Infinite Integration of Fick's Second Law

[DiffUser2349]

Hi everyone!

Recently, I've been trying to understand how the error function pertains to solving for concentration in a non-steady state case (with a constant diffusivity D), but I've been having some trouble with the initial assumptions. The source I am currently using (Crank's The Mathematics of Diffusion) claims that, for a the case of a plane source,

C = A/sqrt(t) * exp(-(x^2)/4Dt)

Where C is the concentration (with respect to position and time), x is the position (assuming one dimension only), t is the time, and A is an arbitrary constant, which is a solution for Fick's Second Law (dC/dt = D (d2C/dx2)). Crank (as well another source I've been using <http://www.eng.utah.edu/~lzang/images/lecture-4.pdf>) claim that this is solvable by integrating Fick's Second Law, but whether I am making a mistake or otherwise not understanding the concept, I can't seem to get this result to work. Could someone help me with this, either by providing the math, or a source which has this derivation? Thanks again.

 

[Chestermiller]

Hi everyone!

Recently, I've been trying to understand how the error function pertains to solving for concentration in a non-steady state case (with a constant diffusivity D), but I've been having some trouble with the initial assumptions. The source I am currently using (Crank's The Mathematics of Diffusion) claims that, for a the case of a plane source,

C = A/sqrt(t) * exp(-(x^2)/4Dt)

Where C is the concentration (with respect to position and time), x is the position (assuming one dimension only), t is the time, and A is an arbitrary constant, which is a solution for Fick's Second Law (dC/dt = D (d2C/dx2)). Crank (as well another source I've been using <http://www.eng.utah.edu/~lzang/images/lecture-4.pdf>) claim that this is solvable by integrating Fick's Second Law, but whether I am making a mistake or otherwise not understanding the concept, I can't seem to get this result to work. Could someone help me with this, either by providing the math, or a source which has this derivation? Thanks again.

Substitute $C(x,t)=\frac{A}{\sqrt{t}}f(\eta)$ into the partial differential equation for Fick's second law, where
$$\eta=\frac{x}{2\sqrt{Dt}}$$
By doing this, the partial differential equation should reduce to an ordinary differential equation to solve for f and a function of $\eta$. This yields a so-called similarity solution.

I think a better book to use than Crank would be Transport Phenomena by Bird, Stewart, and Lightfoot. You may have to look in the chapters on heat transfer, since diffusion problems using Ficks second law are mathematical analogs of unsteady state conductive heat transfer problems.