Showing papers on "Population balance equation published in 1995"

New discretization procedure for the breakage equation

Abstract: A new discretization method, applicable for both batch and continuous systems, is developed for the breakage equation. The problem of intrainterval interactions due to discretization is accounted for by matching the zeroth and first moments of the continuous population balance equation with the corresponding moments of the discretized equation, thereby guaranteeing conservation of mass and total number of particles. Without loss of generality, the use of this method is demonstrated with a power law form of the specific rate of breakage, and with both theoretical and empirical breakage functions. The systematic method requires minimum computational efforts by allowing the user to choose either geometric size intervals with any geometric ratio or equal-size intervals for the particle size range. Simulation results show that the new method significantly improves predictions of the particle size distribution over the discretization method currently in use.

Evaluation of numerical methods for simulating an evolving particle size distribution in growth processes

Abstract: The particle growth term renders hyperbolic the dynamic population balance equation. Problems associated with the numerical solution of hyperbolic partial differential equations with stationary grid methods are well known. Moreover in the common case of combined molecular particle growth and coagulation, the convolution integral of the coagulation terms makes the moving grid methods computationally intractable. To cope with practical problems, this work is focused on numerical solution methods, of the population balance equation, characterized by relatively small computational cost and fair accuracy - comparable to that of relevant experimental data. For this purpose, previous work on appropriate discretization of the coagulation terms is extended for the growth terms. Several numerical methods are systematically evaluated and further extended. Recommendations are made concerning the best method, by taking into account the nature of the problem, the prevailing physical conditions, and the main quantity of...

Kinetics of heterogeneous magnetic flocculation using a bivariate population-balance equation

Abstract: The discrete bivariate population-balance equation is formulated and solved to describe the kinetics of heterogeneous magnetic flocculation of colloidal paramagnetic particles in a uniform magnetic field. The particles are allowed to have various sizes and values of magnetic susceptibility. Computations show the importance of particle size and magnetic susceptibility on the flocculation rate and the transient bivariate (size/magnetic susceptibility) density function. The particle size distribution of certain magnetic-susceptibility particles and the magnetic-susceptibility distribution of certain size particles are calculated as functions of time and initial and operatingconditions. The composition of a floe at any time depends on magnetic, van der Waals, double layer, and hydrodynamic forces among pairs of particles. The magnetic force is a function of the particle size, magnetic susceptibility, and strength of the magnetic field. Results are presented for various initial conditions of particles...

Transient hydrodynamics, mass transfer, and reaction in bubble columns : CO2 absorption into NaOH solutions

Abstract: We investigate the dynamic interactions of hydrodynamics, mass transfer and reaction in a bubble column reactor by means of numerical simulation and experimental observation. The model reactor was operated in the homogeneous flow regime. The chemisorption of carbon dioxide in sodium hydroxide was chosen as a fast model reaction. As a consequence of the enhanced mass transfer, bubble sizes and gas holdup decrease significantly in the axial direction. A one-dimensional, dynamic model was developed in order to predict the transient behaviour of the reaction system in the bubble column. The change of the bubble size due to carbon dioxide consumption has been incorporated by means of a population balance equation. This allows for the prediction of local time-dependent values of gas holdup and of the interfacial area. The numerical simulation was compared with experimental results. A good quantitative agreement was obtained

Simulation of vaporization height by considering the size distribution of drobbles in drobble columns

Abstract: A model for drobbles in drobble columns, based on the concept of phase space and the population balance equation, is presented. This model allows a consideration of the growth, breakage and coalescence of two-phase drobbles. The splitting step method is employed in the numerical calculation while the Monte Carlo method is used for handling the breakage and coalescence of drobbles. The vaporization height is a very important parameter in the engineering design of drobble columns with a vaporizing dispersed phase. Obviously this height has statistical characteristics. The model has been used for the simulation of vaporization height in this kind of system. The simulation results are in good agreement with experimental data not only qualitatively but also quantitatively. Detailed information about the variation of distribution density functions with column height is given both without and with consideration of the breakage and coalescence of drobbles.