Solved: A nonreactive tracer at concentration c0 is continuously introduced into a

Chapter 11, Problem 28

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A nonreactive tracer at concentration c0 is continuously introduced into a steady flow at the upstream end of a column of length L packed with a homogeneous granular medium. Assuming that the tracer concentration in the column is initially zero, the boundary value problem that models this process is ct + vcx = Dcxx, 0 < x < L, t > 0, c(0, t) = c0, cx(L, t) = 0, t > 0, c(x, 0) = 0, 0 < x < L, where c(x, t), v, and D are as in 27. (a) Assume that c(x, t) = c0 + u(x, t) and find the boundary value problem satisfied by u(x, t). (b) Proceeding as in 27, find u(x, t) in terms of an eigenfunction expansion. (c) Let v = 1, D = 0.5, c0 = 1, and L = 10. Plot c(x, t) versus x for several values of t, and also plot c(x, t) versus t for several values of x. (d) Describe in a few words how the solution evolves with time. For example, about how long does it take for the steady-state solution to be essentially attained?