Consider the scale$-$up of fermentation from a $10L$ to $10000L$ vessel. The small fermenter has a high$-$to$-$diameter ratio of $3.$ The impeller diameter is $30\%$ of the tank diameter. Agitator speed is $500rpm$ and three Rushton impellers are used. Assuming geometric similarity, determine the dimensions of the large fermenter and agitator speed for:

a$.$ constant $\frac{P}{V}$

b$.$ constant impeller tip speed

c$.$ constant Reynolds number