Convolution solutions for transport through multiple layers and variable input boundary conditions
The transport solutions (FPTD, SCD, etc.) allow determination and prediction of breakthrough curves at specified distances from the inlet boundary. These solutions are valid as long as the overall behavior of the flow field does not change, so that β remains constant. Moreover, the solutions have been developed for the case of either a pulse or step function tracer input condition. Clearly, in many experimental systems, either the input boundary condition is different (e.g., a decaying step input), or transport occurs through regions with distinctly different properties (e.g., through two or more sedimentary layers). Conveniently, convolution techniques can be used in a straightforward manner to deal with these situations.
We define first an input function, F(t), and a response function for the medium G(t), which corresponds to the FPTD solution for a pulse input. The convolution of F(t) with G(t), H(t), is defined formally as
where H(t) is the resulting breakthrough curve. This flexible and convenient formulation allows us to consider both of the situations outlined above. In the case of transport through two characteristically different layers, we can analyze tracer transport through each of the layers individually, defining two different β values (and thus two FPTD solutions, F(t) with G(t). The overall breakthrough curve that develops after transport through the two layers is given by the convolution of these two solutions. For the second case of a variable inlet boundary condition, G(t) represents the FPTD solution with a specific β value (assuming a pulse inlet boundary condition), while F(t) represents the functional form of the tracer input. Applications of convolution methods in this context have been presented by Kosakowski et al. (2000), and are demonstrated in Section 4.6.