Diffusion and surface reaction in porous cubical catalyst: A mathematical approach

2021 ◽  
Vol 15 ◽  
Author(s):  
Sourav Ghosh ◽  
Dipankar Pal
Keyword(s):  
Differential Equation ◽  
Small Volume ◽  
Three Dimensional ◽  
Steady State Condition ◽  
Thiele Modulus ◽  
First Order ◽  
Diffusion Limited ◽  
Catalyst Pellet ◽  
Rate Of Reaction ◽  
Physical Boundary

Background: Catalysts are the most vital part of any chemical industry. Catalyst is a substance that affects the rate of reaction, but the catalyst itself does not take part in the reaction. Catalysts offer different pathways of reaction by diffusing the reactant inside it to provide a large surface area within a small volume, thus, lowering the activation energy of molecules for reaction. Most of the catalytic reactions take place in liquid-solid or gas-solid interface where catalysts are mostly porous in nature. Spherical and cubic-shaped catalyst particles are commonly used in different industries. Methods: In the first phase of the present study, the physics behind the diffusion inside the catalyst pellet has been discussed. In the second part, governing differential equations have been established at a steady-state condition. For solving the differential equation, the equation is made dimensionless. Physical boundary conditions were used to solve the diffusion equation. The assumption of writing the differential equation of the reaction is elementary. Then the Thiele modulus is derived in terms of the reaction and geometrical parameter (Length) Results and Conclusion: In the third part, the differential equation is solved for first-order reaction with some constant values of the Thiele modulus and three-dimensional plots are obtained using numerical analysis. After that, the obtained Thiele modulus and effectiveness factor plot are compared to draw the conclusion of reaction rate limited and internal diffusion limited.

A Fast and Reliable Dynamic Tyre-Road Contact Model for ABS Braking Manoeuvre Simulations

2006 ◽  
Author(s):  
Francesco Braghin ◽  
Federico Cheli ◽  
Emiliano Giangiulio ◽  
Federico Mancosu ◽  
Daniele Arosio
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Order Differential Equation ◽  
Transient Behavior ◽  
Slip Angle ◽  
Relaxation Length ◽  
Steady State Condition ◽  
Tyre Model ◽  
First Order ◽  
Significant Parameter

Due to the dimensions of the tyre-road contact area, transients in a tyre last approximately 0.1s. Thus, in the case of abrupt maneuvers such as ABS braking, the use of a steady-state tyre model to predict the vehicle’s behavior would lead to significant errors. Available dynamic tyre models, such as Pacejka’s MF-Tyre model, are based on steady-state formulations and the transient behavior of the tyre is included by introducing a first order differential equation of relevant quantities such as the slip angle and the slippage. In these differential equations the most significant parameter used to describe the transient behavior is the so-called relaxation length, i.e. the distance traveled by the tyre to settle to a new steady–state condition once perturbated. Usually this parameter is assumed to be constant.

scholarly journals CONSEQUENCES OF THE FUNDAMENTAL CONJECTURE FOR THE MOTION ON THE SIEGEL–JACOBI DISK

2012 ◽  
Vol 10 (01) ◽  
pp. 1250076 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Three Dimensional ◽  
Quantum Evolution ◽  
Siegel Disk ◽  
Classical Motion ◽  
First Order ◽  
Jacobi Group ◽  
Upper Half Plane ◽  
The One

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain [Formula: see text] as the sum of the Kähler two-form on ℂ and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a linear Hamiltonian in the generators of the Jacobi group [Formula: see text] is described by a Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂ, where H1 denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space [Formula: see text], where [Formula: see text] denotes the Siegel upper half-plane.

scholarly journals DEFORMATIONS OF SURFACES FROM STATIONARY RICCI TENSOR

2020 ◽  
Vol 2 (2) ◽  
pp. 51-62
Author(s):  
T Podousova ◽  
◽  
N Vashpanova ◽  
Keyword(s):  
Differential Equation ◽  
Ricci Tensor ◽  
Three Dimensional ◽  
Small Degree ◽  
Dimensional Euclidean Space ◽  
Tensor Fields ◽  
Regular Surface ◽  
First Order ◽  
Order Deformation ◽  
Nonzero Mean

In this paper, we consider infinitesimal (n. m.) first-order deformations of single-connected regular surfaces in three-dimensional Euclidean space. The search for the vector field of this deformation is generally reduced to the study and solution of a system of four equations (among them there are differential equations) with respect to seven unknown functions. To avoid uncertainty, the following restriction is imposed on a given surface: the Ricci tensor is stored (mainly) on the surface. A mathematical model of the problem is created: a system of seven equations with respect to seven unknown functions. Its mechanical content is established. It is shown that each solution of the obtained system of equations will determine the field of displacement n. m. deformation of the first order of the surface of nonzero Gaussian curvature, which will be an unambiguous function (up to a constant vector). It is proved that each regular surface of nonzero Gaussian and mean curvatures allows first-order n. m. deformation with a stationary Ricci tensor. The tensor fields are found explicitly and depend on two functions, which are the solution of a linear inhomogeneous second-order differential equation with partial derivatives. The class of rigid surfaces in relation to the specified n. m. deformations. Assuming that one of the functions is predetermined, the obtained differential equation in the General case will be a inhomogeneous differential Weingarten equation, and an equation of elliptical type. The geometric and mechanical meaning of the function that is the solution of this equation is found. The following result was obtained: any surface of positive Gaussian and nonzero mean curvatures admits n. m of first-order deformation with a stationary Ricci tensor in the region of a rather small degree. Tensor fields will be represented by a predefined function and some arbitrary regular functions. Considering the Dirichlet problem, it is proved that the simply connected regular surface of a positive Gaussian and nonzero mean curvatures under a certain boundary condition admits a single first-order deformation with a stationary Ricci tensor. The strain tensors are uniquely defined.

The CPML for Hybrid Implicit-Explicit FDTD Method Based on Auxiliary Differential Equation

2014 ◽  
Vol 989-994 ◽  
pp. 1869-1872 ◽  
Author(s):  
Yun Fei Mao ◽  
Pu Hua Huang ◽  
Li Guo Ma
Keyword(s):  
Differential Equation ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Fdtd Method ◽  
Three Dimensional ◽  
Complex Frequency ◽  
Numerical Examples ◽  
First Order ◽  
Contour Plots ◽  
Difference Time

In this paper, an implementation of the complex-frequency-shifted perfectly matched layer (CPML) is developed for three-dimensional hybrid implicit-explicit (HIE) finite-difference time-domain (FDTD) method based on auxiliary differential equation (ADE). Because of the use of the ADE technique, this method becomes more straightforward and easier to implement. The formulations for the HIE-FDTD CPML are proposed. Numerical examples are given to verify the validity of the presented method. Results show that, both HIE-CPML and FDTD-CPML have almost the same reflection error, while their reflection error is about 30 dB, which is less than HIE Mur’s first-order results. The contour plots indicate that the maximum relative reflection as low as-72 dB is achieved by selecting and .

scholarly journals Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 354
Author(s):  
Zouhair Diab ◽  
Juan L. G. Guirao ◽  
Juan A. Vera
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Differential System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Periodic Trajectories ◽  
First Order ◽  
Three Dimensional Space

The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive.

scholarly journals Superconformal boundaries in 4 − ϵ dimensions

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Aleix Gimenez-Grau ◽  
Pedro Liendo ◽  
Philine van Vliet
Keyword(s):  
Three Dimensional ◽  
Spacetime Dimension ◽  
Perfect Agreement ◽  
Bootstrap Analysis ◽  
First Order ◽  
Free Theory ◽  
First Order Correction ◽  
Free Parameters ◽  
Superconformal Theories ◽  
Zumino Model

Abstract Boundaries in three-dimensional $$ \mathcal{N} $$ N = 2 superconformal theories may preserve one half of the original bulk supersymmetry. There are two possibilities which are characterized by the chirality of the leftover supercharges. Depending on the choice, the remaining 2d boundary algebra exhibits $$ \mathcal{N} $$ N = (0, 2) or $$ \mathcal{N} $$ N = (1) supersymmetry. In this work we focus on correlation functions of chiral fields for both types of supersymmetric boundaries. We study a host of correlators using superspace techniques and calculate superconformal blocks for two- and three-point functions. For $$ \mathcal{N} $$ N = (1) supersymmetry, some of our results can be analytically continued in the spacetime dimension while keeping the codimension fixed. This opens the door for a bootstrap analysis of the ϵ-expansion in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we prove that in the free theory limit two-point functions of chiral (and antichiral) fields are unique. The first order correction, which already describes interactions, is universal up to two free parameters. As a check of our analysis, we study the Wess-Zumino model with a super-symmetric boundary using Feynman diagrams, and find perfect agreement between the perturbative and bootstrap results.

scholarly journals An exact expression of positive periodic solution for a first-order singular equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yun Xin ◽  
Xiaoxiao Cui ◽  
Jie Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bounds ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Positive Periodic Solution ◽  
Exact Expression ◽  
Upper And Lower Bounds ◽  
Singular Equation ◽  
First Order

Abstract The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

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