High-resolution Finite Volume Research Articles

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.

A FVCOM study of the potential coastal flooding in apponagansett bay and clarks cove, Dartmouth Town (MA)

A high-resolution Finite-Volume Coastal Ocean Model (FVCOM) inundation model has been developed for Dartmouth Town near Apponagansett Bay and Clarks Cove. Series of modeling experiments were conducted for the purpose of: (1) Assess the potential impacts of the climate-induced Sea Level Rise (SLR) on the storm-induced coastal inundation in Dartmouth Town; (2) Compare the current patterns, wave fields and surge distributions under different dynamic forces including winds in different directions and wave-current interaction; (3) Evaluate the impact of the bank on the flooding protection. Results show that under the hundred-year nor’easter storm condition, the climate-induced SLR could significantly enlarge possible flooding areas with the percent area enlargement of approximately 60% per foot of SLR. The directions of wind essentially determine the feature of the current patterns, wave and surge distributions. The northeasterly and easterly winds mainly threaten the western coast of the bay and the estuarine areas, and the southerly and southeasterly winds endanger the regions around the inner part of the bay. Wave-current interaction can change the current pattern nearshore, including formation of eddies and narrow alongshore currents, greatly enhancing the strength and complexity of the currents near the mouth of the bay. In addition, wave-induced surge tends to accumulate in the bay and near the estuary and coastal regions. The bank blocks a large amount of flooding current and waves into the bay and improves the local current and wave condition effectively near the mouth and in the bay.

Analysis of gradient elution chromatography using the transport model

A transport model is considered to describe gradient elution in liquid chromatography in packed beds with the linear isotherm dependent on the mobile phase modulator. By applying a coordinate transformation, the model is solved analytically using the Laplace transform approach. The moment generating property of the Laplace domain solution is used to derive analytical expressions for the first three moments of the response to rectangular injections. These moments are instructive for analyzing the retention time, band broadening and asymmetry of elution profiles. Compared to isocratic elution, the derivation of analytical solutions and moments for gradient elution is more complicated, because the retention behavior of the solutes depends on the varying mobile phase modulator. Several case studies are evaluated theoretically. To gain confidence on the derived analytical results, a high-resolution finite volume scheme is also applied to solve the same model equations numerically. The analytical solutions and moments provided are utilized to predict the effects of starting and ending times of gradient, magnitude of modulator concentration variation, gradient slopes, and mass transfer coefficient on retention and peak shape. The analytical moment expressions derived can be used to determine retention and mass transfer parameters from experimental peaks and to predict elution behaviors if these parameters are known.

Numerical approximation of a two-dimensional model of non-isothermal reactive liquid chromatography involving slow rates of adsorption–desorption kinetics

A two-dimensional general rate model of non-isothermal reactive column chromatography is formulated considering homogenous and heterogeneous reaction rates, slow rates of adsorption–desorption kinetics, and enthalpies of adsorption and reaction. The model is expressed by a system of six nonlinear partial differential equations (PDEs) coupled with algebraic expressions for the adsorption and reaction rates. The nonlinearity of adsorption isotherm and reaction term hinders the derivation of analytical solutions. For that reason, a flux-limiting high-resolution finite volume scheme is suggested to numerically approximate the model equations. The effects of several kinetic and thermodynamic parameters are rigorously analyzed on the reactant conversion and components separation.

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.

Two-dimensional general rate model for non-isothermal liquid chromatography considering finite rates of adsorption–desorption kinetics

In this work, a two-dimensional general rate model for nonisothermal liquid chromatography is formulated considering adsorption saturation capacity, enthalpy of adsorption, mass and energy dispersion coefficients, conductivity and diffusivity of heat, heat transfer between the stationary and mobile phases of the column and finite rates of adsorption–desorption kinetics. Semi-analytical solutions of the model are derived by linearizing the adsorption and desorption rates and considering the Danckwerts boundary conditions with inner and outer annular regions of sample injection. These two injection regions are helpful to trigger temperature and concentration gradients along the cylindrical coordinate of the column. In deriving the semi-analytical solutions, we have successively applied the Hankel and Laplace transforms one after the other together with the Eigen-decomposition technique. The validity of the semi-analytical solutions is checked by comparing with the numerical solutions obtained by applying a flux limiting high resolution finite volume scheme of Koren. Several numerical test problems are carried out to inspect the key parameters that influence the chromatographic process performance. The numerical values of parameters obtained in this work will play an important role in the future theoretical developments of column liquid chromatography.

Numerical solution of nonlinear and non-isothermal general rate model of reactive liquid chromatography

A nonlinear general rate model (GRM) of liquid chromatography is formulated to analyze the influence of temperature variations on the dynamics of multi-component mixtures in a thermally insulated liquid chromatographic reactor. The mathematical model is formed by a system of nonlinear convection–diffusion reaction partial differential equations (PDEs) coupled with nonlinear algebraic equations for reactions and isotherms. The model equations are solved numerically by applying a semi-discrete high-resolution finite volume scheme (HR-FVS). Several numerical case studies are conducted for two different types of reactions to demonstrate the influence of heat transfer on the retention time, separation, and reaction. It was found that the enthalpies of adsorption and reaction significantly influence the reactor performance. The ratio of density time heat capacity of solid and liquid phases significantly influences the magnitude and velocity of concentration and thermal waves. The results obtained could be very helpful for further developments in non-isothermal reactive chromatography and provide a deeper insight into the sensitivity of chromatographic reactor operating under non-isothermal conditions.

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 evaluation of particle breakage facilitated process intensification for Mixed-Suspension-Mixed-Product-Removal (MSMPR) crystallization

This work studied a system of single stage mixed-suspension-mixed-product-removal (MSMPR) crystallization with particle breakage controlled by wet milling. Conventional single stage MSMPR crystallizer is limited by its degrees of freedom, which makes the process performance largely depend on steady state crystallization rate only tunable by residence time and supersaturation. Breakage brought by wet milling is demonstrated in this work as a supersaturation-independent variable to manipulate steady state conditions. The simulation results show that wet milling can act as a promising process intensification tool that increases process yield and broadens attainable region of particle size. Breakage model is adopted from dry milling and incorporated to high resolution finite volume population balance model. Four cases that cover a wide range of nucleation and growth rates have been tested, and the process yield and particle size are studied as functions of residence time and wet milling tip speeds.

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.

Diurnal Fluctuation of Shallow-Water Acoustic Propagation in the Cold Dome Off Northeastern Taiwan in Spring

The diurnal fluctuation of acoustic propagation in a cold dome region was studied by using the output of the high-resolution Finite Volume Coastal Ocean Model. The 3-D acoustic propagation in this region was studied first, and the results suggest that the diurnal fluctuation of the ocean environment results in the fluctuation of acoustic propagation. However, the variation pattern cannot be determined with only few outputs, and comprehensive analysis was needed. Then, a feature model for the thermal front on the edge of the cold dome was proposed. By analyzing the output of the ocean model, fluctuations were found to mainly occur in the two aspects of the feature model: first, horizontal excursion of the thermal front and second, vertical motion of the isothermal lines in the cold dome. The variation pattern of acoustic propagation over the two kinds of fluctuations was quantified. Results suggest that both kinds of fluctuations of model parameters result in great transmission loss (TL) variation in some cases. Typically, acoustic propagation with high frequency and deep source experience great influence from the fluctuations. The TL versus range usually experience influence from the fluctuations of the first kind from an approximately 40-km range and from the start for the second kind. The reason is that the vertical motion of the isothermal lines of the cold dome induces variation over the temperature profile from the start until the end, whereas the horizontal excursion of the thermal front brings influence mainly over the temperature profiles near the thermal front.

Investigation of irreversible reactive liquid chromatography considering linear general rate model

A two-component model of reactive liquid chromatography is presented consid?ering M ? N type reaction. The model incorporates surface and pore diffusions in the adsorbates, axial dispersion, interfacial mass transfer, first order chemical reactions in the liquid and particle phases, and two sets of boundary conditions. The model contains a system of four coupled PDE describing the dynamics of reactants and products in both phases. The Laplace transformation and eigen-decomposition technique are jointly applied to solve the model equations analytically. An efficient and accurate numerical Laplace inversion technique is utilized to retrieve back solutions in the original time domain. The developed semi-analytical results are verified against the numerical results of a high resolution finite volume scheme. A good agreement between the solutions not only confirms the accuracy of semi-analytical results but also validates the accuracy of proposed numerical scheme. This study extends and generalizes our previous analysis on heterogeneous reactions in the liquid chromatography. In order to analyze the behavior of a chromatographic reactor, different case studies are presented showing the effects of model parameters on the process.

Mesoscopic-based finite volume solutions for waterhammer flows

ABSTRACTIt is becoming evident that high resolution finite volume (FV) numerical schemes for multi-dimensional waterhammer problems are needed for the development of an accurate transient-based condition assessment of pipelines. As a prerequisite, FV methods for one-dimensional waterhammer flows need to be developed. Such models can then be applied to multi-dimensional problems via directional splitting. In this paper, two FV schemes (one uses Bhatnagar–Gross–Krook Boltzmann (BGK) and the other uses kinetic flux vector splitting (KFVS) in the flux approximation) for one-dimensional waterhammer problems are formulated and applied. It is found that the KFVS and BGK schemes correctly capture discontinuity fronts in a classical reservoir-pipe-valve system. An oscillation-free collision time formulation has been proposed, tested and found to be robust. The stability of the proposed schemes is guaranteed when . Comparison between the BGK, KFVS, fixed-grid MOC, and first and second order Godunov schemes reveals that the second-order Godunov performs best, followed closely by the BGK scheme. However, the BGK should not be quickly dismissed since it is known that its real power becomes evident when dealing with complex physics, multi-scale and multi-dimensional problems.

Nonlinear model of liquid chromatography considering finite rates of adsorption-desorption kinetics and core-shell adsorbents

A nonlinear general rate model is numerically approximated to simulate fixed-bed chromatographic columns packed with core-shell particles. The model incorporates explicitly the effect of finite rates of adsorption and desorption at the adsorption sites, typically assumed to be very fast compared to the rates of the various transport processes. Using core-shell particles as a stationary phase can have advantages over applying a fully-porous stationary phase, such as higher efficiencies and better resolution of the sample components. A high resolution finite volume scheme is extended and applied to approximate the model equations. Ranges of the kinetic parameters in which limited rates of the intrinsic adsorption and desorption steps needs to be taken into account are estimated. A few case studies of predicting the elution of single-component and two-component mixtures are considered to evaluate the effects of adsorption and desorption rate constants, core-radius fraction, axial-dispersion coefficient, film mass transfer, and intraparticle diffusion on the elution profiles. Furthermore, it is demonstrated that optimum values of the inert core radius can be obtained by evaluating a typical criterion for process performance.

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.

Irregular wave propagation with a 2DH Boussinesq-type model and an unstructured finite volume scheme

The application and validation, with respect to the transformation, breaking and run-up of irregular waves, of an unstructured high-resolution finite volume (FV) numerical solver for the 2D extended Boussinesq-type (BT) equations of Nwogu (1993) is presented. The numerical model is based on the combined FV approximate solution of the BT model and that of the nonlinear shallow water equations (NSWE) when wave breaking emerges. The FV numerical scheme satisfies the desired properties of well-balancing, for flows over complex bathymetries and in presence of wet/dry fronts, and shock-capturing for an intrinsic representation of wave breaking, that is handled as a shock by the NSWE. Several simulations and comparisons with experimental data show that the model is able to simulate wave height variations, mean water level setup, wave run-up, swash zone oscillations and the generation of near-shore currents with satisfactory accuracy.

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.