Shamsul Qamar

A comparative study of high resolution schemes for solving population balances in crystallization

Abstract This article demonstrates the applicability and usefulness of high resolution finite volume schemes for the solution of population balance equations (PBEs) in crystallization processes. The population balance equation is considered to be a statement of continuity. It tracks the change in particle size distribution as particles are born, die, grow or leave a given control volume. In the population balance models, the one independent variable represents the time, the other(s) are “property coordinate(s)”, e.g. the particle size in the present case. They typically describe the temporal evolution of the number density functions and have been used to model various processes. These include crystallization, polymerization, emulsion and cell dynamics. The high resolution schemes were originally developed for compressible fluid dynamics. The schemes resolve sharp peaks and shock discontinuities on coarse girds, as well as avoid numerical diffusion and numerical dispersion. The schemes are derived for general purposes and can be applied to any hyperbolic model. Here, we test the schemes on the one-dimensional population balance models with nucleation and growth. The article mainly concentrates on the re-derivation of a high resolution scheme of Koren (Koren, B. (1993). A robust upwind discretization method for advection, diffusion and source terms. In C. B. Vreugdenhill, & B. Koren (Eds.), Numerical methods for advection–diffusion problems, Braunschweig: Vieweg Verlag, pp. 117–138 [vol. 45 of notes on numerical fluid mechanics, chapter 5]) which is then compared with other high resolution finite volume schemes. The numerical test cases reported in this paper show clear advantages of high resolutions schemes for the solution of population balances.

Efficient and accurate numerical simulation of nonlinear chromatographic processes

Abstract Models for chromatographic processes consist of nonlinear convection-dominated partial differential equations (PDEs) coupled with some algebraic equations. A high resolution semi-discrete flux-limiting finite volume scheme is proposed for solving the nonlinear equilibrium dispersive model of chromatography. The suggested scheme is capable to suppress numerical oscillations and, hence, preserves the positivity of numerical solutions. Moreover, the scheme has capability to accurately capture sharp discontinuities of chromatographic fronts on coarse grids. The performance of the current scheme is validated against other flux-limiting schemes available in the literature. The case studies include single-component elution, two-component elution, and displacement chromatography on non-movable (fixed) and movable (counter-current) beds.

Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.

Adaptive high-resolution schemes for multidimensional population balances in crystallization processes

Abstract This article focuses on the application of adaptive high-resolution finite volume schemes for solving multidimensional population balance models (PBM) in crystallization processes. For the mesh redistribution, we use the moving mesh technique of Tang and Tang [Tang, H.-Z. & Tang, T. (2003). Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM Journal of Numerical Analysis , 41 , 487–515] which they have developed for hyperbolic conservation laws in conjuction with finite volume schemes. In this technique, an iterative procedure is used to redistribute the mesh by moving the spatial grid points. The corresponding numerical solution at the new grid points is obtained by solving a linear advection equation. The method avoids the usual unsatisfactory, interpolation procedure for updating the solution. The finite volume schemes were originally derived for compressible fluid dynamics. The schemes have already shown their accuracy and efficiency in resolving sharp peaks and shock discontinuities. The accuracy of these schemes has been improved further by using the adaptive meshing techniques. The application of these high-resolution schemes for multidimensional crystallization processes demonstrates their generality, efficiency, and accuracy. The numerical test cases presented in this article show the clear advantage of finite volume schemes and show further improvements when combined with a moving mesh technique.

Kinetic schemes for the ultra-relativistic Euler equations

We present a kinetic numerical scheme for the relativistic Euler equations, which describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic approach is very simple in the ultra-relativistic limit, but may also be applied to more general cases. The basic ingredients of the kinetic scheme are the phase-density in equilibrium and the free flight. The phase-density generalizes the nonrelativistic Maxwellian for a gas in local equilibrium. The free flight is given by solutions of a collision free kinetic transport equation. The scheme presented here is an explicit method and unconditionally stable. We establish that the conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. For that reason we obtain weak admissible Euler solutions including arbitrarily complicated shock interactions. In the numerical case studies the results obtained from the kinetic scheme are compared with the first order upwind and centered schemes.

Kinetic schemes for the relativistic gas dynamics

Summary.A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature β. In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration τM, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times tn=nτM, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time tn>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at tn. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit τM→0 we obtain the weak solutions of Euler’s equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach.

A discontinuous Galerkin method to solve chromatographic models

This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection-diffusion-reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.

Analytical solutions and moment analysis of chromatographic models for rectangular pulse injections.

This work focuses on the analysis of two standard liquid chromatographic models, namely the lumped kinetic model and the equilibrium dispersive model. Analytical solutions, obtained by means of Laplace transformation, are derived for rectangular single solute concentration pulses of finite length and breakthrough curves injected under linear conditions. In order to analyze the solute transport behavior by means of the two models, the temporal moments up to fourth order are calculated from the Laplace-transformed solutions. The limiting cases of continuous injection and negligible mass transfer limitations are evaluated. For validation, the analytical solutions are compared with the numerical solutions of models using the discontinuous Galerkin finite element method. Results of different case studies are discussed for linear and nonlinear adsorption isotherms. The discontinuous Galerkin method is employed to obtain moments for both linear and nonlinear models numerically. Analytically and numerically determined concentration profiles and moments were found to be in good agreement.

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.

Numerical solution of a multi-dimensional batch crystallization model with fines dissolution

In this article a mathematical model for two-dimensional batch crystallization process with fines dissolution is presented. The fines dissolution is useful for improving the quality of a product and facilitates the downstream process like filtration. The crystals growth rates can be size-dependent and a time-delay in the recycle pipe is incorporated in the model. The high resolution finite volume schemes, originally derived for general systems in divergence form, are used to solve the resulting model. The schemes have already been used for the simulation of complex problems in gas dynamics and were found to be computationally efficient, accurate, and robust. The numerical test problems of this manuscript verify the capability of the proposed schemes for solving batch crystallization models.