scholarly journals An Exact Solution of Boltzmann's Equation for a Rigid Sphere Gas

1968 ◽  
Vol 21 (5) ◽  
pp. 543 ◽  
Author(s):  
PI Brooker ◽  
HS Green
Keyword(s):  
Transport Coefficients ◽  
Velocity Distribution Function ◽  
Zero Order ◽  
Rigid Sphere ◽  
Stream Velocity ◽  
Approximation Techniques ◽  
The Third ◽  
Boltzmann's Equation ◽  
Good Agreement ◽  
Boltzmann’S Equation

The second approximation to Boltzmann's equation, in Chapman and Cowling's form, is solved exactly for a rigid sphere gas by using appropriate transformations to reduce the integral equation to a differential equation, which is solved numerically. The values of the transport coefficients, calculated directly from this solution for the velocity distribution function, are in good agreement with those obtained from the usual approximation techniques. A solution for part of the third approximation, with a term of zero order in V, the velocity of a molecule relative to the stream velocity, is also obtained analytically.

scholarly journals Comment on FTI Method and Transport Coefficient Definitions for Charged Particle Swarms in Gases

1995 ◽  
Vol 48 (4) ◽  
pp. 677 ◽  
Author(s):  
RE Robson
Keyword(s):  
Charged Particle ◽  
Path Integral ◽  
Transport Coefficients ◽  
Space Distribution ◽  
Particle Swarms ◽  
Integral Methods ◽  
Time Integral ◽  
Definition Of ◽  
Boltzmann's Equation ◽  
Boltzmann’S Equation

The kinetic theory of charged particle swarms in gases is based upon solution of the space and time dependent Boltzmann's equation for the phase space distribution function f(r, c, t). Hydrodynamic transport coefficients are defined in connection with a density gradient expansion (DGE) of f(r, c, t) and it is believed that these are the quantities measured in experiment. On the other hand, Ikuta and coworkers start with the spatially independent form of the Boltzmann equation, which they solve iteratively as in path-integral methods, and define transport coefficients in terms of the 'starting rate distribution', rather than f itself. Ikuta's procedure has come to be known as the 'flight time integral' (FTI) method and the discrepancies between numerical calculations based upon this and the more commonly known DGE procedure have generated a deal of controversy in recent times. The purpose of this paper is to point out that the respective definitions of the transverse diffusion coefficient DT coincide only for light swarm particles undergoing collisions for which the differential cross section is isotropic, and that the particular technique used for solving Boltzmann's equation, be it a path-integral or a multi-term method, has nothing to do with the numerical discrepancies which are observed when scattering is anisotropic. In particular, it is shown that Ikuta's definition of DT is inconsistent with even the well established result for constant collision frequency.

scholarly journals An Investigation of the Accuracy of Numerical Solutions of Boltzmann's Equation for Electron Swarms in Gases with Large Inelastic Cross Sections

1979 ◽  
Vol 32 (3) ◽  
pp. 231 ◽  
Author(s):  
Ivan D Reid
Keyword(s):  
Monte Carlo Simulation ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Transport Coefficients ◽  
Distribution Functions ◽  
Monte Carlo Simulation Technique ◽  
Energy Distribution Functions ◽  
Boltzmann's Equation ◽  
Term Approximation ◽  
Boltzmann’S Equation

A Monte Carlo simulation technique has been used to test the accuracy of electron energy distribution functions and transport coefficients calculated using conventional numerical solutions of Boltzmann's equation based on a two-term approximation. The tests have been applied to a number of model gases, some of which have characteristics close to those of real gases, and include cases where the scattering is anisotropic. The results show that, in general, previous application of the numerical solution to real gases has been valid.

scholarly journals Generalizations of Boltzmann's Equation

1974 ◽  
Vol 27 (6) ◽  
pp. 773 ◽  
Author(s):  
GR Anstis
Keyword(s):  
Diffusion Coefficient ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
The Self ◽  
One Dimensional ◽  
Point Particles ◽  
Self Diffusion ◽  
Self Diffusion Coefficient ◽  
Boltzmann's Equation ◽  
Boltzmann’S Equation

A scheme is proposed for systematically generalizing Boltzmann's equation in order to describe the non-equilibrium behaviour of an arbitrarily dense gas. The method avoids the divergences that arise from considering the dynamics of groups of isolated particles by introducing appropriate damping terms. Transport coefficients are obtained from the kinetic equations by using the autocorrelation formulae. For a one-dimensional gas of impenetrable point particles, approximations to the coefficient of self-diffusion may be obtained readily from the proposed generalization. A first correction to Boltzmann's equation yields the self-diffusion coefficient to within 1 % of its exact value.

Spectroscopic Study of Photosensitized Oxidation of Polyisoprene

1977 ◽  
Vol 50 (4) ◽  
pp. 704-713 ◽  
Author(s):  
M. A. Golub ◽  
M. L. Rosenberg ◽  
R. V. Gemmer
Keyword(s):  
Zero Order ◽  
Double Bonds ◽  
Photosensitized Oxidation ◽  
The Third ◽  
Type Process ◽  
Zero Order Kinetics ◽  
Hydroperoxide Formation ◽  
Cis And Trans ◽  
In Cis ◽  
Suitable Measure

Abstract The microstructural changes which occur in cis- and trans-1,4-polyisoprenes and in squalene during photosensitized oxidation were investigated with the aid of infrared and proton and carbon-13 NMR spectroscopy. The singlet oxygenation of these isoprenic compounds resulted in allylic hydroperoxides with shifted double bonds, according to the expected “ene”-type process. In contrast to trans-1,4-polyisoprene and squalene, which displayed the three possible double bond shifts, cis-1,4-polyisoprene showed essentially two of the shifts (to di- and trisubstituted double bonds) and very little of the third (to exomethylene groups). A suitable measure of the extent of hydroperoxidation was afforded by the absorbance ratio, A3400/A1440≡A′. Similar correlations of A′ with oxygen uptake were obtained for the three isoprenic compounds, using chlorophyll or methylene blue as sensitizer. The use of rose bengal gave erratic results indicative of some autoxidation accompanying the hydroperoxide formation. The singlet oxygenation followed zero-order kinetics, the relative rates for cis- and trans-1,4-polyisoprenes being approximately 1.0:1.5.

New Developments for the Description of Oil Leakages by Advective Migration From Submarine Pipelines

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Renan Martins Baptista ◽  
Ricardo Antonio Francisco Machado ◽  
Marintho Bastos Quadri ◽  
Ariovaldo Bolzan ◽  
André Lourenço Nogueira ◽  
...  
Keyword(s):  
Horizontal Plane ◽  
Immiscible Displacement ◽  
Preliminary Evaluation ◽  
Leakage Detection ◽  
New Developments ◽  
Gravitational Flow ◽  
The Third ◽  
Alternative Approach ◽  
High Standards ◽  
Good Agreement

The significant growth in offshore operations increases the risk of a pipeline rupture, even considering the high standards of safety involved. Throughout a submarine leakage, four different amounts of oil may be accounted. The first one is the oil volume released until the leakage detection. The second one is the volume leaked throughout mitigation initiatives (e.g., pump shutdown and valve closure). The third parcel is the amount released by gravitational flow. Finally, the fourth and last amount of oil is released due to the water-oil entrainment, generally known as advective migration. Normally, a considerable amount of oil is released in this step. It begins just after the internal pipeline pressure becomes equal to the external one. The present work continues to introduce a mathematical alternative approach, based on the theories of perturbation and unstable immiscible displacement, to accurately estimate the leakage kinetics and the amount of oil released by the advective migration phenomenon. Situations considering different hole sizes and thicknesses were tested experimentally and through simulations. Additional experiments were accomplished using smooth and rough edge surfaces, besides different slopes (using the horizontal plane as reference). Those experiments permitted a preliminary evaluation of the importance of these factors. The results obtained with the model showed good agreement with the experimental data in many situations considered.

scholarly journals Electron Transport in Argon - Hydrogen Mixtures

1980 ◽  
Vol 33 (6) ◽  
pp. 975 ◽  
Author(s):  
GN Haddad ◽  
RW Crompton
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Cross Section ◽  
Transport Coefficients ◽  
Velocity Distribution Function ◽  
Significant Error ◽  
Cross Section Data ◽  
Section Data ◽  
Hydrogen Mixtures ◽  
Legendre Expansion

The transport coefficients υdr and D⊥/μ have been measured in mixtures of 0.5 % and 4 % hydrogen in argon. All measurements were made at 293 K. It is shown that for these mixtures the use of the solution of the Boltzmann equation based on the two-term Legendre expansion of the velocity distribution function introduces no significant error in the analysis of the transport data. All the experimental data have been predicted to within � 3.5 % using previously published cross section data.

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