three-dimensional fiber orientation in complex flows. The fiber orientation state is specified in terms of orientation tensors, which are used in several constitutive models. This method is applied to quasi-steady state Hele-Shaw flows in order to predict the flow-induced fiber orientation during injection molding at zero volume fraction limit. At the inlet, a number of fibers are introduced at a specified rate into the flow and each fiber location is traced during the mold filling. Along these determined paths, the independent components of fourth order orientation tensors are solved, describing the orientation state. The numerical grid generation technique, which is suitable for complex mold shapes, is employed for the flow solution. Orientation ellipsoids are calculated from the second order tensors and are used to present the fiber orientation results. The numerical solutions are obtained for channel and converging flows. Planar, longitudinal, and transverse orientation results are generated from the orthogonal projections of the orientation ellipsoids.     

Computer Simulation of a Novel Photocatalytic Reactor Using Distributive Computing Environment

In this paper, the numerical simulation for the in situ degradation of toxic water pollutants in the presence of semiconductor catalyst particles is considered. The computer simulation is in search of an efficient photocatalytic reactor design to achieve better fluid-catalyst contacting to minimize mass transfer limitation. In addition, in the present work, a state-of-the-art efficient computational technique, namely, distributed computing, has been employed to expedite the speed of calculation and overcome the memory bottleneck present in a single workstation. A comparative study has also been made of the computational effort required for a serial, parallel and distributed computing environment. The results show that the distributed computing technique is an economical and efficient method for overcoming difficulties, such as the memory bottleneck present in single workstations and the long computation time, associated with computer simulations of reacting flows.     

Effect of Mixing on the Nucleation and Growth of Titania Particles

Aerosol processes that produce titania particles by reacting gaseous precursors (such as titanium tetrachloride) initially must mix the precursor into the oxidizer at elevated temperatures to initiate the formation of product. Oftentimes the rate of reaction is sufficiently large as to be mixing limited. Thus the rate of mixing of the reacting species will control the chemistry and morphological properties of the particles that are produced. The interplay between mixing, nucleation, and growth in these systems is difficult to observe experimentally due to the small time scales that are involved and the spatial limitations of most diagnostics. An alternative approach is direct numerical simulation (DNS). DNS refers to a class of numerical solutions of the three-dimensional time-dependent governing equations for a particular system in which no turbulence modeling assumptions are made. To within the precision of the numerical algorithm, DNS can be thought of as a numerical experiment. Here we apply DNS to the mixing, reaction, nucleation, and growth of titania particles formed from the reaction of titanium tetrachloride with oxygen. The simulation solves for the velocity, species concentration, and eight moments of the particle size distribution using a combination of a pseudospectral method (for the velocity) and a compact finite difference scheme (for all of the scalars). The results show that increasing the rate of mixing increases the rate of particle formation while decreasing the variance in the particle size distribution. However, for a given extent of reaction, poorer mixing leads to larger mean particle sizes and larger standard deviations. The results are most easily interpreted in terms of the reaction volume between the unmixed reactants, where most of the reaction occurs. Based on this analysis, we present rules of thumb for controlling the particle size distribution in aerosol reactors.     

Large Eddy simulation for lean premixed combustion

In comparison to previous numerical studies interested in the ORACLES benchmark (One Rig for Accurate Comparisons with Large Eddy Simulations), the present study demonstrates the advantages of LES‐WALE model in both inert and reacting flows using the Fluent‐CFD. So, the confirmation is based on the experimental research effort that was involved in the European Union‐funded research program MOLECULES (Modelling of Low Emissions Combustors Using Large Eddy Simulations), for three parameters: longitudinal velocity, longitudinal velocity fluctuation, and length of recirculation zone. In line with what was observed by the experimental reference study, the dynamic model (LES‐WALE) predicts, respectively, as well as the asymmetry and the symmetry, for both inert and reacting flows. In addition, the simulation succeeds to predict the zones of recirculation and shows the differences between the two cases, inert and reacting flows. Moreover, results have been compared with those of the k–ε model performed by Kurenkov and Obserlack [Kurenkov and Obserlack, Flow Turbulence Combustion 74, 387–407 (2005)] study. © 2012 Canadian Society for Chemical Engineering     

Random noise and pole-dynamics in unstable front propagation

The problem of flame propagation is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise and also analytic arguments that explain these results in terms of the noisy pole dynamics.     

不可压缩湍流反应流动的直接模拟和RANS-SOM燃烧模型的检验

The second-order moment combustion model, proposed by the authors is validated using the direct numerical simulation （DNS） of incompressible turbulent reacting channel flows. The instantaneous DNS results show the near-wall strip structures of concentration and temperature fluctuations. The DNS statistical results give the budget of the terms in the correlation equations, showing that the production and dissipation terms are most important. The DNS statistical data are used to validate the closure model in RANS second-order moment （SOM） combustion model. It is found that the simulated diffusion and production terms are in agreement with the DNS data in most flow regions, except in the near-wall region, where the near-wall modification should be made, and the closure model for the dissipation term needs further improvement. The algebraic second-order moment （ASOM） combustion model is well validated by DNS.     

含反应节点过程数据校正的扩展独立物流法

Among the techniques developed for bilinear data reconciliation problems, the method based on independent flows is well known in terms of both accuracy and efficiency. However, the independent flow method is not effective when reactor units are present in the process. In this paper, an extended independent flow method is proposed for the data reconciliation of the process with chemical reaction. By the new method, the independent flows finding algorithm is adjusted to avoid the difficulties caused by the reactors in the process, and the reaction constraints are introduced into the objective function of data reconciliation. As a result, the new method can be applied to the process with chemical reaction while retaining high solution accuracy. To test the performance, the new method and the most typical Crowe‘s projection method are used in the data reconciliation of a typical industrial process. The results show that the new method can effectively accomplish the data reconciliation of the muhicomponent process with chemical reaction and gives more accurate estimates than the Crowe‘s method.     

Analytic solution for polydispersed aerosol dynamics by a wet removal process

An analytic solution for polydispersed aerosol dynamics by wet deposition of droplets was obtained. The collection efficiency used was based on the results of Jung and Lee (Aerosol Sci. Technol. 29 (1998) 389). The derived efficiency was compared with the existing formula of Slinn (Atmospheric Science and Power Production—1979, Division of Biomedical Environmental Research, U.S. Department of Energy, Washington DC, USA, 1983 (Chapter 11)) and showed good agreement. The particle size distribution was assumed to have a log-normal function, and the Cunningham slip correction factor was simplified to introduce the moment relationship. Both analytic and numerical solutions were obtained and compared with each other. The results show that the analytic solution can simulate the scavenging of polydispersed aerosol by wet deposition for the diffusion-dominant range well. The particle diameter of minimum collection efficiency was obtained analytically. It was found that the minimum collection efficiency diameter increases with decreasing velocity and decreasing packing density.