Orthogonal collocation for the analysis of immobilized enzyme systems

Orthogonal collocation method is applied to the analysis of nonlinear ordinary differential equations containing Michaelis—Menten kinetics. The solution is expanded in a series of Lagrange interpolation polynomials and Gauss—Jacobi quadratures are used in calculating effectiveness factors. A set of nonlinear algebraic equations resulting from collocation approximation is conveniently solved by the Gauss—Seidel iterative method, but its convergence path is not monotonous. Although the rate of convergence depends on system parameters, orthogonal collocation is more efficient than the Runge—Kutta method in solving boundary value problems even at a high Thiele modulus.     

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.     

Digital simulation of industrial steam reformers for natural gas using heterogeneous models

A model is developed for industrial natural gas steam reformers with top-fired and side-fired furnance design. The one-dimensional heterogeneous model for catalyst tubes takes into account the intra-particle diffusion resistances. Two approximations are used for the computation of the effective diffusivities via the Stefan-Maxwell equations. The two-point boundary value differential equations for the catalyst pellets are solved by a novel efficient and modified orthogonal collocation technique. The performance of the model has been checked in the case of two industrial reformers used in the ammonia and methanol industries in Egypt and Saudi Arabia.     

Application of block pulse functions in dynamic simulation

Block pulse functions are used to obtain piecewise-constant solutions of stiff ordinary differential equations, two-point boundary value problems, and simultaneous first-order partial differential equations. The advantage and limitation of the application of block pulse functions are discussed.     

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.     

Evaluation and improvement of dynamic optimality in electrochemical reactors

A systematic approach for the dynamic optimization problem statement to improve the dynamic optimality in electrochemical reactors is presented in this paper. The formulation takes an account of the diffusion phenomenon in the electrode/electrolyte interface. To demonstrate the present methodology, the optimal time-varying electrode potential for a coupled chemical-electrochemical reaction scheme, that maximizes the production of the desired product in a batch electrochemical reactor with/without recirculation are determined. The dynamic optimization problem statement, based upon this approach, is a nonlinear differential algebraic system, and its solution provides information about the optimal policy. Optimal control policy at different conditions is evaluated using the best-known Pontryagin's maximum principle. The two-point boundary value problem resulting from the application of the maximum principle is then solved using the control vector iteration technique. These optimal time-varying profiles of electrode potential are then compared to the best uniform operation through the relative improvements of the performance index. The application of the proposed approach to two electrochemical systems, described by ordinary differential equations, shows that the existing electrochemical process control strategy could be improved considerably when the proposed method is incorporated.     

The moment method for one-dimensional dynamic reactor models with axial dispersion

A polynomial approximation method for calculating state profiles for plug-flow reactors is extended to one-dimensional reactor models that include axial dispersion. The method is based on the conservation of reactor state profile moments along the spatial dimension. The moments are then transformed analytically into a polynomial approximation at each timestep. The boundary conditions of the parabolic partial differential equation are given special attention. It is shown that the Danckwerts boundary conditions are an appropriate set of boundary conditions for flow problems with axial dispersion in closed-closed geometries. A significant feature of the present method is that boundary conditions of the partial differential equation model to be solved are implicitly satisfied via the moment transformation, while the polynomial profile in the numerical approximation does not have to satisfy the boundary conditions exactly. The method is tested in two cases: startup of a tubular reactor and fixed-bed adsorber involving axial dispersion.     

Thermal pulse-response of reversible liquid reaction in a tubular reactor(I)

The transient action of a tubular reactor, in which the liquid mixtures react reversibly, in response to the pulse input establishing a set of nonlinear partial differential equations has been analysed. Using collocation with B-spline basis piecewise polynomial approximation, the solution protocols of the system has been formulated in order to estimate parameters, such as transport, kinetic and thermodynamic parameters, from stochastic temperature responses, instead of compositions, in the reactor. The thermal pulse-response technique has proved to be one of the efficient and plausible candidates for the fields of estimations with the exception of the activation energy of the reaction.     

A fourth-order accurate, three-point compact approximation of the boundary gradient, for electrochemical kinetic simulations by the extended Numerov method

The fourth-order accurate, three-point compact (extended Numerov) finite-difference scheme of Chawla [J. Inst. Math. Appl. 22 (1978) 89] has been recently found superior (in terms of accuracy and efficiency) to the conventional second-order accurate spatial discretisation commonly used in electrochemical kinetic simulations. However, the two-point compact boundary gradient approximation, accompanying the scheme, is difficult to apply in the case of time-dependent kinetic partial differential equations, because it introduces unwanted second temporal derivatives into calculations. The conventional five-point gradient formula is free from this drawback, but it is also not very convenient, owing to the locally increased bandwidth of the matrix of linear equations arising from the spatio-temporal discretisation. A new three-point compact boundary gradient approximation derived in this work, avoids the above inconveniences and economically re-uses expressions utilised by the extended Numerov discretisation. The fourth-order accuracy of the new approximation is proven theoretically and verified in computational experiments performed for examples of kinetic models.     

MODIFIED METHODOLOGY FOR DISTILLATION MODELLING BY ORTHOGONAL COLLOCATION

A major shortcoming of polynomial approximation in the medelling of distillation columns isthe difficulty encountered while choosing the number and location of collocation points,which are usually doneby rule of the thumb,inevitably giving rise to high dimensionality and longer computation time for the resultingmodel.In order to take full advantage of polynomial approximation in the modelling of complicatedmulticomponent distillation columns,modifications must be made to the model reduction procedure originallyproposed by Cho.This is achieved by putting in special polynomials to each of the variable profiles.Furthermore,the number and location of the collocation points can be determined by the optimization of anappropriate objective function.This would bring about less dimensionality and less computation time for theresulting reduced--order model as compared with Cho's procedure while its accuracy is still kept excellent.Theeffectiveness of such modifications is illustrated by two simulation examples.Bot     

Solution of moving boundary problems for gas—solid noncatalytic reactions by orthogonal collocation

An integral transformation, a coordinate transformation for immobilization of the moving boundary, and orthogonal collocation are used to reduce a nonlinear initial-boundary value problem in time and space to a set of ordinary differential equations in time with given initial conditions. The method is developed for solution of models for gas—solid noncatalytic reactions and is especially useful for moving boundary, two-stage reaction problems. The method represents an advantage and an alternative to the available finite difference techniques. Results of various gas—solid reaction models are analyzed.     

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.     

Solution of nonlinear boundary value problems—VII A novel method: differentiation with respect to boundary condition

A simple numerical method for construction of the dependence of solutions to nonlinear boundary value problem on a paremeter will be developed. The set of differential equations is differentiated with respect to the boundary condition chosen and the resulting partial differential equations are solved by a finite-difference method. The method is illustrated by an example of heat and mass transfer in a porous catalyst.     

HYDROMAGNETIC BOUNDARY LAYER FLOW AND HEAT TRANSFER IN VISCOELASTIC FLUID OVER A CONTINUOUSLY MOVING PERMEABLE STRETCHING SURFACE WITH NONUNIFORM HEAT SOURCE/SINK EMBEDDED IN FLUID-SATURATED POROUS MEDIUM

Analytical study for the problem of flow and heat transfer of electrically conducting viscoelastic fluid over a continuously moving permeable stretching surface with nonuniform heat source/sink in a fluid-saturated porous medium has been undertaken. The momentum and thermal boundary layer equations, which are partial differential equations, are converted into ordinary differential equations, by using suitable similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved analytically assuming exponential solution, and similarly thermal boundary layer equations are solved exactly by using power series method, with the solution obtained in terms of Kummer's function. The results are shown with graphs and tables. The effect of various physical parameters like viscoelastic parameter, porosity parameter, Eckert number, space, and temperature-dependent heat source/sink parameters enhances the temperature profile, whereas increasing the values of the suction parameter and Prandtl number decreases the temperature profile. The results have technological applications in liquid-based system involving stretchable materials.     

Electroanalytical investigations—IV. use of orthogonal collocation for the simulation of quasireversible electrode processes under potential scan conditions

The method of orthogonal collocation is used to discretize the differential equations describing quasireversible electrode processes under conditions of potential scanning. The resulting set of ordinary differential equations is solved to give numerical simulations. The numerical values are compared to literature data. A method for optimization of the dimensionless parameter β is given.     

Studies in approximation methods—III: Boundary value ordinary differential equations

A numerical comparison is made between the orthogonal collocation and Runge-Kutta techniques for solving boundary value ordinary differential equations. Using various forms of the catalyst effectiveness factor system, it is shown that orthogonal collocation is the computationally superior method; this result also holds for a specific class of initial value problems. A multiple element approach in which the elements are located in an optimal manner is also developed. This optimal procedure holds promise for efficiently solving complicated boundary value problems.     

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

Hydromagnetic non‐Darcy flow,heat and mass transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

A numerical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics on steady two‐dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non‐Darcy porous medium in the presence of thermal radiation and viscous dissipation. The governing partial differential equations are convected into a system of nonlinear ordinary differential equations by similarity transformation and are solved numerically by using the Successive linearisation method, together with the Chebyshev pseudo‐spectral collocation method. The effects of various parameters on the velocity, temperature, and concentration fields as well as on the skin‐friction coefficient are presented graphically and in tabular forms.     

ANALYSIS OF SLOW GAS-LIQUID REACTIONS IN CONTINUOUS STIRRED TANK REACTORS

Generalized criteria are developed for identifying the various reaction-mass transfer regimes that might be encountered in the analysis of gas-liquid reactions in continuous stirred-tank reactors. These criteria are based on approximations to the governing differential equations (boundary value problem). The approximate criteria are shown to be in good agreement with the exact criteria obtained by solving the boundary value problem. It is not necessary to solve the boundary value problem to evaluate the approximate criteria, so numerical work can usually be avoided. Calculation of the utilization factor in the various reaction-mass transfer modes is discussed. The approximation of van Krevelen and Hoftijzer³ is applicable in all regimes.