scholarly journals Dead-Core Solutions to Simple Catalytic Reaction Problems in Chemical Engineering

2019 ◽  
pp. 29
Author(s):  
F. Sabit ◽  
M. Shakipov ◽  
P. Skrzypacz ◽  
B. Golman
Keyword(s):  
Catalytic Reaction ◽  
Chemical Engineering ◽  
A Priori ◽  
Approximate Solutions ◽  
Diffusion Reaction ◽  
Porous Space ◽  
Orthogonal Collocation ◽  
Reaction Term ◽  
Dead Core ◽  
The Dead

The catalytic chemical reaction is usually carried out in a pellet where the catalyst is distributed throughout its porous structure. The selectivity, yield and productivity of the catalytic reactor often depend on the rates of chemical reactions and the rates of diffusion of species involved in the reactions in the pellet porous space. In such systems, the fast reaction can lead to the consumption of reactants close to the external pellet surface and creation of the dead core where no reaction occurs. This will result in an inefficient use of expensive catalyst. In the discussed simplified diffusion-reaction problems a nonlinear reaction term is of power-law type with a small positive reaction exponent. Such reaction term represents the kinetics of catalytic reaction accompanied by a strong adsorption of the reactant. The ways to calculate the exact solutions possessing dead cores are presented. It was also proved analytically that the exact solution of the nonlinear two-point boundary value problem satisfies physical a-priori bounds. Furthermore, the approximate solutions were obtained using the orthogonal collocation method for pellets of planar, spherical and cylindrical geometries. Numerical results confirmed that the length of the dead core increases for the more active catalysts due to the larger values of the reaction rate constant. The dead core length also depends on the pellet geometry.

Surface waves on shear currents: solution of the boundary-value problem

1993 ◽  
Vol 252 ◽  
pp. 565-584 ◽  
Author(s):  
Victor I. Shrira
Keyword(s):  
Boundary Value Problem ◽  
Growth Rates ◽  
Wave Motion ◽  
Boundary Value ◽  
A Priori ◽  
Approximate Solutions ◽  
Short Wave ◽  
Capillary Waves ◽  
Potential Motion ◽  
Approximate Expressions

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

scholarly journals Spatial Profile of the Dead Core for the Fast Diffusion Equation with Dependent Coefficient

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Zhengce Zhang ◽  
Biao Wang
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Single Point ◽  
Fast Diffusion ◽  
Spatial Profile ◽  
Fast Diffusion Equation ◽  
Dead Core ◽  
Core Profile ◽  
The Dead ◽  
Core Problem

We consider the dead-core problem for the fast diffusion equation with spatially dependent coefficient and obtain precise estimates on the single-point final dead-core profile. The proofs rely on maximum principle and require much delicate computation.

Distribution of Modern Salt-marsh Foraminifera from the Eastern Mississippi Sound, U.s.a.

2019 ◽  
Vol 49 (1) ◽  
pp. 29-47 ◽  
Author(s):  
Christian Haller ◽  
Christopher G. Smith ◽  
Pamela Hallock ◽  
Albert C. Hine ◽  
Lisa E. Osterman ◽  
...  
Keyword(s):  
A Priori ◽  
Abundant Species ◽  
Environmental Data ◽  
Tidal Marshes ◽  
High Marsh ◽  
Mississippi Sound ◽  
Pascagoula River ◽  
Flooding Frequency ◽  
The Dead ◽  
The Relationship

Abstract This study documented surface distributions of live and dead foraminiferal assemblages in the low-gradient tidal marshes of the barrier island and estuarine complex of the eastern Mississippi Sound (Grand Bay, Pascagoula River, Fowl River, Dauphin Island). A total of 71,833 specimens representing 38 species were identified from a gradient of different elevation zones across the study area. We identified five live assemblages and nine biofacies for the dead assemblages from estuarine, low marsh, middle marsh, high marsh, and upland transition environments. Although dissolution of calcareous tests was observed in the dead assemblages, characteristic species and abundance patterns dependent on elevation in the intertidal zone were similar between living assemblages and dead biofacies. The assemblages from the eastern Mississippi Sound estuaries were dominated by Ammonia tepida, Cribroelphidium poeyanum, C. excavatum, and Paratrochammina simplissima. The low marshes were dominated by Ammotium salsum, Ammobaculites exiguus, and Miliammina fusca. The dominant species in the middle marshes was Arenoparrella mexicana. The most abundant species in the high marshes was Entzia macrescens. The upland–marsh transition zones were dominated by Trochamminita irregularis and Pseudothurammina limnetis. Canonical correspondence analysis was applied to assess the relationship between a priori defined biofacies and measured environmental data (elevation, grain size, organic matter, and salinity) to test the hypothesis that distribution of foraminiferal assemblages is driven by elevation and hence flooding frequency. Salinity was the second most important explanatory variable of dead assemblages. Riverine freshwater from the Pascagoula River markedly influenced the live and dead assemblages in the Pascagoula River marsh, which was represented by low diversity and densities and dominance by Ammoastuta inepta. The relationship between the measured environmental variables and assemblage distributions can be used in future Mississippi Sound paleo-environmental studies.

scholarly journals Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations

2004 ◽  
Vol 2004 (37) ◽  
pp. 1973-1996 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Wide Class ◽  
Noise Level ◽  
A Priori ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.

Smoothing operators for waveform tomographic imaging

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1646-1654 ◽  
Author(s):  
Steve Cardimona ◽  
Jan Garmany
Keyword(s):  
Fresnel Zone ◽  
A Priori ◽  
Approximate Solutions ◽  
Velocity Model ◽  
Forward Problem ◽  
Nonlinear Inversion ◽  
Approximation Solution ◽  
Waveform Data ◽  
Sensitivity Functions ◽  
Priori Information

The analytic kernel in the space‐time domain for the Frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the Born approximation solution to the integral equation of waveform scattering. Preconditioning operators in the solution of this forward problem, which may incorporate a priori information and approximate solutions, are smoothing operators in the imaging problem, the first iteration of a nonlinear inversion for the slowness model. Some preconditioning operators are determined for solutions to the parabolic wave equation, and then used to create new sensitivity functions that retain appropriate characteristics of the true Frechet kernel in forward calculations. The new sensitivity functions define near‐source, near‐receiver and far‐field kernels, as well as kernels that exhibit an amplitude decay off the ray yielding ray‐perpendicular sensitivity that scales with the Fresnel zone size. A sample calculation from a synthetic cross‐well imaging experiment shows the usefulness of introducing physically appropriate model smoothing directly into the sensitivity function of the forward problem, helping to obtain a geologically reasonable image of the velocity model when ray coverage is insufficient.

Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides

2019 ◽  
Vol 53 (4) ◽  
pp. 1191-1222 ◽  
Author(s):  
Seungil Kim
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Piecewise Linear ◽  
A Priori ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Two Parameters

In this paper, we study finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is defined in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an artificial boundary rather than a Dirichlet condition for cutoff modes that waveguides may allow. In the finite element analysis for the PML problem, we have to deal with two difficulties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic finite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cutoff mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that confirm the convergence analysis will be presented.

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