Finite Element Collocation Research Articles

Dynamic simulation of crystallization processes: Adaptive finite element collocation method

AbstractAn adaptive orthogonal collocation on finite elements method with adaptively varied upper bound of the relevant size interval is developed for numerical solution of population balance equation of crystallizers. Nucleation producing monosized and heterosized nuclei, size‐dependent crystal growth, seeding, classified product removal and fines removal with dissolution are included into the model. Adaptation of the number and length, as well as the distribution of finite elements over the variable length computational interval is carried out forming a number of adaptation rules, based on ordering the finite elements of size coordinate according to the maxima of first derivatives of the population density function. The approximation is obtained using the Lagrange interpolation polynomials. The method is used for solving the mixed set of nonlinear ordinary and partial differential equations, forming a detailed dynamical model of continuous crystallizers with product classification and/or fines removal. The program can be used efficiently for simulation of stationary and dynamic processes of crystallization systems, computing either transients or long‐time oscillating steady statesgenerated by different nonlinear phenomena, and internal and external feedbacks. © 2007 American Institute of Chemical Engineers AIChE J, 2007

A finite element collocation method for two-phase incompressible immiscible problems

Two-phase, incompressible, immiscible flow in porous media is governed by a coupled system of nonlinear partial differential equations. The pressure equation is elliptic, whereas the concentration equation is parabolic, and both are treated by the collocation scheme. Existence and uniqueness of solutions of the algorithm are proved. A optimal convergence analysis is given for the method.

Iterative solution of elliptic collocation systems on a cognitive parallel computer

The performance of parallel algorithms implementing the block SOR iterative method, used for the solution of linear systems arising from the discretization of elliptic BVPs by the finite element collocation method, is investigated in this paper. By making use of a recently introduced block tridiagonal partitioning of the collocation matrix, which yields faster rates of convergence, and its associated modified red-black ordering of unknowns and equations, we succeeded to increase the scalability of the problem. The study of the communicational and computational needs of the problem on a virtual parallel machine made possible the mapping of the resulted SOR-algorithm on a fixed size distributed memory parallel system, utilizing a simple pipeline architecture among its processors, in an optimum way. The proposed parallel schemes are being realized on a Parsytec cognitive computer via an SPMD programming model. Speedup measurements are used to reveal the efficiency of our implementation.

Optimal control of bioreactors: a simultaneous approach for complex systems

The inherent features of dynamic bioprocesses prevent the application of conventional optimization algorithms, hence making necessary the development of tailored methods and strategies. On the other hand, the optimization of biotechnological processes may generate significant improvements in operating conditions and policies. Fed-batch bioprocesses are specific examples where complexity and difficulty depend on the model characteristics, the operating limits (path constraints) and the production target (objective function). We propose the use of orthogonal collocation into a simultaneous optimization approach to solve these problems. Initially, the methodology is applied to a simplified model for the biosynthesis of penicillin from glucose. Then, it is applied to a cybernetic structured model for the fermentative production of polyhydroxyalkanoates (PHAs). Results show that the discretization of differential-algebraic equation (DAE) systems by orthogonal collocation in finite elements efficiently transforms dynamic optimization problems into nonlinear programming (NLP) problems, thus enabling to solve complex problems with several control variables satisfying the approximation error tolerance.

Optimization of the VARTM process for enhancement of the degree of devolatilization of polymerization by-products and solvents

Devolatilization of the polymerization by-products and the impregnation solvent during Vacuum Assisted Resin Transfer Molding (VARTM) of the polyimide polymers is analyzed using a combined continuum hydrodynamics/chemical reaction one-dimensional model. The model which consists of seven coupled partial differential equations is solved using a finite element collocation method based on the method of lines. The results obtained reveal that the main process parameters which give rise to lower gas-phase contents in the VARTM-processed polymer matrix composites are the vacuum pressure and the tool-plate heating rate. Lower tool-plate heating rates are found to be beneficial since hey promote devolatilization of the impregnation solvent at lower temperatures at which the degree of polymerization and, hence, resin viscosity are low.

Simultaneous optimization of dynamic bioprocesses

It is known that the optimal control may introduce significant economical benefits into production processes, thus being an important and challenging research area with practical relevance. The modeling and optimization of biotechnological processes has been object of research and their related results have generated improvements in operating conditions and strategies, however, the inherent features of dynamical bioprocesses prevent the application of conventional optimization algorithms, hence making necessary the development of tailored methods and strategies. The objective of this work is to develop mathematical programming strategies for simultaneous optimization of dynamic systems and evaluate their computational performance. Simultaneous optimization with orthogonal collocation is applied to a simplified model for biosynthesis of penicillin from glucose, which was studied by Cuthrell and Biegler (1989). The results show that discretization of differential equations systems (DAE) by orthogonal collocation in finite elements efficiently transforms dynamic optimization problems into nonlinear programming (NLP) problems, enabling to solve complex problems with several control variables and minimizing the approximation error.

Polyacrylic gel beads reinforced by a polyamide membrane: application to copper removal in aqueous media

Polyacrylic acid gel is known for its ion chelating properties. Its poor mechanical resistance hinders its use in extraction operations. Crosslinking can improve the mechanical resistance but lowers the chelating capacity. A gel encapsulation technique has been developed to obtain a material constituted by a low cross linked polyacrylic gel coated by a polyamide membrane and having improved mechanical resistance with respect to the corresponding gel. Extracting properties of various metal ions (Cu, Ni, Co and Zn) carried out in “batch” and “column” processes are presented. Mass transfer modeling studies have been carried out by using the mass action law for the ion exchange equilibrium and by describing the ion flow with the Nernst—Planck equation. Collocation in finite elements was used to solve the model equations. Results indicate that the capsules can be compared with ion exchange resins.

Asymptotic behaviour of the Galerkin and the finite element collocation methods for a parabolic equation

The asymptotic convergence of the solution of a parabolic equation is proved. The proof is based on two methods namely, the Galerkin method expressed in terms of linear splines and the Finite Element Collocation method expressed by cubic spline basis functions. Both methods are considered in continuous time. The asymptotic rate of convergence for the two methods is found to be of order O( h 2).

A Maxwell−Stefan Model of Bidisperse Pore Pressurization for Langmuir Adsorption of Gas Mixtures

The dynamics of bidisperse pore pressurization with multicomponent mixtures is studied using the Maxwell−Stefan approach to describe the mass-transfer processes in macro- and micropores. The solid network is described by the branched micropore−macropore model, allowing for adsorption to occur at both levels. The adsorption equilibrium is described with the extended Langmuir isotherm. Numerical solutions of partial pressure and surface coverage profiles are obtained using orthogonal collocation in finite elements. The effects of micropore surface diffusivity and feed composition on the controlling resistance for the separation of a binary mixture of inert and adsorbable gases are studied. A criterion is suggested for identification of the controlling resistance that takes into account the effect of composition on convective transport and micropore surface diffusion. The effect of macropore resistance on a kinetics-based separation is analyzed for a binary mixture of two adsorbable components. The separation of a ternary mixture is also addressed, in a study of the effects of the inert gas concentration in the feed on the system dynamics for a mixture of an inert and two adsorbable gases with different affinities.

Stress formation during heating in supercritical drying

Silica aerogel preparation is analyzed and two different supercritical drying techniques, autoclave drying of the solvent and drying following carbon dioxide exchange of the solvent, are compared in terms of stress formation during the heating step. The model used is a partial differential equation that relates stress to thermal expansion and flow of pore liquid for a radially bounded cylindrical geometry and the model is modified to see the effect of condensation reactions that result in the contraction of the body. The finite element collocation method is used to solve the model for different drying techniques, at several heating rates, and shear moduli. The carbon dioxide exchange method is found to be more advantageous as far as the stress formation in the heating step of supercritical drying is concerned.

An Integral approach to dynamic data rectification

Because industrial process is often under changes, it is necessary to reconcile the measurement of the dynamic process. There are many problems in the present dynamics data reconciliation methods, such as low calculation efficiency, difficulty of treatment for the reconciliation of input variables. Based on the analysis of the characteristic of dynamic data reconciliation, an integral approach is presented which integrates finite element collocation method, filtering technique and robust method. The finite element collocation method can reduce the amount of discrete model constrained equations to decrease the problem complexity without any loss of measurement information. The filtering technique can eliminate random errors in input variables effectively and no lag or signal distortion will be introduced. Monte Carlo method is used to test the efficiency of the robust method for gross error detection. The calculation results show that the integral approach can improve the calculation efficiency greatly and can deal with the gross error of abnormal type properly.

Adsorption and Diffusion in Bidisperse Pore Structures

Transient diffusion and adsorption processes of one component in a bidisperse pore system are studied. The solid network is described by the branched micropore-macropore model. A criterion to identify the controlling mechanism, macropore diffusion, micropore diffusion, or both, is suggested. This criterion based on diffusional information and pellet geometry is compared with the criterion for bidisperse systems with homogeneous diffusion at the microscale level. The effects of the nature of the adsorption isotherm (linear, Langmuir, and irreversible) in concentration profiles and in uptake are analyzed. The mathematical model is analytically solved for linear systems, and solutions for the macropore or micropore diffusion control cases are also obtained. Numerical solutions of concentration in macro- and micropores for the three isotherms are obtained using orthogonal collocation in finite elements. For irreversible systems an analytical expression for uptake is derived on the basis of a simplified model. The time dependence of the uptake for the three possible mechanisms is also addressed.

Model predictive control algorithms and their application to a continuous fermenter

In many continuous fermentation processes, the control objective is to maximize productivity per unit time. The optimum operational point in the steady state can be obtained by maximizing the productivity rate using feed substrate concentration as the independent variable with the equations of the static model as constraints. In the present study, three model-based control schemes have been developed and implemented for a continuous fermenter. The first method modifies the well-known dynamic matrix control (DMC) algorithm by making it adaptive. The other two use nonlinear model predictive control algorithms (NMPC, nonlinear model predictive control) for calculation of control actions. The NMPC1 algorithm, which uses orthogonal collocation in finite elements, acted similar to NMPC2, which uses equidistant collocation. These algorithms are compared with DMC. The results obtained show the good performance of nonlinear algorithms.

Determination of pore size distributions by regularization and finite element collocation

The problem of extracting pore size distributions from characterization data is solved here with particular reference to adsorption. The technique developed is based on a finite element collocation discretization of the adsorption integral, with fitting of the isotherm data by least squares using regularization. A rapid and simple technique for ensuring non-negativity of the solutions is also developed which modifiesthe original solution having some negativity. The techniqueyields stable and converged solutions, and is implementedin a package RIDFEC. The package is demonstrated to berobust, yielding results which are less sensitive to experimental error than conventional methods, withfitting errors matching the known data error. It is shownthat the choice of relative or absolute error norm in the least-squares analysisis best based on the kind of error in the data.

Dynamic behaviour of a fixed-bed biofilm reactor: analysis of the role of the intraparticle convective flow under biofilm growth

The influence of the intraparticle convective flow on the dynamic response of a fixed-bed biofilm reactor is evaluated theoretically for a stepwise increase in substrate concentration in the feed line. Dimensionless transient mass balance equations for the substrate within the bulk liquid, biofilm and support phases are derived considering a substrate inhibition kinetic model (Haldane equation) and biofilm growth. Governing equations are simultaneously solved for phenol biodegradation by finite element collocation methods to yield the bed response, in terms of dimensionless phenol concentration at outlet against a reduced time (breakthrough curves), for two different bulk flow rates. The results reported in this study, for different intraparticle velocities, show that the major beneficial effect of the intraparticle convective flow on the production of a good effluent quality is expressed by the occurrence of later breakthrough curves, due to the improvement of intraparticle mass transfer by convection, particularly when higher diffusional limitations are present inside the biofilm and support. Thus, bioreactors must be operated under such conditions that liquid flow occurs within biofilms.

Solution of adsorption problems involving steep moving profiles

The moving finite element collocation method proposed by Kiil et al. (1995)Chem. Engng Sci. 51 (4), 2793–2799 for solution of problems with steep gradients is further developed to solve transient problems arising in the field of adsorption. The technique is applied to a model of adsorption in solids with bidisperse pore structures. Numerical solutions were found to match the analytical solution when it exists (i.e. when the adsorption isotherm is linear). The method is simple yet sufficiently accurate for use in adsorption problems, where global collocation methods fail.

On the cibstryctuib of discretizations of elliptic partial differential equations

Algorithmic aspects of a class of finite element collocation methods for the approximate numerical solution of elliptic partial differential equations are described Locall for each finite element the approximate solution is a polynomial. polynomials corresponding toadjacent finite elements need not match continuously but their values and noumal derivatives match at a discrete set of points on the common boundary.High order accuracy can be attained by increasing the number of mathching points and the number of colloction points for each finite element.Forlinear equations the collocation methods can be equivalently definde as generlized finite difference methods. The linear (or linearzed )equations that arise from the discretization lend themselves well to solution by the methods of the methods nested dissection.An implememtation is described and some numerical results are givevn.

Dynamics of Simulated Moving-Bed Adsorption Separation Processes

Two mathematical models, namely the continuous moving-bed model (steady-state model) and the intermittent moving-bed model (transient model), have been used to describe simulated moving-bed adsorption separation processes. Four-column and 12-column simulated moving-bed systems for the separation of optical isomers with nonlinear adsorption isotherms were studied with the two models, respectively. The model equations were solved by using the numerical method of orthogonal collocation in finite elements. For the 4-column simulated moving bed, the differences between the simulation results from the two mathematical models are very obvious. Process dynamics, i.e., evolution of the species concentrations in the extract and raffinate and the axial profiles at different times, of the above systems in one switching period at the cyclic steady state are presented and discussed. They were numerically simulated with the intermittent moving bed model. The computations in a Cray J916 computer were very time consuming for the 12-column system.

On direct methods for the discretization of a heat-conduction equation using spline functions

The finite element collocation method is sometimes equivalent to the Galerkin method. The two methods are therefore considered together here. The nonhomogeneous heat-conduction equation with nonhomogeneous boundary condition is chosen for consideration. The equation is transformed into a variational form by continuous time Galerkin method and expressed in terms of linear spline basis function. On the other hand, the collocation method is applied on the heat equation with cubic splines as the basis functions. The two methods of discretization transform the heat equation to systems of ordinary differential equations in time, respectively. The corresponding stiffness matrices for the two systems of equations turn out to be the same. These matrices are sparse, banded and symmetric, and are therefore convenient for computer application.

Modeling of biodegradation/adsorption combined processes in fixed-bed biofilm reactors: Effects of the intraparticle convective flow

A mathematical model for a fixed-bed biofilm reactor is developed in this work, in order to investigate the roles of reaction, adsorption and mass transfer on substrate consumption. Mass transfer throughout the biofilm and support is described by simultaneous Fickian diffusion and convective flow. Numerical solutions were obtained for phenol biodegradation by finite element collocation methods. In terms of substrate removal and production of a good effluent quality, the results show the beneficial role introduced by adsorption in a biodegradation process, as well as the beneficial effects of the convective flow through both the biofilm and support, particularly when a higher intraparticle convective velocity is assumed to occur through the biofilm.