Dirichlet conditions coefficient

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

The mass diffusivity coefficient of isobutane blowing agent from LDPE foam was found using a onedimensional diffusion model of two concentric cylinders with Dirichlet boundary conditions. An average mass diffusivity coefficient was used to calculate the mass of isobutane remaining in the foam for different boundary conditions. The influence of temperature and additives on diffusion was also examined. The use of the mass diffusivity coefficient in assessing the flammability of PE foam in the post-extrusion period is discussed. 2 refs. USA... 

Boundary value — A boundary value is the value of a parameter in a differential equation at a particular location and/or time. In electrochemistry a boundary value could refer to a concentration or concentration gradient at x = 0 and/or x = oo or to the concentration or to the time derivative of the concentration at l = oo (for example, the steady-state boundary condition requires that (dc/dt)t=oo = 0). Some examples (dc/ dx)x=o = 0 for any species that is not consumed or produced at the electrode surface (dc/dx)x=o = -fx=0/D where fx=o is the flux of the species, perhaps defined by application of a constant current (-> von Neumann boundary condition) and D is its diffusion coefficient cx=o is defined by the electrode potential (-> Dirichlet boundary condition) cx=oo, the concentration at x = oo (commonly referred to as the bulk concentration) is a constant. 

Figures 1 is an example for (non-dimensionalized) thermal stresses (von Mises stress) under steady-state heat conduction in a rectangular body defined over (a ,j/),0 < x < 1,0 < y < 1 subject to the Dirichlet type homogeneous boundary condition (T = 0 and Ui — 0) on the boundary with a uniform internal heat generation [5]. The thermal conductivity, elastic modulus and thermal expansion coefficient are all assumed to vary in the form of ko -f kix + k2y where fc, s are constants. For the purpose of illustrations, the values of all the material properties are taken to be unity.

The solution in Equation 8.26 is inconvenient for several reasons. Each term in the series contains two coefficients w and A ) which require numerical calculation. In the case of a linear wall reaction, these quantities depend on the wall kinetic parameter, and this relationship is recently obtained in a simple and explicit manner by Lopes et al. [40]. In addition, whenever this slowly convergent series is used to describe the inlet region, a large number of terms may be required so that a satisfactory result is obtained. The efficient evaluation of the terms in Graetz series has been the object of many studies. Housiadas et al. [51] presented a comparative analysis between several methods to estimate these terms, remarking the numerical issues associated with the rigorous calculation of these quantities. However, this was done for uniform wall concentration (Dirichlet boundary condition), excluding the important case of finite reaction rates. 

Note, in particular, one feature in the behavior of the approximate solutions of boundary value problems with a concentrated source. It follows from the results of Section II that, in the case of the Dirichlet problem, the solution of the classical finite difference scheme is bounded 6-uniformly, and even though the grid solution does not converge s-uniformly, it approximates qualitatively the exact solution e-uniformly. But now, in the case of a Dirichlet boundary value problem with a concentrated source, the behavior of the approximate solution differs sharply from what was said above. For example, in the case of a Dirichlet boundary value problem with a concentrated source acting in the middle of the segment D = [-1,1], when the equation coefficients are constant, the right-hand side and the boundary function are equal to zero, the solution is equivalent to the solution of the problem on [0,1] with a Neumann condition at x = 0. It follows that the solution of the classical finite difference scheme for the Dirichlet problem with a concentrated source is not bounded e-uniformly, and that it does not approximate the exact solution uniformly in e, even qualitatively. 

We consider a diffusion/adsorption test of cesium (Cs), and the data provided are as follows The size of a clay platelet is 100 x 100 x 1. Based on experimental data of Baeyens and Bradbury (1997) the cross-section area of clay edges, which are adsorption sites, are assumed to be 35 m /g. Since the atomic radius of cesium is 3.34 A, and because of the monolayer adsorption, we set the layer thickness of the edge domain where cesium ions are adsorbed as 0.67 nm. The maximum amount of cesium adsorbed in this domain is 6.51 x 10 mol/g. The molecular number of interlayer water is given as n=2.5. The diffusion coefficient of cesium ions in the bulk water is 2x10 cm /s, and in the interlayer water is 2.62x 10 cm /sec, which is obtained from the MD simulation. The concentration of cesium at the upstream boundary, i.e., the l.h.s. surface of Fig. 9.2 (Dirichlet boundary condition) is given as Case (1) 10 mol/1, Case (2) 10 mol/1 and Case (3) 10 " mol/1. 

Pressure solution. Next, consider the corresponding pressure field. We recall from Equations 12-2 and 12-4a that g(x,y,z) = p(x,y,z) Vk(x,y,z) satisfies 9 g/9 + g/9y + g/9z = 0. If we assume that both the permeabilities and pressures are known at all well positions and boundaries, it follows that g = pVk can be prescribed as known Dirichlet boundary conditions. Then, the numerical methods devised in Chapter 7 for elliptic equations can be applied directly on the other hand, analytical separation of variables methods can be employed for problems with idealized pressure boundary conditions. The general approach in this example is desirable for two reasons. First, the analytical constructions devised for the permeability function (see Equations 12-5b, 12-10, and 12-11) allow us to retain full control over the details of small-scale heterogeneity. Second, the equation for the modified pressure g(x,y,z) (see Equation 12-4a) does not contain variable, heterogeneity-dependent coefficients. It is, in fact, smooth thus, it can be solved with a coarser mesh distribution than is otherwise possible. 

The boundary conditions imposed are a zero flux condition at the outermost surface of the solder and a Dirichlet zero condition at the interface solder/metallization (Au sink). The diffusion coefficient for Au in Sn-Pb was determined by other investigators for several annealing temperatures. Comparison of diffusion coefficients of Au in Sn-Pb to those of Au in single crystal Sn (a... 

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