Diffusion into a Semi-Infinite Medium

Consider the case of the semi-infinite medium x 0 in which the concentration Co is rmiform throughout and which is exposed at time f x 0 and the position X = 0 to a constant surface concentration C. The solution to this problem can be given as a terse analytical expression and takes the form 

For a medium initially devoid of solute, C = 0, the equation reduces to 

Both of these expressions make frequent appearances in the literature. They contain, as do some source problems, an error function, but lack the preexponential factor we have seen there. As a result, the concentration distributions that arise in this case are a function of only one dimensionless parameter, x / 2- jDt. It follows from this that  

The distance of penetration of any given concentration is proportional to the square root of time. 

Mass Transfer and Separation Processes Principles and Applications 

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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... 

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 ... 

The penetration theory in its simplest form represents the case of transient molecular diffusion into a semi-infinite medium It can be applied to real situations if hydrodynamic conditions exist for which such an assumption is approximately valid This would be the case if flow close to the interface is laminar, concentration profiles there are practically nonaal to the interface and time of contact of the phases is reasonably short ... 

Model 1 (Semi-Infinite Medium). For linear diffusion into a semi-infinite medium where the concentration at X = 0 is of the constant value Co for aU times t, we find the solution for the concentration field to be... 

These simple examples show that the case of diffusion into a semi-infinite medium can yield rapid answers in a relatively straightforward fashion. Furthermore, the geometry is not trivial. It can often be used to approximate finite geometries, particularly if the diffusion process is a slow one, as it is in liquid or solid media. Penetration will then be confined to short distances from the surface, at least initially, and the medium can consequently be regarded as a semi-infinite one for the short period imder consideration (note the similarity to Illustration 4.1). 

Here we calculate the amount of material per unit area that has diffused into a semi-infinite medium to a certain point in time t. We proceed in two steps by first calculating the rate cf dijfusion at the base plane and then integrating that rate over time. 

Transient Semi-Infinite Diffusion The simplest transient diffusions problems are generally those that involve semi-infinite or infinite boundary conditions. Consider, for example, the situation illustrated in Figure 4.6, which represents diffusion of a substance from a surface into a semi-infinite medium. 

The solution of the problem of diffusion from a semi-infinite medium into another semiinfinite medium, is generally expressed in terms of the error-function complement [7] ... 

The self-diffusion coefficient of calcium in single crystals of calcium oxide was measured at 1465 to 1760C using vapor-deposited thin films of radioactive Ca O and boundary conditions for diffusion from a plane source into a semi-infinite medium. The temperature dependence of the diffusivity was expressed as ... 

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. 

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. 

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 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]. 

The exponential expression has the form of the Gaussian normal distribution curve. If an amount s of diffusing substance diffuses only into one side of a semi-infinite medium -that is, if the diffusing material is placed on the end of a rod with all the other above conditions applying - then the solution (5-40) needs only be multiplied by a factor 2. (The superposition principle for solutions of linear differential equations has been used here.)... 

The geometry considered is a semi-infinite medium bounded by a plane surface. An object of finite dimensions (sphere, disk, cylinder) is embedded at a distance x from the surface and is assumed to have a constant surface concentration C2. Solute diffuses into the surrounding space and ultimately reaches the bounding surface, which is held at a constant concentration (Figure 2.10). Q is often near zero because of dispersion into a flowing fluid or the atmosphere. The release rate when = 0 exceeds that of any other practical configuration and sets an upper limit to diffusion from a source of constant concentration. It can therefore be used to estimate the maximum performance to be expected in finite spaces. 

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. 

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 ... 

The diffusion distance concept is best defined for infinite and semi-infinite media diffusion problems. In these cases, C depends on x 4Dt), so if at time ti the concentration is Ci at Xi, then at time tz = 4fi the concentration is Ci at X2 = 2xi (because = Xi/(4Dfi) = 2 = 2/(4T>f2) ). This fact is often referred as the square root of time dependence. That is, the distance of penetration of a diffusing species is proportional to the square root of time. In other words, the concentration profile propagates into the diffusion medium according to square root of time. It can also be shown that the amount of diffusing substance entering the medium per unit area increases with square root of time. The square root dependence is often expressed as... 

When solving Pick s second law for any specific problem, the first step is always to specify the boundary and initial conditions. For the semi-infinite diffusion process illustrated in Figure 4.6 as an example, the concentration of species i is initially constant everywhere inside the medium at a uniform value of c°. At time f = 0, the surface is then exposed to a higher concentration of species i (c ), which causes i to begin to diffuse into the medium (since c > c"). It is assumed that the surface concentration of species i is held constant at this new higher value c during the entire transient diffusion process. Based on this discussion, we can mathematically specify the boundary and initial conditions as follows ... 

FIGURE 4.6 Schematic illustration of the transient semi-infinite diffusion of a species i from the surface into the hulk of a medium. The concentration of species i at the surface of the medium is assumed to he held fixed at c while the initial concentration of species i within the bulk of the medium is assumed to be c°. As time elapses, species i diffuses deeper and deeper into the medium from the surface. Since the medium is semi-infinitely thick, this process can proceed indefinitely and the concentration of species i never reaches c anywhere inside the medium except at the surface. This figure assumes that c > c° however, the reverse situation (which would involve out-diffusion of i from the bulk) could be similarly modeled. 

Fig. 4.1.7 Scattering model in thermodynamic equilibrium. An opaque slab is placed over an isothermal semi-infinite partially scattering medium. Both the slab and fte medium are held at the same temperature 7a. Radiation from the slab is incident on the medium in all downward directions, and a component of intensity h is diffusely reflected by the medium into the direction jx. The intensity of radiation thermally emitted by the medium into the direction /u. is /r. Because the space between the slab and medium is equivalent to a blackbody cavity (see Section 1.7), the sum of h and /r is the Planck intensity B Ta).

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