The course of consecutive reactions inside a nonisotropic catalyst particle, affected by internal difusion

1986 ◽  
Vol 51 (1) ◽  
pp. 54-65
Author(s):  
Jiří Hanyka ◽  
Alena Fialová
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Catalyst Surface ◽  
Spherical Geometry ◽  
Catalyst Particle ◽  
Diffusion Equations ◽  
System Of Differential Equations ◽  
Reaction Selectivity ◽  
Consecutive Reactions ◽  
Reaction Orders

A system of differential equations for consecutive reactions inside a nonisotropic catalyst particle under conditions of internal diffusion is solved. The system of diffusion equations for the spherical geometry of the catalyst grain is numerically solved by using the collocation method. The solution is sought for various radial activity profiles across the catalyst particle and for various values of Thiele's modulus for the two consecutive reactions. The effect of the reaction orders with respect to the reactants on the degree of utilization of the internal catalyst surface and on the reaction selectivity is examined.

Natural Frequencies of the Bias Tire

1976 ◽  
Vol 4 (2) ◽  
pp. 86-114 ◽  
Author(s):  
M. Hirano ◽  
T. Akasaka
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Composite Structure ◽  
Natural Frequencies ◽  
Variable Coefficients ◽  
System Of Differential Equations ◽  
Inflation Pressure ◽  
Point Collocation ◽  
Membrane Shell ◽  
Wide Range

Abstract The lowest natural frequencies of a bias tire under inflation pressure are deduced by assuming the bias tire as a composite structure of a bias-laminated, toroidal membrane shell and rigorously taking three displacement components into consideration. The point collocation method is used to solve a derived system of differential equations with variable coefficients. It is found that the lowest natural frequencies calculated for two kinds of bias tire agree well with the corresponding experimental results in a wide range of inflation pressure.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Kolmogorov and classic diffusion equations

1988 ◽  
Vol 53 (6) ◽  
pp. 1181-1197
Author(s):  
Vladimír Kudrna
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Parabolic Type ◽  
Diffusion Equations ◽  
Stochastic Motion ◽  
Energy Component ◽  
Classic Form

The paper presents alternative forms of partial differential equations of the parabolic type used in chemical engineering for description of heat and mass transfer. It points at the substantial difference between the classic form of the equations, following from the differential balances of mass and enthalpy, and the form following from the concept of stochastic motion of particles of mass or energy component. Examples are presented of the processes that may be described by the latter method. The paper also reviews the cases when the two approaches become identical.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

scholarly journals Outflowing Envelopes From Stars at Arbitrary Optical Depths a New Approach to the Problem

1998 ◽  
Vol 11 (1) ◽  
pp. 381-381
Author(s):  
A.V. Dorodnitsyn
Keyword(s):  
Differential Equations ◽  
Massive Star ◽  
System Of Differential Equations ◽  
Continuous Transition ◽  
Satisfactory Description ◽  
Sonic Point ◽  
New Approach ◽  
Star Evolution ◽  
Matter Flux ◽  
Massive Star Evolution

We have considered a stationary outflowing envelope accelerated by the radiative force in arbitrary optical depth case. Introduced approximations provide satisfactory description of the behavior of the matter flux with partially separated radiation at arbitrary optical depths. The obtained systemof differential equations provides a continuous transition of the solution between optically thin and optically thick regions. We analytically derivedapproximate representation of the solution at the vicinity of the sonic point. Using this representation we numerically integrate the system of equations from the critical point to the infinity. Matching the boundary conditions we obtain solutions describing the problem system of differential equations. The theoretical approach advanced in this work could be useful for self-consistent simulations of massive star evolution with mass loss.

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