(583f) Simulation and Development of an Estimation Technique for a Bi-Component Aggregation Systems

In chemical engineering, material science and biology there are many systems that require modeling the evolution of particle population. Modeling efforts usually result in deriving integro-differential equations. The particles in these systems, depending on the system, are characterized by one or more properties such as type, size or composition. However, the governing equation obtained from the population balance is not easy to solve especially for multivariate systems. One of the simplest examples with a bivariate distribution function is coagulation in pharmaceutical applications. In such systems, an excipient (solvent) binds the drug particles (solute) to make larger granules.  We would like to estimate the output’s unmeasurable properties i.e. statistical information, for control purposes. This work focuses on two-component coagulation processes and estimates the output distribution of particles in time.

The rate of coagulation is quantified by a kernel. In literature, there are analytical solutions for population balance equations in which the kernels are constant or additive [1-4]. However, the dynamic of agglomeration is more complicated and its rate is described by a Brownian kernel. Constant-number Monte Carlo (cNMC) is an algorithm to simulate these more complicated processes [5,6]. In this work, we modified this algorithm for a continuous coagulation process. Although extremely useful, Monte Carlo simulations are computationally demanding and impractical in online applications. The computations are necessary for real-time state estimation during the evolution process. The method of moments is an approach to reduce the original integro-differential equation to a set of ordinary differential equations (ODEs). The state variables of these ODEs are the moments of particles’ distribution. To ensure moment closure, we use Taylor expansion to approximate some moments with fractional index with respect to the considered ones. This approach has already been employed but only for single variable distribution functions [7]. However, to the best knowledge of authors, there is no effort in the literature for reduction of the original integro-differential equations in multivariate systems. The results show the probabilistic moments obtained by method of moments in time are in agreement with cNMC simulation results.

The measurement of number of particles in the outflow stream is possible, whereas measuring the other output properties online which specify the further moments is not easy. The goal in this work is to estimate these probabilistic moments by measuring the output population of particles. In addition, we assume the feed flow concentration is affected by a white uniformly distributed noise signal. As a result, to obtain the statistical information, we need to employ an estimator which deals with uncertainties in the system. Moving Horizon Estimation (MHE) is a successful estimation technique for unmeasurable states in presence of noise in dynamics or output measurements. Although this method has many advantages over other alternative methods, it needs to solve a dynamic optimization at every sampling time which makes it slow. To deal with this issue, we use a new design proposed in our previous work [8,9]. This approach exploits Carleman linearization to give a more accurate approximation compared to the traditional linearization. This approach not only provides the analytical solution of the system, but also derives the KKT matrix which decreases the dynamic optimization computations significantly. Simulation results show this approach reduces the simulation running time significantly while the performance is at the same level as using the traditional MHE.

References:

[1] Lushinkov AA. Evolution of coagulation systems: III. Coagulating mixtures. J of Colloid & interface Sci. 1976;54:94-101.

[2] Gelbard FM, Seinfeld JH. Coagulation and growth of a multicomponent aerosol. J of Colloid & Interface Sci. 1978;63:472-479. 

[3] Krapivsky PL, Ben-Naim E. Aggregation with multiple conservation laws. Phys. Rev. E. 1996;53:291-298. 

[4] Vigil RD, Zi RM. On the scaling theory of two-component aggregation. Chem Eng Sci. 1998;53:1725-1729.

[5] Matsoukas T, Lee K, & Kim T. Mixing of components in two-component aggregation, American Institute of Chemical Engineers. 2006;52, 3088-3099. 

[6] Marshall Jr. C. L, Rajniak P, & Matsoukas T. Multi-component population balance modeling of granulation with continuous addition of binder, Powder Technology. 2013; 236, 211-220. 

[7] Yu M, Lin J & Chan T. A new moment method for solving the coagulation equation for particles in Brownian motion. Aerosol Sci. & Technol. 2008;42:705-713. 

[8] Hashemian N & Armaou, A. Fast moving horizon estimation of nonlinear processes via Carleman linearization. American Control Conference, Chicago, IL, 2015. 

[9] Hashemian N & Armaou A. Moving horizon estimation using Carleman linearization and sensitivity analysis. Annual AIChE Meeting, Atlanta, GA, 2014.