Droplets Population Balance in a Rotating Disc Contactor: An Inverse Problem Approach A. Vikhansky and M. Kraft Dept. of Chemical Engineering, University of Cambridge, Cambridge CB2 3RA, U.K. M. Simon, S. Schmidt, and H.-J. Bart Dept. of Mechanical and Process Engineering, Technical University Kaiserslautern, 67653, Kaiserslautern, Germany DOI 10.1002/aic.10735 Published online December 16, 2005 in Wiley InterScience (www.interscience.wiley.com). An inverse problems method is applied to a two-phase liquid–liquid system in a rotating disc contactor (RDC). The dispersed phase is modeled by population balance equations, which are solved by a Monte Carlo method, together with the equations for the parametric derivatives of the solution with respect to the parameters of the model. The best-fitting problem is solved by a gradient search method. Because the inverse problem is ill-posed, the iteration procedure is augmented by an appropriate termination criterion to stabilize the calculations. The parametric derivatives of the solution can be used to quantify the relative importance of different parameters of the model. It is shown that the model’s parameters, which are identified on one set of the experimental data, adequately describe the behavior of the system under another unfitted operation condition, that is, the proposed method can be applied to scale-up problems. © 2005 American Institute of Chemical Engineers AIChE J, 52: 1441–1450, 2006 Keywords: two-phase systems, sensitivity analysis, inverse problems, Monte Carlo, pop- ulation balances, weighted particles method Introduction The common framework for the description and analysis of two-phase systems consists of mass, momentum, and energy balances, which require some additional closure relationships to make the equations mathematically tractable. A difficulty arises from the relationship between the available experimental data and the information that is needed for an analytical de- scription of multiphase flows. The experimentally accessible quantities such as gas holdup or interphase area characterize the dispersed system as a whole, whereas the mathematical modeling requires more detailed information about single drop- let behavior and droplet– droplet interaction. Therefore, an additional mathematical treatment is necessary to extract this information from the measurements, that is, the solution of the inverse problem is a necessary part of a reliable modeling strategy. Inverse problems for population balances have attracted much attention in recent years. 1-6 The solution of inverse prob- lems is often aided by the self-similar behavior of many prac- tically important dispersed systems. 2,4 In more general cases a mathematical programming procedure has to be applied to find the best fit of the experimental data. Note that inverse problems are highly sensitive to errors in the experimental data. 7 Because the experimental results always contain some noise, a regular- ization scheme has to be implemented to make the numerical algorithm stable. In the present study we consider liquid–liquid flow in a pilot-scale rotating disc contactor (RDC). 8 The parameter-fit- ting problem is solved by a gradient-search algorithm, 9 where a Monte Carlo method is used for the solution of the population balance problem and the calculation of the parametric deriva- tives of the solution. This approach has several important advantages: (1) a Monte Carlo method can easily be extended Correspondence concerning this article should be addressed to M. Kraft at mk306@cam.ac.uk. © 2005 American Institute of Chemical Engineers AIChE Journal 1441 April 2006 Vol. 52, No. 4