EXAMPLE $24\mathrm{.}2-4.$ Filtration in a Continuous Rotary$-$Drum Filter A rotary$-$vacuum$-$drum filter having a $33\%$ submergence of the drum in the slurry is to be used to filter a $CaC{O}_3$ slurry as given in Example $24\mathrm{.}2-1$ using a pressure drop of $67\mathrm{.}0$kPa. The solids concentration in the slurry is $c_x=0\mathrm{.}191kg$ solid $\frac{?}{kg}$ slurry and the filter cake is such that the $kg$ wet cake $\frac{?}{kg}$ dry cake $=m=2\mathrm{.}0.$ The density and viscosity of the filtrate can be assumed to be those of water at $298\mathrm{.}2K.$ Calculate the filter area needed to filter $0\mathrm{.}778kg$ slurry$/$s$.$ The filter$-$cycle time is $250s.$ The specific cake resistance can be represented by $=(4\mathrm{.}37{10}^9)(-p{)}^{0\mathrm{.}3},$ where $-p$ is in $Pa$ and $$ is in $\frac{m}{k}g.$

Solution: From Appendix A$.2$ for water, $=996\mathrm{.}9k\frac{g}{m^3},=0\mathrm{.}8937{10}^{-3}Pa*$ s$.$ From Eq$.(24\mathrm{.}2-10),$

$c_s=\frac{c_x}{1-m{c}_x}=\frac{996\mathrm{.}9(0\mathrm{.}191)}{1-(2\mathrm{.}0)(0\mathrm{.}191)}=308\mathrm{.}1kg$ solids $\frac{?}{m^3}$ filtrate

Solving for $,=(4\mathrm{.}37{10}^9)(67\mathrm{.}0{10}^3{)}^{0\mathrm{.}3}=1\mathrm{.}225{10}^{11}\frac{m}{k}g.$ To calculate the flow rate of the filtrate,

$\frac{V}{t_c}=0\mathrm{.}778\frac{c_x}{c_s}$

$=(0\mathrm{.}778\frac{kg\text{s}\text{l}\text{u}\text{r}\text{r}\text{y}}{s})\frac{0\mathrm{.}191\frac{kg\text{s}\text{o}\text{l}\text{i}\text{d}}{kg\text{s}\text{l}\text{u}\text{r}\text{r}\text{y}}}{\frac{1}{308\mathrm{.}1(kg)soli\frac{d}{m^3}\text{f}\text{i}\text{l}\text{t}\text{r}\text{a}\text{t}\text{e}}}$

$=4\mathrm{.}823{10}^{-4}{m}^3$ filtrate $\frac{?}{s}$

Substituting into Eq$.(24\mathrm{.}2-24),$ neglecting and setting $B=0,$ and solving,

$\frac{V}{t_c}=\frac{4\mathrm{.}823{10}^{-4}}{A}=[\frac{2(0\mathrm{.}33)(67\mathrm{.}0{10}^3)}{250(0\mathrm{.}8937{10}^{-3})(1\mathrm{.}225{10}^{11})(308\mathrm{.}1)}{]}^{\frac{1}{2}}$

Hence, $A=6\mathrm{.}60m^2.$

$24\mathrm{.}2-9.$ Throughput in a Continuous Rotary$-$Drum Filter. A rotary$-$drum filter having an area of $2\mathrm{.}20m^2$ is to be used to filter the $CaC{O}_3$ slurry given in Example $24\mathrm{.}2-4.$ The drum has a $28\%$ submergence and the filter$-$cycle time is $300s.$ A pressure drop of $62\mathrm{.}0k\frac{N}{m^2}$ is to be used. Calculate the slurry feed rate in $kg$ slurry$/$s for the following cases:

a$.$ Neglect the filter$-$medium resistance.