Multi-dimensional Population Balance Equations Research Articles

Process Optimization of Para-xylene Crystallization Separation Process via Morphology Approach, Multi-dimensional Population Balance Equation, and Equation-Oriented Models.

An activity coefficient-based model was proposed to predict pertinent saturated concentrations in organic solid-liquid equilibrium, and the binary parameters of xylene mixtures were experimentally obtained. Also, a novel monocular 3D reconstruction technique was developed to measure crystal size and applied to derive the kinetics of nucleation and growth of para-xylene crystals. Subsequently, a multi-dimensional population balance equation was used to predict the particle size distribution in the crystallizer and an algorithm was designed to simulate and optimize the economic benefit of the crystallization separation process. Consequently, it became possible to predict the optimal coolant flowrate and inlet temperature, as well as the feed flowrate for a crystallization process with given operating conditions and device parameters.

Modeling and Optimization of the Para-Xylene Continuous Suspension Crystallization Separation Process via a Morphology Technique and a Multi-Dimensional Population Balance Equation

In this study, we carried out a para-xylene crystallization experiment at constant temperature and concentration levels. Throughout the process, the kinetics of nucleation, growth, breakage, and aggregation of para-xylene particles were measured and built using a morphological approach. An additional a three-stage continuous suspension crystallization separation experiment was carried out, the process for which was simulated using the population balance model based on correlated kinetic equations. The population balance equation was solved using an extended moment of classes algorithm, and the solving process was implemented in MATLAB. In this case, the predicted particle size distribution of the products matched well with the experiment. In order to provide references for the optimization of the industrial para-xylene crystallization process, a three-stage suspension crystallization separation experiment was designed and conducted, in which each crystallizer had a distinct operating temperature and mean residence time. The effects of operating parameters on the final product were investigated further. The proposed models and algorithms can also be applied in other cases and provide an alternative approach for optimizing continuous crystallization processes.

Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation.

The application of multi-dimensional population balance equations (PBEs) for the simulation of granulation processes is recommended due to the multi-component system. Irrespective of the application area, numerical scheme selection for solving multi-dimensional PBEs is driven by the accuracy in (size) number density prediction alone. However, mixing the components, i.e., the particles (excipients and API) and the binding liquid, plays a crucial role in predicting the granule compositional distribution during the pharmaceutical granulation. A numerical scheme should, therefore, be able to predict this accurately. Here, we compare the cell average technique (CAT) and finite volume scheme (FVS) in terms of their accuracy and applicability in predicting the mixing state. To quantify the degree of mixing in the system, the sum-square parameter is studied to observe the deviation in the amount binder from its average. It has been illustrated that the accurate prediction of integral moments computed by the FVS leads to an inaccurate prediction of the parameter for a bicomponent population balance equation. Moreover, the cell average technique (CAT) predicts the moments with moderate accuracy; however, it computes the mixing of components parameter with higher precision than the finite volume scheme. The numerical testing is performed for some benchmarking kernels corresponding to which the analytical solutions are available in the literature. It will be also shown that both numerical methods equally well predict the average size of the particles formed in the system; however, the finite volume scheme takes less time to compute these results.

Sonofragmentation of Rectangular Plate-like Crystals: Bivariate Population Balance Modeling and Experimental Validation

The use of multidimensional population balance equations in predicting ultrasound assisted sonocrystalization processes are curbed due to the unavailability of physically motivated macroscopic math...

Hybrid finite volume and Monte Carlo method for solving multi-dimensional population balance equations in crystallization processes

A novel hybrid framework that combines our recently proposed event driven constant number Monte Carlo (MC) method and a finite volume discretization scheme is developed to solve population balance equation (PBE) involving nucleation, growth, and breakage events for a batch crystallization process. At each time step, the nucleation and growth terms of the PBE are solved using high resolution finite volume method (HRFV), and then the breakage term is solved using the MC approach. This integrated approach is especially advantageous for solving multi-dimensional PBE involving nucleation, growth, and breakage events. The efficiency of the proposed approach is demonstrated by solving both one-dimensional (1D) and two-dimensional (2D) PBE from published literature. Also, a set of seeded batch cooling crystallization of plate-like pyrazinamide crystals were performed in presence of ultrasound (US) and the proposed hybrid solution scheme was used successfully to model and simulate this system characterised by 2D PBE.

A deterministic model for positional gradients in copolymers

A deterministic modelling approach is developed to predict the internal structure of gradient copolymer chains. A key innovation of the modelling approach is the introduction of a positional variable that gives direct access to quantitative gradient characteristics: the ensemble average composition and the gradient deviation. This positional variable is used to develop multi-dimensional population balance equations that can be solved numerically to calculate gradient quality measures. The methodology is illustrated using the gradient copolymerisation of ethylene and 1-octene via coordinative chain transfer mechanism, which is representative of a variety of polymerisation schemes for gradient copolymers. Simulation results are validated with those obtained by stochastic simulations which, until now, were the only means of predicting detailed gradient quality.

Non-isothermal generalized coalescence/redispersion model for micromixing

The non-isothermal coalescence/redispersion micromixing model is presented and analysed using the population balance approach. Development of the model is based on the concept of diffusive mass and heat exchange interactions of the fluid elements identified as the Kolmogorov-scale eddies. The model is formulated as a non-linear, multidimensional population balance equation describing deterministic chemical reactions and temperature variations inside the fluid elements and stochastic mass and heat exchange processes between the fluid elements by their impacts. The population balance equation is solved by applying a hybrid continuous time/Monte Carlo method. The behaviour and properties of the model is analysed by numerical experimentation using an adiabatic continuous stirred tank coalescence/redispersion reactor model. Simulation results are presented for second order uni-molecular and quasi-linear consecutive–competitive chemical reactions. It is shown that non-isothermal micromixing in a reactor can be evaluated and characterized by applying appropriately chosen concentration and temperature configurations of partially premixed or segregated feeds measuring the mean concentrations and temperature of the reaction mixture both in dynamic and steady state processes. In well-designed conditions measurement of the mean temperature in itself provides satisfactory information on the mixing state of reactors at microscale.

A class of exact solutions for population balances with arbitrary internal coordinates

We develop a novel transformation that maps the linear, nonhomogeneous, multidimensional population balance equation (PBE) into an advection equation that is readily solved using the method of characteristics. The PBEs targeted by this transformation exclude aggregation, breakage, and time dependent birth and death rates. In addition, internal coordinates are assumed to grow independently of each other. The ensuing general formulation is then used to recover closed‐form analytical solutions for problems with monosurface and bulk‐diffusion growth‐rates as well as Gaussian‐type nucleation. For completeness, we derive the multidimensional Green's functions for our approach. This is followed by a brief discussion on how the proposed framework may be used for code verification of moment methods such as the quadrature method of moments and the direct quadrature method of moments. Finally, a sample Mathematica code is provided to derive analytical solutions for the single‐internal‐coordinate case given user‐specified growth, birth, and death rates. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1691–1698, 2015

SIMULATION AND CONTROL OF MULTIDIMENSIONAL CRYSTALLIZATION PROCESSES

In this article, solving the population balance equation (PBE) and controlling the final particle size for crystallization problems have been addressed. For solving the general form of multidimensional PBE, a numerical method called conservation element and solution element (CE/SE) has been used. By applying this method to one- and two-dimensional crystallization problems, it has been shown that this technique can handle all types of source terms in the PBE. Model accuracy has been checked with experimental data reported in the literature and also with the analytical solution of PB-type equations. Control of final particle size was formulated in an optimization framework. To obtain a desired final size distribution for one-dimensional crystallization, the optimal temperature trajectory was achieved for size-dependent growth rate case with the agglomeration effect. For the two-dimensional case, the aspect ratio was controlled and the crystallization batch time minimized. To track the optimal temperature trajectories, PI controllers were used. Simulation results indicate that the desired final size distribution for the one-dimensional case and the aspect ratio for the two-dimensional case were approached very closely.

A computationally efficient technique for the solution of multi-dimensional PBMs of granulation via tensor decomposition

Multi-dimensional population balance equations (PBEs) are commonly used to describe the dynamics of particulate processes such as granulation. Such a class of equations are numerically complex and computationally intensive to solve due to the multiple internal coordinates involved. A computationally efficient model reduction technique would overcome the computational overheads associated with the solution of multi-dimensional PBEs. Moreover, this enables the process model to be used efficiently in process control and optimization. This study is concerned with the development of a novel reduced order model for a three-dimensional population balance model (PBM) for granulation, using a tensor decomposition technique in combination with separation of variables and singular value decomposition. These techniques were used to decompose the complex aggregation and breakage integrals. The developed model is faster by two orders of magnitude, requires less memory allocation for the storage of variables and results in negligible error when compared with the full model.

Discretized n-Dimensional Population Balance for Agglomeration

In this article, a new method for solving multidimensional population balance (PB) equations for agglomeration is presented. First, discretized 2-dimensional PB (DPB) is derived and by applying probability approach, this method is extended to n-dimensional PB. The discretization of size intervals is in the form of v(j) = 2j-i v(i) and the volume based PB is used. Because of conservative nature of volume, all internal coordinates are converted to some independent volumes. A simple and clear procedure for solving multidimensional PB is proposed using probability concept. In addition, various types of birth and death mechanisms involved in PB are analyzed separately. Moreover, results of obtained DPB in 2 and 3-dimensional are compared with time variations of zeroth moment of continuous 1D PB and excellent agreement is found. Finally, the developed method is applied for the simulation of granulation processes and the predicted results are presented and discussed.

On the solution of bivariate population balance equations for aggregation: X–discretization of space for expansion and contraction of computational domain

A new structured discretization of 2D space, named X-discretization, is proposed to solve bivariate population balance equations using the framework of minimal internal consistency of discretization of Chakraborty and Kumar [2007, A new framework for solution of multidimensional population balance equations. Chem. Eng. Sci. 62, 4112–4125] for breakup and aggregation of particles. The 2D space of particle constituents (internal attributes) is discretized into bins by using arbitrarily spaced constant composition radial lines and constant mass lines of slope −1. The quadrilaterals are triangulated by using straight lines pointing towards the mean composition line. The monotonicity of the new discretization makes is quite easy to implement, like a rectangular grid but with significantly reduced numerical dispersion. We use the new discretization of space to automate the expansion and contraction of the computational domain for the aggregation process, corresponding to the formation of larger particles and the disappearance of smaller particles by adding and removing the constant mass lines at the boundaries. The results show that the predictions of particle size distribution on fixed X-grid are in better agreement with the analytical solution than those obtained with the earlier techniques. The simulations carried out with expansion and/or contraction of the computational domain as population evolves show that the proposed strategy of evolving the computational domain with the aggregation process brings down the computational effort quite substantially; larger the extent of evolution, greater is the reduction in computational effort.

Lattice Boltzmann method for multi-dimensional population balance models in crystallization

In this work, lattice Boltzmann method (LBM) is developed for efficient and accurate solution of multi-dimensional population balance equations (PBEs) used to model crystallization processes with growth and nucleation. Detailed derivation of LBM for multi-dimensional advection equation is presented, where the velocity is a function of space coordinates. The developed scheme is subsequently applied to solve multi-dimensional PBEs with size-dependent growth rate by drawing an analogy between the advection equation and the PBE. The computational advantage of LBM is shown by solving several benchmark examples taken from the literature and comparing the results with those obtained using well-established high resolution (HR) method. It is found that LBM provides at least as accurate solution as HR method, while requiring lower computation time.

Dimension reduction of bivariate population balances using the quadrature method of moments

Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.

A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles

In this work, we show that straight forward extensions of the existing techniques to solve 2-d population balance equations (PBEs) for particle breakup result in very high numerical dispersion, particularly in directions perpendicular to the direction of evolution of population. These extensions also fail to predict formation of particles of uniform composition at steady state for simultaneous breakup and aggregation of particles, starting with particles of both uniform and non-uniform compositions. The straight forward extensions of 1-d techniques preserve 2 n properties of non-pivot particles, which are taken to be number, two masses, and product of masses for the solution of 2-d PBEs. Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112–4125] have recently proposed a new framework of minimal internal consistency of discretization which requires preservation of only ( n + 1 ) properties. In this work, we combine a new radial grid [proposed in 2008. part I, Chemical Engineering Science 63, 2198] with the above framework to solve 2-d PBEs consisting of terms representing breakup of particles. Numerical dispersion with the use of straight forward extensions arises on account of formation of daughter particles of compositions different from that of the parent particles. The proposed technique eliminates numerical dispersion completely with a radial distribution of grid points and preservation of only three properties: number and two masses. The same features also enable it to correctly capture mixing brought about by aggregation of particles. The proposed technique thus emerges as a powerful and flexible technique, naturally suited to simulate PBE based models incorporating simultaneous breakup and aggregation of particles.

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112–4125] to solve n-dimensional population balance equations (PBEs) with preservation of ( n + 1 ) instead of 2 n properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311–1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.

The cell average technique for solving multi-dimensional aggregation population balance equations

In this paper, the cell average discretization [Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., & Mörl, 2006a. Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique. Chemical Engineering Science, 61, 3327–3342] is extended to solving multi-dimensional population balance equations. Similar to the one-dimensional case, the scheme is based on an accurate prediction of certain moments of the population. The formulation is quite simple to implement, computationally not expensive and highly accurate. Numerical diffusion is a common problem with many numerical methods when applied on coarse grids. The presented technique nearly eliminates numerical diffusion and predicts four moments (zeroth, first, first cross and second) of the distribution function with high accuracy. The technique may be implemented on any type of grid. The accuracy of the scheme has been analyzed by comparing analytical and numerical solutions of three test problems. The numerical results are in excellent agreement with the analytical results and show the ability to predict higher moments very precisely. Additionally, an extension of the proposed technique to higher dimensional problems is discussed.

A new framework for solution of multidimensional population balance equations

A new framework is proposed in this work to solve multidimensional population balance equations (PBEs) using the method of discretization. A continuous PBE is considered as a statement of evolution of one evolving property of particles and conservation of their n internal attributes. Discretization must therefore preserve n + 1 properties of particles. Continuously distributed population is represented on discrete fixed pivots as in the fixed pivot technique of Kumar and Ramkrishna [1996a. On the solution of population balance equation by discretization—I. A fixed pivot technique. Chemical Engineering Science 51(8), 1311–1332] for 1-d PBEs, but instead of the earlier extensions of this technique proposed in the literature which preserve 2 n properties of non-pivot particles, the new framework requires n + 1 properties to be preserved. This opens up the use of triangular and tetrahedral elements to solve 2-d and 3-d PBEs, instead of the rectangles and cuboids that are suggested in the literature. Capabilities of computational fluid dynamics and other packages available for generating complex meshes can also be harnessed. The numerical results obtained indeed show the effectiveness of the new framework. It also brings out the hitherto unknown role of directionality of the grid in controlling the accuracy of the numerical solution of multidimensional PBEs. The numerical results obtained show that the quality of the numerical solution can be improved significantly just by altering the directionality of the grid, which does not require any increase in the number of points, or any refinement of the grid, or even redistribution of pivots in space. Directionality of a grid can be altered simply by regrouping of pivots.

A stochastic technique for multidimensional granulation modeling

AbstractRecent granulation modeling research has produced compelling evidence that simple one‐dimensional (1‐D) models will not suffice when describing the dynamics of particle growth. During the mixing process particles gradually become more saturated due to the loss of air in particles resulting from collisions with the surroundings. This process, called consolidation, influences not only granule pore saturation and binder layer thickness, but also particle coalescence properties. Due to the consolidation behavior observed, a simple size dependent model is typically not adequate. The solution of a multidimensional population balance equation (PBE) is performed using a multiscale solution technique. A microscale model which utilizes a discrete element method (DEM) simulation is used to determine granular flow characteristics and coalescence efficiency. Statistical analysis of data from interactions at the microscale is then used to produce a multidimensional coalescence kernel. Since analytical solutions to PBEs with more than one internal dimension typically do not exist, a constant number kinetic Monte‐Carlo (MC) method is used to solve the PBE. This solution technique is used in a test case to model growth of a lactose/starch/HPC and water mixture. The simulated results corresponded well with experimental data. This technique, while in its infancy, proves to be a unique and effective method of bridging multiple scales of the granulation process. © 2006 American Institute of Chemical Engineers AIChE J, 2006

Challenges in particulate polymerization reactor modeling and optimization: A population balance perspective

In the present article, recent developments on modeling and optimization of particulate polymerization processes are reviewed. A unified population balance approach is described to follow the time evolution of molecular and morphological polymer properties in batch and continuous polymerization reactors. The numerical methods as well as the computational issues related with the solution of the dynamic population balance equation are critically assessed. The orthogonal collocation on finite elements (OCFE) method and the fixed-pivot technique (FPT) are then applied to a free-radical batch polymerization reactor to calculate the dynamic evolution of the molecular weight distribution (MWD). Moreover, theoretical and experimental results are shown on the dynamic evolution of particle size distribution (PSD) in a suspension polymerization reactor. Recent advances in on-line monitoring of “polymer quality” are briefly discussed in the context of available hardware and software sensors. The problem of real-time optimization of polymerization processes under parametric uncertainty is also examined. Finally, new issues related with the modeling, numerical solution and control of multidimensional population balance equations are conferred.