Specialty steel for the core of power transformers must be loaded with silicon up to a prescribed

content. This is accomplished by exposing a steel sheet with negligible silicon content to an

atmosphere of $SiC{l}_4,$ which dissociates to $Si(g)$ and $C{l}_2(g)$ at high temperatures. The silicon gas

then dissolves into the steel. The treatment is carried out at atmospheric pressure and $1255K$; at

this temperature the diffusivity of silicon in steel is $8\mathrm{.}2*{10}^{-12}\frac{m^2}{s}.$ The concentration of $Si$ in the

bulk of the gas phase is $30$ mole percent; at equilibrium this would result in a silicon concentration

in the steel of $2wt.\%.$

A$)$ If a steel plate of $1mm$ thickness is exposed to the $SiC{l}_4$ treatment on one side, while the

other side of the plate rests on an impermeable surface, determine how long it will take

before the bottom of the sheet reaches a silicon concentration of $1\mathrm{.}5wt\%.$ Assume

negligible surface resistance.

B$)$ In reality, it turns out that it takes $1\mathrm{.}5$ times longer than calculated in $\frac{A}{?}$ to achieve the

desired concentration of $1\mathrm{.}5wt.\%$ at the bottom of the sheet due to the presence of surface

resistance. Estimate the value of the convective mass transfer coefficient under these

process conditions.