On the solution and parameter estimation problem of a nonlinear intergrodifferential population balance

Abstract

The prediction of crystal size distribution from a continuous crystallizer at steady state is important for the simulation, operation and design of crystallizers. In this research, we consider integrodifferential population balance equations (PBE) describing the crystal size distribution for a crystallizer with random growth dispersion and particle agglomeration. We first develop numerical schemes to solve the initial value problem after we establish the well-posedeness of this problem. We then test the performance of these schemes on examples with known solutions. The numerical results from the first scheme we offer are in excellent agreement with the analytical solutions. However, the other variations we examined appear to be inferior. We then examine the analytical solution of the physical model and study its convergence, positivity and monotonicity under certain conditions. We then address the problem of parameter identification in the constant parameters case. The nature of the equation is that the solutions are oscillatory for certain ranges of the parameters that are of interest. In order to carry out the identification of parameters that would be physically meaningful, we are required to solve a boundary value problem coupled with an optimization procedure. We solve the BVP by the shooting method employing our nurnerical scheme for the IVP. We then couple it with the Marquadt-Levenberg algorithm to obtain optimal estimates for the parameters. We found out that the shooting method is rather sensitive to the initial guesses. To ensure the speedy convergence of the optimization, it is well known that "good" initial guesses are necessary. For this purpose, we derive the moments of the PB Ii' and use them to obtain good initial estimates. Our algorithm performed well when applied on a made-up example, and to the physical problem for small parameter values. For large parameter values, or for larger domains of the independent variable, obtaining physically meaningful results was not possible due to the oscillating nature of the solution. This suggests that the implementation of a more refined shooting method is necessary.