Consider a process in which metal ions in the electrolyte solution are plated on a rotary disk$-$shaped electrode. The plating rate depends on the rate at which metal ions diffuse to the surface of the rotary disk. This process is affected by the process variables below. Use dimensional analysis to express the relationship between the dimensionless numbers $($groups$).$

$($The dimensional analysis problem is for six given process variables. There are no other process variables. Therefore, it is the dimensionless number that subtracts the number of

basic dimensions from here.$)$

$[$Process variables and units$]$

k: mass transfer coefficient $[$m$/$s$]$

D: Diffusion coefficient $[\frac{m^2}{s}]$

a: the radius of a disk $[$m$]$

w$($omega$)$ : the rotational angular velocity of a disk$[$s$^(-1)]$

rho : density of electrolyte $[$m$^(3)/$s$]$

mu : viscosity of electrolyte$[$kg$/$m$^3]$

A meaningful correct answer can be obtained by organizing the dimensionless number by selecting the diffusion coefficient D and the exponent of viscosity mu$($mu$).$ On the left side, write down the dimensionless number containing the mass transfer coefficient k$.$

$.$

$.$

$.().$

$[]$

$k$ :$[\frac{m}{s}],D$ :$[\frac{m^2}{s}],a$ :$[m],$

$w$ :$[s^{-1}],$

$$ :$[\frac{m^3}{s}],$ :$\frac{kg}{m^3}$