Consider electrodeposition of nickel on a nickel cathode immersed in molten lithium silicate at $1350C.$ The cathode is a vertical nickel rotating cylinder at $500rpm.$ The melt contains $2wt\%$NiO. Assume the rate to be controlled by mass transfer of NiO to the cathode.

$($a$)$ Determine the limiting current density.

$($b$)$ Calculate the overpotential at the cathode at a current density of $15A*m^{-2}$

Given:

i$.$ Density $()$ and viscosity $()$ of the melt are $2\mathrm{.}2\frac{g}{c}m3$ and $1\mathrm{.}0$ poise respectively.

ii$.$ Cylinder diameter $(d)$ is $0\mathrm{.}5cm.$

iii. Diffusivity of NiO$(D_{\text{N}\text{i}\text{O}})$ in the melt is $2{10}^{-6}c\frac{m^2}{s}$

iv$.S{h}_d=\frac{k_{m}d}{D_{\text{N}\text{i}\text{O}}}=0\mathrm{.}0791R{e}^{0\mathrm{.}7}S{c}^{0\mathrm{.}356}(Re=\frac{vd}{})$