An inorganic salt is to be crystallized from an aqueous solution in a continuous cooling MSMPR crystallization operation at $10C.$ At this temperature, the solubility of the salt is $0\mathrm{.}01g$ solute $\frac{?}{mL}$ solution.

The dissolved feed solution is at a temperature of $40C$ and contains $0\mathrm{.}05k\frac{g}{L}$ of the salt.

If it is required to produce crystals with a mass$-$based average size of $400m,$ estimate the required residence time.

Data:

The growth kinetics of the solute are given by

$G=3\mathrm{.}2c^{1\mathrm{.}1}$

where $G=\frac{m}{s},$ and $c=kg$ solute $\frac{?}{L}$ solution

The nucleation kinetics of the solute are given by

$B=1{10}^{12}{c}^3$

where $B=\frac{\text{\#}}{m^3}*s,$ and $c=kg$ solute$/$L solution

The salt crystals have a density of $1,500k\frac{g}{m^3}$ and can be assumed to have a volumetric shape

factor of $0\mathrm{.}6.$