$[20$ points$]$

a$)$ Calculate the total single collector collision efficiency for the mechanisms identified in the diagram below. The particle has a diameter of $3\mathrm{.}5m$ and a density of $5\mathrm{.}3\frac{g}{c}m3,$ in a deep bed filter with a grain size of $0\mathrm{.}04mm.$ The influent flow has a velocity of $0\mathrm{.}003\frac{m}{s}.$ Assume the temperature is $25C$ and that the density and viscosity of water are $0\mathrm{.}997\frac{g}{c}m3$ and $8\mathrm{.}910-3\frac{g}{c}m-s,$ respectively.

b$)$ What is the dominant mechanism?

c$)$ What is the ratio of the settling velocity to the approach fluid velocity?

$[25$ points$]$

Taste and odor compounds are large molecular weight molecules that can be removed from water by filtration and then biologically degraded within the filter. This process can be modeled as:

$\frac{dC}{dZ}=-C$

where $C$ is the mass concentration of taste and odor compounds, $z$ is the distance into the filter bed, and lambda is the filter coefficient with a constant value of $20m^{-1}$ at a constant approach velocity, $U,$ of $200\frac{m}{d}.$ The accumulation of taste and order mass on the filter, $($grams per cubic meter of bed$),$ is limited by first order decay with a rate constant $k_1=0\mathrm{.}1d{d}^{-1}.$

$($a$)$ Show that the differential equation for the mass accumulation is given by

$\frac{d}{dt}=-U\frac{dC}{dz}-k_1$

$($b$)$ Under steady state conditions, what is the expression for the profile of accumulated taste and ordor mass within the filter $(z)?$ Assume the influent concentration is $C_{in}.$

$($c$)$ How deep of a filter is required for $95$ percent removal of taste and odor compounds?