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
A general CFD-PBE(computational fluid dynamics-population balance equation) solver for gas–liquid poly-dispersed flows of both low and high gas volume fractions is developed in OpenFOAM(open-source field operation an...A general CFD-PBE(computational fluid dynamics-population balance equation) solver for gas–liquid poly-dispersed flows of both low and high gas volume fractions is developed in OpenFOAM(open-source field operation and manipulation) in this work. Implementation of this solver in OpenFOAM is illustrated in detail. The PBE is solved with the cell average technique. The coupling between pressure and velocity is dealt with the transient PIMPLE algorithm, which is a merged PISO-SIMPLE(pressure implicit split operator-semi-implicit method for pressure-linked equations) algorithm. Results show generally good agreement with the published experimental data, whereas the modeling precision could be improved further with more sophisticated closure models for interfacial forces, the models for the bubble-induced turbulence and those for bubble coalescence and breakage.The results also indicate that the PBE could be solved out the PIMPLE loop to save much computation time while still preserving the time information on variables. This is important for CFD-PBE modeling of many actual gas–liquid problems, which are commonly high-turbulent flows with intrinsic transient and 3 D characteristics.展开更多
A new method for the solution of population balance equations(PBE) describing the micro-processes such as nucleation,growth,aggregation of particle swarms in a multiphase system is proposed.The method is based on the ...A new method for the solution of population balance equations(PBE) describing the micro-processes such as nucleation,growth,aggregation of particle swarms in a multiphase system is proposed.The method is based on the fixed pivot moment and allows arbitrary number of moments to be tracked si-multaneously.By expressing PBEs for both batch and continuous operations in a general form,and using weighted residual method to derive the moment equations,different moments can be tracked directly.The numerical density function is assumed to be a summation of several weighted Dirac Delta functions,and the integral and derivative terms in PBEs are transformed to a summation in order to reduce computational cost.Simulations of a batch nucleation-growth process and a continuous ag-gregation-growth process have demonstrated good agreement with the corresponding analytical solu-tions,with relative errors less than 108%.Simulation of a combined nucleation-growth-aggregation process,which does not have an analytical solution,is also included,so as to reproduce the mi-cro-behaviors of such a complex system,demonstrating the feasibility and reliability of this method.展开更多
The breakage of brittle particulate materials into smaller particles under compressive or impact loads can be modelled as an instantiation of the population balance integro-differential equation.In this paper,the emer...The breakage of brittle particulate materials into smaller particles under compressive or impact loads can be modelled as an instantiation of the population balance integro-differential equation.In this paper,the emerging computational science paradigm of physics-informed neural networks is studied for the first time for solving both linear and nonlinear variants of the governing dynamics.Unlike conventional methods,the proposed neural network provides rapid simulations of arbitrarily high resolution in particle size,predicting values on arbitrarily fine grids without the need for model retraining.The network is assigned a simple multi-head architecture tailored to uphold monotonicity of the modelled cumulative distribution function over particle sizes.The method is theoretically analyzed and validated against analytical results before being applied to real-world data of a batch grinding mill.The agreement between laboratory data and numerical simulation encourages the use of physics-informed neural nets for optimal planning and control of industrial comminution processes.展开更多
The nucleation and growth kinetics of benzoic acid were determined in a population balance model,describing the seeded batch antisolvent crystallization process.The process analytical technologies(PATs)were utilized t...The nucleation and growth kinetics of benzoic acid were determined in a population balance model,describing the seeded batch antisolvent crystallization process.The process analytical technologies(PATs)were utilized to record the evolution of chord length distributions(CLDs)in solid phase together with the concentration decay in liquid phase,which provided essential experimental information for parameter estimation.The model was solved using standard method of moments based on the moments calculated from CLDs and solute concentration.A developed model,incorporating the nucleation and crystal growth as functions of both supersaturation and solvent composition,has been constructed by fitting the zeroth moment of particles and concentration trends.The determined kinetic parameters were consequently validated against a new experiment with a different flow rate,indicating that the developed model predicted crystallization process reasonably well.This work illustrates the strategy in construct a population balance model for further simulation,model-based optimization and control studies of benzoic acid in antisolvent crystallization.展开更多
Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna...Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.展开更多
Brownian coagulation is the most important inter-particle mechanism affecting the size distribution of aerosols. Analytical solutions to the governing population balance equation （PBE） remain a challenging issue. In...Brownian coagulation is the most important inter-particle mechanism affecting the size distribution of aerosols. Analytical solutions to the governing population balance equation （PBE） remain a challenging issue. In this work, we develop an analytical model to solve the PBE under Brownian coagulation based on the Taylor-expansion method of moments. The proposed model has a clear advantage over conventional asymptotic models in both precision and efficiency. We first analyze the geometric standard deviation （GSD） of aerosol size distribution. The new model is then implemented to determine two analytic solu- tions, one with a varying GSD and the other with a constant GSD, The varying solution traces the evolution of the size distribution, whereas the constant case admits a decoupled solution for the zero and second moments, Both solutions are confirmed to have the same precision as the highly reliable numerical model, implemented by the fourth-order Runge-Kutta algorithm, and the analytic model requires significantly less computational time than the numerical approach. Our results suggest that the proposed model has great potential to replace the existing numerical model, and is thus recommended for the study of physical aerosol characteristics, especially for rapid predictions of haze formation and evolution,展开更多
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.
基金Supported by the National Key Research and Development Program(2016YFB0301702)National Natural Science Foundation of China(21776284,21476236)+1 种基金Key Research Program of Frontier Sciences,CAS(QYZDJ-SSW-JSC030)Jiangsu National Synergetic Innovation Center for Advanced Materials
文摘A general CFD-PBE(computational fluid dynamics-population balance equation) solver for gas–liquid poly-dispersed flows of both low and high gas volume fractions is developed in OpenFOAM(open-source field operation and manipulation) in this work. Implementation of this solver in OpenFOAM is illustrated in detail. The PBE is solved with the cell average technique. The coupling between pressure and velocity is dealt with the transient PIMPLE algorithm, which is a merged PISO-SIMPLE(pressure implicit split operator-semi-implicit method for pressure-linked equations) algorithm. Results show generally good agreement with the published experimental data, whereas the modeling precision could be improved further with more sophisticated closure models for interfacial forces, the models for the bubble-induced turbulence and those for bubble coalescence and breakage.The results also indicate that the PBE could be solved out the PIMPLE loop to save much computation time while still preserving the time information on variables. This is important for CFD-PBE modeling of many actual gas–liquid problems, which are commonly high-turbulent flows with intrinsic transient and 3 D characteristics.
基金Supported by the National Basic Research Program of China (Grant No.2004CB720208)National Natural Science Foundation of China (Grant Nos.40675011 and 10872159)Key Laboratory of Mechanics on Disaster and Environment in Western China (Grant No.200707)
文摘A new method for the solution of population balance equations(PBE) describing the micro-processes such as nucleation,growth,aggregation of particle swarms in a multiphase system is proposed.The method is based on the fixed pivot moment and allows arbitrary number of moments to be tracked si-multaneously.By expressing PBEs for both batch and continuous operations in a general form,and using weighted residual method to derive the moment equations,different moments can be tracked directly.The numerical density function is assumed to be a summation of several weighted Dirac Delta functions,and the integral and derivative terms in PBEs are transformed to a summation in order to reduce computational cost.Simulations of a batch nucleation-growth process and a continuous ag-gregation-growth process have demonstrated good agreement with the corresponding analytical solu-tions,with relative errors less than 108%.Simulation of a combined nucleation-growth-aggregation process,which does not have an analytical solution,is also included,so as to reproduce the mi-cro-behaviors of such a complex system,demonstrating the feasibility and reliability of this method.
基金supported in part by the Ramanujan Fellowship from the Science and Engineering Research Board,Government of India(Grant No.RJF/2022/000115)。
文摘The breakage of brittle particulate materials into smaller particles under compressive or impact loads can be modelled as an instantiation of the population balance integro-differential equation.In this paper,the emerging computational science paradigm of physics-informed neural networks is studied for the first time for solving both linear and nonlinear variants of the governing dynamics.Unlike conventional methods,the proposed neural network provides rapid simulations of arbitrarily high resolution in particle size,predicting values on arbitrarily fine grids without the need for model retraining.The network is assigned a simple multi-head architecture tailored to uphold monotonicity of the modelled cumulative distribution function over particle sizes.The method is theoretically analyzed and validated against analytical results before being applied to real-world data of a batch grinding mill.The agreement between laboratory data and numerical simulation encourages the use of physics-informed neural nets for optimal planning and control of industrial comminution processes.
基金supported by National Natural Science Foundation of China (grant Nos.22108061,22178054,and 22068002)Natural Science Foundation of Hebei Province (grant No.B2022407009)Academic and Technical Leader Training Program for Major Disciplinessin Jiangxi Province (grant No.20212BCJ23001).
文摘The nucleation and growth kinetics of benzoic acid were determined in a population balance model,describing the seeded batch antisolvent crystallization process.The process analytical technologies(PATs)were utilized to record the evolution of chord length distributions(CLDs)in solid phase together with the concentration decay in liquid phase,which provided essential experimental information for parameter estimation.The model was solved using standard method of moments based on the moments calculated from CLDs and solute concentration.A developed model,incorporating the nucleation and crystal growth as functions of both supersaturation and solvent composition,has been constructed by fitting the zeroth moment of particles and concentration trends.The determined kinetic parameters were consequently validated against a new experiment with a different flow rate,indicating that the developed model predicted crystallization process reasonably well.This work illustrates the strategy in construct a population balance model for further simulation,model-based optimization and control studies of benzoic acid in antisolvent crystallization.
文摘Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.
基金the Alexander von Humboldt Foundation(Grant No.1136169)the Open Foundation of State Key Laboratory of Loess and Quaternary Geology for financial supports+2 种基金the joint support of the National Natural Science Foundation of China(Grant Nos.11372299 and 11132008)the Sino-German Research Project (Grant No.GZ971)ZJNSF(Grant No.LY13E080007)
文摘Brownian coagulation is the most important inter-particle mechanism affecting the size distribution of aerosols. Analytical solutions to the governing population balance equation （PBE） remain a challenging issue. In this work, we develop an analytical model to solve the PBE under Brownian coagulation based on the Taylor-expansion method of moments. The proposed model has a clear advantage over conventional asymptotic models in both precision and efficiency. We first analyze the geometric standard deviation （GSD） of aerosol size distribution. The new model is then implemented to determine two analytic solu- tions, one with a varying GSD and the other with a constant GSD, The varying solution traces the evolution of the size distribution, whereas the constant case admits a decoupled solution for the zero and second moments, Both solutions are confirmed to have the same precision as the highly reliable numerical model, implemented by the fourth-order Runge-Kutta algorithm, and the analytic model requires significantly less computational time than the numerical approach. Our results suggest that the proposed model has great potential to replace the existing numerical model, and is thus recommended for the study of physical aerosol characteristics, especially for rapid predictions of haze formation and evolution,
