Please answer these $2$ questions in detailed solution and answer. Thank you!

$1.$ The dimensions of a laboratory$-$scale disc centrifuge are $500mm$ in diameter and $300mm$ in height. It is utilized to process $20$ liters of a liquid mixture containing solid particles with a density of $2600k\frac{g}{m^3},$ while the liquid density is $800k\frac{g}{m^3}.$

The centrifuge operates at $1800$ revolutions per minute. If the average particle size

is $5m$ and the viscosity of the liquid is $0\mathrm{.}008$ Pa$.$s$,$ calculate $($a$)$ the time required

to separate the solid particles from the liquid mixture. $($b$)$ Evaluate how long it would

take for these particles to settle by gravity alone in the same centrifuge.

$2.$ In a biochemical engineering laboratory, students are given a task to design a

mixing system for a bioreactor used in the production of pharmaceutical$-$grade

enzymes. The bioreactor has a capacity of $1000$ liters and operates under sterile

conditions. The process requires the thorough mixing of the enzyme precursor and

several nutrient solutions to ensure optimal enzyme yield and activity. Design a

suitable mixing system for this bioreactor. Choose between impellers, turbines, or

magnetic stirrers, and justify your choice based on the specific requirements of

biochemical reactions.