(287p) Towards the Optimal Reconstruction of a Distribution from Its Moments

In the last 10 years, a renewed interest can be observed in the scientific community dealing with chemical engineering applications concerning moment-based determination of Particle-Size Distributions (PSD). This is in particular due to the fact that, when external features like turbulent flow properties play an important role for the process under investigation, fast and numerically efficient methods must be employed to describe the population interacting with this flow. Acceptable computational costs are typical for standard moment methods and for related evolutions like the quadrature method of moments (QMOM) and its direct alternative (DQMOM). For all such methods, only a finite number of moments associated with the real distribution are finally determined by the numerical procedure. Therefore, after having computed these moments, it is necessary to reconstruct in the best possible way the full, real distribution corresponding to the resulting PSD. Since PSD generally constitute the key result to judge the quality of the process, the high importance of the reconstruction procedure appears clearly. This inverse problem is well-known and has been often considered during the last hundred years, not only in the field of chemical engineering. Different solutions have been proposed and tested. Nevertheless, no really satisfactory method can be found in the literature up to now. For most researchers in chemical engineering dealing with fundamental aspects, this is due to the fact that, mathematically, all the moments up to infinity are requested in order to obtain a really accurate reconstruction. As such, the problem considered in this work would not have any practicable solution. At the other end, for chemical engineers dealing with very applied questions, one possible solution concerning the reconstruction, often used in practice, is to assume a priori the shape of the distribution (gaussian, log-normal, beta function...) and to find a best fit of the small number of parameters needed to fully determine this assumed distribution. Building on top of this experience methods are developed in this work, which should be easily usable in practice to obtain the best possible PSD, but without imposing a priori the shape of the distribution. In this way, complex PSD can be computed naturally and at acceptable numerical costs using the described procedure. This is made possible in particular by revisiting the mathematical conditions underlying this inverse reconstruction problem, while explicitly taking into account the supplementary information concerning the constraints underlying any real PSD found in chemical engineering. By combining those with optimal methods of applied mathematics and algorithms, new possibilities appear for moment-based reconstruction. In the paper, the available literature on this subject is first considered, going back to Smoluchowski (1917) up to present time [1,2], to show that both approaches (mathematical view: ?this is impossible?; practical view: ?just use an assumed shape?) offer more possibilities that what has been used up to now. In the mathematical literature, the finite-moment problem has been mainly studied analytically, e.g. proving the existence of solutions for certain classes of functions. Nevertheless, a thorough treatment of this problem from the point of view of Numerical Mathematics is still missing.

Examples of results obtained using a priori shapes for the PSD will be presented in order to illustrate the advantages (fast and easy computations) and drawbacks (limitation to simple shapes, a priori knowledge needed about the solution) of this approach. In a second step, a discrete method based on a time-dependent update of the distribution together with the computation of the moments will be presented. This method is very fast, numerically efficient and easy to implement, but cannot always be extended to all possible physical processes encountered in chemical engineering. In this case, by solving numerically the moment equations, the trajectories of the system are determined and can be used to recover the time-dependent birth and nucleation profiles during the process. Knowing the initial particle distribution, a simple numerical algorithm is implemented to add new particles and shift appropriately the existing ones. In a final step, a highly flexible solution based on general functions is presented to solve the reconstruction problem. For this, either a Laplace transformation or alternatively a spline-based reconstruction are introduced to describe the distribution, without any a priori hypothesis concerning its shape. Using the information contained in the moments, a very good approximation of the real distribution can be obtained at an acceptable computational cost. The practical numerical procedure underlying these approaches will be presented in detail. To finally illustrate the accuracy of all these approaches (advantages, problems and limitations), the results of two test-cases will be presented and discussed. We first consider a given, three-peaked PSD as a test problem (Fig.1). In a second step these methods are tested on a real engineering problem concerning crystal growth, demonstrating the interest of these developments for an accurate reconstruction of the Crystal Size Distribution.

[1] R.B. Diemer and J. H. Olson, A moment methodology for coagulation and breakage problems: Part 2-moment models and distribution reconstruction, Chem. Eng. Sci. 57:2211-2228 (2002) [2] A. Giaya and R. W. Thompson, Recovering the crystal size distribution from the moment equations, AIChE J. 50(4):879-882 (2004)

Legend of Fig.1: Example of moment-based reconstruction. Thick solid line: exact solution, used to determine the moments. Dashed line, left figure: reconstruction using 3 beta-functions based on 3x3 moments. Dash-dotted line, right figure: automatic spline-based reconstruction using 8 moments.