High-resolution Finite Volume Research Articles

A two-stage mechanistic reduced-order model of pharmaceutical tablet dissolution: Population balance modeling and tablet wetting functions

We propose a two-stage reduced-order model (ROM) of pharmaceutical tablet dissolution that is comprised of (i) a mechanistic dissolution function of the active pharmaceutical ingredient (API) and (ii) a tablet wetting function. The former is derived from a population balance model, using a high-resolution finite volume algorithm for a given API crystal size distribution and dissolution rate coefficient. The latter is obtained from the mechanistic understanding of water penetration inside a porous tablet, and it estimates the rate at which the API is exposed to the buffer solution for a given formulation and the dimensions of the tablet, contact angle, and surface tension between the solid and liquid phases, liquid viscosity, and mean effective capillary radius of the pore solid structure. In turn, the two-stage model is mechanistic in nature and one-way coupled by means of convolution in time to capture the start time of the API dissolution process as water uptake, swelling, and disintegration take place. The two-stage model correlates dissolution profiles with critical process parameters (CPPs), critical material attributes (CMAs), and other crucial critical quality attributes (CQAs). We demonstrate the model’s versatility and effectiveness in predicting the dissolution profiles of diverse pharmaceutical formulations. Specifically, we formulate and fabricate acetaminophen and lomustine solid tablets using different API content and size distributions, characterize their dissolution behavior, and estimate capillary radius as a function of tablet porosity. The estimations generated by the proposed models consistently match the experimental data across all cases investigated in this study.

Theoretical Investigation of the Nonlinear General Rate Model with the Bi-Langmuir Adsorption Isotherm Using Core-Shell Adsorbents.

Core-shell particles enable the separation of intricate mixtures in a highly efficient and rapid manner. The porous shell particles increased the separation efficiency with expedited flow rates due to an abatement in the pore volume accessible for longitudinal diffusion and a decrease in diffusion path length. This study focuses on the numerical approximation of a nonlinear isothermal general rate model applied to stationary bed columns that are replete with inert core adsorbents featuring double adsorption sites. The transport of solute in heterogeneous porous media can be modeled by a nonlinear convection acquiescent partial differential equation system together with a specific nonlinear algebraic relation: the bi-Langmuir adsorption isotherm. Therefore, it is important to develop accurate and reliable numerical techniques that can perform accurate numerical simulations of these models. We extended and implemented a second-order, semidiscrete, high-resolution finite volume method to simulate the governing equations of the model. Single solute flow and multi component mixture flows are assessed through a series of numerical experiments to theoretically illustrate the repercussions of intraparticle diffusion, film mass resistance, axial dispersion, and the size of the inert core radius upon simulated elution curves. Standard performance criteria are assessed to determine the optimal core radius fraction range for optimizing the separation performance.

Multi-dimensional population balance modeling of sonocrystallization of pyrazinamide with systematic estimation of kinetic parameters based on uncertainty and sensitivity analyses

Application of ultrasound has several beneficial effects on both crystallization process and product crystal properties. This work focuses on experimental study and population balance modeling of ultrasound-assisted unseeded batch cooling crystallization of an important polymorphic pharmaceutical. The δ-polymorph of pyrazinamide, a drug for mycobacterium tuberculosis, exhibits plate-like morphology, and such crystals are best described by atleast two dimensions: length and width. A multi-dimensional population balance model for batch cooling sonocrystallization of pyrazinamide from its 1,4-dioxane solution is developed and validated. A series of experiments are performed with different ultrasound amplitudes (silent, 10%, and 50%) to study the effect of ultrasound on nucleation rate, crystal size, and polymorphism. The ultrasound is applied till the point of nucleation and an enhanced nucleation rate is introduced in the model in the presence of ultrasound irradiation. The various kinetic parameters are estimated following a systematic methodology based on uncertainty quantification and sensitivity analysis using experimental observations related to solute concentration and crystal size distribution. The Polynomial Chaos Expansion (PCE) is used to quantify the uncertainties in model prediction due to kinetic parameter uncertainties and sensitivity analysis is performed using total order Sobol’s sensitivity indices derived from PCE model. The kinetic parameters are estimated in two steps. First, a global search technique is used to simultaneously estimate all the kinetic parameters. Next, the less sensitive parameters are kept fixed and the more significant parameters are fine-tuned using a gradient-based local optimizer. Out of total nine kinetic parameters, only four parameters appeared to be significant for both silent and sonicated cases. The high-resolution finite volume scheme is used to solve the multi-dimensional population balance model and the simulation results agree very well with experimental data related to both concentration and crystal size distributions. The results revealed that compared to silent cases, sonication increased the nucleation rate by 102 times and 104 times for 10% and 50% amplitude, respectively. The growth rate was also enhanced significantly with increase in ultrasonic amplitude. All the experiments yielded the desired δ-polymorph of pyrazinamide.

Simulation of Fixed-Bed Chromatographic Processes Considering the Nonlinear Adsorption Isotherms.

This paper presents the numerical approximation of a nonlinear equilibrium-dispersive (ED) model of multicomponent mixtures for simulating single-column chromatographic processes. Using Danckwerts boundary conditions (DBCs), the ED is studied for both generalized and standard bi-Langmuir adsorption isotherms. Advection-diffusion partial differential equations are used to represent fixed-bed chromatographic processes. As the diffusion term is significantly weaker than the advection term, sophisticated numerical techniques must be applied for solving such model equations. In this study, the model equations are numerically solved by using the Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The technique is designed to handle sudden changes (sharp discontinuities) in solutions and to produce highly accurate results. The method is tested with several case studies considering different parameters, and its results are compared with the high-resolution finite volume scheme. One-, two-, and three-component liquid chromatography elutions on fixed beds are among the case studies being considered. The dynamic model and its accompanying numerical case studies provide the initial step toward continuous monitoring, troubleshooting, and effectively controlling the chromatographic processes.

Analysis of Liquid Chromatography Considering a Linear Single-Component Heterogeneous-Type Reactive General Rate Model.

This paper elaborates on the significance of liquid chromatography for a single-component reactive linear general rate model. The model equations consist of a set of two coupled partial differential equations, which include diffusion, interfacial mass transfer, axial dispersion, external and intraparticle pore diffusivity, and heterogeneous chemical reaction of the first order with two sets of boundary conditions. The model equations are solved by the Laplace transformation. The actual time domain solution is obtained by numerical Laplace inversion, as analytical inversion cannot be obtained. The graphical sketch of different physical parameters is presented to analyze the dynamics of the elution profiles. The result indicates that the chromatographic reactor works more efficiently on increasing the value of the heterogeneous-type first-order reaction. To check the analytical results, a second-order high-resolution finite volume scheme is used. Both results are in good agreement and indicate the correctness of the numerical scheme. The current work is also compared with the previously available numerical schemes, which shows that the proposed numerical scheme is better for elaborating the chromatographic reactor performance. A comparison table is also presented to compute error analysis and computational run time for analyzing the efficiency of the reactor. A graphical sketch of the numerical temporal moment analysis is also presented, which gives significant information about the performance and the shape of the concentration profiles.

Numerical Study of the Influence of the Geometric Parameters of a Radial Crystalizer Using a Multiscale CFD Model

This work investigates how the configuration of the geometric parameters of a radial crystallizer influences the results of the crystallization of lovastatin by antisolvent and using a multi-scale computational fluid dynamics (CFD) model. The OPENFOAM open-source software uses macro and micromixing expressions for flow, and complete energy and population equilibrium equations during nucleation and crystal growth. The model is based on the Reynolds-Averaged-Navier-Stokes (RANS) equation, along with a multi-environment probability density function (PDF) model and the spatially semi-discretized population equilibrium equation, operating a high-resolution finite volume method. The variation crystallizer construction parameters provided another crystallizer design, and analyses demonstrated improved performance and effects on crystal distribution.

High resolution finite volume method for kinetic equations with Poisson brackets

Simulation of plasmas in electromagnetic fields requires numerical solution of a kinetic equation that describes the time evolution of the particle distribution function. In this paper we propose a finite volume scheme based on integral relation for Poisson brackets to solve the Liouville equation, the most fundamental kinetic equation. The proposed scheme conserves the number of particles, maintains the total-variation-diminishing (TVD) property, and provides high-quality numerical results. Other types of kinetic equations may be also formulated in terms of Poisson brackets and solved with the proposed method including the transport equations describing the acceleration and propagation of Solar Energetic Particles (SEPs), which is of practical importance, since the high energy SEPs produce radiation hazards. The proposed scheme is demonstrated to be accurate and efficient, which makes it applicable to global simulation systems analyzing space weather.

Evolutionary neural architecture search for surrogate models to enable optimization of industrial continuous crystallization process

Optimal performance of the crystallization process is of utmost importance for industries handling bulk commodity chemicals to pharmaceuticals. Such an optimization exercise becomes extremely time expensive as the mathematical models mimicking such complex processes involve the solution of Integro-Differential Population Balance Equations using High Resolution Finite Volume Methods. In order to build a fast and robust data based alternative model, a surrogate assisted approach using Artificial Neural Networks has been proposed here. To overcome the heuristics-based estimation of the hyper-parameters in ANNs, we aim to contribute a novel Neural Architecture Search strategy for the auto-tuning of hyper-parameters integrated with sample size determination techniques. While solving a multi-objective optimization of crystallization process ensuring maximum productivity, the results from surrogates are compared with those of a high-fidelity physics driven model, which reports five order of magnitude speed improvement without sacrificing much on accuracy.

A Novel Numerical Treatment of Nonlinear and Nonequilibrium Model of Gradient Elution Chromatography considering Core-Shell Particles in the Column

An extended method of semidiscrete high-resolution finite volume is used in this paper to obtain numerical solutions for a formulated nonlinear lumped kinetic model of liquid chromatographic process to examine the effect of chromatographic column overloading gradient elution considering core-shell particles. The model constitutes linear solvent strength (LSS), Henry’s constant, coefficient of nonlinearity, and coefficient of axial dispersion. The effects of modulator concentration changes for the elution of single and two components are analyzed. The advantages of introducing gradient elution against isocratic elution in terms of core radius fraction are investigated intensively. Numerical temporal moments are generated from the solutions obtained for a more in-depth examination of the considered model. Moreover, multiple forms of a single- and two-component mixture are generated to analyze the influences of core radius fractions on gradient elution. For example, the obtained results are utilized to investigate the effects of the slope of gradient, concentration of modulator, solvent strength parameter, coefficient of nonlinearity, coefficient of mass transfer, and coefficient of axial dispersion on the profiles of concentration in order to improve the process performance using optimal core radius fraction parameter values.

Numerical prediction of erosion due to a cavitating jet

We numerically investigate the erosion potential of a cavitating liquid jet by means of high-resolution finite volume simulations. As thermodynamic model, we employ a barotropic equilibrium cavitation approach, embedded into a homogeneous mixture model. To resolve the effects of collapsing vapor structures and to estimate the erosion potential, full compressibility is considered. Two different operating points featuring different cavitation intensities are investigated and their erosion potential is estimated and compared. Different methods are used for this purpose, including collapse detection (Mihatsch et al., 2015), maximum pressure distribution on the wall, and a new method of generating numerical pit equivalents. The data of numerical pit equivalents is analyzed in detail and compared with experimental data from the literature. Furthermore, a comprehensive grid study for both operating points is presented.

A high-resolution finite volume scheme based on optimal spectral properties of the fully discrete scheme with minimized dispersion and adaptive dissipation

For the compressible flows with broadband length scales, a high-resolution scheme based on the finite volume method with minimized dispersion and adaptive dissipation is proposed, which is capable of capturing the small-scale flow details and discontinuities. Firstly, the dispersion property is optimized for the fully discrete scheme, and the minimized dispersion parameters for different Courant numbers are obtained. Then, the dispersion-dissipation condition and a scale sensor are employed to get the critical dissipation parameter which not only can suppressed the numerical oscillations but also can keep the dissipation as low as possible to capture more flow details. Moreover, the critical dissipation isosurface is constructed to develop the functional relationship between the dissipation and the estimated scaled wavenumber. To have the capability of capturing shock waves, the proposed scheme is blended with the optimized WENO-Z scheme to form the hybrid scheme. Finally, several benchmark test cases are employed to verify the high-resolution properties and the robustness of the scheme proposed in the present paper. Futhermore, more high-fidelity solutions obtained by the proposed scheme for some numerical simulations with complex geometric configurations are expected.

A new framework and a hybrid method for one-dimensional population balance modeling of batch thermocycling crystallization

The hybrid method (HM) presented in this article assembles the high-resolution finite volume method (HRFVM) and the standard method of moments (SMOM) with an accurate inversion technique for determining the crystal size distribution (CSD), providing a computationally efficient solution with reliable accuracy for one-dimensional population balance modeling in batch thermocycling crystallizations. Two chemical systems, paracetamol and l-ascorbic acid, are chosen to study the accuracy and computational efficiency of the approach in the unseeded and seeded cases, respectively. Comparisons between the HRFVM and the HM are made for different temperature profiles in both cases. In addition, the compiled C/Mex files (MXFs) are implemented to further minimize the run time of the HM, and the benefits of the compiled files are discussed compared to the MATLAB files (MAFs).

DREAM: A fluid-kinetic framework for tokamak disruption runaway electron simulations

Avoidance of the harmful effects of runaway electrons (REs) in plasma-terminating disruptions is pivotal in the design of safety systems for magnetic fusion devices. Here, we describe a computationally efficient numerical tool, that allows for self-consistent simulations of plasma cooling and associated RE dynamics during disruptions. It solves flux-surface averaged transport equations for the plasma density, temperature and poloidal flux, using a bounce-averaged kinetic equation to self-consistently provide the electron current, heat, density and RE evolution, as well as the electron distribution function. As an example, we consider disruption scenarios with material injection and compare the electron dynamics resolved with different levels of complexity, from fully kinetic to fluid modes. Program summaryProgram Title:DreamCPC Library link to program files:https://doi.org/10.17632/vs3yvnrzg6.1Developer's repository link:https://github.com/chalmersplasmatheory/DREAMLicensing provisions: MITProgramming language: C++, PythonNature of problem: Self-consistently simulates the plasma evolution in a tokamak disruption, with specific emphasis on runaway electron dynamics. The runaway electrons can be simulated either as a fluid, fully kinetically, or as a mix of the two. Plasma temperature, current density, electric field, ion density and charge states are all evolved self-consistently, where kinetic non-thermal contributions are captured using an orbit-averaged relativistic electron Fokker-Planck equation, which couples to the plasma evolution. In the typical use case, the electrons are represented by two distinct populations: a cold fluid population and a kinetic superthermal population.Solution method: The system of equations is solved using a standard multidimensional Newton's method. Partial differential equations—most prominently the bounce-averaged Fokker–Planck and current diffusion equations—are discretized using a high-resolution finite volume scheme that preserves density and positivity.

Numerical approximation of nonlinear and non-equilibrium model of gradient elution chromatography

A nonlinear lumped kinetic model of liquid chromatography is formulated and solved numerically to theoretically investigate the effect of column overloading on gradient elution. Linear solvent strength (LSS) model is utilized for Henry’s constant, non linearity coefficient and axial dispersion coefficient. A semi-discrete high-resolution finite volume scheme is extended and applied to obtain the approximate solutions of the governing model equations. The effects of changing modulator concentration are examined on the single and two-component elution. The benefits of gradient elution over isocratic elution are thoroughly discussed. The influences of optimizing free-parameters available in gradient chromatography are analyzed on the efficiency of the column and on the production of targeted components. For instance, the results obtained are used to study the effects of gradient slope, modulator concentration, solvent strength, nonlinearity coefficient, mass transfer coefficient, and axial dispersion coefficient on the concentration profiles.

Improved Weighted Essentially Non-Oscillatory Schemes by Mixed Stencils

Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping method. The restriction that conventional ENO or WENO schemes only use r order stencils, is removed. Higher resolution can be achieved by introducing the r + 1 order stencils. This method is verified by three cases, i.e. the interaction of a moving shock with a density wave problem, the interacting blast wave problem and the double mach reflection problem. The numerical results show that the Improved Weighted Essential Non-oscillatory method is a stable, accurate high-resolution finite volume scheme.

A linearized non-isothermal general rate model for studying thermal effects in liquid chromatographic columns

A linearized non-isothermal general rate model is formulated and analytically solved to quantify the effects of temperature variations in fixed-bed chromatographic columns. The model contains a set of four coupled PDE accounting for energy transfer resistances, inner and outer particle-pore diffusions, and interfacial mass and axial dispersion. The Laplace transform, the eigenvalue-decomposition technique, and a conventional technique for the solutions of ODE are jointly employed for the solution of the model equations. A few numerical test studies are considered to assess the impact of system parameters on the performance of packed-bed adsorption columns. To access the range of applicability and to get the scope of the appropriateness of calculated analytical results, the numerical results are also obtained by applying a high resolution finite volume scheme. The analytical solutions obtained can be used as an invaluable tool for analyzing, optimizing, and upgrading the non-isothermal liquid chromatographic procedures.

The recurring impact of storm disturbance on black sea bass (Centropristis striata) movement behaviors in the Mid-Atlantic Bight.

Storm events are a significant source of disturbance in the Middle Atlantic Bight, in the Northwest Atlantic, that cause rapid destratification of the water column during the late summer and early fall. Storm-driven mixing can be considered as a seasonal disturbance regime to demersal communities, characterized by the recurrence of large changes in bottom water temperatures. Black sea bass are a model ubiquitous demersal species in the Middle Atlantic Bight, as their predominantly sedentary behavior makes them ideal for tagging studies while also regularly exposing them to summer storm disturbances and the physiological stresses associated with thermal destratification. To better understand the responsiveness of black sea bass to storm impacts, we coupled biotelemetry with a high-resolution Finite Volume Community Ocean Model (FVCOM). During the summers of 2016-2018, 8-15 black sea bass were released each year with acoustic transponders at three reef sites, which were surrounded by data-logging receivers. Data were analyzed for activity levels and reef departures of black sea bass, and fluctuations in temperature, current velocity, and turbulent kinetic energy. Movement rates were depressed with each consecutive passing storm, and late-season storms were associated with permanent evacuations by a subset of tagged fish. Serial increases in bottom temperature associated with repeated storm events were identified as the primary depressor of local movement. Storm-driven increases in turbulent kinetic energy and current velocity had comparatively smaller, albeit significant, effects. Black sea bass represents both an important fishery resource and an indicator species for the impact of offshore wind development in the United States. Their availability to fisheries surveys and sensitivity to wind turbine impacts will be biased during periods of high storm activity, which is likely to increase with regional climate change.

Effect of temperature variations on non-equilibrium and non-isothermal two-component liquid chromatography in cylindrical columns

The effect of temperature variations in liquid chromatographic columns of cylindrical geometry are studied by formulating a non-isothermal and non-equilibrium two-dimensional (2D) lumped kinetic model (LKM) simulating the flows of two-component mixtures. The developed models contain systems of coupled convection-dominated partial differential, differential, and algebraic equations for mass and energy balances in the mobile and stationary phases. For linearized isotherms, semi-analytical solutions are obtained by utilizing a sequential application of finite Hankel and Laplace transformations along with Tschirnhaus-Vieta and eigen-decomposition techniques. Temporal moments are also generated numerically from the derived semi-analytical solutions for a deeper understanding of the process execution. For nonlinear isotherms, a high-resolution finite volume scheme (HR-FVS) is employed to numerically simulate the model equations and also utilized to benchmark the precision scopes of derived semi-analytical solutions. A few case studies of linear and nonlinear chromatography are conducted. The connectivity of thermal waves with the concentration peaks is examined via numerical simulations and essential model parameters that affect the chromatographic column performance are identified. The results of this study will be very useful for optimizing and improving the non-isothermal and non-equilibrium liquid chromatography involving two-component mixtures and cylindrical-shaped columns.

Numerical approximation of non-linear chromatographic models considering Bi-Langmuir isotherm

In this research article, two standard models of liquid chromatograophy, namely the dispersive equilibrium model and the kinetic lumped model are approximated numerically. We studied the transport of multi components in a single column of chromatography considering non-linear adsorption thermodynamics. The models are analyzed for standard bi-Langmuir and generalized bi-Langmuir types adsorption equilibrium isotherms using Danckwert boundary conditions. Mathematically, the model equations form a non-linear system of PDE accounting for the phenomena of advection and diffusion, paired with an algebraic equation or a differential equation for adsorption isotherm. An extended semi-discrete high resolution finite volume scheme is employed to obtain the approximate solutions of the governing model equations. The method has second to third order accuracy. Several test case studies are conducted to examine the influence of various critical parameters on the process performance. The contemplated case studies incorporate the elution process of liquid chromatography with an increasing number of components. In particular, single component, two component, and three component mixtures are considered for the assessment of process performance. The formulated numerical algorithm provide an efficacious mechanism for investigating the retention behavior and the influence of mass transfer kinetics on the shapes of elution profiles.

Numerical approximation of a two-dimensional non-equilibrium model of non-isothermal liquid chromatography considering core-shell particles

A nonlinear and non-isothermal two-dimensional lumped kinetic model (2D LKM) is solved numerically to simulate the transport of multi-component mixtures in fixed-bed liquid chromatographic columns of cylindrical geometry packed with core-shell particles. The model considers axial and radial dispersions, interfacial mass, and heat transfers, nonlinear adsorption, and rectangular pulse injections. It is formed by a nonlinear system of partial differential equations coupled with differential and algebraic equations. A semi-discrete high-resolution finite volume scheme (HR-FVS) is extended and applied to approximate the model equations. A few case studies related to single and multi-component elution are considered to illustrate the effects of important parameters, such as core radius fraction, enthalpy of adsorption, radial and axial-dispersion coefficients, equilibrium constant, as well as film mass and heat transfer coefficients. A typical performance criterion is adopted to quantify the influence of thermal effects on the process performance and to evaluate the optimum value of inert core radius. The results obtained in various case studies are useful for understanding and upgrading liquid chromatographic processes considering non-isothermal conditions and core-shell particles.