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.

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.

Equilibrium and diffusion equations using cylindroidal coordinates

2002 ◽  
Vol 216 (1) ◽  
pp. 61-72 ◽  
Author(s):  
D B Dooner
Keyword(s):  
Diffusion Equation ◽  
Equilibrium Equation ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Spherical Coordinates ◽  
Cartesian Coordinates ◽  
And Diffusion

Presented is a system of curvilinear coordinates based on a cylindroid. Cartesian, cylindrical and spherical coordinates are special scenarios that emerge from a system of cylindroidal coordinates. A system of cylindroidal coordinates was originally proposed to parameterize toothed bodies or generalized hyperboloidal gear elements, and consequently certain fundamental relations for conjugate hyperboloidal pitch surfaces in mesh are presented. Two applications of cylindroidal coordinates are presented. The first application addresses the equilibrium equation and the second application addresses the diffusion equation for a differential cylindroidal element. It is demonstrated that the developed diffusion equation degenerates into established diffusion relations for cylindrical, spherical and Cartesian coordinates.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Modelling of processes by means of stochastic differential equations

1988 ◽  
Vol 53 (7) ◽  
pp. 1500-1518 ◽  
Author(s):  
Vladimír Kudrna
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Transport Equations ◽  
Diffusion Equations ◽  
Stochastic Diffusion ◽  
Direct Use

The paper points at certain problems associated with direct use of stochastic differential equations for description of chemical engineering processes or with the use of corresponding diffusion equations. It is shown that on the basis of various definitions one can write down three types of stochastic differential equations which might, in principle, describe the same process. One of these types is at the same time equivalent to the classic transport equations common in chemical engineering. A method is described removing these inconsistencies.

Certain Problems with the Application of Stochastic Diffusion Processes for the Description of Chemical Engineering Phenomena. Relations between Different Types of Diffusion Equations

1992 ◽  
Vol 57 (6) ◽  
pp. 1248-1261 ◽  
Author(s):  
Vladimír Kudrna ◽  
Daniel Turzík
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Stochastic Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Point Of View ◽  
Diffusion Equations ◽  
Complex Diffusion ◽  
Different Types ◽  
Individual Particles

The dependence is discussed between the "clasical" diffusion equation commonly used in chemical engineering and the stochastic differential equations which describe this diffusion from the point of view of micromotion of individual particles. The resulting equations can be useful above all for the modelling of more complex diffusion processes.

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 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.

Non-standard finite-differences schemes for reaction–diffusion equations in curvilinear coordinates

2009 ◽  
Vol 33 (1) ◽  
pp. 277-286 ◽  
Author(s):  
Jose Alvarez-Ramirez ◽  
Francisco J. Valdes-Parada ◽  
Jesus Alvarez ◽  
J. Alberto Ochoa-Tapia
Keyword(s):  
Finite Differences ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Curvilinear Coordinates ◽  
Diffusion Equations

scholarly journals MOCVD Equipment for Recent Developments towards the Blue and Green Solid State Laser

1996 ◽  
Vol 1 ◽  
Author(s):  
H. Jürgensen ◽  
D. Schmitz ◽  
G. Strauch ◽  
E. Woelk ◽  
M. Dauelsberg ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Growth Rate ◽  
Mass Transport ◽  
Stokes Equations ◽  
Chemical Species ◽  
Heterogeneous Reactions ◽  
Radiative Properties ◽  
Coupled Flow ◽  
Device Structure ◽  
Group Iii

For the growth of an electrically pumped lasing nitride emitter, the development of the MOCVD equipment and the process are mutually dependent. Most important is the implementation of the rapid temperature changes that are required between the growth of the different layers of a device structure. Equally important is to provide a reaction chamber that develops a stable gas phase at all growth temperatures used in the process. In this paper we will give insight in the technology and the relationship between processes and equipment. The development of the reation chamber was supported by mathematical modeling that formed the basis for the selection of appropriate process parameters for growth of group-III nitrides. The modeling consists of the numerical solution of the Navier-Stokes equations coupled with heat transfer and mass transport of the chemical species. The modeling of radiative heat transfer takes into account the effect of changing surface radiative properties. These changes result from the coating of the reactor inner surfaces during the growth run. Coupled flow dynamics and chemistry including homogeneous and heterogeneous reactions play an important role for predicting growth rate distributions on the susceptor area. At the practically used high temperatures, group-III metalorganics turn out to be almost entirely decomposed and it is the mass transport of these decomposition products to the growing layer that is assumed to control the growth rate in accordance with experimental observations.AIXTRON GmbH

Curvilinear Coordinates and Physical Components: An Application to the Problem of Viscous Flow and Heat Transfer in Smoothly Curved Ducts

1996 ◽  
Vol 63 (4) ◽  
pp. 985-989 ◽  
Author(s):  
C. J. Bolinder
Keyword(s):  
Heat Transfer ◽  
Viscous Flow ◽  
Time Derivative ◽  
Curvilinear Coordinates ◽  
Flow And Heat Transfer ◽  
Curved Ducts ◽  
Physical Components

Expressions are derived for the gradient, divergence, Laplacian, curl, and material time derivative in terms of general curvilinear coordinates using physical components of all vector quantities. The results are conveniently expressed in terms of two new coefficients, involving physical and reciprocal base vectors. An application to the problem of viscous flow and heat transfer in arbitrarily smoothly curved ducts is presented. In particular, helical ducts are considered.

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