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.

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.

Stochastic model of the nonideal flow mixer

1979 ◽  
Vol 44 (4) ◽  
pp. 1094-1115
Author(s):  
Vladimír Kudrna
Keyword(s):  
Stochastic Model ◽  
Chemical Engineering ◽  
Dispersion Model ◽  
Model Based ◽  
Proposed Model

The flow mixer is described by use of the unidimensional stochastic model based on the earlier published general approach. It is demonstrated that on basis of the proposed model two individual models result which are most frequently used in chemical engineering for description of mixers and reactors: the dispersion model and model of the cascade of ideal mixers. On the basis of the model proposed it is also possible to derive the usual relation for calculation of the reactor conversion at macroflow.

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.

One-Dimensional Steady Inviscid Flow Through a Stenotic Collapsible Tube

1990 ◽  
Vol 112 (4) ◽  
pp. 444-450 ◽  
Author(s):  
David N. Ku ◽  
Marvin N. Zeigler ◽  
J. Micah Downing
Keyword(s):  
Flow Rate ◽  
Inviscid Flow ◽  
One Dimensional ◽  
Constant Stiffness ◽  
Coupled Equations ◽  
Collapsible Tubes ◽  
A Value ◽  
Clinical Consequences ◽  
Flow Through ◽  
Stenotic Arteries

A one-dimensional inviscid solution for flow through a compliant tube with a stenosis is presented. The model is used to represent an artery with an atherosclerotic plaque and to investigate a range of conditions for which arterial collapse may occur. The coupled equations for flow through collapsible tubes are solved using a Runge-Kutta finite difference scheme. Quantitative results are given for specific physiological parameters including inlet and outlet pressure, flow rate, stenosis size, length and stiffness. The results suggest that high-grade stenotic arteries may exhibit collapse with typical physiological pressures. Critical stenoses may cause choking of flow at the throat followed by a transition to supercritical flow with tube collapse downstream. Greater amounts of stenosis produced a linear reduction of flow rate and a shortening of the collapsed region. Changes in stenosis length created proportional changes in the length of collapse. Increasing the stiffness of the stenosis to a value greater than the nominal tube stiffness caused a greater amount of flow limitation and more negative pressures, compared to a stenosis with constant stiffness. These findings assist in understanding the clinical consequences of flow through atherosclerotic arteries.

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 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.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Diffusional Change of Solid Particle Size

1996 ◽  
Vol 61 (4) ◽  
pp. 536-563
Author(s):  
Vladimír Kudrna ◽  
Pavel Hasal
Keyword(s):  
Particle Size ◽  
Solid Particle ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Granular Systems ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Diffusion Term ◽  
Population Balances ◽  
And Diffusion

To the description of changes of solid particle size in population, the application was proposed of stochastic differential equations and diffusion equations adequate to them making it possible to express the development of these populations in time. Particular relations were derived for some particle size distributions in flow and batch equipments. It was shown that it is expedient to complement the population balances often used for the description of granular systems by a "diffusion" term making it possible to express the effects of random influences in the growth process and/or particle diminution.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Diffusion equations in curvilinear coordinates

1991 ◽  
Vol 56 (3) ◽  
pp. 602-618
Author(s):  
Vladimír Kudrna
Keyword(s):  
Heat Transfer ◽  
Mass Transport ◽  
Probability Density ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Spherical Coordinates ◽  
Transformation Procedure

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

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