Shanks Transformation Research Articles

ENHANCEMENT FACTORS FOR SIMULTANEOUS ABSORPTION OF TWO GASES

Abstract The process of simultaneous absorption of two gases in an inert liquid is examined. It is shown that by a suitable combination of the asymptotic solutions for small and large values of M, (a dimensionless reaction rate constant), the enhancement factors of the two gases can be calculated very accurately for alt reaction rates. Using a [5/5] Pade approximant and an e 5 −-Shanks transformation, we have also extended our previous results (Sen, 1988) for slow and intermediate reaction regimes. A detailed comparison of the results obtained by the Pade approximants. Shanks transformations, matched asymptotic expansions and numerical solutions is presented for a wide range of diffusivities and reaction rates.

An application of the Adomian decomposition method to the transient behavior of a model biochemical reaction

An approximate analytical solution for the transient phase of the Michaelis-Menten reaction is derived using the Adomian decomposition method. The analytical solution, which is given in the form of a power series, is found to be highly accurate in predicting the behavior of the reaction in the very early stages. To accelerate the convergence of the power series solution and extend its region of applicability throughout the entire transient phase, we have used (a) the method of Padé approximants and (b) the iterated Shanks transformation. Both the Padé approximant and the Shanks transformation are shown to converge rapidly throughout and beyond the transient period and yield very accurate results. A comparison of the various analytical approximations and a direct numerical solution of the nonlinear initial value problem is also presented.

PERTURBATION ANALYSIS OF GAS ABSORPTION WITH CHEMICAL REACTION: A NONVOLATILE LIQUID REACTANT

Enhancement factors for gas absorption in a liquid containing a nonvolatile reactant are calculated using a regular perturbation technique. The perturbation solutions are found to be quite accurate in the slow reaction regime but lead to erroneous results for moderate and large reaction rates. To extend these solutions to the intermediate and fast reaction regimes, we have used the method of Padé approximants and Shanks transformations. The results obtained by these acceleration techniques are shown to be in excellent agreement with the direct numerical solutions for the enhancement factors in the intermediate and fast reaction regimes.

Enhancement factors for gas absorption with chemical reaction: Approximate analytical solutions

In this paper we examine the problem of absorption of two gases in an inert liquid with simultaneous chemical reaction. In the slow reaction regime, a regular perturbation solution is derived for the enhancement factors of the solute gases. Using the methods of Padé approximants and Shanks transformations, the regular perturbation solution (which is strictly valid in the slow reaction regime) is extended so that the results are applicable in the intermediate reaction regime and, in many cases, in the fast reaction regime. The results obtained by Padé approximants and Shanks transformations are found to be in excellent agreement with the direct numerical solutions of the enhancement factors. By virtue of their analytical nature and high accuracy, the present results may be used as a valuable tool in setting forth design criteria for many mass transfer operations in gas-liquid systems.

Determination of the hydrodynamic characteristics of a partially filled cavity containing a pendulum

Equations are given for the small oscillations of a viscous liquid-filled cavity containing a pendulum and the dissipative and inertial coefficients are determined on the basis of a systematic application of the finite element method (FEM). In order to improve the accuracy of the values obtained for these coefficients the use of a nonlinear Shanks transformation is proposed. This makes it possible to achieve the required accuracy using much less machine time and memory. The properties of the inertial hydrodynamic characteristics associated with an ideal fluid are studied in relation to cavities lacking a pendulum but fitted with annular ribs. It is shown that as a result of the presence of these rings the estimate given in [6] for the generalized mass of the fundamental mode of the fluid oscillations is inaccurate and must be modified.

Forced convection along a longitudinal cylinder embedded in a saturated porous medium

The thermal boundary layer along an isothermal cylinder in a porous 3edium is studied numerically by a finite difference scheme and also using the method of extended perturbation series. The series in terms of the transverse curvature parameter ξ extended to seven terms and is subsequently improved by applying the Shanks transformation twice and thrice, respectively. Results for heat transfer characteristics are found in very good agreement.

Use of nonlinear convergence accelerators for the efficient evaluation of GTO molecular integrals

Different methods for accelerating convergence of known series expansions for the auxiliary functions Fm(t) encountered in molecular integrals over Gaussian-type orbitals have been tested and compared. We have investigated the numerical properties of the Shanks transformation and the Levin transformations. It turns out that Levin’s t transformation is the best accelerator, whereas the Shanks transformation, which nearly yields the same results as the continued-fraction method, is less effective.

PERTURBATION THEORY AND MOMENTS FOR NONLINEAR FREUNDLICH DESORPTION IN A CONTINUOUS-FLOW WELL-STIRRED VESSEL

A regular perturbation expansion is applied to desorption governed by a Freundlich isotherm in a continuous-flow well-stirred vessel. A power series solution is presented, whose convergence is improved by the Shanks transformation. With a symbolic manipulation computer program, the first few temporal moments are derived exactly in closed form from the perturbation scheme.

Transformation approximation for the plasma dispersion function and application to electrostatic waves

Simple algebraic approximations for the plasma dispersion function Z(s) and its derivative are obtained by the Aitken's Δ2 or Shanks transformation. Using the approximate function Z'(s), higher-order dispersion relations for the electron wave and ion-acoustic wave are derived.

Analytical solutions for diffusion-controlled adsorption kinetics with non-linear adsorption isotherms

A regular perturbation series method provides a solution to the diffusion equation when the boundary condition is a non-linear adsorption isotherm. For adsorption at the interface the Freundlich and Langmuir isotherms yield power series in the square root of time. Convergence of the power series solutions is improved by applying the Shanks transformation. The solutions are compared to limiting cases and to published numerical solutions. The results are most accurate for small time where the numerical finite difference method is least reliable.

Improved perturbation solutions for laminar natural convection on a vertical cylinder

The method of extended perturbation series is applied to solve for laminar natural convection from an isothermal, thin vertical cylinder. The series in terms of the transverse curvature parameterξ extended to five terms and is subsequently improved by applying the Shanks transformation twice. The validity of the solution is extended up toξ=10 and possibly even beyond. Up toξ=10, the results for wall shear as well as the local and average Nusselt numbers agree very closely with those of local nonsimilarity and finite difference solutions. The ease of computation coupled with high accuracy makes the present approach far more attractive than the currently popular local nonsimilarity and finite difference methods. Its success with the present problem should motivate applications to a host of nonsimilar boundary layer flows.

Extended perturbation series analysis and improvement applied to a porous catalyst slab

In this paper techniques available for the analysis and improvement of extended regular perturbation series are applied to an example problem of chemical reaction inside a porous catalyst slab of infinite length. The techniques used include the extraction of singularity, Euler transformation, Pade approximants, Shanks transformation, Levin transformations and series reversion. The example illustrates the utility of these techniques in improving the convergence of regular perturbation series. In general, the Pade approximants given the best results for the present problem.

ANALYSIS OF HEAT TRANSFER PROBLEMS WITH COMPUTER-EXTENDED PERTURBATION SERIES

The method of computer extension of perturbation series and its subsequent improvement are applied to four problems in nonlinear heat conduction. The nonlinearities involve temperature-dependent thermal properties, temperature-dependent heat generation, and simultaneous conduction-radiation interaction. The series are improved by techniques such as the extraction of singularity. Fade approximants, and Euler and Shanks transformations. The improvement achieved highlights the usefulness of the approach.

Analysis Of Heat Transfer Problems With Computer-Extended Perturbation Series

The method of computer extension of perturbation series and its subsequent improvement are applied to four problems in nonlinear heat conduction. The nonlinearities involve temperature-dependent thermal properties, temperature-dependent heat generation, and simultaneous conduction-radiation interaction. The series are improved by techniques such as the extraction of singularity. Fade approximants, and Euler and Shanks transformations. The improvement achieved highlights the usefulness of the approach.

Scaling and Pade approximants in perturbation theory

Attention is drawn to the connection between the scaling (or (2/1) Pade) approximation for improving the convergence of perturbation series, recently discussed by Wilson (1979) and Aitken's (1939) delta 2 transformation. The invariance with respect to scaling of the (n+1/n) Pade approximant is shown to follow from its equivalence to a generalised Aitken (1939) or Shanks (1955) transformation. An algorithm for generating this series of Pade approximants is given.

The RF vector potential in a gapped tunnel

The expressions given by Kosmahl and Branch for the RF fields in a gapped tunnel are combined and integrated to give the RF vector potential. The problem of evaluating the resulting expressions at the nodes of a suitable mesh is considered in detail. Different methods are used in different regions, and Shanks's transformation is introduced to deal with the difficult gap-edge point. Buneman's fast routine INTRA is used to evaluate the radial and axial fields at an arbitrary point, by nine-point interpolation and differencing of the potential matrix. The method is free from any paraxial approximations.

A practical method for the numerical evaluation of Sommerfeld integrals

Sommerfeld integrals, which appear frequently in dipole radiation problems, often must be evaluated numerically. Gauss-Laguerre quadrature is an effective integration method, provided the horizontal distance \rho from the source to the receiver is less than their vertical separation \zeta . A complementary method is to use Romberg adaptive quadrature to integrate over the positive and negative half-cycles of the integrand, then from a sequence of approximations by summing the contributions from successive half-cycle subintegrals, accelerating the convergence of this sequence using the Shanks transformation implemented via Wynn's algorithm. A comparison of the efficiencies of these techniques favors Gauss-Laguerre quadrature for \rho/\zeta and the Romberg-Shanks method otherwise.

Generalisations de la transformation de shanks, de la table de Pade et de l'ε-algorithme

This paper deals with a generalization of the Shank's transformation for a sequence of elements of a topological vector space. It is showned how this generalization leads to a generalization of the Pade table. A recursive algorithm for effecting this transformation is given. Its properties are studied and some theorems are proved. This algorithm can be considered as a generalization of the e-algorithm of Wynn. A second generalization is also studied and a third one which needs less elements of the initial sequence. The paper ends with a discussion on the vector case.