You must predict the performance of a large industrial mixer under various operating conditions. To obtain the necessary data, you decide to run a laboratory test on a small-scale model of the unit. You have deduced that the power $(P)$ required to operate the mixer depends upon the following variables:

Tank diameter, $D$

Impeller diameter, $d$

Impeller rotational speed, $N$

Fluid density, $ho$

Fluid viscosity, $\mu$

(a) Determine the minimum number of fundamental dimensions involved in these variables and the number of dimensionless groups that can be defined by them.

(b) Find an appropriate set of dimensionless groups such that $D$ and $N$ each appear in only one group. If possible, identify one or more of the groups with groups commonly encountered in other systems.(c) You want to know how much power would be required to run a mixer in a large tank $6 \mathrm{ft}$ in diameter, using an impeller with a diameter of $3 \mathrm{ft}$ operating at a speed of $10 \mathrm{rpm}$, when the tank contains a fluid with a viscosity of $25 \mathrm{cP}$ and a specific gravity of 0.85 . To do this, you run a lab test on a model of the system, using a scale model of the impeller that is $10 \mathrm{in}$. in diameter. The only appropriate fluid you have in the lab has a viscosity of $15 \mathrm{cp}$ and a specific gravity of 0.75 . Can this fluid be used for the test? Explain.

(d) If the preceding lab fluid is used, what size of the tank should be used in the lab, and how fast should the lab impeller be rotated?

(e) With the lab test properly designed and the proper operating conditions chosen, you run the test and find that it takes $150 \mathrm{~W}$ to operate the lab test model. How much power would be required to operate the larger field mixer under the plant operating conditions?