Consider a small river, with average width W and depth H, passing by a large agricultural field as shown in Fig XXXXXXXXXXFor simplicity, assume W and H are constants. During rain storms there is...

Consider a small river, with average width W and depth H, passing by a large agricultural field as shown in Fig. 10.28. For simplicity, assume W and H are constants. During rain storms there is direct runoff (flow rate = q, per unit length along the stream) from the field into the stream, carrying a concentration of some parameter of interest (say, a pesticide) with concentration Cr0 . Upstream of this field the concentration is zero. This contaminant tends to adsorb strongly onto sediment particles, so that settling of particles in the stream represents a loss (sink) of the contaminant from the stream. Assume that particles settle at a rate us, so that the settling flux of the contaminant is usC. In addition, assume that there is an overall decay of the chemical due to biological activity, which may be modeled using a first-order reaction with rate constant k.

(a) Show that the rate of change of stream flow is given by

(b) Develop a one-dimensional time-dependent advection–diffusion model, including initial and boundary conditions, to simulate concentration C in the stream from x = 0 to x = L.

(c) List at least one factor that makes this problem difficult (maybe impossible) to solve using an analytical solution.