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Using nonlinear least-squares techniques and batch experimental data, a mathematical model is obtained for the formation of zeolite A from amorphous sodium aluminosilicate substrate and aqueous sodium hydroxide solution. The effects of operating variables upon the productivity and stability during continuous formation of zeolite A are discussed. Analytical solutions are presented for the steady-state equations. The critical dilution rate and productivity can be increased by recycling a proper portion of crystals produced.

By using curve fitting techniques, mathematical formulas are obtained for the catalyst effectiveness factors for nonisothermal first-order and second-order irreversible chemical reactions. The effects of intraparticle mass and heat diffusion upon the concentration and temperature profiles in adiabatic and nonadiabatic catalytic reactors are examined. A relation between the maximum interstitial adiabatic temperature and the maximum intraparticle temperature in the reactor is obtained. For some sets of parameters in a nonadiabatic reactor, the concentration and temperature and, consequently, the reactor effluent are sensitive to small changes in the properties of catalyst particles.

A finite-difference method is developed for numerical solution of parabolic partial differential equations. This technique is explicit and stable. It is shown that the present method is more accurate and faster, in terms of computer time, than the Crank-Nicholson method. A method of handling nonlinear problems is also presented. Two examples are given to illustrate the present technique. The first problem is a linear diffusion equation. The second problem deals with two simultaneous nonlinear parabolic partial differential equations with Neumann boundary conditions describing the steady state of a packed-bed catalytic reactor with radial mixing.

The design of chemical reactors for polymerization and degradation processes requires the consideration of the kinetics of reaction systems which may contain several hundred or even thousand consecutive and simultaneous reactions. The problem is further complicated by the fact that the kinetic mechanisms for these processes are not well established. The present paper is a theoretical analysis of addition polymerization, copolymerization and degradation systems occurring in both continuous stirred tank and batch reactors for a number of kinetic models reported in the literature. Analytical solutions are derived for the steady state continuous process. In the batch process a steady state is not assumed and approximately 200 simultaneous first order differential equations for species concentrations are solved numerically. The paper is divided into three parts. Addition polymerization is discussed in the first part for each of the special cases of monomer, spontaneous, combination and disproportionation termination. For the continuous process, the steady state concentrations of the polymers arc obtained and the molecular weight distribution function and the optimum isothermal operating temperature are discussed. For the batch process, the rate equations are solved numerically by the Runge-Kutta method on a digital computer and the effects of the system parameters on the monomer concentration profile and the molecular weight distribution are examined. By the use of numerical methods with a digital computer it is possible to obtain the concentration of each of a large number of polymer species during the course of polymerization. The result of computation shows that the steady state assumption for active polymer species is not accurate, especially in early stage of reaction, and as well, is inaccurate also for high molecular weight active species. In the case of spontaneous termination, the rate of monomer consumption is slower than that in the case of monomer termination, because the monomer is reproduced by the termination process of the active polymer, P1. The profiles of monomer concentration and molecular weight distribution are the same for the cases of no termination and combination termination. Essentially the same treatment is made for copolymerization in the second part. This time the two simultaneous algebraic equations for the monomer concentrations are solved by the Newton-Raphsom method and these are then used to obtain the steady state concentrations of the copolymer species as functions of the system parameters. The analysis of the batch case involves the numerical solution of 194 simultaneous nonlinear first order differential equations. It is shown that the steady state approximation for the active copolymer concentrations cannot be made. There is a little delay in the formation of the dead species relative to that of the corresponding active copolymer. This is expected, because the dead species are produced by the termination reactions of the corresponding active species. In the third part, degradation is considered as random scission, as a chain reaction, and as a reverse polymerization. The rate equations describing the random scission process in a batch reactor arc shown to be linear so that they may be solved by methods of straightforward integration and by matrixes, while the chain reaction and reverse polymerization mechanism require the same numerical techniques as used for polymerization.

Systems of simultaneous second-order nonlinear ordinary differential equations with boundary conditions at two points are solved by a new numerical scheme. By adding fictitious accumulation terms, ordinary differential equations become parabolic partial differential equations. A stable numerical method based upon Saul'yev's technique is used to integrate the parabolic partial differential equations. At an early stage of integration only a few mesh points are required and later the number increases gradually as the steady-state is approached. Since one is only interested in obtaining the steady-state solution to the partial differential equations, one can change the values of coefficients of accumulation terms and also use relatively large time increments. Two examples are given to illustrate this technique. The first problem is three simultaneous ordinary differential equations describing two reactions within porous catalyst particles under nonisothermal conditions. The second problem deals with Taylor diffusion in tubular reactors. Results are obtained within short computer times and one does not require a close initial guess to the steady-state solution.

Anti-American propaganda, as typified by the contents of a Peking newspaper, has a functional purpose in Communist Chinese foreign policy and in bulwarking the regime internally. The authors analyze its use in the crucial year of 1959.

The purpose of this paper is to propose a method for the calculation of molecular-weight distributions for addition polymerization. The method involves the use of a generating function which reduces the infinite number of differential equations to two differential equations and the evaluation of an integral. The method has the advantage that the concentration of any particular n -mer may be calculated without calculating all of its precursors. Other methods do not have this property. An example is worked out for first-order initiation and monomer termination although other initiation and termination steps may be easily treated. It is again verified that the steady state hypothesis may not be valid.