Finite diffusion medium

More solutions in finite diffusion medium may be found in Appendix 3. 

For one-dimensional diffusion, if diffusion starts in the interior and has not reached either of the two ends yet, the diffusion medium is called an infinite medium. An infinite diffusion medium does not mean that we consider the whole universe as the diffusion medium. One example is the diffusion couple of only a few millimeters long (discussed later). In an infinite medium, there is no boundary, but one often specifies the values of C x= and C f=oo as constraints that must be satisfied by the solution. These constraints mean that the concentration must be finite as x approaches or +oo, and the concentrations at +oo or —CO must be the same as the respective initial concentrations. These obvious conditions often help in simplifying the solutions. 

In experimental studies of diffusion, the diffusion-couple technique is often used. A diffusion couple consists of two halves of material each is initially uniform, but the two have different compositions. They are joined together and heated up. Diffusive flux across the interface tries to homogenize the couple. If the duration is not long, the concentrations at both ends would still be the same as the initial concentrations. Under such conditions, the diffusion medium may be treated as infinite and the diffusion problem can be solved using Boltzmann transformation. If the diffusion duration is long (this will be quantified later), the concentrations at the ends would be affected, and the diffusion medium must be treated as finite. Diffusion in such a finite medium cannot be solved by the Boltzmann method, but can be solved using methods such as separation of variables (Section 3.2.7) if the conditions at the two boundaries are known. Below, the concentrations at the two ends are assumed to be unaffected by diffusion. 

When the medium is finite, there will be two boundaries in the case of onedimensional diffusion. This finite one-dimensional diffusion medium will also be referred as plate sheet (bounded by two parallel planes) or slab. The standard method of solving for such a diffusion problem is to separate variables x and t when the boundary conditions are zero. This method is called separation of variables. As will be clear later, the method is applicable only when the boundary conditions are zero. 

Two impedance arcs, which correspond to two relaxation times (i.e., charge transfer plus mass transfer) often occur when the cell is operated at high current densities or overpotentials. The medium-frequency feature (kinetic arc) reflects the combination of an effective charge-transfer resistance associated with the ORR and a double-layer capacitance within the catalyst layer, and the low-fiequency arc (mass transfer arc), which mainly reflects the mass-transport limitations in the gas phase within the backing and the catalyst layer. Due to its appearance at low frequencies, it is often attributed to a hindrance by finite diffusion. However, other effects, such as constant dispersion due to inhomogeneities in the electrode surface and the adsorption, can also contribute to this second arc, complicating the analysis. Normally, the lower-frequency loop can be eliminated if the fuel cell cathode is operated on pure oxygen, as stated above [18],... 

Reynolds, L. Johnsrm, C.C. Ishijima, A., Diffuse Reflectance from a Finite Blood Medium Applications to the Modeling of Fiber Optie Catheters (TE), AppL Opt. (1976), 15, 2059-2067. 

Due to the complexity of the mathematical treatment for cylindrical systems that include phenomena such as the presence of a diffusion boundary layer, a membrane that laminates the device surface and/or finite external medium, analytical solutions are difficult to obtain. Consequently, the study of drug release from cylindrical matrix systems using numerical methods is a common practice. Zhou and Wu analyzed in detail the release from cylindrical monolithic dispersion devices by using the finite element method [189]. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

The exponential term which represents the effect of a point source is sometimes called the influence function or Green function of this diffusion problem. The method of sources and sinks easily produces solutions for an infinite medium or for systems of finite dimension when their boundary is kept at zero concentration. Different boundary conditions require a more elaborate formulation (Carslaw and Jaeger, 1959). 

Steady state may be reached in a diffusion problem proceeding for a long time in a finite medium. Steady state means that the concentration at any point does not change with time any more, i.e.,... 

For one-dimensional diffusion in an infinite medium with constant D, if the initial condition is an extended source, meaning C is finite in a region ( 8, 8), and 0 outside the region (Figure 3-6a) ... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

Diffusion in a finite medium (0, L) with a constant D can be nondimensionalized by letting... 

In his experiment the piston underwent several velocity increments, thus propagating a series of consecutive small shock waves thru the medium, each succeeding wave having higher velocity than the one in front of it. Eventually then, the waves all catch up with the first one. The result is a disturbance of finite amplitude with a very steep front. The last of these diagrams (c), represents the front as a discontinuous change in pressure in real substances, the processes of diffusion and heat conditions make this impossible, and a finite pressure slope is maintained. 

The treatment of the diffuse double layer outlined in the last section is based on an assumption of point charges in the electrolyte medium. The finite size of the ions will, however, limit the inner boundary of the diffuse part of the double layer, since the centre of an ion can only... 

Characteristic times are a key factor in formulating conduction or diffusion models, because they determine how fast a system can respond to changes imposed at a boundary. In other words, if the temperature or concentration is perturbed at some location, it is important to estimate the finite time required for the temperature or concentration changes to be noticed at a given distance from the original perturbation. The time involved in a stagnant medium is the characteristic time for conduction or diffusion, therefore this is the most widely used characteristic time in transport models [3, 6],... 

The system we consider here is a semi-infinite porous medium having a uniform porosity f, and a uniform permeability k, a finite thickness L (the smaller lateral dimension) in the direction, and a finite width hL in the direction. The gravity acts in the z[ direction. The magnitude of the fluid velocity has a uniform value vj over the injection face. We will consider two cases in which the injection fluid is well mixed and diffusion upstream of the injection face can be neglected. Either the composition at the injection face is or it increases linearly with time from... 

At this point, a distinction should be made between cellular and finite difference/element models. The latter are finite approximations of continuous equations [e.g., Eq. (11)], with the implicit assumption that the width of the reaction zone is larger than other pertinent length scales (diffusion, heterogeneity of the medium, etc.). However, no such assumptions need to be made for cellular... 

C In transient mass dilliision analysis, can we treat the diffusion of a solid into another solid of finite thickness (such as the diffusion of carbon into an ordinary steel component) as a diffusion process in a semi-infinite medium Explain. 

Example 2.4 Diffusion in a Finite Domain The steady-state conservation equation for diffusion of species i in an finite medium can be expressed as... 

As an exercise, the reader can verify that equation (2.73) satisfies both real and imaginary parts of equation (2.70). This development represents the starting point for both the Warburg impedance associated with diffusion in a stationary medium of infinite depth and the diffusion impedance associated with a stationary medium of finite depth. 

The oscillating concentration variables are given in Figure 11.3 as a function of dimensionless position. The behavior of 9i ( ) for high frequencies, e.g., Kj = 100, resembles that obtained for a stagnant medium of infinite dimension, whereas the behavior at low frequencies, e.g., K, < 10, is influenced by the finite extent of the diffusion layer. 

Statement of the problem. Following [367, 368], let us consider stationary diffusion to a particle of finite size in a stagnant medium, which corresponds to the case Pe = 0. We assume that the concentration on the surface of the particle and remote from it is constant and equal to Cs and C), respectively. The concentration field outside the particle is described by the Laplace equation... 

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