Analysis of diffusion equation with the second and third boundary conditions based on steady state approximation

One dimensional diffusion equation is widely used to describe mass transfer in particles or droplets in a reactor. The length of the definition domain of the one-dimension Fick equation is limited, because it is determined by the scale of the particles or the droplets. The diffusion equation with a certain length of definition domain has no analytic solution unless series solution. So, to obtain approximate solutions of diffusion equation is of theoretical significance and practical significance. Firstly, assumption of constant concentration variance ratio is used instead of assumption of constant concentration frequently used in kinetics of process metallurgy. Secondly, a detail process to deal with diffusion equation based on steady state approximation is given, and the approximate solutions of the diffusion equation at certain conditions are obtained at the same time. By comparing the approximate solutions with the numerical solutions, it is concluded that the diffusive process of non-steady state is considerably well predicted by the approximate solutions, and approximate solutions accord with the situations being close to the final steady state a little better than accord with the situations being close to the begin of the diffusion, and it fairly satisfies the total mass balance.     

Analysis of diffusion equation based on steady state assumption：Ⅱ.approximate analytic solutions

Based on the analytic solutions obtained by steady state approximation,the approximate analytic solutions on a closed interval is completed. It is concluded by some theoretical analysis and some comparisons with numerical solutions that the diffusive proc     

Diffusion Behavior of Chloride Ions in Concrete Pipe Piles

Concrete pipe piles, whose internal surface is usually closed and external one is exposed to chloride environment, are widely used in marine environment. Based on Fick second law and the boundary condition of pipe pile, the analytical solution to chloride diffusion equation was deduced with the method of separation of variables. The solution comprises two parts: a stationary solution consisting of Bessel functions, and a transient solution. This is different from the traditional analytical solution that is based on semi-infinite boundary condition. Applying this result to a typical PHC pipe pile, the diffusion of chloride concentration was analyzed. Furthermore, the effects of the ratio of outer radius and inner radius, diffusion coefficient and protective concrete cover thickness on diffusion of chloride concentration were also discussed. When the ratio of outer radius and inner radius equals 1.5, the decay rate of secondary segment of transient solution is 4 times larger than that of the primary part. This example provides a computational basis for the corrosion of steel bar and some reference for durability design of pipe piles as well.     

Interaction of Phonons and Yhermoequilibrium of a Crystal

The interaction of phonons and the thermoequilibrium of a crystal are analyzed with Boltzmann equation and Landau equation.The thermoequilibrium of a crystal is the result of interaction of phonons.There is no interaction of phonons in the harmonic approximation. A cbystal can not reach the state of thermoequilibrium in the harmonic approximation.     

Finite Element Method of 3D Stable Heat Conduction of Steel Fluid in Continuous Casting Moulds

The heat transfer of steel fluid in continuous casting mould is a stable process and can be depicted with three-dimensional stable heat conduction equation depending on tension speed. The corresponding finite element equation, including the first, second and third boundary conditions, is deduced out with Galerkin residual method. The coded FEM program is used to analyze the temperature distribution of Q235 steel in continuous billet casting mould. The method proposed is a foundation of thermo-mechanical coupled analysis for the formation of solidified shells and stress in the shells in continuous casting.     

Boundary Element Method for Time-dependent Chloride Diffusion and the Chloride Distribution in Concrete

The process of chloride diffusion in concrete is time-dependent.The boundary element method (BEM) with a time-dependent diffusion coefficient is presented for chloride diffusion in concrete based on the suitable transformation of variables.The fundamental solution of the partial differential equation for time-dependent chloride diffusion in concrete is developed,and the compensation length of the diffusion field is defined as well as the compensation coefficient.The scheme of BEM with a time-dependent diffusion coefficient is developed.Two examples are given to demonstrate the accuracy and efficiency of the presented method and the rationality and the importance of the compensation length for the method.     

Kinetics Characteristics of Resolving of Rock Salt of Changshan in Dynamic State

Kinetics characteristics of resolving of rock salt of Changshan in dynamic state have been studied by using conductance method and equipment made by ourself. The results show that the quantitative relationship between conductance(L) or concentration (c) of resolving solution in the process of resolving of rock salt and time t ( kinetics equation )is analogous to Langmuir equation for different flows in ambient temperature.Constant E or a increases with the increase of the flow( V ), whereas constant F or b constant decreases. The above 4 constants in dynamic state are smaller than those in static state.     

Existence of Periodic Solutions for Reaction-diffusion Systems with Time Delays

Periodic solutions of reaction-diffusion systems with time delays are investigated. The upper and lower control function of nonmonotone reaction term is constructed. It is showed that the function satisfies a global Lipschitz condition and quasimonotone. A sort of effective method of studying differential equation with nonmonotone reaction term is gained. It is shown that periodic solutions of this system exist when reaction-term is not monotone and the boundary value system has a pair of coupled to-upper and lower solutions.     

Boundary Element Methods with Overlapping Domain Decomposition for Parabolic Problem

Boundary element method is a numerical method for solving partial differential equations. There are several formulations of boundary element method (BEM) applied to solve a parabolic differential equation.The approach,which employs time- dependent fundamental solution,allows longer time steps in time integration than other approaches,and this can cut down on time for computer implementation with high precision.Domain decomposition method,which decompose the domain that a given problem is to be solved into subdomains,has the advantages of reducing the large problem into smaller ones and reducing the complex problem into simpler ones,and allows parallel computing.An overlapping domain decomposition method is applied combining a boundary element formulation with time-dependent fundamental solution to solve a diffusion equation. Firstly, by domain decomposition, the problem divided into two problems on subdomains, and then the initial-Boundary problems are solved by boundry element method on each subdomain.Some numerical examples are presented to illustrate feasibility and efficiency of the method. The numerical experiments show that the convergence rate of the method is dependent with the overlapping degree of the subdomains.     

Numerical Analysis for Torsion of Prismatic Bars with Sectorial Cross-section

In mechanical design and application, in order to calculate torsion angle and shear stresses of prismatic bars with seetorial cross-section undergoing extemal couples, the numerical method is presented. The torsion equation is nonhomogeneous partial differential equation. First, using the method of separation of variables, torsion stress function is acquired in polar coordinate. Then, the method of boundary collocation is improved to calculate the undetermined parameters. Finally,approximate numerical solutions of stress function and shear stresses in cross-section are obtained. It is given the several calculation results of shear stresses of prismatic bars with different vertex angles. These results show that the method has some precision and application feasibility in engineering design. The method of separation of variables is combinied with the method of boundary collocation simplified calculation process.     

Galerkin Approximate Solutions of Displacement Including the Effect of Dead Loads for Two Kinds of Rectangular Plates

Based on the strain energy considering the effect of dead loads, the general form static equilibrium differential equations were formulated by means of the potential energy principle. The approximate solutions of live load or later load displacement including the effect of dead loads for the clamped rectangular plate and the simply supported rectangular plate were generated by the Galerkin method. These formulas are simple and clear, and their physical concepts are explicit. The correctness of these formulas was verified by the finite element method. The dead loads and other factors that influence the effect of dead loads are shown by these formulas. The effect of dead loads on the two different boundary condition rectangular plate were analyzed by these approximate solutions. The effect of dead loads improves the bending stiffness of plate and minimizes the displacement of live loads or later loads. The key physical factors governing the effects of dead loads on plates are the dead load, the ration of span to thickness and boundary condition etc. This effect of dead loads is not negligible, especial in thinner plates or plates with smaller stiffness, more attention should be paid to the effect of dead load in the calculation and analysis for plates.     

Galerkin Boundary Element Method for Laplace Equation

Galerkin method based on the variation principle is used to solve differential and integral equations. The boundary problem of Laplace equation is changed into the variational equation which is equivalent to the boundary integral equation. Using linear element, it is solved by Galerkin boundary element method. In computation of stiffness matrix, the exactly integral formula is used in the first order integral expression, The numerical integral formula is used in the second order integral expression. Thus the problem of calculation of double singular integral is carried out. The numerical experiments also prove this method is reliable. The error of Galerkin boundary element is tested with numerical experimentation.     

Galerkin Boundary Element Method for Two-dimension Laplace Equation of Neumman Condition

The authors apply the Galenkin variational equation to solve the integral equation with hyper singularity, which can be deduced from the double layer solution for Neumann problem of Laplace equation. The scheme of partial integration in the sense of distributions is introduced to reduce the hyper singularity integral into a weak one with the boundary rotation of unknown function. The numerical implementation with linear boundary elements is presented. The numerical examples illustrate the feasibility and efficiency of the method.     

Mix Domain Decomposition Method with Boundary Element Method For Stokes Equations and Application

As the BEM was used in the knaggy domain,its approach error was big.Domain Decomposition Method change the problem in the knaggy domain to the problem in several convex domains which do not wrap.The whole solution is gained by alternating boundary conditions between Dirichlet condition and Neumann condition on the common boundary.Domain Decomposition Method for Stokes equations in the anomalistic knaggy domain is presented,the example of Domain Decomposition Method in the flow of the lake in GuiYang City is given.     

Boundaty Element Method for Analysis of Thick Plates on Two- Parameter Elastic Foundations

This paper discusses the Reissner's plates on elastic foundation.The elastic foundation is considered as two- paramerer and its effect to thick plates are taken into account by a set of govering differential equations. According to the foundamental solutions for bending problem on two- parameter foundation derived, three boundary integral equations denoted by generalized displacement functions are established. This method is suitable to solve the bending problem of thin or thick plates on two- parameter foundation with arbitary boundary condition, arbitary shape and arbitary load conditions     

Solution for One-Dimensional Contaminant Diffusion in Layered Clayey Soils Using Laplace Transforms

Contaminant diffusion in layered soil is very complex. The diffusion process is simplified as a one-dimensional diffusion mode. The Laplace transformation and Laplace numerical inverse transformation are adopted. The proposed solution can be degraded to a single layer case. The obtained results are well consistent with those obtained by the classic theory. The correctness of the presented solution is verified by comparing the calculation results with the results generated by existing numerical software. One case study is conducted and the calculation results have a good agreement with measured data, which prove the reliability of the proposed method. The presented method can be used to solve more complex boundary conditions of contaminant migration.     

Analysis on the Moisture Migration of Subgrade Soil with Constant Temperature Condition

A moisture migration test for unsaturated clayey loess along Shenyang-Haerbin Express Railway for Passenger Line was carried out to investigate the water movement of soil in an open system with constant temperature condition.It is shown that moisture transfer is mainly caused by matrix potential and gravitational potential energy.The water content by volume gradually decreases from the bottom to the top of specimen.Over time the capillary water height increases by the power function and eventually reaches a stable value.The water flux decreases with time by the logarithm function.Based on the model test,finite element method is adopted to simulate the moisture transfer process in soil under isothermal condition.It is found that the numerical results are in good agreement with the experimental data.The model is thereby used to simulate the rise of groundwater in the subgrade soil.During the whole process of moisture movement,pore water pressure gradually decreases from the bottom to the top of sample.The rapid change of pore water pressure in the early process contrasts with slow change in the late process.The pore water pressure has a linear change along the specimen when the seepage reaches steady state.     

Singularly Perturbation of the Nonlinear Three Species Prey-predator Reaction Diffusion System

A class of nonlinear three species in the biomathematics is considered with the prey-predator singularly perturbed problems for the reaction diffusion system method of differential inequalities. Under suitable conditions, the existence and asymptotic behavior of solution for the initial boundary value problems are studied with the theory of differential inequalities. The theory of differential inequalities is really to constructing two auxiliary functions, which are super and lower solutions of the system respectively. And then the super and lower solutions are satisfied corresponding inequalities. Finally, there is a solution of the system between the super solution and lower solution. Then the existence of the solution for the system is proved and the asymptotic estimation is obtained.     

Garlerkin Boundary Element Method with Hyper Singular integral kernel

A Galerkin Boundary Elements was applied to solve the first kind of integral equation with hyper-singularity, which can be deduced from the direct boundary integral formula for the Neumann problem of Laplace equation. The concept of integration by parts in the sense of distributions was used. When boundary rotation is introduced, the two order derivatives of singular kernel are shifted to the boundary rotation of unknown function in the Galerkin variational formulation. While linear boundary elements are used for 2-dimensional problems, the boundary rotation on each element can be discretized into a constant vector, so that the integration can be performed in a simple way and the difficulty of numerical calculation for hyper-singularity is overcome. The results of numerical examples demonstrate that the scheme presented is practical and effective.     

Experimental Analysis on Time dependent Law of Surface Chloride Ion Concentration of Concrete

Immersion in sodium chloride solution was adopted to periodically test chloride ion concentration at the surface of concrete and regression analysis was completed.And time dependent law of surface chloride ion concentration of concrete with time was investigated.Influences of sodium chloride solution concentration, water to cement ratio and fly ash content on size and accumulation rate of surface chloride ion concentration were also analyzed.It is found that surface chloride ion concentration gradually grows with the of time increase the of time and reaches a steady state finally.With the increase of sodium chloride solution concentration, surface chloride ion concentration reaches the steady state more quickly and then increases numerically.Bigger water to cement ratio means quicker accumulation of surface chloride ion concentration and greater surface chloride ion concentration.Fly ash replacement could fasten accumulation of surface chloride ion concentration and surface chloride ion concentration increases with the increase of fly ash content.An improved model was proposed by comparison and correction of current models.