Constant Planar Source Diffusion

Figure 44 The concentration distribution in case of constant planar source diffusion, where concentration at X = 0 remains constant at all later times (a) the x-dependence of concentration at three time instants and (b) the time dependence of concentration at four streamwise locations. Here, Lq is the length scale and Tq = C /D is the time scale...

The other variation of diffusion problem is when a constant planar source is diffused. The initial concentration is equal to zero. The geometry is the same as the previous problem, but the boundary condition is different, that is. 

Figure 4.4 shows the diffusion of a constant planar source, where at time f = 0 a source fills the half space such that the density or concentration at boundary plane x = 0 remains constant at all later times. Figure 4.4(a) shows that with an increase in time, the concentration font diffuses to a larger x-location. Figure 4.4(b) shows the variation of concentration at a particular x-location as a function of elapsed time. For large time, the concentration reaches the initial concentration, Cq, and the time to reach the equihbrium concentration, Cq, increases with distance, x, from the source location. 

Figure 6.23 illustrates a variety of situations . We will meet further cases, in Chapter 7, in which the fluxes and hence the gradients are kept constant at the boundary. Naturally the one-dimensional solutions are not restricted to one-dimensional systems (cf. Fig. 6.22). In a suitable pseudo one-dimensional experiment is the application of a diffusion source in the form of a thin strip (e.g. gold) onto a thin film (e.g. a thin sheet of silver) (chemical diffusion). It can also be a thin strip of the same but now radioactive material (different isotope), or the exposure of a slit-shaped opening of the otherwise sealed thin oxide film to a (radioactive or chemically modified) gas atmosphere (tracer diffusion). The analogue in D is the sandwich technique or the planar application of the diffusion source onto the surface (as already considered in Fig. 6.24). If the diffusion source is a gas phase, again sealing is necessary unless the aspect ratio is very favourable, i.e. if the extension is sufficiently small in the direction of diffusion compared with the other directions in space . Otherwise the three-dimensional solution has to be considered.

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