Problem $1$: Industrial Scaling of Flavored Water: You are in charge of the industrial scaling. You need to size a $5$ m$^3$ stirred tank to generate an emulsion from distilled water, boldo essential oil, and the surfactant Tween $80.$ For this purpose, very high shear efforts are required. It has been informed that the process objective was achieved using a hand blender in a beaker without baffles filled with $5$ L of solution and shape factors S$1=1/3,$ S$6=0\mathrm{.}9.$ Consider the hand blender as an R$-2$ type turbine agitator operating at $786$ RPM$,$ the solution density equal to $1000$ kg$$m$3,$ and the solution viscosity equal to $0\mathrm{.}079$ Pa$$s$.$

a$)$ Calculate the Reynolds number for the experiment and determine the flow regime.

b$)$ Find the power required by the experimental agitator and the power per unit volume ratio.

c$)$ Using the criterion of constant power per unit volume and geometric similarity, find the diameter of the industrial agitator and the agitation speed N$2.$ Remember that geometric similarity implies that the shape factors in both tanks are the same.

d$)$ Your boss requests that the mixing time at industrial scale, t$2,$ must not be more than $20$ times the experimental mixing time, t$1.$ That is$,$ tt$1.$

Using the graph of mixing time versus Reynolds Number from Figure $1,$ respond:

e$)$ Find the mixing time in the experimental tank.

f$)$ Determine the flow regime and the mixing time in the industrial tank.

g$)$ Evaluate whether the scaling obtained in c$)$ produces a mixing time within the limit$.$

h$)$ What is the minimum agitation speed in the industrial tank that ensures the mixing time is within the limit$?$

i$)$ Due to the high power consumption of the industrial agitator, suggest two research and development alternatives to achieve the same process objective while reducing power consumption.