Collocation method orthogonal

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

Applying orthogonal collocation method, the above equations become... 

Simulation results of one-dimensional and two-dimensional models by orthogonal collocation method... 

Cuthrell and Biegler (1989) considered the orthogonal collocation method which is described below. Two Lagrange polynomials one for the state variable (x) and one for the control variable (u) can be written as ... 

Discretization of the partial differential equation system in axial (z) and radial (r) direction by means of the orthogonal collocation method (7) leads to the following system of ordinary differential equations. 

When radial dispersion is included, even the steady state equations are partial differential equations — in the axial and radial space variables. The dispersion model equations can be numerically solved by finite-difference schemes, or more efficiently, by orthogonal collocation methods (14, 15). 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

The orthogonal collocation method using piecewise cubic Her-mite polynomials has been shown to give reasonably accurate solutions at low computing cost to the elliptic partial differential equations resulting from the inclusion of axial conduction in models of heat transfer in packed beds. The method promises to be effective in solving the nonlinear equations arising when chemical reactions are considered, because it allows collocation points to be concentrated where they are most effective. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

Some of the important research results that led to the material in this book were developed during the senior author s collaboration with former PhD student Jan Sorensen. This extensive research pioneered the development of orthogonal collocation methods for the solution of modeling and... 

The approach proposed in this paper belongs to this class and uses the orthogonal collocation method on finite elements [1, 2, 3] to convert the DAEs... 

The problem of decreased catalyst activity due to irreversible thiophene adsorption was solved numerically using an orthogonal collocation method with three internal points. The numerical results were compared with experimental data obtained by... 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

The same complex kinetics gives multiple steady states isothermally in thin or thick porous electrocatalysts (418). A simple, graphical orthogonal collocation method (422) can show the existence of multiple solutions for concentration within a certain potential range (418). If ohmic losses in the pores cause a potential change within the electrode structure, multiplicity can also arise with respect to potential, even with simpler rate expressions... 

Zn(II) as presented in Table 24.2 [6], or Cr(VI), more than 99% of which was removed from industrial electroplating wastewater [20], The modeling of the experimental breakthrough of lead (II) onto activated carbon fibers in a fixed bed, using axial dispersion and diffusion equations solved by the orthogonal collocation method, demonstrated that the intraparticle and external mass transfer is not the rate-controlling step, due to the short diffusion path for the adsorbate in activated carbon fibers [21]. 

The catalyst tube partial differential equations are reduced to a set of ordinary differential equations using the orthogonal collocation method. For the combustion chamber the governing differential equations describing the temperature distribution in the furnace are to be also solved, using the collocation method. 

This method is based on the choice of a suitable trial series to represent the solution. The coefficients of the trial series are determined by making the equation residual vanish at a set of points, called collocation points, in the solution domain. The orthogonal collocation method (Villadsen and Stewart, 1967 Finlayson, 1972) provides a systematic basis for choosing the collocation points. Figures 5.4 and... 

Numerical methods of solution Hansen (1971) used the orthogonal collocation method to solve both the steady state and transient equations of six different models of the porous particle of increasing complexity. He found that only 8 collocation points were necessary to obtain accurate results. This leads to a considerable saving in computing time compared to the conventional finite difference methods such as the Crank-Nicolson method. 

Non-isothermal ejfectiveness factor The non-isothermal effectiveness factor can be obtained numerically only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method. 

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