Two methods for sensitivity analysis of coagulation processes in population balances by a Monte Carlo method Alexander Vikhansky, Markus Kraft ∗ Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, CB2 3RA, UK Abstract We consider two stochastic simulation algorithms for the calculation of parametric derivatives of solutions of a population balance equation, namely, forward and adjoint sensitivity methods. The dispersed system is approximated by an N-particle stochastic weighted ensemble. The infinitesimal deviations of the solution are accounted for through infinitesimal deviation of the statistical weights that are recalculated at each coagulation. In the forward method these deviations of the statistical weights immediately give parametric derivatives of the solution. In the second method the deviations of the statistical weights are used to calculate a finite-mode approximation of the linearized version of the population balance equation. The linearized equation allows for the calculation of the eigenmodes and eigenvalues of the process, while the parametric derivatives of the solution are given by a Lagrange formalism.  Keywords: Population balances; Monte Carlo; Sensitivity analysis 1. Introduction Equations of population balances give the general mathemat- ical framework for the description and modelling of particulate systems (Ramkrishna, 2000). Starting from the famous Boltz- mann equation (Bird, 1976), equations of population balances span over a wide range of physical, technological and envi- ronmental applications such as mixing (Spielman and Leven- spiel, 1965; Pope, 1985), liquid/liquid dispersion (Tsouris and Tavlarides, 1994; Hounslow and Ni, 2004), nanoparticle for- mation (Goodson and Kraft, 2002; Balthasar and Kraft, 2003), dynamics of atmospheric aerosols (Piskunov and Petrov, 2002), breakage and agglomeration of powders (Smith and Matsoukas, 1998; Lin et al., 2002; Maisels et al., 2004; Zhao et al., 2005; Mort, 2005), growth of microbial cell population (Henson, 2003), polymerization (Immanuel and Doyle III, 2003) and crystallization (Bermingham et al., 2003). The general area of applicability implies that the equations of population balances account for the most basic physical ∗ Corresponding author. Tel.: +44 1223 762784; fax: +44 1223 334796. E-mail address: mk306@cam.ac.uk (M. Kraft).  principles (mass, momentum and energy conservation), i.e., they provide only a framework, which has to “be filled in” by the physical information specific to the system under consideration. The general form of the equation of population balance reads m(t, x ; ) t = B(m(t, x ; ); ) - D(m(t, x ; ); ) ≡ L(m(t, x ; ); ), (1) where m(t, x ; ) is mass density of the particles with mass x (in the case of multicomponent particles x can be vector containing the masses of the components), B(m(t, x ; ); ) and D(m(t, x ; ); ) are birth and death rates of the particles due to collision and breakage, respectively, and  is a parameter.  can be a scalar, a vector or, even a function, which contains the empirical information about the system. Thus, a researcher who starts modelling a particulate system has to specify  which is unknown a priori, i.e., the following questions have to be addressed: (1) In many cases the dimension of the model is not obvious i.e., we do not know a priori how many parameters are necessary to describe the shape of the particles, or how This is the computational modelling group's latest version of the publication. For the published version please refer to doi:10.1016/j.ces.2006.03.009 at http://dx.doi.org.