You must design a transfer system to feed a coal slurry to a boiler. However, you don't know the slurry properties, so you measure them in the lab using a cup and bob (Couette) viscometer. The cup has a diameter of $10 \mathrm{~cm}$ and a bob diameter of $9.8 \mathrm{~cm}$, and the length of the bob is $8 \mathrm{~cm}$. When the cup is rotated at a rate of $2 \mathrm{rpm}$, the torque measured on the bob is $2.4 \times 10^{4}$ dyn $\mathrm{cm}$, and at $20 \mathrm{rpm}$, it is $6.5 \times 10^{4} \mathrm{dyn} \mathrm{cm}$.

(a) If you use the Bingham plastic model to describe the slurry properties, what are the values of the yield stress and the limiting viscosity?

(b) If the power law model were used instead, what would be the values of the flow index and consistency?(c) Using the Bingham plastic model for the slurry, with a value of the yield stress of $35 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $35 \mathrm{cP}$, and a density of $1.2 \mathrm{~g} / \mathrm{cm}^{3}$, what horsepower would be required to pump the slurry through a $1000 \mathrm{ft}$ long, $3 \mathrm{in}$. ID sch 40 pipe at a rate of $100 \mathrm{gpm}$ ?