You must transport a sludge product from an open storage tank to a separations unit at $1 \mathrm{~atm}$, through a 4 in. sch 40 steel pipeline that is $2000 \mathrm{ft}$ long, at a rate of $250 \mathrm{gpm}$. The sludge is $30 %$ solids by weight in water and has a viscosity of $50 \mathrm{cP}$ with Newtonian properties. The solid particles in the sludge have a density of $3.5 \mathrm{~g} / \mathrm{cc}$. The pipeline contains 4 gate valves and 6 elbows.

(a) Determine the pump head (in feet) required to do this job. You can select any pump with the characteristics given in Appendix $\mathrm{H}$, and you must find the combination of motor speed, motor horsepower, and impeller diameter that should be used.

(b) You want to install a long radius venturi meter in the line to monitor the flow rate, and you want the maximum pressure drop to be measured to be equal to or less than $40 \mathrm{in}$. of water. What should the diameter of the venturi throat be?

(c) At the separations unit, the sludge is fed to a settling tank. The solids settle in the tank, and the water overflows the top. What should the diameter of the tank be if it is desired to limit the size of the particles in the overflow to $100 \mu \mathrm{m}$ or less?

(d) If the sludge is fed to a centrifuge instead of the settling tank, at what speed (rpm) should the centrifuge operate to achieve the same separation as the settling tank, if the centrifuge dimensions are $L=D=1 \mathrm{ft}, R_{1}=R_{2} / 2=0.25 \mathrm{ft}$.

(e) Suppose the sludge is fed to a rotary drum filter instead, which removes all of the solids from the stream. The drum operates at a vacuum of $6 \mathrm{psi}$, has dimensions $L=D=4 \mathrm{ft}$, and operates with $30 %$ of the surface submerged. A lab test is performed on the sludge using $1 \mathrm{ft}^{2}$ of the same filter medium as on the drum, operating at a vacuum of $500 \mathrm{~mm} \mathrm{Hg}$. In this test, it is found that $8 \mathrm{gal}$ passes the filter after $2 \mathrm{~min}$, and a total of $20 \mathrm{gal}$ passes through after 10 minutes. What speed (rpm) should the rotary drum filter be operated?