Integration and dynamic inversion of population balance equations with size-dependent growth rate

Two novel solution schemes for integration and dynamic inversion of a class of population balance equations with size-dependent growth rate are contributed in this article. The proposed methods are developed for population balance systems that, in addition to an advective and a birth rate term, include an external variable which may be fixed (referring to the integration problem), or is to be computed for some prespecified evolution of the population system (referring to the dynamic inversion problem). A unique diffeomorphism for the independent time and internal (property) coordinates is introduced, which transforms the original nonlinear partial differential equation into a linear one with straight lines as equation characteristics. The evolution of the density function in the time and internal property coordinates is then computed by transporting the initial and boundary density functions in the transformed domain. For the integration of the temporal behavior of the boundary density function, a generalization of the standard method of moments is introduced, resulting in a closed integro-differential structure driven by convolution and correlation integrals. While the correlation integrals refer to the given initial density function and, hence, can be a priori computed, the convolution integrals involve the boundary density function and have to be integrated a posteriori online. The solution of the dynamic inversion problem, on the other hand, turns out to become simpler, as it converts to an algebraic equation after pre-computation of the correlation/convolution integrals for the given initial and boundary density function. In a next step, we introduce the concept of internal or eigenmoments which is useful for the representation of the original physical moments in terms of an infinite series. The dynamics of eigenmoments exhibits a closed ODE structure, which we refer to as the internal model. From the perspective of the systems theory this turns out to be an infinite dimensional flat system. Hence, in addition to providing a highly simple structure – basically, a chain of integrators – for the integration of the moments and the density function, the internal model allows for a direct solution of the dynamic inversion problem due to its flatness property. However, the ease and elegance of the method, in general, come at the price of an approximation of the infinite dimensional problem by a finite one. The usability of both proposed solution methods is illustrated on a batch crystallization process with size-dependent growth rate kinetics. The proposed methods are compared in terms of efficiency and accuracy with a state-of-the-art high-resolution finite volume scheme using a numerical example.