Problem 54735. Solve an ODE: concentrations predicted by the cells-in-series model
Chris, It's really quite interesting how that model develops into a closed-form solution! I am supposing that this model describes a river that is more-or-less a series of "ponds", and that a more continuously flowing river would be more like "plug flow", in which the contaminants would just translate uniformly in time while decaying at rate k.
The standard starting point for 1D mixing in a river is the advection-dispersion-reaction equation C_t + U C_x = K C_xx - k C, where C is concentration, U is mean velocity, and K is a dispersion coefficient. Subscripts denote partial derivatives. Dispersion, or spreading, arises from several mechanisms, including trapping in recirculation zones and differences in velocity over the cross section. When the parameter UL/K (where L is a distance downstream) is large, the transport is approximately plug flow.
This model has several shortcomings. One is that it does not reproduce persistent skewness in tracer-response curves observed in field experiments. Plots of concentration vs. time show long tails that give a skewness of about 1.1, whereas skewness predicted by the ADR model above goes to zero far downstream.
I learned about the cells-in-series model from a section in a book on alternative models of longitudinal mixing. The CIS model might be more physically appealing for a stream with pools and riffles, but for more uniform channels, it has some drawbacks. Like the ADR model, it also predicts vanishing skewness. You can also show that the movement downstream and the spreading are connected: in other words, they cannot be specified independently.
I've used a modified version of the CIS model to study movement of a tracer between a bay and a tidal river (the Hudson River north of Poughkeepsie)--see DOI 10.1016/j.jmarsys.2003.05.004. In that case, the detail in the measurements we had didn't warrant using a more complex model.
Thanks for filling in the background. I anticipated that this would be heavily affected by the variation in flow rate across the cross-section, caused for example by the boundary conditions on the floor and sides of the channel -- but the persistent skewness is quite and interesting feature of the real-life data.
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