Mechanical design failure models for buckling : Abhijit Dasgupta and Henry W. Haslach Jr, IEEE Transactions on Reliability, 42(1), 9 (1993)

Abstract

A method based on the asymptotic joining of expansions is developed for solving the two-point boundary value problem for a system of equations with a single small parameter (high Péclet number) that occur in the theory of isothermal reactors with nonhomogeneous fluidized bed. External analytic solution in the form of a series containing three terms is derived for the case of small coefficient of exchange between bubbles and medium. An analytic solution of the problem is presented. It is uniformly valid over the whole segment and accurate to within quadratic terms containing the small parameter, which appear in the two-point boundary value problem.Models of reactors with a nonhomogeneous fluidized bed (see, e. g. [1–4]) are based on the concept of the fluidized bed as a two-phase system consisting of a continuous phase moving in the reactor at the initial fluidization rate, and of a discrete phase which is the excess of the fluidizing agent which moves through the bed in the form of bubbles. The computation of such type of reactor in the case of a reaction of an arbitrary order is complicated even in the one-dimensional case and can only be carried out by numerical methods. Because of this, approximate methods of nonlinear mechanics are of particular interest, for instance, for the computation of the steady mode of reactor operation.A method of solving the steady state equation for the isothermal reactor at high Péclet numbers was developed, based on the joining of asymptotic expansions. Later, a method similar to that of joining asymptotic expansions was proposed in [7] for solving equation of the isothermal reactor. However the extension of that method to systems of equations presents some difficulties [8–10]. Due to the nonlineari ty of equations of the nonisothermal reactor it was not possible to obtain an external analytic solution in the zero and subsequent approximations.