Fixed Planar Source Diffusion

Let us first consider a case with a constant source, diffusing inside a fluid domain. The initial condition states that the concenttation is zero, that is, at t = 0 c = 0. Subsequently, a source of Hq number of molecules is introduced, which can be represented as delta function, that is, 

At any time, t, the total concentration of the source material is constant, which is given as 

The solution of the diffusion equation (4.42) with the above boundary condition will provide the concentration distribution as a function of space (x) and time (t). Let us define a new variable (f) as 

We can write the individual terms of the diffusion equation by applying chain rule as 

indicates the derivative with respect to C- Substituting in the governing equation. 

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Note that equation (4.61) is similar to equation (4.54), where the plug region is moving at speed Vq and x is replaced by x - V t. The Taylor dispersion flow is analogous to the diffusion of fixed planar source with change in frame of reference, that is, the solvent flows at average velocity, Vg. 

Equation (7.37) can be solved for certain boundary conditions that can be approximated experimentally. For example, a common technique to measure D is to deposit a very thin film of a radioactive isotope (or mass isotope) on a plane surface of a thick sample and, after annealing for fixed times, determine the concentration of the diffusing species as a function of distance. Our experimental system approximates to the case of diffusion of a thin planar source in a semiinfinite solid. If the initial thickness of the layer of radioisotope is small compared to the distance over which the radioisotope diffuses, the solution of Eq. (7.37) is... 

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