$(25$ Points$)$ A batch fluidized bed reactor holds $1750kg$ of solids with a particle density $2000k\frac{g}{m^3}$ and with an initial size distribution shown below:

$\backslash$table$[[$Size Range $(m),$Mass Fraction in bed initially$],[10-30,0\mathrm{.}1],[30-55,0\mathrm{.}25],[55-80,0\mathrm{.}3],[80-100,0\mathrm{.}35]]$

The bed is fluidized by a gas with density of $1\mathrm{.}5k\frac{g}{m^3}$ at a superficial gas velocity of $0\mathrm{.}5\frac{m}{s}.$ The fluid bed has a cross$-$sectional area of $1\mathrm{.}5m^2.$

Use a discrete time interval of $3$ mins to determine the following:

a$)$ The size distribution of the bed at $15$mins and $30$mins.

b$)$ The total mass of solids lost at those times.

c$)$ The maximum solids loading at the exit.