$[$Filtration$](40$ points$)$

Engineers without Borders at the University of Toledo has been assigned to build a tri$-$media filter in a water treatment plant in a small city in Venezuela.

The filter has a $0\mathrm{.}5$ m crushed anthracite coal layer $($specific gravity$=1\mathrm{.}20),$ a $0\mathrm{.}3$ m sand layer $($specific gravity$=2\mathrm{.}65),$ and a $0\mathrm{.}1$ m garnet layer $($specific gravity$=4\mathrm{.}20).$ The average sizes of the particles in the three layers are $1\mathrm{.}5mm,0\mathrm{.}8mm,$ and $0\mathrm{.}30$ mm $,$ respectively. The shape factor $($sphericity$)$ values are $0\mathrm{.}9$ and the porosities of the three layers are $0\mathrm{.}47$ for anthracite, $0\mathrm{.}40$ for sand, and $0\mathrm{.}50$ for garnet. The filtration rate is $3\mathrm{.}40\frac{L}{s}-m^2$ and the water temperature is $10C($Note: treat each layer as a bed of uniform particles$).$

Using the Rose equation, determine the total clean bed head loss for the tri$-$media filter.