model reduction of bi-component aggregation systems via laguerre polynomials

PowerPoint Presentation

Negar HashemianAntonios ArmaouChemical Engineering DepartmentPennsylvania State UniversityModel reduction of bi-component aggregation systems via Laguerre polynomials

www.personal.psu.edu/users/s/u/suh245/

Hi, Im Negar Hashemian from Penn State University and my advisor is Dr. Armaou. Today, I am going to present a new method to reduce wet granulation models in Wurster unit. There are many dynamic systems in nature which consist of particles. Two-component granulation which is widely used in pharmaceutical applications is an example of these kind of processes. The particles in this system are distinguished by their size and composition. And a population balance study results in a complex mathematical model. In this presentation, Im going to talk about modeling the bi-component granulation processes, using orthogonal projections and the method of moments. Also, For more information you may read our papers listed at my webpage.

2This talk focuses on simulation and estimation of bi-component granulation:

Bi-component granulation

Model Reduction

N. Hashemian & A. Armaou

n this presentation, first, I explain the granulation process in a Wurster unit operation. Afterward, I will explain Constant-number Monte Carlo method to simulate this process. This method is reliable but very slow. And, we cant use this approach in granulation control or estimation. So, Im going to introduce a new method to reduce the original model down to an ordinary differential equation set employing Laguerre Polynomials and method of moments. The state variables of this reduced model are moments of the outputs distribution.

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The Wurster unit operation is used for granulation in the pharmaceutical industries. This unit has two concentric cylinders, the insert and the annulus. 3

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An air stream flows into the unit from the bottom.4

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The particles are loaded in the annulus. 5

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The particles from the annulus pass through the partition gap and then are pushed into the insert. The solvent is sprayed on the particles flowing in the insert. Then particles collide with each other and form bigger ones. The particles slow down and fall back into the annulus. The recirculation is continued until the desired size and composition are achieved.

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The population balance results in an integro-differencial equation:

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Each particle in this process has two components: The total mass of a particle is denoted by p, and s represents themass of the binder content in the particle. Also, f(p; s) describes the population distribution of the particles. In this small range, the number of particles may increase because of particle formation from the smaller ones. Or they might decrease because of collision with other particles and bigger particle formation. This integro-differential equation considers these two phenomena to obtain the rate of population distribution change, where k is the kernel of formation of particles from particles r1 and r2. This kernel function is dependent on size and composition. 7

8Coagulation kernel is a discontinuous function:Bonding ProbabilityGeometrical FactorPhysical FactorType of particles

Considers Particle Type

Considers Composition

ratio of the binder to the total mass

Considers Size


The agglomeration kernel is proportional to type of particles, physical and geometrical factor. In Wurster unit, no coagulation occurs between two binder drops. Also, the coagulation rate constant between a particle and a binder droplet is 100 times faster than two arbitrary particles. The Stokes number is used to quantify the behavior of the particles.The collision between two particles is successful when the Stokes number calculated for the pair is below a critical value, $St^*$. Therefore, $\psi$ returns one when the Stokes number of colliding particles is less than or equal to $St^*$ and is otherwise zero. Also geometrical factor relates the coagulation rate to the amount of accessible binder. Where yi is the ratio of the binder to the total mass

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Constant-Number Monte Carlo considers a simulation box with fixed number of particles:

Generate a random number

Pick two particles i and j, randomly

Noi and j are bonded Pick a particle randomly and duplicate

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One way to solve this intrgro-differential equation in time is constant Monte Carlo simulation (cNMC). In the cNMC method, there should always be a constant number of particles in the simulation box". At every sampling time we randomly pick two particles. Then we generate a random number. The ratio of the corresponding kernel to average value for kernels shows the agglomeration probability of the selected pair. When rho is bigger than this ratio, the pair is rejected and the random pair selection is repeated. Otherwise, the pair is accepted to form a bigger particle. After an agglomeration the number of particles in the box decreases. So, we randomly pick a particle in the box to duplicate. To keep the real concentration, the box must become larger:9

Stochastic SimulationMicroscopic behavior, more preciseSingle Component

Deterministic SimulationBulk statistics, fasterConstant-Number Monte Carlo Simulation Smith & Matsoukas (1998)Taylor approximation in Method of Moments Yu et al. (2008)Bi-Component

Constant-Number Monte Carlo Simulation Matsoukas et al. (2006)Taylor approximation in Method of Moments for a two-component coagulation process Hashemian & Armaou(2015)10The literature offers different solution methods for PBE:ACC 2016: Model Reduction of Bi-Component Processes for a more general Kernel using Laguerre Polynomials


Smith and Matsoukas introduced Constant-number Monte Carlo simulation for single component processes. This method is reliable but slow for applications such as state estimation or control. Yu el al. employ Taylor expansions to approximate the distribution moments of the system and derive a deterministic model. However, in agglomeration process the composition of particles is important as well. So, Matsoukas extends his stochastic simulation method for bi-component coagulation. In a previous work we exploited the idea in Yu el al.'s work but for bi-component coagulation processes to obtain a deterministic reduced order model. However, the Taylor expansions of moments cannot be used for problems with a composition-dependent kernel function. As a result in this work we use Lagurre polynomials to approximate the implicit governing equation with an ODE set.

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Model Reduction using Method of Moments:

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For the model reduction we define mixed moment variables: Therefore M00 is the number of particles per volume unit;M10 is the total mass per volume unit and M01 is total solute mass per volume unit. As we discussed before the governing equation appears in an integro-differential form. Then, we employ Laguerre polynomials Lns and Lms to approximate the population function, f. Plug the approximate function in the original integro-differential. After some simplifications this results in an ODE set, in which the moments are the variables.11

12Model Reduction using Method of Moments:

mono-component coagulation

Hulbert & Katz (1964)

Bi-component coagulation Hashemian et al (2016)


Hulbert exploit reduction technique for a mono-component coagulation process. The $n^{th}$ order Laguerre polynomial is defined as. And this is the weight function.The gamma distribution is used in the construction of these orthogonal polynomials. The support for the gamma distribution is limited to positive numbers. This makes these polynomials the preferred choice. We extend their work by considering the addition of a binder using 2D Laguerre polynomials. where $\lambda , \gamma , c$ and $d$ are the parameters of the Laguerre polynomials and are required to be positive. In order to obtain the coefficients $\kappa_{n,m}$, we use orthogonality properties and we obtain, knm12

Method of Moments +Simplifications13

Model Reduction using Method of Moments:

Original Integro-Differential Equation

Reduced Order Model


Using definition of moments and after some simplifications of the initial IDE we obtain the following expressionwhere $A( r):=p^i s^j$ and $\dot u_{i,j}$ is the $ij$ mixed moment of the input binder flow. In theory, all the moments are required for reconstruction of the particle population. However, for practicality issues, we limit our attention to only a finite set of the population moments. Also, the right-hand side of Equation needs to be numerically integrated over an unbounded four dimensional polyhedron. In practice there is an upper bound on the total mass and the binder mass of the particles in the system. This upper bound generally depends on the duration and the initial population distribution.13

The reduced order model tracks the Monte Carlo simulation results properly:14N. Hashemian & A. Armaou

To perform the evaluation, we use the result obtained by cNMC simulation as a benchmark and compare it with the reduced-order model results. These figures show the moments using cNMC method. And the black lines show the result from the reduced system.14

In summary, the reduced order model is reliable and applicable in online estimation:The models for granulation in the literature are in integro-differential forms. Method of moments using Laguerre polynomials is a useful method to reduce these models.The simulation result shows the reduced model can be used for on-line estimation of distribution moments of the system.

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N. Hashemian & A. Armaouwww.personal.psu.edu/users/s/u/suh245/

In summary, The models for granulation in the literature are in integro-differential forms. Method of moments using Laguerre polynomials is a useful method to reduce these models.The simulation result shows the reduced model can be used for on-line estimation of distribution moments of the system. For more information please read our papers list at my webpage.

We finally would like to acknowledge NSF for financial support of our research.