Correction of kinetic data in non-isothermal reactions with non-uniform temperatures: analytical treatments for spherical reactant masses

1981 ◽  
Vol 378 (1772) ◽  
pp. 27-60 ◽  
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Spherical Geometry ◽  
Close Relation ◽  
Infinite Cylinder ◽  
Central Temperature ◽  
Temperature Position ◽  
Special Cases ◽  
Satisfactory Precision ◽  
Endothermic Reactions

Exothermic reactions normally proceed non-isothermally, and to treat them quantitatively allowance must be made for variations in internal temperature. Except for the special cases of the infinite slab and infinite cylinder, this has so far had to be done solely by numerical computation. The sphere and other finite, three-dimensional geometries have eluded analytical solution, even in the stationary state. The present paper presents two useful treatments for spherical geometry. They provide families of compact analytical expressions of very satisfactory precision for all the important aspects, errors rarely being greater than 2% right up to criticality. The aspects studied include temperature-position profiles, central temperature-excesses and surface temperature-gradients. (Both stable and unstable subcritical conditions as well as critical cases can be treated.) From these there follow values for rate-constant correction-factors f (and their reciprocals, the effectiveness factors η) and, in turn, the means of correcting errors in Arrhenius activation energies and in reaction orders - again in expressive, simple forms. In addition, explicit equations have been set out for the first time to cover the whole range of boundary conditions (arbitrary Biot number, β ) from Frank-Kamenetskii ( β → ∞) to Semenov ( β ═ 0) extremes. This is a major development, since mere tabulation of numerical solutions has hitherto been a formidable task. Calculating procedures are detailed in an Appendix. Endothermic reactions show self-cooling and self-repression instead of self-stimulation. These aspects are all expressively encapsulated in the change of sign of single coefficients in tidy equations. The two routes - quintic approximation (q. a.) and second-order reversion (r2) - can both be applied most conveniently in parametric form (this is necessary for r2 but not for q. a). The parameters l and x that appear naturally in the two treatments bear a close relation to corresponding integration constants encountered in one- and two-dimensional geometry. Each has striking physical significance, l being most directly related to the effectiveness factor and x to the central temperature excess.

Direct Time Integration Methods for Structural Acoustics and Far Field Scattering Computations

1995 ◽  
Vol 117 (4) ◽  
pp. 488-492 ◽  
Author(s):  
J. P. Wright
Keyword(s):  
Numerical Solutions ◽  
Time Integration ◽  
Three Dimensional ◽  
Spherical Geometry ◽  
Far Field ◽  
Structural Acoustics ◽  
Explicit Integration ◽  
Orthogonal Function ◽  
Preliminary Validation ◽  
Time Integration Methods

Procedures are described for doing: (a) transient three dimensional structural acoustics computations, using a combination of modal and finite element techniques in space and explicit integration in time; (b) far field scattering computations, using orthogonal function expansions and the method of characteristics. These methods are discussed and preliminary validation of the approach is accomplished by comparing numerical solutions with available analytic results for some problems in spherical geometry.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sheng Yu-ming ◽  
Li Chao ◽  
Xia Ming-yao ◽  
Zou Jin-feng
Keyword(s):  
Finite Element ◽  
Principal Stress ◽  
Stress Condition ◽  
Strain Softening ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Strength Criterion ◽  
Surrounding Rock ◽  
Flow Rule ◽  
Finite Element Software

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

scholarly journals Application of temporary reinforcement technology in insufficient construction of the understructure bearing capacity of viaduct

2019 ◽  
Vol 136 ◽  
pp. 04080
Author(s):  
Guohui Cao ◽  
Reqiang Liu ◽  
Jing Liu ◽  
Xiang Gao ◽  
Peng Wang
Keyword(s):  
Bearing Capacity ◽  
Three Dimensional ◽  
Social Benefits ◽  
Construction Technology ◽  
Project Cost ◽  
Special Cases ◽  
Project Construction ◽  
Lower Structure ◽  
Traditional Construction

The complex three-dimensional traffic construction often occur when the lower structure cannot bear the construction load and other special cases, indicating the need for temporary reinforcement of the lower structure. In this paper, combined with a project construction example, various temporary reinforcement technologies are adopted to solve the insufficient bearing capacity during understructure construction, which poses a serious danger, to ensure synchronous construction of the understructure and viaduct. Compared with the traditional construction technology, the temporary reinforcement technology proposed in this paper features the advantages of saving project cost and time and has achieved better economic and social benefits.

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

1992 ◽  
Vol 237 ◽  
pp. 413-434 ◽  
Author(s):  
Jae Min Hyun ◽  
Jun Sang Park
Keyword(s):  
Mach Number ◽  
Compressible Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Direction ◽  
Ekman Layer ◽  
Infinite Cylinder ◽  
Full Time ◽  
Viscous Effects ◽  
Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

Preliminary results of numerical simulation of submarine landslide-generated waves

2021 ◽  
Author(s):  
Ramtin Sabeti ◽  
Mohammad Heidarzadeh
Keyword(s):  
Numerical Simulation ◽  
Numerical Modelling ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Source Region ◽  
Three Dimensional ◽  
Terminal Velocity ◽  
Sensitivity Analyses ◽  
Physical Experiments ◽  
Set Up

<p>Landslide-generated waves have been major threats to coastal areas and have led to destruction and casualties. Their importance is undisputed, most recently demonstrated by the 2018 Anak Krakatau tsunami, causing several hundred fatalities. The accurate prediction of the maximum initial amplitude of landslide waves (<em>η<sub>max</sub></em>) around the source region is a vital hazard indicator for coastal impact assessment. Laboratory experiments, analytical solutions and numerical modelling are three major methods to investigate the (<em>η<sub>max</sub></em>). However, the numerical modelling approach provides a more flexible and cost- and time-efficient tool. This research presents a numerical simulation of tsunamis due to rigid landslides with consideration of submerged conditions. In particular, this simulation focuses on studying the effect of landslide parameters on <em>η<sub>max</sub>.</em> Results of simulations are compared with our conducted physical experiments at the Brunel University London (UK) to validate the numerical model.</p><p>We employ the fully three-dimensional computational fluid dynamics package, FLOW-3D Hydro for modelling the landslide-generated waves. This software benefit from the Volume of Fluid Method (VOF) as the numerical technique for tracking and locating the free surface. The geometry of the simulation is set up according to the wave tank of physical experiments (i.e. 0.26 m wide, 0.50 m deep and 4.0 m). In order to calibrate the simulation model based on the laboratory measurements, the friction coefficient between solid block and incline is changed to 0.41; likewise, the terminal velocity of the landslide is set to 0.87 m/s. Good agreement between the numerical solutions and the experimental results is found. Sensitivity analyses of landslide parameters (e.g. slide volume, water depth, etc.) on <em>η<sub>max </sub></em>are performed. Dimensionless parameters are employed to study the sensitivity of the initial landslide waves to various landslide parameters.</p>

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Sofia Sarraf ◽  
Ezequiel López ◽  
Laura Battaglia ◽  
Gustavo Ríos Rodríguez ◽  
Jorge D'Elía
Keyword(s):  
Boundary Element Method ◽  
Linear System ◽  
Boundary Element ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Computational Time ◽  
Creeping Flows ◽  
Order Of Magnitude ◽  
Element Method

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

Taxonomic description and 3D modelling of a new species of myzostomid (Annelida, Myzostomida) associated with black corals from Madagascar

Zootaxa ◽  
2017 ◽  
Vol 4244 (2) ◽  
pp. 277 ◽  
Author(s):  
LUCAS TERRANA ◽  
IGOR EECKHAUT
Keyword(s):  
New Species ◽  
Phylogenetic Analyses ◽  
Three Dimensional ◽  
Close Relation ◽  
Ventral Side ◽  
Middle Part ◽  
The Body ◽  
Taxonomic Description ◽  
Black Corals ◽  
Organ Systems

Eenymeenymyzostoma nigrocorallium n. sp. is the first species of myzostomid worm associated with black corals to be described. Endoparasitic specimens of E. nigrocorallium were found associated with three species of antipatharians on the Great Reef of Toliara. Individuals inhabit the gastrovascular ducts of their hosts and evidence of infestation is, most of the time, not visible externally. Phylogenetic analyses based on 18S rDNA, 16S rDNA and COI data indicate a close relation to Eenymeenymyzostoma cirripedium, the only other species of the genus. The morphology of E. nigrocorallium is very unusual compared to that of the more conventional E. cirripedium. The new species has five pairs of extremely reduced parapodia located on the body margin and no introvert, cirri or lateral organs. Individuals are hermaphroditic, with the male and female gonads both being located dorsally in the trunk. It also has a highly developed parenchymo-muscular layer on the ventral side, and the digestive system lies in the middle part of the trunk. A three-dimensional digital model of this worm’s body plan has been constructed whereby the external morphology and in toto views of the observed organ systems (nervous, digestive and reproductive) can be viewed on-screen: http://doi.org/10.13140/RG.2.2.17911.21923. 

scholarly journals A Nonlinear Inverse Design Problem for a Pipe Type Heat Exchanger Equipped with Internal Z-Shape Lateral Fins and Ribs

Energies ◽  
2020 ◽  
Vol 13 (23) ◽  
pp. 6424
Author(s):  
Cheng-Hung Huang ◽  
Chih-Yang Kuo
Keyword(s):  
Heat Exchanger ◽  
Thermal Performance ◽  
Design Problem ◽  
Direct Problem ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Design Variables ◽  
Non Linear ◽  
Inverse Design Problem ◽  
Η Value

A non-linear three-dimensional inverse shape design problem was investigated for a pipe type heat exchanger to estimate the design variables of continuous lateral ribs on internal Z-shape lateral fins for maximum thermal performance factor η. The design variables were considered as the positions, heights, and number of ribs while the physical properties of air were considered as a polynomial function of temperature; this makes the problem non-linear. The direct problem was solved using software package CFD-ACE+, and the Levenberg–Marquardt method (LMM) was utilized as the optimization tool because it has been proven to be a powerful algorithm for solving inverse problems. Z-shape lateral fins were found to be the best thermal performance among Z-shape, S-shape, and V-shape lateral fins. The objective of this study was to include continuous lateral ribs to Z-shape lateral fins to further improve η. Firstly, the numerical solutions of direct problem were solved using both polynomial and constant air properties and then compared with the corrected solutions to verify the necessity for using polynomial air properties. Then, four design cases, A, B, C and D, based on various design variables were conducted numerically, and the resultant η values were computed and compared. The results revealed that considering continuous lateral ribs on the surface of Z-shape lateral fins can indeed improve η value at the design working condition Re = 5000. η values of designs A, B and C were approximately 13% higher than that for Z-shape lateral fins, however, when the rib numbers were increased, i.e., design D, the value of η became only 11.5 % higher. This implies that more ribs will not guarantee higher η value.

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