Question $1.$ Diffusion through cylindrical film $(23$ marks$)$

A steel wire, $0\mathrm{.}8mm$ in diameter, is slowly pulled through a bath containing an aqueous solution of $40\frac{g}{L}$ of $N{i}^{2+}.$ The length of the wire immersed in solution is $1m.$ An electric current is run through the wire, causing the $Ni$ ions in solution to become electroplated onto the wire, gradually building up a surface coating. Under steady$-$state operating conditions, the rate of electroplating on the wire surface is $S_N=100mgN\frac{i}{m^2}\frac{?}{s}$ throughout, and the laminar layer between the wire surface and the bulk liquid is $0\mathrm{.}5mm$ thick.

a$)$ Make$-$a sketch of the system with all fluxes and boundaries.

$[2]$

b$)$ Determine the molar flux of $Ni,N_{A,\text{s}\text{u}\text{s}\text{f}},$ at the wire surface.

$[1]|$

c$)$ Show that the diffusive flux of $N{i}^{2+}$ from the bulk to the wire surface can be described by: uarr

$N_{A,\text{s}\text{u}\text{r}\text{f}}\frac{\text{d}\text{e}\text{l}\text{r}}{r}=-\frac{D_{AB}}{R_0}\frac{del{c}_A}{(1-\frac{c_A}{c})}$cdots

d$)$ Show by calculation that this system can be considered dilute and show how this simplifies above equation.

e$)$ Given the operating parameters of the system, determine the concentration of $Ni-$at the wire surface.

f$)$ Show that the rate of growth of the wire radius as a function of time can be determined through the following differential equatiop:$$

$\frac{\text{d}\text{e}\text{l}\text{r}}{\text{d}\text{e}\text{l}\text{t}}=-\frac{N_{A,\text{s}\text{u}\text{r}\text{f}}}{\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}({)}_A}$

g$)$ If the target thickness of the $Ni$ coating is intended to be $0\mathrm{.}01mm,$ determine the speed at which the wire should be drawn through the electroplating bath.

NB: Be careful to distinguish between molar and mass based quantities in this problem! $$

Additional$-$Data:

Molecular$-$mass of $Ni$:$,,M_{Ni}=58\mathrm{.}7\frac{g}{m}ol$

Solid density of Ni metal: $,{}_{Ni}=9800k\frac{g}{m^3}$

Aqueous diffusivity of $N{i}^{2+}$:$,,D_{AB}=2*2*{10}^{-9}\frac{m^2}{s}||$