Model Study Of Micro-mixing Based On Population Balance Framework

Micro-mixing is a common phenomenon in the chemical industry,which generally refers to,after the turbulent dispersion,the homogenization of materials from the smallest scale(Kolmogorov scale)to the molecular scale.The micro-mixing process is the key to realize fast chemical reaction system.If the micro-mixing rate is slower than the intrinsic rate of chemical reaction,the material may not be dispersed uniformly before the reaction is completed,resulting in frequent occurrence of side reactions,which ultimately have an important impact on the reaction yield,conversion rate,selectivity,and product quality.Consequently,how to deeply study the mechanism of micro-mixing process and construct its corresponding theoretical model has been the focus and difficulty in the field of international chemical engineering.By analyzing the previous work,it is concluded that the existing micro-mixing model with the assumption that "the collision frequency between fluid elements is constant" is not universal.Therefore,in this paper,based on the fluid mechanics principle of collision between fluid elements,starting from the derivation of the collision frequency equation,a population balance framework with random parameters is established for the mixing behavior of fluid elements in the reactor by using the statistical method.Then,the methods of standard moment are used to solve the population balance model.Besides,combined with the physical behavior of the fluid elements,the cumulant-neglect closure method is used to close the unclosed chemical reaction terms,so that we can obtain a micro-mixing model which describes the process of coalescence and dispersion of fluid elements.Next,ground on the proposed model,the mathematical expressions of quasi-linear bi-molecular chemical reaction,parallel-competitive chemical reactions and consecutive-competitive chemical reactions are established respectively.Different from the previous models,the micro-mixing model based on population balance framework is extended and applied to more experimental comparisons in this paper to verify the rationality of the model.Furthermore,the experimental data of the parallel-competitive reaction system reported in the literature are validated by the model,and the results illustrate that the segregation index simulated by the model in this paper is in good agreement with the experimental results.Meanwhile,it shows that the segregation index decreases with the prolonging of the feeding time.In the stirred tank reactor,when the feeding time is certain,the impeller stirring speed increases,the degree of turbulence in the reactor increases(dissipation rate increases),the time for the material to disperse into fluid elements is shortened,and the initial scale of the fluid elements reduces,leading to a decrease of the segregation index finally.It is proved that the higher the intensity of the turbulence where the feeding point is set at(discharge stream region of the impeller),the smaller the segregation index becomes.The reliability of the model in this paper was further verified by comparing the predicted results of the model with the experimental data of the consecutive–competitive chemical reactions reported in the literature.The simulation results show that the increase of the initial concentration and the volume ratio of the reactants can cause the increase of the segregation index.Although there are some differences between the simulation results and the experimental data,the overall trend is consistent.In addition,under the condition of semi-continuous operation,the prediction results of the model in this paper are in better agreement with the experimental data than those of the slab model and the engulfment model.With the increase of viscosity,the scale of fluid elements increases,and the mass exchange between fluid elements weakens,and the time to reach the micro-mixing uniformity is prolonged.As the reaction time is determined,the reaction will end before uniform mixing on the molecular scale has been reached,thereby the increase of viscosity will increase the segregation index.