$[$Problem #$4$ from the handout on Spheres and Forces.$]$ A knock$-$out drum is a simple and widely used device that separates a $2-$phase, gas$-$liquid mixture into its component phases. A $2-$phase mixture is fed to the drum $($diameter $D,$ height $L$ with $L>D$ with $\frac{L}{D}=3)$ at a mass flow rate $m^{}$ and vapor $($mass$)$ fraction $$ and the drum acts to separate, the vapor phase from the liquid phase based on the differential effect of gravity $($i$.$e$.,$ the liquid portion exits from the bottom of the drum while the vapor phase exits from the top of the drum$).$

As the phases separate, droplets of the liquid phase can form. If the vapor$-$phase velocity through the drum is large enough, those droplets could be carried over into the vapor outlet rather than leaving with the liquid phase. That is$,$ some of the liquid phase will be entrained in the vapor phase. This carry$-$over reduces the efficiency of the separation step and so it is important to design the drum to minimize the carryover.

a$.$ Create a clear and well$-$labelled diagram that summarizes the information for the problem.

b$.$ For a critical liquid droplet size $d_c,$ develop an expression to estimate the minimum vapor$-$phase velocity in the drum, $v_f,$ that will entrain droplets of this size.

c$.$ Use this vapor$-$phase velocity to provide an design equation for sizing the knockout drum $($i$.$e$.,$ for determining the $D$ and $L$ dimensions of the drum$)$ given the mass flow rate and a gas phase fraction of $.$

You may assume that Stokes Law will be valid for the calculation and you can reasonably assume that the difference in density between the fluid and vapor phases is large enough so that you can neglect the density of the vapor phase.