Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system.A population balance equation(PBE),a non-linear hyperbolic equation o...Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system.A population balance equation(PBE),a non-linear hyperbolic equation of the number density function,is usually employed to describe the micro-behavior(aggregation,breakage,growth,etc.) of a disperse phase and its effect on particle size distribution.Numerical solution is the only choice in most cases.In this paper,three different numerical methods(direct discretization methods,Monte Carlo methods,and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation,computational load and numerical accuracy.Special attention is paid to the relatively new and superior moment methods including quadrature method of moments(QMOM),direct quadrature method of moments(DQMOM),modified quadrature method of moments(M-QMOM),adaptive direct quadrature method of moments(ADQMOM),fixed pivot quadrature method of moments(FPQMOM),moving particle ensemble method(MPEM) and local fixed pivot quadrature method of moments(LFPQMOM).The prospects of these methods are discussed in the final section,based on their individual merits and current state of development of the field.展开更多
The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time...The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time(the particle k-th moments),and the other is a function of dimensionless volume(self-preserving size distribution).In this paper,a simple iterative direct numerical simulation(iDNS)is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment,which has been solved with the Taylor-series expansion method of moment(TEMOM)in our previous work.The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature,and then the method is extended to the field of Brownian agglomeration over the entire size range.The results show that the difference between the lognormal function and the self-preserving size distribution is significant.Moreover,the thermodynamic constraint of the algebraic mean volume is also investigated.In short,the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship;thus,a complete method to solve the Smoluchowski coagulation equation asymptotically is established.展开更多
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...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(Hy QMOM). 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.展开更多
基金Supported by the National Basic Research Program of China (Grant No. 2004CB720208)the National Natural Science Foundation of China (Grant Nos. 40675011 & 10872159)the Key Laboratory of Mechanics on Disaster and Environment in Western China
文摘Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system.A population balance equation(PBE),a non-linear hyperbolic equation of the number density function,is usually employed to describe the micro-behavior(aggregation,breakage,growth,etc.) of a disperse phase and its effect on particle size distribution.Numerical solution is the only choice in most cases.In this paper,three different numerical methods(direct discretization methods,Monte Carlo methods,and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation,computational load and numerical accuracy.Special attention is paid to the relatively new and superior moment methods including quadrature method of moments(QMOM),direct quadrature method of moments(DQMOM),modified quadrature method of moments(M-QMOM),adaptive direct quadrature method of moments(ADQMOM),fixed pivot quadrature method of moments(FPQMOM),moving particle ensemble method(MPEM) and local fixed pivot quadrature method of moments(LFPQMOM).The prospects of these methods are discussed in the final section,based on their individual merits and current state of development of the field.
基金This research was funded by the National Natural Science Foundation of China with grant numbers 11972169 and 11902075.
文摘The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time(the particle k-th moments),and the other is a function of dimensionless volume(self-preserving size distribution).In this paper,a simple iterative direct numerical simulation(iDNS)is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment,which has been solved with the Taylor-series expansion method of moment(TEMOM)in our previous work.The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature,and then the method is extended to the field of Brownian agglomeration over the entire size range.The results show that the difference between the lognormal function and the self-preserving size distribution is significant.Moreover,the thermodynamic constraint of the algebraic mean volume is also investigated.In short,the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship;thus,a complete method to solve the Smoluchowski coagulation equation asymptotically is established.
文摘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(Hy QMOM). 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.
