Solution of final stages of polyethylene terephthalate reactors using orthogonal collocation technique

The formation of polyethylene terephthalate (PET) has been modeled to have reactions with monofunctional compounds, redistribution, and cyclization reactions in addition to the usual polycondensation step. In the final stages, the overall polymerization is mass-transfer controlled and solution of the reactor performance equations have been determined through the orthogonal collocation technique. This technique is found to be considerably more efficient for PET reactors compared to the finite difference method; the use of ten collocation points gives results which are close to the exact solution.     

Propylene polymerization — a computer efficient algorithm with orthogonal collocation

This paper presents a new approach to the modelling of solid catalyzed Ziegler-Natta polymerization of propylene, using orthogonal collocation technique. It is shown that this model can predict the width of the molar mass distribution (MWD) of the product polymer. Its computation time is much shorter compared to other models. An interesting feature of the model is that it considers the discrete nature of the microcatalyst particles and takes advantage of solving a smaller number of stiff differential equations by choosing the number of internal collocation points judiciously. In addition, pseudo-steady state approximation (QSSA) is applied to some of the moment generating equations.     

An implementation variant of the polynomial finite difference method with orthogonal collocation and adjustable element length

The polynomial finite difference method, an easy-to-use variant of the finite difference method for the numerical solution of differential and differential–algebraic equations, has been recently presented [Wu, B., & White, R.E. (2004). Computers & Chemical Engineering, 28, 303–309]. In this work, it is shown that the polynomial finite difference method can be seen as a collocation method with finite elements of equal size with uniform distribution of collocation points within each element. We show that the same type of implementation can be improved if one uses orthogonal distribution of collocation points, without significantly affecting the computational effort. The suggested method is further improved with the use of Michelsen's technique for step-size adjustment to solve stiff differential equations with a semi-implicit third order method. Several examples that show improvements of one or two orders of magnitude of the proposed approach over the implementation by Wu and White are presented.     

ORTHOGONAL COLLOCATION IN CHEMICAL REACTION ENGINEERING

The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first approximation to gain insight into a problem and we refine the calculations to obtain numerical convergence to the exact result. In this later view the method is similar to and in direct competition with finite difference methods, and some of the references listed in the next section discuss the relative advantages of the orthogonal collocation method.     

ORTHOGONAL COLLOCATION IN CHEMICAL REACTION ENGINEERING

The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first approximation to gain insight into a problem and we refine the calculations to obtain numerical convergence to the exact result. In this later view the method is similar to and in direct competition with finite difference methods, and some of the references listed in the next section discuss the relative advantages of the orthogonal collocation method.     

Adsorption in finite bath and countercurrent flow with systems having a nonlinear isotherm

Isothermal countercurrent, cocurrent and finite bath adsorption has been modelled mathematically. The model equations have been solved numerically by using the method of orthogonal collocation. The equilibrium between the solid and the fluid phase is assumed to be nonlinear and is described by either the Langmuir or the Freundlich type of equation. The transport mechanisms in the particles are assumed to be pore diffusion, solid diffusion or a combination thereof. The effect of film resistance is included. The above three adsorption cases are mathematically described by the same equations.Computed results are shown for various flowratios and parameters in the Langmuir and Freundlich equations. The results may be of help for evaluating coefficients of diffusion from finite bath experiments and for designing continuous countercurrent absorbers.The equations, their parameters and the diagrams showing the results are given in what is hoped to be a comprehensive way to facilitate the comparison of various mechanisms and isotherms.     

Film‐surface diffusion during the adsorption of acid dyes onto activated carbon

A mass transport model has been developed and applied to the adsorption of three acid dyes onto activated carbon in three single component systems. The mass transfer model is based on two rate controlling mass transfer steps, namely external film mass transfer and homogeneous solid‐phase surface diffusion (HSD). Almost all previous film‐HSD models have been based on numerical solutions to the diffusion equation using orthogonal collocation or Crank–Nicolson finite difference solutions. However, in the present model a semi‐analytical solution to the solid surface diffusion equation is presented, yielding a sophisticated solution of the differential equations. The solutions provide a good correlation between the experimental concentration–time decay curves by incorporating the Langmuir equilibrium isotherm to describe the solid phase surface dye concentrations. However, the surface diffusivities show a dependence on the carbon particle surface coverage and these diffusivities have been correlated using a Darken relationship. Copyright © 2004 Society of Chemical Industry     

Helical Flow of a Sutterby Model Fluid

The problem of helical flow of a Sutterby model fluid has been analyzed numerically using a finite difference technique. Sutterby model fluid is one of the most important non-Newtonian fluids representing constitutive equations for high polymer aqueous solutions of CMC, HEC, and MC, and hence the problem is of practical significance. Computations have been made for the case when the inner cylinder rotates about the common axis with constant angular velocity and the outer cylinder is at rest, while a constant pressure gradient is acting on the axis of rotation. The dimensionless velocity profiles, volumetric flow rate, torque, and pressure distribution have been calculated for various values of the model parameters and the radius ratio.     

填充床电化学反应器普遍化模型方程数值求解方法

以有限差分法与正交配置法联用对填充床电化学反应器的模型方程进行了求解，结果表明该法是一种快速、有效地求解具有较为复杂的边界条件的偏微分方程的计算方法。     

Dynamic behaviour and stability of non-porous catalyst particles

The stability and dynamic behaviour of a model of an exothermic chemical reaction occurring on a non-porous catalyst particle are studied. The model accounts for the effects of external film heat and mass transfer resistances and intraparticle heat conduction.An orthogonal collocation technique is shown to be superior to the more conventional finite difference methods in obtaining solutions of the system equations.It is found that the thermal conductivity as well as the heat capacity of the particle can have a profound effect on stability and dynamic behaviour. Two simpler models which neglect the effects of thermal conductivity and accumulation of reactants are shown to give very limited insight into the dynamic behaviour of the system.     

Packed bed reactor analysis by orthogonal collocation

The equations governing a packed bed reactor with radial temperature and concentration gradients are solved using the orthogonal collocation method. The method is shown to be faster and more accurate than finite difference calculations. Using the orthogonal collocation method it is straightforward to extend one-dimensional (lumped parameter) models to the two-dimensional models needed when radial temperature and concentration gradients are important.     

模拟移动床色谱手性分离过程的非线性非理想模型

A non-linear non-ideal model,taking into account non-linear competitive isotherms,axial disperison,film mass transfer,intraparticle diffusion,and port periodic switching,was developed to simulate the dynamics of simulated moving bed chromatography(SMBC),The model equations were solved by a new efficient numerical technique of orthogonal collocation on finite elements with periodical movement of conceantration vector,The simulated SMBC performance is in accordance with the experimental results reported in the literature for separation of 1,1‘‘‘‘‘‘‘‘-bi-2-naphtol enantiomers using SMBC,This model is useful for design,operation ,optimization and scale-up of non-linear SMBC for chiral separations with significant non-ideal effects,especially for high solute concentration and small intraparticle diffusion coefficient or large chiral stationary phase particle.     

Mathematical modeling of bio-physico chemical processes in activated carbon columns

Predictive mathematical models which describe the adsorption and biological phenomena in the plug-flow stationary solid phase column and completely mixed recycle fluidized bed were developed. The models incorporate liquid film transfer, biodegradation and diffusion in the biofilm, adsorption onto the adsorbent, as well as biofilm growth. The model equations are solved by a combination of orthogonal collocation and finite difference techniques. Computer simulations were conducted to compare the performance of the two models. Sensitivity tests were also performed to determine the effects of physical and biological parameters on model profiles.     

Steady‐State Modeling and Simulation of an Axial‐Radial Ammonia Synthesis Reactor

A two‐dimensional model was developed for an axial‐radial ammonia synthesis reactor of the Shiraz petrochemical plant. In this model, momentum and continuity equations as well as mass and energy balance equations are solved simultaneously by orthogonal collocation on the finite element method to obtain pressure, velocity, concentration and temperature profiles in both axial and radial directions. For the catalyst particle, the effectiveness factor is calculated by solving a two‐point boundary value differential equation. The boundary conditions for the Navier‐Stokes and continuity equations are obtained by using equations representing the phenomena of gases splitting or joining in different streams and going through holes in a thin wall. The results of the mathematical model have been compared with the plant data and a good agreement is obtained.     

Mathematical modeling for chemical vapor deposition in a single-wafer reactor: Application to low-pressure deposition of Tungsten

A mathematical model for low pressure chemical vapor deposition in a single-wafer reactor in stagnation point flow has been developed to investigate the reactor performance. The transient transport equations for a simulated reactor include continuity, momentum, energy, and gaseous species balances. The model equations are simultaneously solved by using a numerical technique of orthogonal collocation on finite element method. Simulation studies have been performed to gain an understanding of tungsten low pressure chemical vapor deposition process. The model is then used to optimize the deposition rate and uniformity on a wafer, and the effects of operating conditions on deposition rate are studied to examine how system responses are affected by changes in process parameters. Deposition rate and uniformity calculated at the steady state are observed to be very sensitive to both temperature and total pressure. In addition, the model predictions for tungsten deposition from hydrogen reduction of tungsten hexafluoride have been compared with available experimental data in order to demonstrate the validity of the model.     

Bifurcation points computation of the diffusion—reaction equations

This paper investigates the computation of the simple bifurcation points, the double bifurcation points and the Hopf bifurcation points for diffusion—reaction equations discretized with orthogonal collocation. Both the direct and indirect methods are used. The test problems selected are well-known diffusion—reaction equations (catalyst pellet and 2-D thermal explosion). The results show that orthogonal collocation can be used successfully in solving bifurcation problems.     

Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation,Galerkin, tau and least-squares methods

In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study.Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods.The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.     

Nodal positioning analysis (NPA)—A numerical method for determining flow behavior of thermoplastic materials

A finite difference method is presented for the solution of two dimensional flow problems in polymer processing. The method is applicable to narrow gaps of any shape and variable thickness. NPA was developed for analyzing the filling stage of the injection molding cycle, but it could be used in extrusion, blown film, and other polymer processing operations. In NPA the position of the flow front is calculated at the end of each time increment, and an axial node is placed at the newest location of the flow front. Each axial node is then divided into a determined number of radial nodes. The velocity and temperature profiles are obtained from the simultaneous solution of the momentum and energy equations. The use of finite differences transforms the continuity, momentum, and energy equations into a system of linear equations which can be solved by any direct or iterative technique. The procedure is repeated until axial nodes have been placed throughout the whole flow channel or until the flow front stops due to polymer solidification. The main advantage of this technique, when compared to the use of a fixed finite difference grid, is that computation time is saved by considering only nodes filled with the fluid. Empty nodes are not considered and corrections for partially filled nodes are not needed. No complications due to the parabolic-shape of the flow front profile are introduced because the axial nodes are placed at average front locations determined by the average velocity at the particular time interval under consideration.     

Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods

In this paper, we analyse some properties of the orthogonal collocation in the context of its use for reducing PDE (partial differential equations) chemical reactor models for numerical simulation and/or control design. The approximation of the first order derivatives is first considered and analysed with respect to the transfer of the stability properties of the transport component from the PDE model to its approximated ODE (ordinary differential equations) model. Then the choice of the collocation points as zero of Jacobi polynomial is analysed and interpreted as an optimal choice with respect to a weighted norm. Finally, some guidelines for the use of orthogonal collocation are proposed and the results are illustrated on a simulation example.     

Analysis of non-isothermal catalytic reactions. Limiting effect of non-key component

An improved method for predicting effectiveness factors for pseudo first order catalytic reactions with limiting non-key reactant has been derived. The method takes account of intraparticle temperature and concentration profiles in addition to interphase effects and solution of the system equations is obtained using the orthogonal collocation technique which is simple and involves only algebraic equations. The depth of penetration of reactant is also determined by this method. The results are discussed and compared with earliler work.