Infinite medium

A key parameter in determining the possibiUty of a self-sustained chain reaction is the value of k for an infinite medium, k. In the four-factor formula,... 

The ultrasonic relaxation loss may involve a thermally activated stmctural relaxation associated with a shifting of bridging oxygen atoms between two equihbrium positions (169). The velocity, O, of ultrasonic waves in an infinite medium is given by the following equation, where M is the appropriate elastic modulus, and density, d, is 2.20 g/cm. ... 

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,... 

An important variation of the self-consistent model is the three-phase model, introduced by Kerner 20), according to which the inclusion is enveloped by a matrix annulus, which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. 

Prove that for equimolecular counterdiffusion from a sphere to a surrounding stationary, infinite medium, the Sherwood number based on the diameter of the sphere is equal to 2. 

The rise velocity, Uo, in general depends on the bubble size, or the bubble Reynolds number but as bubble size increases, as in two-phase upflow, Ua approaches an asymptotic value that is independent of Reynolds number. The following expressions have been accepted for a single bubble rising in an infinite medium, and for one rising in a swarm of surrounding bubbles, respectively (Duckler and Taitel, 1991b) ... 

A reasonably consistent universal velocity profile is obtained by plotting (UJr - Ujb)/(Ujm - U.b) vs. r/r1/2 in Figs. 16-18, comparable with the Tollmien solution for a circular homogeneous jet in an infinite medium (Abramovich, 1963 Rajaratnam, 1976). 

The thermally thin case holds for d of about 1 mm. Let us examine when we might approximate the ignition of a solid by a semi-infinite medium. In other words, the backface boundary condition has a negligible effect on the solution. This case is termed thermally thick. To obtain an estimate of values of d that hold for this case we would want the ignition to occur before the thermal penetration depth, <5T reaches x d. Let us estimate this by... 

Wallis, G.B., The terminal speed of liquid drops and bubbles in an infinite medium, International Journal ofMultiphase Flow, l,pp. 491-511 (1974). 

Parallel flux the instantaneous point source in the infinite medium... 

The infinite medium with one-dimensional diffusion and constant diffusion coefficient can be treated easily with the point source theory. Let us first assume that two half-spaces with uniform initial concentrations C0 for x < 0 and 0 for x > 0 are brought into contact with each other. The amount of substance distributed per unit surface between x and x + dx is just C0dx. From the previous result, at time t the effect of the point source C0 dx located at x on the concentration at x will be... 

The infinite medium with a layer of uniform initial concentration... 

The method of point sources can be extended to any form of the initial distribution C0(x) in the infinite medium. The amount of substance distributed per unit surface between x and x1 + dx is just C0(x )dx. Summing over all the point sources at x from — oo to +oo gives the concentration distribution... 

Figure 8.15 The infinite medium with a layer of thickness 2X and uniform initial concentration C0 [equation (8.5.6)]. Interface concentration at x = Z stays nearly constant and equal to C0/2 for StJgO.5. The curves are labeled for different values of the parameter St.

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). 

The infinite medium with C0(x) being a periodic function of x A useful result is obtained for the initial distribution C0(x) given by... 

The semi-infinite medium with constant surface concentration... 

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 ... 

The force due to the movement of the liquid surrounding the bubble is m (dt>ldt). For a sphere moving in an infinite medium of an inviscid fluid, the mass of the liquid m is equal to half the mass of the displaced liquid. The authors, however, assumed merely a direct proportionality between m and the mass of the displaced fluid, instead of the above relationship, because they considered their flow not to be irrotational. 

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

In Chapter 3, equation 3.40 was proposed for the calculation of the free falling velocity of a particle in an infinite medium(61). This equation which was shown to apply over the whole range of values of Ga of interest takes the form ... 

Richardson and Zaki(11) found that m, corresponded closely to u0, the free settling velocity of a particle in an infinite medium, for work on sedimentation as discussed in Chapter 5, although u, was somewhat less than n0 in fluidisation. The following equation for fluidisation was presented ... 

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 ... 

A plot of the terminal velocity of a drop moving in an infinite medium vs. drop size will show the features shown in Fig. 5. To exhibit all of these features, both drop and field liquids must be of very high purity. Using the pertinent fluid properties, a plot of Cd vs. Re will appear as in Fig. 6. In this plot the length term used is the very convenient De, in... 

Many of the data on the gross terminal velocity of drops have been taken in vertical cylindrical glass tubes of limited size. To interpret such data in terms of a drop moving in an infinite medium, a wall correction factor is necessary. 

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.

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. 

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. 

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. 

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 ... 

Equations 3-45a to 3-45d, in conjunction with the following superposition principle, are powerful in deriving solutions for the diffusion equation with infinite medium. 

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