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# Absorption with Chemical Reaction in a Semi-Infinite Medium

Consider the unsteady-state absorption with a chemical reaction in a semi-infinite medium. The governing equation is:
,
where and are the diffusion coefficient and first-order reaction rate constant, respectively.
The initial and boundary conditions are:
, ,
, ,
, ,
where is the saturation concentration and is the position.
This problem admits an analytical solution [4] given by:
.
The rate of absorption is given by [4]:
.
This Demonstration plots the solution , as well as the rate of absorption versus time. The numerical solution obtained using the Chebyshev orthogonal collocation is given by the red dots. The analytical solution is given by the blue curve. The numerical rate of absorption is shown with a red curve. The analytical rate of absorption is given by the blue dashed curve. Excellent agreement between both solutions is observed.
You can vary the values of , , and as well as the number of Chebyshev collocation points, .

### DETAILS

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind, .
The Chebyshev derivative matrix at the quadrature points is an matrix given by
, , for , and for , , and ,
where for and .
The matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .
References
[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.
[2] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
[3] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley & Sons, 2002.
[4] P. V. Danckwerts, "Absorption by Simultaneous Diffusion and Chemical Reaction," Transactions of the Faraday Society, 46, 1950 pp. 300–304. doi:10.1039/TF9504600300.

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