Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Dynamic Stochastic Model of Flow Mixer

1994 ◽  
Vol 59 (7) ◽  
pp. 1551-1570
Author(s):  
Vladimír Kudrna ◽  
Libor Vejmola ◽  
Pavel Hasal
Keyword(s):  
Time Distribution ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Stochastic Diffusion ◽  
One Dimensional ◽  
Simplifying Assumptions ◽  
Dynamic Stochastic ◽  
The Common ◽  
The One

The one-dimensional stochastic diffusion model of a continuous flow mixer is proposed incorporating (contrary to commonly used diffusion models) a distribution of velocities of diffusing particles. Simplifying assumptions enabled us to derive an analytical expression for the liquid residence time distribution and concentration profile inside the mixer. For extreme values of parameters, the model becomes identical with the common idealized models usually adopted in chemical engineering.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Stochastic Model of Non-Isothermal Flow Chemical Reactor

1996 ◽  
Vol 61 (2) ◽  
pp. 242-258 ◽  
Author(s):  
Vladimír Kudrna ◽  
Libor Vejmola ◽  
Pavel Hasal
Keyword(s):  
Stochastic Model ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Dispersion Model ◽  
Chemical Reactor ◽  
Segregation Index ◽  
One Dimensional ◽  
A Value ◽  
Isothermal Case ◽  
Flow Through

Recently developed stochastic model of a one-dimensional flow-through chemical reactor is extended in this paper also to the non-isothermal case. The model enables the evaluation of concentration and temperature profiles along the reactor. The results are compared with the commonly used one-dimensional dispersion model with Danckwerts' boundary conditions. The stochastic model also enables to evaluate a value of the segregation index.

Numerical investigation on a melt-blowing die with internal stabilizers

2019 ◽  
pp. 152808371986693 ◽  
Author(s):  
Changchun Ji ◽  
Yudong Wang ◽  
Yafeng Sun
Keyword(s):  
Average Velocity ◽  
Fiber Diameter ◽  
Measurement Data ◽  
Melt Blowing ◽  
One Dimensional ◽  
Temperature Decay ◽  
Slot Die ◽  
The Common ◽  
The One ◽  
Peak Value

In order to decrease the fiber diameter and reduce the energy consumption in the melt-blowing process, a new slot die with internal stabilizers was designed. Using computational fluid dynamics technology, the new slot die was investigated. In the numerical simulation, the calculation data were validated with the laboratory measurement data. This work shows that the new slot die could increase the average velocity on the centerline of the air-flow field by 6.9%, compared with the common slot die. Simultaneously, the new slot die could decrease the back-flow velocity and the rate of temperature decay in the region close to the die head. The new slot die could reduce the peak value of the turbulent kinetic energy and make the fiber movements more gradual. With the one-dimensional drawing model, it proves that the new slot die has more edge on the decrease of fiber diameter than the common slot die.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

scholarly journals The development of biofilm architecture

2016 ◽  
Vol 472 (2188) ◽  
pp. 20150798 ◽  
Author(s):  
A. C. Fowler ◽  
T. M. Kyrke-Smith ◽  
H. F. Winstanley
Keyword(s):  
Bacterial Biofilm ◽  
Governing Equations ◽  
Biofilm Growth ◽  
One Dimensional ◽  
Direct Numerical Solution ◽  
The Common ◽  
Common Observation ◽  
The Stability ◽  
The One ◽  
A Free Boundary Problem

We extend the one-dimensional polymer solution theory of bacterial biofilm growth described by Winstanley et al . (2011 Proc. R. Soc. A 467 , 1449–1467 ( doi:10.1098/rspa.2010.0327 )) to deal with the problem of the growth of a patch of biofilm in more than one lateral dimension. The extension is non-trivial, as it requires consideration of the rheology of the polymer phase. We use a novel asymptotic technique to reduce the model to a free-boundary problem governed by the equations of Stokes flow with non-standard boundary conditions. We then consider the stability of laterally uniform biofilm growth, and show that the model predicts spatial instability; this is confirmed by a direct numerical solution of the governing equations. The instability results in cusp formation at the biofilm surface and provides an explanation for the common observation of patterned biofilm architectures.

scholarly journals Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Augusto Beléndez ◽  
Enrique Arribas ◽  
Tarsicio Beléndez ◽  
Carolina Pascual ◽  
Encarnación Gimeno ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Elliptic Integral ◽  
First Integral ◽  
Jacobi Elliptic Function ◽  
Complete Elliptic Integral ◽  
One Dimensional ◽  
The Common ◽  
The One ◽  
Two Parameters

Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at x=0 are also considered.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Formulation of Boundary Conditions in Flow Systems

1994 ◽  
Vol 59 (2) ◽  
pp. 345-358
Author(s):  
Vladimír Kudrna ◽  
Pavel Hasal ◽  
Libor Vejmola
Keyword(s):  
Boundary Conditions ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Dispersion Model ◽  
Point Of View ◽  
General Point ◽  
Engineering Systems ◽  
Boundary Conditions For Diffusion ◽  
Flow Through ◽  
The One

Problems associated with the formulation of the boundary conditions for diffusion equations describing flow-through chemical-engineering systems from the point of view of stochastic process theory are discussed. An approach to modelling such systems is presented, allowing the one-dimensional diffusion (dispersion) model of a continuous flow mixer, commonly used in chemical engineering, to be reassessed from a rather general point of view.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Stochastic Model of Isothermal Continuous Flow Chemical Reactor

1994 ◽  
Vol 59 (8) ◽  
pp. 1772-1787
Author(s):  
Vladimír Kudrna ◽  
Libor Vejmola ◽  
Pavel Hasal
Keyword(s):  
Continuous Flow ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Chemical Reactor ◽  
Point Of View ◽  
General Point ◽  
Stochastic Description ◽  
One Dimensional ◽  
Reactor Operating ◽  
Parameter Values

A model of an isothermal one-dimensional continuous flow chemical reactor operating at the steady state was derived using a stochastic description of motion of the reacting molecules. The model enables evaluation of the conversion of the reacting components. At the limiting parameter values the model yields results identical to those of the simplified models conventionally used in chemical reactor engineering. The model also enables the applicability of Danckwerts' boundary conditions to be assessed from a more general point of view.

Coherence and Continuity: A Study in Comparative Codification

1987 ◽  
Vol 22 (2) ◽  
pp. 184-218 ◽  
Author(s):  
Celia Wasserstein Fassberg
Keyword(s):  
Common Law ◽  
Characteristic Mode ◽  
Historical Process ◽  
One Dimensional ◽  
Conceptual Coherence ◽  
Legal Provisions ◽  
The Common ◽  
Internal Coherence ◽  
The One ◽  
Linguistic Fact

The English common law is frequently referred to as a seamless web; continental lawyers tend rather to think of law in terms of internal coherence and consistency. This is not merely a linguistic fact, and the terms are not simply interchangeable. Each reflects the characteristic mode of thought and of development in its respective system: the common law constantly and gradually emerging as a cumulative historical process; continental law stemming from, and in every case ultimately resting on interpretation of, codes, the product of a moment in history. Thus, although they are both capable of denoting the same idea of wholeness, each term has a slightly different connotative emphasis, the one stressing historical coherence and the other emphasising conceptual coherence.This is but one example of the proposition that institutions can not be imported wholesale, that foreign legal provisions, and terms of thought and reference, have to be evaluated beyond their immediate superficial appearance before they may be adopted or used as measures for local purposes. All such institutions have both a historical and a contextual significance which makes comparison on the level of one-dimensional questions such as, “Which is the better rule or the more attractive term?” meaningless.

scholarly journals BIC in waveguide arrays

2021 ◽  
Vol 255 ◽  
pp. 07001
Author(s):  
Vladimír Kuzmiak ◽  
Jiří Petráček
Keyword(s):  
Bound State ◽  
Coupled Mode Theory ◽  
One Dimensional ◽  
Waveguide Arrays ◽  
Coupled Mode ◽  
Simple Theoretical Model ◽  
The Common ◽  
The One ◽  
Symmetry Protected ◽  
The Continuum

We propose a simple theoretical model based on the coupled-mode theory which allows to calculate the spectral properties and transmittance of the one-dimensional waveguide structures. The model was verified on the common coupled-waveguide array in which the existence of the symmetry-protected bound state in the continuum (BIC) was confirmed experimentally by Plotnik et al. [Phys. Rev. Lett. 107, 28-31 (2011)]. The method can be extended to topologically nontrivial lattices to explore the properties of the BICs protected by time-reversal symmetry.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Numerical Simulation of One-Dimensional Diffusion Process

1996 ◽  
Vol 61 (4) ◽  
pp. 512-535 ◽  
Author(s):  
Pavel Hasal ◽  
Vladimír Kudrna
Keyword(s):  
Numerical Simulation ◽  
Diffusion Coefficient ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Stochastic Integral ◽  
Random Motion ◽  
One Dimensional ◽  
Dimensional Diffusion

Some problems are analyzed arising when a numerical simulation of a random motion of a large ensemble of diffusing particles is used to approximate the solution of a one-dimensional diffusion equation. The particle motion is described by means of a stochastic differential equation. The problems emerging especially when the diffusion coefficient is a function of spatial coordinate are discussed. The possibility of simulation of various kinds of stochastic integral is demonstrated. It is shown that the application of standard numerical procedures commonly adopted for ordinary differential equations may lead to erroneous results when used for solution of stochastic differential equations. General conclusions are verified by numerical solution of three stochastic differential equations with different forms of the diffusion coefficient.

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