Problem $3(53$ pts$)$ A transformed carrot cell line is used to make a biologic drug. The carrot cells grow as individual cells in suspension with an average diameter of $23m.$ A $1\mathrm{.}75-$L bioreactor equipped with four baffles and an A$310$ impeller $($a type of turbine$)$ was used for bench$-$scale experiments. The liquid height to tank diameter was $1$:$1$ and the ratio of tank impeller to tank diameter was $1$:$3.$ Bench$-$scale experiments were conducted using stir speeds of $350$ rpm and $900$ rpm $.$ Growth was observed to be poor at the $900$ rpm stir speed. A geometrically similar $1500-$L bioreactor will be used for production. Even though this is a plant cell line, it is suspected that the cells are shear sensitive$,$ so you have been asked to scale up the process by constant shear. Culture Broth: density $1075k\frac{g}{m^3},$ viscosity $4\mathrm{.}8$ cP $.$

a$)$ What agitation$/$stir speed is needed for the $1500-L$ bioreactor, if scaled up by constant shear $(17$ pts $)?$ Hint: In class, we equated PowerNolume for the smaller and larger vessels. Here, equate constant shear.

b$)$ What is the ratio of mixing times $(t_T)$ for the two vessels $($large$/$small ratio$),$ when constant shear was used for scale$-$up $(36$ pts$)?$ State in words what your calculated mixing time ratio means in the space below $($not in the box$).$