High-resolution Finite Volume Method Research Articles

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.

Deep‐learning based in situ image monitoring crystal polymorph and size distribution: Modeling and validation

AbstractIn situ monitoring and closed‐loop control of the solution crystallization process are the modern trends for pharmaceutical development, in which the critical process parameters (CPPs) as well as the product critical quality attributes (CQAs) can be regulated and guaranteed during the manufacturing process. In this study, an in situ image monitoring methodology based on a state‐of‐the‐art deep‐learning model was developed to track the CQAs such as polymorph ratio, two‐dimensional crystal size, and crystal shape in a solvent‐mediated polymorphic transformation (SMPT) process. Coupled with the multidimensional process information, a 2D population balance model (PBM) was developed and validated using the results of the in situ image‐based CQAs analysis. The 2D‐PBM was solved using a high‐resolution finite volume method (HR‐FVM) which could provide a high dimensional particle‐size distribution. Through the validation between the process image analysis and the 2D‐PBM, the accuracy of image analysis was discussed, and the potential and challenges of in situ image analysis were proposed. This work aims to integrate the crystal polymorphism and two‐dimensional crystal size distribution (2D‐CSD) information in the SMPT process using intelligent microscopic image analysis and then to validate the results of neural network processing by solving the numerical solution of the multidimensional PBM.

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 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).

Consistent high resolution interface-capturing finite volume method for compressible multi-material flows

Compressible multi-material flows are characterized by complex flow structures with a broad range of length scales and discontinuities associated with material interfaces and shock waves. High order and high resolution numerical methods are required to capture material interface as sharply as possible and to increase the resolution of complex structures along the material interface. This paper will present the application of the high resolution finite volume (FV) method based on the minimized dispersion and controllable dissipation (MDCD) reconstruction to compressible multi-material flows. The MDCD scheme with independent dispersion and dissipation provides a flexible mechanism to control the numerical dissipation. The adjustment of dissipation of the MDCD reconstruction will not affect the consistency, which is required for interface-diffusion capturing methods to prevent spurious oscillations. Several one- and two dimensional multi-material numerical simulations have indicated that the high resolution FV method based on MDCD reconstruction can capture material interfaces free of spurious oscillations and provide better resolved small-scale features than other numerical schemes on the same grid resolution.

Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.

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.

Analytical solution of non-isothermal two-dimensional general rate model of liquid chromatography

A non-isothermal two-dimensional general rate model is formulated and analytically solved to analyze the effects of temperature changes inside liquid chromatographic columns of cylindrical geometry. The model equations form a system of convection-diffusion partial differential equations. The finite-Hankel transformation, the Laplace transformation, the eigen-decomposition technique and a conventional solution technique of ordinary differential equations are used to solve the equations of the model. The coupling between concentration and temperature fronts is demonstrated and important parameters that affect the performance of the column are evaluated. To find the ranges of validity of our analytical results, a semi-discrete high resolution finite volume method is applied to solve the same system of equations for both linear and nonlinear isotherms. The results of this contribution can be helpful to optimize non-isothermal liquid chromatographic processes in which both radial and axial gradients occur.

Model-based analysis and quality-by-design framework for high aspect ratio crystals in crystallizer-wet mill systems using GPU acceleration enabled optimization

The simultaneous control of crystal size and shape is particularly important in pharmaceutical crystallization processes. These two quantities not only influence the dissolution rate and bioavailability of final drug products, but also contribute to the manufacturability and efficiency of downstream operations. The manipulation of crystal shape, however, is difficult since it requires decoupled growth rate control of individual crystal faces. The aim of this work is to analyze and evaluate the effectiveness of wet milling for control of bivariate size distribution of high-aspect ratio crystals during solution crystallization. Three configurations are compared: the pure batch crystallizer, the crystallizer with internal wet mill (immersion mill) and crystallizer with external wet mill. Population balance models (PBMs) are employed to describe bivariate size density function dynamics, with stirring rate dependent breakage to enable the direct optimization of wet mill operation. The generated system of hyperbolic partial differential and ordinary differential equations is solved by a fully discretized high-resolution finite volume method (HR-FVM), involving graphical processing unit (GPU) acceleration, which brings up to two orders of magnitude speed-up by compared to the serial C implementation. The simulation studies revealed that wet milling significantly extends the achievable crystal size and shape domain, but it also broadens the product size distribution. The analysis of the optimized operating profiles, which include temperature, wet mill rotation (RPM) and recirculation rate, enabled the generalization of optimal operation, which can efficiently guide a QbD approach by considerably reducing the design space without compromising the process performance.

Modeling and Simulation Studies of a Novel Coupled Plug Flow Crystallizer for the Continuous Separation of Conglomerate-Forming Enantiomers

Separation of enantiomers is a major concern in pharmaceutical industries due to the different therapeutic activities exhibited by the enantiomers. Preferential crystallization is an attractive means to separate the conglomerate-forming enantiomers. In this work, a simulation study is presented for a proposed novel preferential crystallization configuration that involves coupled plug flow crystallizers (PFCs). The PFCs are coupled through liquid phase exchange which helps the enrichment of the preferred enantiomer in the liquid phase. A set of coupled population balance equations (PBEs) are used to describe the evolution of the crystal size distribution (CSD) in the PFCs. The PBEs and the relevant mass balance equations are solved using the high-resolution finite-volume method. The simulation results predict that the proposed configuration has higher productivity compared to the currently used crystallization configurations while maintaining the same level of purity. Moreover, the effect of process variables, such as the extent of liquid phase change and the location of the PFC where liquid phase exchange occurs, are studied. The insights obtained from this simulation study will be useful in design, development, and optimization of such novel crystallization platforms.

Study of Thermal Effects in Two-Component Nonisothermal Liquid Chromatography Considering Thermally Insulated Columns

A two-component nonisothermal model of linear chromatography is formulated to study heat effects and interference between components in thermally insulated columns. The Laplace transform and decoupling techniques are jointly applied to solve the model equations analytically. The first and second analytical temporal moments are derived which simply vectorized the moments of a single-solute nonisothermal model. To confirm ranges of applicability of the derived analytical solutions, a high resolution finite volume method is additionally applied to solve the nonlinear nonisothermal model equations numerically. Various test problems are considered. The influence of heat transfer on the retention times and shape of the profiles are studied, as both the thermodynamics and kinetics of adsorption processes are functions of temperature. The enthalpy of adsorption essentially influences the thermal behavior of the column. It was found that the ratio of density times specific heat of solid and liquid phases has a significant influence on the magnitude and speed of the thermal waves. The results obtained could be helpful to understand and optimize competitive adsorption chromatography.

OpenCrys: Open-Source Software for the Multiscale Modeling of Combined Antisolvent and Cooling Crystallization in Turbulent Flow

The open-source software, called openCrys, is provided for the multiscale simulation of antisolvent and combined antisolvent-cooling crystallization. It simulates the macro- and micromixing scales, and the complete energy and population balance equations during crystal nucleation and growth. The model is based on the Reynolds-Averaged-Navier–Stokes equation, coupled with a three-environment presumed probability density function model, and the spatially varying population balance equation semidiscretized using a high resolution finite-volume method. openCrys is implemented in C++ object oriented programming language using the open-source CFD package OpenFOAM. The software is used to compare the performance of dual impinging jet, coaxial, and radial crystallizers. It is shown that improving the micromixing does not necessarily result in a narrower crystal size distribution when temperature effects are taken into account. The complex interplay of crystallizer kinetics and momentum, mass, and heat transfer makes the selection of the best mixer for a particular application to be nonobvious, which motivates the development and application of high-fidelity multiscale simulations for the design of antisolvent crystallizers.

Aspect Ratio Distribution and Chord Length Distribution Driven Modeling of Crystallization of Two-Dimensional Crystals for Real-Time Model-Based Applications

Two-dimensional (2D) crystals, for which the shape is described by two linear sizes, are common in fine chemical and pharmaceutical industries. Since the crystal size and shape are directly related to the performance of active pharmaceutical ingredients, the simultaneous size and shape distribution control is of paramount importance in pharmaceutical crystallization engineering. To efficiently achieve simultaneous size and shape control often requires model-based control strategies; however, the increased computational cost of the process simulation and the substantial differences between the simulated and measurable quantities make the implementation of model-based control approaches challenging. This paper addresses the important problem of the real-time simulation of the most likely measurable chord length distribution (CLD) and aspect ratio distribution (ARD) as well as the concentration variations during the crystallization of 2D needle-shaped crystals. This enables the application of focused beam reflectance measurement (FBRM) and particle vision and microscopy (PVM), two routinely applied probes, as quantitative direct feedback control tools. Artificial neural network (ANN)-based FBRM and PVM soft-sensors are developed, which enable the direct and fast transformation of 2D crystal size distribution (CSD) to CLD and ARD on arbitrary 2D grids. The training data for the ANN are generated by a first principle, geometrical model-based simulation of FBRM and PVM for high aspect ratio crystals, although the ANN approach is applicable for any simulated or experimental training data sets. It is also demonstrated that the in situ imaging-based shape measurement underestimates the real aspect ratio (AR) of crystals, for which a simple correction is proposed. From the model-equation solution perspective, the soft-sensors require full population balance solution. The 2D high-resolution finite volume method is applied to simulate the full 2D CSD, which is an accurate, stable, but computationally expensive technique. The real-time applicability is achieved through various implementation improvements including grid optimization and data-type optimized hybrid central processing unit–graphical processing unit calculations.

Numerical study on the turbulent mixing of planar shock-accelerated triangular heavy gases interface

The interaction of a planar shock wave with a triangle-shaped sulfur hexafluoride ( $$\mathrm{SF_6}$$ ) cylinder surrounded by air is numerically studied using a high resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The vortex dynamics of the Richtmyer–Meshkov instability and the turbulent mixing induced by the Kelvin–Helmholtz instability are discussed. A modified reconstruction model is proposed to predict the circulation for the shock triangular gas–cylinder interaction flow. Several typical stages leading the shock-driven inhomogeneity flow to turbulent mixing transition are demonstrated. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum in late time manifest the turbulent mixing transition for the present case. The analysis of variable-density energy transfer indicates that the flow structures with high wavenumbers inside the Kelvin–Helmholtz vortices can gain energy from the mean flow in total. Consequently, small scale flow structures are generated therein by means of nonlinear interactions. Furthermore, the occasional “pairing” between a vortex and its neighboring vortex will trigger the merging process of vortices and, finally, create a large turbulent mixing zone.

Turbulent mixing and energy transfer of reshocked heavy gas curtain

The interaction between the shock wave and the air-SF6-air gas curtain and the associated instability of the reshocked interface are numerically studied using a high-resolution finite volume method with minimum dispersion and controllable dissipation reconstruction. The evolution of flow structures shows that, although the initial perturbation of the interface is two-dimensional, the reshock process will lead to an asymmetrical instability and enhance the mixing of fluids dramatically. The exponential scaling laws for the growth of mixing width are observed both before and after the reshock. The nearly homogeneous fluids’ mixing at the late stage after the reshock is confirmed by the temporal asymptotic behaviors of mixing parameters such as local anisotropy and density-specific volume correlation. Both the decoupled length scales and the broadened inertial range of the turbulent kinetic energy spectrum and density fluctuation spectrum imply that the turbulent mixing transition is triggered at the late stage. The analysis of sub-grid stress energy transfer manifests the energy cascade in physical space. Furthermore, the investigation of the role of coherent structures in sub-grid stress energy transfer indicates that the backscatter of sub-grid stress energy is most prominent in the regions with washing motion induced by nearly two-dimensional vortical structures. However, the intense forward scatter of sub-grid stress energy occurs in the regions where two washing motions collide with each other leading to highly three-dimensional motions.

A non-negative and high-resolution finite volume method for the depth-integrated solute transport equation using an unstructured triangular mesh

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green–Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green–Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values.

Liquid droplet impingement erosion on groove roughness

A study of erosion mechanism during the liquid droplet impingement (LDI) on a rough surface was conducted using both numerical simulation and experiments. The numerical simulation was carried out for the LDI on single groove roughness utilizing the two-phase full Eulerian approach based on the high-resolution finite volume method to understand the evolution of the droplet deformation and the wall pressure distribution. The numerical results for the LDI on a large groove roughness showed that the first and the second impacts occurred close to the first contact location of the droplet on the wall, which was similar to the LDI on a smooth surface. The largest impact occurred by the third impact at the groove bottom, which was caused by the side-jet focusing and droplet focusing generated from the contact edge of the droplet. It was also found that the maximum wall pressure at the groove bottom increased with the increase in the groove roughness parameter that represented the geometrical ratio of the groove depth to the droplet diameter. In addition to the numerical studies, the erosion behavior of LDI on single groove roughness was observed experimentally using a Scanning Electron Microscope (SEM) after spray jet impingement on an aluminum specimen with various groove roughness parameter. The experimental results showed that the LDI erosion on the large groove roughness started from the groove bottom, while the LDI erosion on the small groove roughness started from the groove edge and the groove bottom. These experimental findings are well correlated with the maximum wall pressure distributions obtained from the numerical results.

Population Balance Modeling and Optimization of an Integrated Batch Crystallizer–Wet Mill System for Crystal Size Distribution Control

In this work, the modeling, simulation, and optimization of an integrated batch crystallizer, wet mill system are presented. It is shown that by coupling the external wet mill to the crystallizer it is possible to increase the overall system flexibility, increase the attainable crystal size distribution (CSD), and provide a significantly better distribution shaping control than the crystallizer alone. The population balance modeling approach with appropriate mechanisms is applied for the description of crystal population in both the crystallizer and wet mill. This description generates a system of partial differential-integral equations, which are solved with a high resolution finite volume method, involving calculations on parallel graphical processing unit for improved solution time. In the batch crystallizer, it is assumed that primary nucleation and crystal growth are the key mechanisms, whereas in the wet mill, attrition and fragmentation of crystals occurs. The nucleation and growth rate kinetics ar...