Fig. P$3\mathrm{.}12$ Drilling mud circulation system.

Fig. P$3\mathrm{.}12$ illustrates the mud$-$circulation system on an oil$-$well drilling rig.

Drilling mud from a mixing tank T flows to the inlet of the pump P $,$ which dis$-$

charges through BD to the inside of the drill pipe DE $.$ During drilling, the mud

flow is to be steady at $Q=100gpm.$ The mud is a Newtonian liquid with $=5\mathrm{.}0$

cP at the average flowing temperature of $70F,$ and its density, due to weighing

agents and other additives, is $=67l\frac{b_m}{f}{t}^3.$

The drill pipe DE$,$ of depth $10,000ft,$ is surrounded by the casing C$.$ At the

bottom, the mud jets out through the drill bit and recirculates back through the

annular space to $F,$ where it is piped back to the tank T$.$ The surface piping has

a total equivalent length $($including all valves, elbows, etc.$)$ of $1,000ft.$ The mild

steel piping has a roughness $=0\mathrm{.}00015ft.$ Other properties of the piping are

given in Table P$3\mathrm{.}12.$

Calculate:

$($a$)$ The flow rate $Q$ in $f\frac{t^3}{s}$ throughout the system.

$($b$)$ The mean velocities $(f\frac{t}{s})$ in the surface piping, the drill pipe, and the annular

space between the casing and the tubing.