Question

The concentration C(x, t) of a pollutant in a semi-infinite pipe diffuses according to the PDE over 0<<o for > 0, where the diffusivity D is a given positive constant. An experiment is performed in which the initial concentration is zero everywhere, so that C(x, 0)0 for all r 20. Once t > 0 a fixed concentration of C(0, t) Co is imposed at x-0, where Co is a constant, and it is also assumed that the concentration remains negligible far away, so C(z. t) → 0 as x → oo for all t >0. The objective here is to predict what will happen to the pollutant concentration C(a, t) at every point in the pipe, for all z 2 0 when t 0, assuming that the values of D and Co are known. 1. State clearly the mathematical problem being solved here, including all relevant equa tions and conditions on C(x, t). Use results from lectures to state an integral expression for the exact solution C(x,t) over 03 < oo, using an appropriate integrand for <0 that ensures the boundary condition at 0 is alway satisfied. Justify your answer clearly. [Hint: If it assists, introduce a temporary dependent variable, say φ = C-C 2. Using appropriate manipulation, simplify the expression for C(x, t) as much as possible so that it is expressed in terms of the error function 2 I exp(-ข้า dr erf(x)- and/or the exponential or other functions

please answer them in detail.

Answer 1