Diffusion semi-infinite medium

Pick MotECUtAR Diffusion—Semi-Infinite Medium... 

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

Let us assume parallel flux in a semi-infinite medium bound by the plane x=0. Diffusion of a given element takes place from the plane x=0 kept at concentration Cint. Introducing a Boltzmann variable u with constant diffusion coefficient such as... 

Diffusion in Matrix. The transport equation for a semi-infinite medium of uniform initial concentration of mobile species, with the surface concentration equal to zero for time greater than zero, is given by Crank (13). The rate of mass transfer at the surface for this model is ... 

Huguenin-Elie et al. (2003) measured the diffusion of P to a resin sink placed in contact with a soil that was either moist, flooded or flooded then moist, and derived values of the diffusion coefficient of P in the soil by fitting to the results the equation for diffusion from a semi-infinite medium to a planar sink ... 

Figure 1-8 Heat and mass diffusion in a semi-infinite medium in which the diffusion profile propagates according to square root of time, (a) The evolution of temperature profile of oceanic plate. The initial temperature is 1600 K. The surface temperature (at depth = 0) is 275 K. Heat diffusivity is 1 mm /s. (b) The evolution of profile in a mineral. Initial in the mineral is l%o. The surface is 10%o. D= 10 m /s.

If diffusion starts from one end (surface) and has not reached the other end yet in one-dimensional diffusion, the diffusion medium is called a semi-infinite medium (also called half-space). There is, hence, only one boundary, which is often defined to be at x = 0. This boundary condition usually takes the form of CU=o = g(t), (dC/dx) x=o=g f), or (dC/dx) x=o + aC x=o=g(t), where u is a constant. Similar to the case of infinite diffusion medium, one often also writes the condition C x=x, as a constraint. 

Because an "infinite" or a "semi-infinite" reservoir merely means that the medium at the two ends or at one end is not affected by diffusion, whether a medium may be treated as infinite or semi-infinite depends on the timescale of our consideration. For example, at room temperature, if water diffuses into an obsidian glass from one surface and the diffusion distance is about 5 /im in 1000 years, an obsidian glass of 50 / m thick can be viewed as a semi-infinite medium on a thousand-year timescale because 5 fim is much smaller than 50 /im. However, if we want to treat diffusion into obsidian on a million-year time-scale, then an obsidian glass of 50 fim thick cannot be viewed as a semi-infinite medium. 

One-dimensional diffusion in infinite or semi-infinite medium with constant diffusivity... 

This section introduces the method of Boltzmann transformation to solve onedimensional diffusion equation in infinite or semi-infinite medium with constant diffusivity. For such media, if some conditions are satisfied, Boltzmann transformation converts the two-variable diffusion equation (partial differential equation) into a one-variable ordinary differential equation. 

A Diffusion in semi-infinite medium with constant surface concentration... 

Solution Heat conduction during aging of the plate (that is, as it moves away from the ocean ridge) can be described by the heat-diffusion problem in a semi-infinite medium. The solution is... 

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

For one-dimensional diffusion in a semi-infinite medium during crystal growth, define the crystal to be on the left-hand side and the melt on the right-... 

Then, an analytical expression that approximates temperature profiles in the semi-infinite medium with variable thermal diffusivity is obtained by inserting Eq. (9.57) with ax = aXav into Eq. (9.50) ... 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-infinite medium. We give here a brief account of... 

Consider the absorption of oxygen from air in the aeration of a lake or the sohd surface diffusion in the hardening of mild steel in a carburizing atmosphere. Both these processes involve diffusion in a semi-infinite medium. Assume that a semi-infinite medium has a uniform initial concentration of CAo and is subjected to a constant surface concentration of CAs. Derive the equation for the concentration profiles for a preheated piece of mild steel with an initial concentration of 0.02 wt% carbon. This mild steel is subjected to a carburizing atmosphere for 2 h, and the surface concentration of carbon is 0.7%. If the diffusivity of carbon through the steel is 1 X 10 11 m2/s at the process temperature and pressure, estimate the carbon composition at 0.05 cm below the surface. 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

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. 

Drops and bubbles are indeed the same mathematical object. However, in marine water studies, the profile analysis of captive (or emerging) bubbles is preferable in respect to the analysis of drops. Actually, from the physical point of view, bubbles exhibit some differences in respect to drops a) diffusion to the air-water interface from a semi-infinite medium (rather than from the small volume confined by the drop) b) limited evaporation c) possibility of observing bubble properties both in quiescent hydrodynamic conditions or in laminar flow regime. Moreover, a captive bubble can be expanded to very large dimensions. 

The concentration within a semi-infinite medium (C2) into which the substance diffuses out of the lower layer (initial concentration C0) may be calculated by the following relationship (24, p. 91 coordinates were changed to conform to the coordinates of the model discussed here)... 

In both cases the samples were weighed periodically and the mass of water absorbed or desorbed, against the square root of the time, t, was plotted. The average diffusion coefficient, D, was calculated from the slope of this graph using the solution to the diffusion equation for a semi-infinite medium (3)... 

Diffusion Equations. Fick s second law can be solved for the special case of a layer of substance of constant (with time) concentration diffusing into a semi-infinite medium which initially contains no solute. Using the method of the Laplace transform (8) the solution yields an expression for the concentration in the medium ... 

Calculated concentrations are based on the diffusion equation for a semi-infinite medium (see text) and are given immediately under the experimental values. Diffusion coefficients are calculated from the total copper absorbed by the laminate stack. 

In the case of short diffusion times (i.e., only near surface penetration), it can be useful to approximate a mineral with a planar boundary as a semi-infinite medium. For the case of diffusion from a well-stirred semi-infinite reservoir at concentration Co into a half space initially at zero concentration, the concentration distribution is given by... 

Typical problems of diffusive transport. Many real examples of diffusion in organs and tissues ean be analyzed in terms of simple solutions to the diffusion equation in rectangular, cylindrical, or spherical coordinates (a) a bolus of molecules is injected into a cylindrical volume of infinite extent (b) a cylindrical source of molecules in an infinite volume (c) a spherical source of molecules in an infinite volume or (d) drug concentration is maintained at a constant value at the surface of a semi-infinite medium. 

Application 2 Drug Penetration in Tissue. The diffusion equation can be used to develop a simple, quantitative method for predicting the extent of drug penetration into a tissue following the introduction of a local source. Consider the simple geometry shown in Figure 3.4d, where drug is maintained at a constant value, cq, at the interface of a semi-infinite medium. From the steady-state solution, Equation 3-58, it is possible to... 

For a semi-infinite medium where the diffusion takes place only in the direction x>0 into a medium initially of zero diffusant concentration and with the concentration at the borderline x=0 instantaneously established on a constant concentration Q for the whole timescale of the diffusion experiment, the concentration at any position x in the medium and at any time t could be given by Eq. (22) [36]. 

Another simple model, the surface renewal model [31], predicts a dependence. In this model, the interface between the air and water is renewed periodically by turbulence eddies, a process that mixes gas that has diffused into the water surface down into the bulk phase (Fig. 3). Jacobs [32] gives the quantity of substance diffusing across a plane between two semi-infinite mediums as ... 

It is seen from Fig. 2.31a and b [109] that the sublimation rate of VCI (G-2) diminishes both in the bulk and in the surface layer of PE films. The kinetic curve of the initial VCI sublimation has a linear character. At the same time, all desorption curves of G-2 from the extruded PE films (Fig. 2.31a) display parabolic dependencies of the mjmo = art kind and obey Boltzmann s solution of the diffusion equation in a semi-infinite medium [110]. Therefore, it is possible to anticipate that the VCI desorption rate from the film carrier is limited by its diffusion. At the initial moment of diffusion, the surface concentration of the diffusant in the film is equal to that in the volume, although a concentration gradient is formed with time. So, diffusion of VCI in the films within a wide time and temperature range is described by the relation... 

Equation 6 together with Eqs. 10 and 11 describe a process of onedimensional diffusion, initiated by a change in the surrounding atmosphere so that the corresponding equilibrium concentration varies from Co to Coo-Equation 10 requires that immediately after the pressure step, the concentration at the boundary (namely for y = 0) assumes the new equilibrium value. This means that the existence of additional transport resistances at the surface of the system is excluded. The second term in Eq. 11 indicates that the process has to proceed as in a semi-infinite medium. This means in particular that the transient adsorption or desorption profiles originating from different crystal faces must not yet have met each other. 

---