7.* (Pollutant dispersal). Consider a chemical pollutant that enters a river that flows with constant speed...

 

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7.* (Pollutant dispersal). Consider a chemical pollutant that enters a river that flows with constant speed u. Often it is assumed that the concentration of the pollutant c(x, t) is lost at a rate proportional to the concentration. a) Show that pollutant conservation is assured if Ctuc₂-ac where ac is the loss per unit volue of water if the concentration is c. b) Suppose that the river is initially pollutant free and a prescribed volume of pollutant is instantaneously released when t = 0 over the interval x € [-L, L]. Determine the concentration levels at later times t. c) Suppose that rather than releasing all the pollutant at the time t = 0 and along part of the river, the pollutant is fed in continuously at the location x = 0 so that c(0, t) = ro for 0 < t < To and zero thereafter. Determine concentration levels downstream from the release point. 7.* (Pollutant dispersal). Consider a chemical pollutant that enters a river that flows with constant speed u. Often it is assumed that the concentration of the pollutant c(x, t) is lost at a rate proportional to the concentration. a) Show that pollutant conservation is assured if Ctuc₂-ac where ac is the loss per unit volue of water if the concentration is c. b) Suppose that the river is initially pollutant free and a prescribed volume of pollutant is instantaneously released when t = 0 over the interval x € [-L, L]. Determine the concentration levels at later times t. c) Suppose that rather than releasing all the pollutant at the time t = 0 and along part of the river, the pollutant is fed in continuously at the location x = 0 so that c(0, t) = ro for 0 < t < To and zero thereafter. Determine concentration levels downstream from the release point.

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a The equation for pollutant conservation is given by ct ucx ac where ct is the time derivative of the concentration of the pollutant cx is the spatia... View the full answer