Difference Solution to Chloride Diffusion in Concrete

2011 ◽  
Vol 99-100 ◽  
pp. 754-757
Author(s):  
Ai Ping Yu ◽  
Hai Bo Lu ◽  
Yan Lin Zhao ◽  
Ke Yu Wei
Keyword(s):  
Differential Equation ◽  
Discrete Model ◽  
Three Dimensional ◽  
Chloride Diffusion ◽  
Second Law ◽  
Continuous Variables ◽  
Convergence Conditions ◽  
One Dimensional ◽  
The Difference ◽  
2D And 3D

Fick’s second law and its analytical solution were usually used for analysis of chloride diffusion in concrete. But discretization of the continuous variables is more appropriate in modeling and discrete model is more suitable for engineering applications. In this paper, Difference Equation equivalent to partial differential equation was established with the difference method, and three-dimensional differential equation of Fick's second law was resolved. Based on this study，the convergence conditions of difference Equations for one-dimensional, 2D and 3D diffusion was given.

Theory of ultrasonic attenuation in one and two dimensional metals

1976 ◽  
Vol 54 (14) ◽  
pp. 1454-1460 ◽  
Author(s):  
T. Tiedje ◽  
R. R. Haering
Keyword(s):  
Ultrasonic Attenuation ◽  
Three Dimensional ◽  
Electronic Systems ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
One Dimensional ◽  
The Difference

The theory of ultrasonic attenuation in metals is extended so that it applies to quasi one and two dimensional electronic systems. It is shown that the attenuation in such systems differs significantly from the well-known results for three dimensional systems. The difference is particularly marked for one dimensional systems, for which the attenuation is shown to be strongly temperature dependent.

The role of the geoid in one-, two-, and three-dimensional network adjustments

CISM journal ◽  
1990 ◽  
Vol 44 (1) ◽  
pp. 9-18 ◽  
Author(s):  
Michael G. Sideris
Keyword(s):  
Three Dimensional ◽  
One Dimensional ◽  
Control Networks ◽  
Deflections Of The Vertical ◽  
Dimensional Network ◽  
3D Network ◽  
Computational Procedures ◽  
The One ◽  
2D And 3D

The geoid and its horizontal derivatives, the deflections of the vertical, play an important role in the adjustment of geodetic networks. In the one-dimensional (1D) case, represented typically by networks of orthometric heights, the geoid provides the reference surface for the measurements. In the two-dimensional (2D) adjustment of horizontal control networks, the geoidal undulations N and deflections of the vertical ξ, η are needed for the reduction of the measured quantities onto the reference ellipsoid. In the three-dimensional (3D) adjustment, N and ξ, η are basically required to relate geodetic and astronomic quantities. The paper presents the major gravimetric methods currently used for predicting ξ, η and N, and briefly intercompares them in terms of accuracy, efficiency, and data required. The effects of N, ξ, η on various quantities used in the ID, 2D, and 3D network adjustments are described explicitly for each case and formulas are given for the errors introduced by either neglecting or using erroneous N, ξ, η in the computational procedures.

ENERGY SPECTRUM OF DISTURBANCE AT TURBULENT TRANSITION VIA ENERGY GRADIENT METHOD

2012 ◽  
Vol 19 ◽  
pp. 293-303 ◽  
Author(s):  
HUA-SHU DOU ◽  
BOO CHEONG KHOO
Keyword(s):  
Energy Spectrum ◽  
Flow Instability ◽  
Three Dimensional ◽  
Wave Number ◽  
Gradient Theory ◽  
Turbulent Transition ◽  
Energy Gradient Theory ◽  
Energy Gradient ◽  
The Difference ◽  
2D And 3D

The energy gradient theory for flow instability and turbulent transition was proposed in our previous work. The theoretical result obtained accords well with some experimental data for pipe and channel flows in the literature. In the present study, the energy gradient theory is extended to examine the effect of disturbance frequency on turbulent transition. Then, the energy spectrum of disturbance at the turbulent transition is obtained, which scales with the wave number by an exponent of –2. This scaling is near to the K41 law of –5/3 for the full developed isentropic homogenous turbulence. The difference for the two energy spectra may be due to the intermittency of turbulence at the transition state. The intermittence causes the distribution of the energy spectrum to take on a steeper gradient (tending to –2 from –5/3). Finally, the flow instability leading to turbulent transition can be classified as two-dimensional (2D) or three-dimensional (3D) in terms of the wave number and the Re. It is found that there is an optimum wave number which separates the 2D and 3D transitions and at which the disturbance energy at transition is minimum.

scholarly journals A Note on One- and Three-Dimensional Infiltration Analysis from a Mini Disc Infiltrometer

Water ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 1783 ◽  
Author(s):  
George Kargas ◽  
Paraskevi Londra ◽  
Konstantinos Anastasiou ◽  
Petros Kerkides
Keyword(s):  
Shape Parameter ◽  
Sandy Loam ◽  
Three Dimensional ◽  
Silty Clay ◽  
Soil Hydraulic Properties ◽  
One Dimensional ◽  
Cumulative Infiltration ◽  
The Difference ◽  
Infiltration Tests ◽  
Disc Infiltrometer

Disc infiltrometers are used to characterize soil hydraulic properties. The purpose of this study was to determine the difference between three- and one-dimensional infiltration and to calculate the infiltration shape parameter γ from a proposed analytical infiltration equation. One- and three-dimensional infiltration tests were done on three repacked soils (loam, sandy loam, and silty clay loam) for two negative pressure heads. A mini disc infiltrometer of a radius of 22.5 mm with suction that ranged from −5 mm to −70 mm was used. The difference between experimental three- and one-dimensional cumulative infiltration was linear with time, which confirmed the proposed equation. In this study, the shape parameter γ seems not to be seriously affected by the soil type and acquires values from 0.561 to 0.615, i.e., smaller than the value γ = 0.75, which is widely used. With these values, the criteria proposed for calculating hydraulic conductivity using three-dimensional infiltration data may be fulfilled in most soils.

Combined Effects of 2D and 3D Inhomogeneities on the Dynamic Susceptibility of Superlattices

2010 ◽  
Vol 168-169 ◽  
pp. 97-100
Author(s):  
V.A. Ignatchenko ◽  
D.S. Tsikalov
Keyword(s):  
Density Of States ◽  
Three Dimensional ◽  
Wave Number ◽  
Dynamic Susceptibility ◽  
Combined Effects ◽  
Two Dimensional ◽  
Simultaneous Presence ◽  
One Dimensional ◽  
The One ◽  
2D And 3D

The dynamic susceptibility and the one-dimensional density of states (DOS) of an initially sinusoidal superlattice (SL) with simultaneous presence of two-dimensional (2D) phase inhomogeneities that simulate the deformations of the interfaces between the SL’s layers and three-dimensional (3D) amplitude inhomogeneities of the layer material of the SL were investigated. An analytical expression for the averaged Green’s function of the sinusoidal SL with 2D phase inhomogeneities was obtained in the Bourret approximation. It was shown that the effect of increasing asymmetry of heights of the dynamic susceptibility peaks at the edge of the Brillouin zone of the SL, which was found in [6] at increasing the rms fluctuations of 2D inhomogeneities, also takes place at increasing the correlation wave number of such inhomogeneities. It was also shown that the increase of the rms fluctuations of 3D amplitude inhomogeneities in the superlattice with 2D phase inhomogeneities leads to the suppression of the asymmetry effect and to the decrease of the depth of the DOS gap.

Numerical Project for the Undergraduate Heat Transfer Course

2017 ◽  
Author(s):  
Dani Fadda
Keyword(s):  
Heat Transfer ◽  
Solid Body ◽  
Three Dimensional ◽  
Engineering Curriculum ◽  
One Dimensional ◽  
Conduction Heat Transfer ◽  
Heat Transfer Simulation ◽  
The Difference ◽  
Dimensional Simulation ◽  
Classroom Time

A numerical simulation project, described in this paper, was assigned in an undergraduate heat transfer course in the mechanical engineering curriculum. This project complemented the heat transfer lecture course and its corresponding heat transfer lab. It was used to help students visualize and better understand the difference between conduction heat transfer which occurs within a three-dimensional solid body and the convection and/or radiation which occur at the surface of the solid body. It also allowed the students to generate and compare results of one dimensional heat transfer calculations to three dimensional simulation results. The project contained well defined deliverables and an open-ended deliverable which allowed students to be creative. It gave the students reason to discuss the course outside the classroom. It allowed students to use SolidWorks heat transfer simulation and manage a MATLAB script without taking classroom time. It was appreciated and enjoyed by the students.

Climate: A Chain of Identical Reservoirs

1991 ◽  
Author(s):  
James C. G. Walker
Keyword(s):  
Differential Equation ◽  
Energy Balance ◽  
Differential Equations ◽  
Climate Model ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Difference ◽  
A Chain ◽  
The One

One class of important problems involves diffusion in a single spatial dimension, for example, height profiles of reactive constituents in a turbulently mixing atmosphere, profiles of concentration as a function of depth in the ocean or other body of water, diffusion and diagenesis within sediments, and calculation of temperatures as a function of depth or position in a variety of media. The one-dimensional diffusion problem typically yields a chain of interacting reservoirs that exchange the species of interest only with the immediately adjacent reservoirs. In the mathematical formulation of the problem, each differential equation is coupled only to adjacent differential equations and not to more distant ones. Substantial economies of computation can therefore be achieved, making it possible to deal with a larger number of reservoirs and corresponding differential equations. In this chapter I shall explain how to solve a one-dimensional diffusion problem efficiently, performing only the necessary calculations. The example I shall use is the calculation of the zonally averaged temperature of the surface of the Earth (that is, the temperature averaged over all longitudes as a function of latitude). I first present an energy balance climate model that calculates zonally averaged temperatures as a function of latitude in terms of the absorption of solar energy, which is a function of latitude, the emission of long-wave planetary radiation to space, which is a function of temperature, and the transport of heat from one latitude to another. This heat transport is represented as a diffusive process, dependent on the temperature gradient or the difference between temperatures in adjacent latitude bands. I use the energy balance climate model first to calculate annual average temperature as a function of latitude, comparing the calculated results with observed values and tuning the simulation by adjusting the diffusion parameter that describes the transport of energy between latitudes. I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands.

scholarly journals Comparison of 2D and 3D Stability Analyses for Natural Slope

2018 ◽  
Vol 7 (4.35) ◽  
pp. 662
Author(s):  
Erly Bahsan ◽  
Rifani Fakhriyyanti
Keyword(s):  
Three Dimensional ◽  
3D Models ◽  
Soil Parameters ◽  
3D Analysis ◽  
Natural Slope ◽  
Stability Analyses ◽  
The Difference ◽  
2D And 3D ◽  
The Impact ◽  
Natural Slopes

Slope stability analyses are performed mostly as a two-dimensional (2D) section under the assumption of plane strain conditions, without much consideration to the impact of three-dimensional (3D) shapes. For natural slopes that have the complexities of slope surfaces, 3D modeling may also be considered since it can represent the more realistic geometry of the slope. However, previous studies show that the factor of safety (FS) as a result of 3D analyses mostly overestimated the FS from 2D analyses. This may lead to a long discussion on whether the 3D analysis is still applicable for the natural slopes, and could it represent the same results as the 2D analysis. This study was conducted using the finite element method for calculating the 2D and 3D FS of Pasir Muncang natural slope in order to observe differences of FS resulted from both analyses. A comparison of the FS from the 2D and 3D analyses, and also verification of sensitivity on several factors that impact the 2D and 3D models have been performed. The results of this study has indicated that some factors such as soil parameters, contour interval, and mesh coarseness greatly affect the results of the 2D and 3D calculations. Having carefully selected the aforementioned factors as the inputs for calculations, the difference between the FS values of 3D and 2D analyses becomes smaller. The final result of FS for this case study from the 3D analysis was still higher than the one from the 2D analysis, with the ratio of FS from 3D to FS from 2D was 1.44. It can be inferred that the use of 3D analyses needs more accurate data selections compared to the 2D analyses.  

scholarly journals DERIVED SCHWARZ MAP OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION AND A PARALLEL FAMILY OF FLAT FRONTS

2008 ◽  
Vol 19 (07) ◽  
pp. 847-863 ◽  
Author(s):  
TAKESHI SASAKI ◽  
KOTARO YAMADA ◽  
MASAAKI YOSHIDA
Keyword(s):  
Differential Equation ◽  
Projective Space ◽  
Hyperbolic Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Flat Surfaces ◽  
Hypergeometric Differential Equation ◽  
The One ◽  
Dimensional Projective Space

In one of the previous papers, we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.

DISTRIBUTION OF CHARGED PARTICLES TRAPPED IN A VARYING STRONG MAGNETIC FIELD (ONE-DIMENSIONAL CASE) WITH APPLICATIONS TO TRAPPED RADIATION

1964 ◽  
Vol 42 (6) ◽  
pp. 1185-1194 ◽  
Author(s):  
Tomiya Watanabe
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Distribution Function ◽  
Strong Magnetic Field ◽  
Three Dimensional ◽  
Charged Particles ◽  
Dimensional Case ◽  
One Dimensional ◽  
Canonical Variables ◽  
Trapped Radiation

The differential equation for the distribution function of charged particles trapped in a strong magnetic field is discussed for the case when the charged particles are constrained in a tube of magnetic lines of force with a small normal cross section (the one-dimensional case). The distribution function is not defined by the six canonical variables, the variables in geometrical space and velocity space, as is done usually when referring to the Boltzmann equation. Instead, it is defined by the space coordinates of the guiding center of a representative particle together with its speed and pitch angle (i.e. five variables, in the three-dimensional case). In some problems, this type of description makes the correspondence between the physical picture and its mathematical description much easier. Several problems relating to trapped radiation are discussed using the differential equation.

Sign in / Sign up

Export Citation Format

Share Document