Tridiagonal matrix algorithm

Abstract: Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satis es the equation Txi = yi, for i = 1; 2; ; n. In this article, we investigate Hilbert-Schmidt interpolation problems for vectors x and y in tridiagonal algebras.

Abstract: This work presents a simplified steady state one dimensional heat transfer model for stabilized premixed flames in porous inert media. Two energy conservation equations describe the heat transfer process in solid and fluid regions of a porous burner. The thermophysical properties are considered constant and a plug flow is adopted. The stabilized premixed flame acts as a heat source in a specified section of the domain. The energy conservation equations are discretized by the finite volume method, using upwind scheme on the convective terms and central difference scheme on the diffusive terms. The linear systems of algebraic equations are solved by Tridiagonal Matrix Algorithm (TDMA). The results are compared with experimental and theoretical data. The effects of the porosity, Peclet number and thermal conductivity ratio between the solid and the fluid on temperature fields are depicted. Furthermore, the results reveal that the model is able to represent superadiabatic flames and the heat recirculation process in the porous burner.

Abstract: Determination of effective diffusivity of a substrate in biocatalyst particles is a key requirement in modeling heterogeneous reactions. The diffusivity is mainly controlled by the molecular-diffusion characteristics of the substrate, the tortuousness of the diffusion path within the biocatalyst, and the void fraction of particle volume available for diffusion. A general numerical model describing reactions in a biocatalyst particle following zero-order, first-order, and Michaelis-Menten kinetics was developed. Finite volume method was used to discretize the nonlinear second-order differential equation, and tridiagonal matrix algorithm was applied to solve the algebraic equation iteratively after discretization. Computer codes were written for calculating the diffusivity under specified boundary conditions. The numerical solution of the diffusion-reaction equations was validated against the experimental data from literatures. The model was further calibrated with experimental data obtained from fungal pellet experiments and then verified using additional data. The results show that the inverse methodology developed in this study was capable of predicting diffusivity in biocatalyst particles. Based on the predicted diffusivity, oxygen consumption by an individual pellet was simulated, oxygen consumptions by small versus large pellets were compared, and the effect of reaction rate on oxygen consumption was evaluated. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Abstract: In this paper, the normative matrices and their doubleLR transformation with origin shifts are defined, and the essential relationship between the doubleLR transformation of a normative matrix and theQR transformation of the related symmetric tridiagonal matrix is proved. We obtain a stable doubleLR algorithm for doubleLR transformation of normative matrices and give the error analysis of our algorithm. The operation number of the stable doubleLR algorithm for normative matrices is only four sevenths of the rationalQR algorithm for real symmetric tridiagonal matrices.