A common device used in various applitations to cl$-$an particleladen air is the reverse$-$flow eyclooe. shown in the figure. Dusty air $($volume flow rate $Q$ and density $)$ enters tangentially through an opening in the side of the eyclone and swirls zroand in the tank. Dust particles are flang outward and fall out the bottom, while clean air is drawn out the lop. The revene$-$flow cycleees being studied are all grometrically similar; bence, diameler $D$ represents the only length seale required to fully specify the entire cyclone geometry. Engineers are concerned about the pressure drop $P$ through the eyclone.

$($a$)$ Carry out the dimensional analysis to determine the appropriate dimensionless relationship berween the pressure drop through the cyclone and the given paramelers.

$($b$)$ If the cyclone sire is doubled, all clse being equal, by what facter will the pressure drop change? If the volume flow rate is doatled, all else beieg equal, by what factor will the pressure drop change?