It is desired to construct a settling chamber to remove particles from an airstream of $120$ m$^3/$min$.$ The temperature of the air is $50\backslash$deg C$,$ and the specific gravity of the particles is $2\mathrm{.}5.$ The chamber is to be strapped to the ceiling of an industrial building, and the space is limited vertically to $2$ m and horizontally to $1\mathrm{.}5$ m$.$ Determine the length required to remove $100$ percent of $50-$m particles.

hint: the shape is cylindar $,$ find the area accordinly $.$ use the follwing equations

Equation $9-1$: $\frac{v_t}{H}=\frac{v_h}{L}$

$v_t$ : Terminal settling velocity, which is the constant speed at which a particle settles in a fluid.

$H$ : Height of the settling chamber, measured in meters $($m$).$

$v_h$ : Horizontal flow$-$through velocity, which is the speed of the gas flow moving horizontally

through the chamber, measured in meters per second $(\frac{m}{s}).$

$L$ : Length of the settling chamber, measured in meters $(m).$

Derived equation: $v_t=\frac{v_{h}H}{L}$

This equation is derived by solving for $v_t$ from Equation $9-1.$

Substitution into Eq$.(7-3)$: $\frac{v_{h}H}{L}=\frac{g({}_p-{}_a)d_p^2}{18}$