A batch reactor is stirred by an impeller. The quality of mixing is mostly determined by the stirring power W $,$ which is the rate at which work is done by the liquid within the reactor by the impeller. Stirring power is affected by the diameter D and rotational speed N $($revolutions$/$s$)$ of the impeller, $\backslash$rho and $\backslash$mu of the liquid, and sometimes gravity $($g$).$

The company you work for is planning on installing a batch reactor mixed by a $3-$m diameter impeller. Before integrating the reactor into the chemical process, engineers would like to know the stirring power at a given rotation rate. Therefore, you are tasked with building a pilot mixer at $1/10$th scale. The reaction mixture is an aqueous solution whose density and viscosity can be taken to the same as that of water at $20\backslash$deg C$,$ so you use room$-$temperature water in your pilot$-$scale mixer as well.

a$.$ Find a set of dimensionless $\backslash$Pi s that fully describes this system. Identify any known and named dimensionless groups that each $\backslash$Pi corresponds to$.$

b$.$ If using $20\backslash$deg C water in both the full$-$scale and pilot$-$scale mixer, is it possible to find a rotation rate N$\_$f of the full$-$scale mixer that would make it completely similar to the pilot$-$scale mixer at rotation$-$rate N$\_$p$?$ Show your work proving that this is or is not possible.

c$.$ In the case of stirred tank reactors, gravitational effects can sometimes be assumed negligible. If that assumption is reasonable for our system, which $\backslash$Pi can now be neglected?

d$.$ At an impeller rotation rate of $2$ s$-1,$ you find the pilot$-$scale mixer to draw $9\mathrm{.}72$ kW of power. Assuming gravitational effects to be negligible, find the rotation rate N$\_$f needed to make the full$-$scale mixer completely similar to the pilot$-$scale mixer at $2$ s$-1.$

e$.$ Estimate the power draw of the full$-$scale mixer at the rotation rate calculated in part d$.$