On the effect of axial diffusion on the residence time distributions in an isothermal one-dimensional reactor where a first order reversible or irreversible reaction takes place

1974 ◽  
Vol 29 (1) ◽  
pp. 91-96 ◽  
Author(s):  
Á. Pethö ◽  
K. Schügerl
Keyword(s):  
Residence Time ◽  
Axial Diffusion ◽  
One Dimensional ◽  
Irreversible Reaction ◽  
First Order ◽  
Residence Time Distributions

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

scholarly journals Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

2021 ◽  
Author(s):  
Amarjot Singh Bhullar ◽  
Gospel Ezekiel Stewart ◽  
Robert W. Zimmerman
Keyword(s):  
Approximate Solution ◽  
Pore Pressure ◽  
Constant Pressure ◽  
Initial Permeability ◽  
Order Perturbation ◽  
Zeroth Order ◽  
One Dimensional ◽  
First Order ◽  
Outflow Boundary ◽  
Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

scholarly journals NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS

2010 ◽  
Vol 12 (01) ◽  
pp. 85-106 ◽  
Author(s):  
S. N. ANTONTSEV ◽  
J. I. DÍAZ
Keyword(s):  
Type Equation ◽  
Parabolic Type ◽  
Parabolic Systems ◽  
Diffusion Term ◽  
One Dimensional ◽  
First Order ◽  
Estimates Of Solutions ◽  
Gradient Type ◽  
Degenerate Type ◽  
Parabolic Type Equation

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

Residence time distributions in SX–EW equipment

1998 ◽  
Vol 54 (3-4) ◽  
pp. 235-242 ◽  
Author(s):  
Hossein Aminian ◽  
Claude Bazin ◽  
Daniel Hodouin
Keyword(s):  
Residence Time ◽  
Residence Time Distributions

Spectrochemical Behavior of Carcinogenic Polycyclic Aromatic Hydrocarbons in Biological Systems. Part II: A Theoretical Rate Model for BaP Metabolism in Living Cells

1996 ◽  
Vol 50 (11) ◽  
pp. 1352-1359 ◽  
Author(s):  
Ping Chiang ◽  
Kuang-Pang Li ◽  
Tong-Ming Hseu
Keyword(s):  
Rate Constant ◽  
Living Cells ◽  
Rate Model ◽  
Number Density ◽  
Order Process ◽  
Irreversible Reaction ◽  
First Order ◽  
Metabolism Rate ◽  
Polycyclic Aromatic ◽  
Kinetics Of

An idealized model for the kinetics of benzo[ a]pyrene (BaP) metabolism is established. As observed from experimental results, the BaP transfer from microcrystals to the cell membrane is definitely a first-order process. The rate constant of this process is signified as k1. We describe the surface–midplane exchange as reversible and use rate constants k2 and k3 to describe the inward and outward diffusions, respectively. The metabolism is identified as an irreversible reaction with a rate constant k4. If k2 and k3 are assumed to be fast and not rate determining, the effect of the metabolism rate, k4, on the number density of BaP in the midplane of the microsomal membrane, m3, can be estimated. If the metabolism rate is faster than or comparable to the distribution rates, k2 and k3, the BaP concentration in the membrane midplane, m3, will quickly be dissipated. But if k4 is extremely small, m3 will reach a plateau. Under conditions when k2 and k3 also play significant roles in determining the overall rate, more complicated patterns of m3 are expected.

System Properties of One-Dimensional Distributed Systems

1977 ◽  
Vol 99 (2) ◽  
pp. 85-90 ◽  
Author(s):  
L. S. Bonderson
Keyword(s):  
Distributed Systems ◽  
Sufficient Conditions ◽  
State Variables ◽  
One Dimensional ◽  
First Order ◽  
Lumped Element ◽  
Order Form ◽  
Power Product ◽  
System Properties ◽  
Necessary And Sufficient

The system properties of passivity, losslessness, and reciprocity are defined and their necessary and sufficient conditions are derived for a class of linear one-dimensional multipower distributed systems. The utilization of power product pairs as state variables and the representation of the dynamics in first-order form allows results completely analogous to those for lumped-element systems.

Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Swati Chauhan ◽  
Antim Chauhan ◽  
Rajan Arora
Keyword(s):  
Shock Waves ◽  
Mass Fraction ◽  
Strong Shock ◽  
Dust Particles ◽  
Excluded Volume ◽  
One Dimensional ◽  
First Order ◽  
Nonideal Gas ◽  
Spherical Shock ◽  
Strong Shock Waves

Abstract In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust particle to the density of the mixture and van der Waals excluded volume.

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