You must determine the horsepower required to pump a coal slurry through an $18 \mathrm{in}$. diameter pipeline, 300 miles long, at a rate of 5 million tons/year. The slurry can be described by the Bingham plastic model, with a yield stress of $75 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $40 \mathrm{cP}$, and a density of $1.4 \mathrm{~g} / \mathrm{cm}^{3}$. For non-Newtonian fluids, the flow is not sensitive to the wall roughness.

(a) Determine the dimensionless groups that characterize this system. Use these to design a lab experiment, from which you can scale up measurements to find the desired horsepower.

(b) Can you use the same slurry in the lab as in the actual pipeline?

(c) If you use a slurry in the lab that has a yield stress of $150 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $20 \mathrm{cP}$, and a density of $1.5 \mathrm{~g} / \mathrm{cm}^{3}$, what size pipe and what flow rate (in gpm) should you use in the lab?

(d) If you run the lab system as designed and measure a pressure drop $\Delta P(\mathrm{psi})$ over a $100 \mathrm{ft}$ length of pipe, show how you would use this information to determine the required horsepower for the actual pipeline.