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$$

D_{AB}\frac{\partial^2 C_A}{\partial x^2} = \frac{\partial C_A}{\partial t}

$$

where $C_A$ is the concentration of the species $A$ diffusing into a medium $B$ and $D_{AB}$ is the mass diffusivity.

Consider steel carburization.

In this high-temperature process carbon is diffused into the steel to achieve certain desirable material properties (e.g., increase tensile strength).

When the carburization is performed at a temperature $1200^{\circ}$C the value of the diffusion coefficient is approximately $D_{AB}\approx 5.6\times 10^{-10}$ $(\text{m}^2/\text{sec})$.

Suppose that a 1cm thick slab initially has a uniform carbon concentration of $C_A = 0.2\%$.

How much time is required to raise the carbon concentration at a depth of 1-mm below the slab surface to a value of 1%?

Is the IC $C_A(x,0) = .2$? What are the boundary conditions?

If I can identify those, I can solve it.