Field$-$scale tests using a completely mixed flow reactor $($CMFR$)$ have been conducted to determine the effectiveness of ozone in oxidizing color$-$producing compounds in a wastewater treatment operation. Ozone is sparged into a tank having an HRT of $10$ min $.$ The following empirical rate expression is thought to best describe the process:

$r=-k{C}^{0\mathrm{.}5}$

where C is a concentration of color$-$causing compounds expressed in terms of intensity as color units. The color units entering and leaving the ozone contactor have been measured in three separate tests with the results shown below.

$\backslash$table$[[$Test number,Color Units In$,$Color Units Out$],[1,8,1\mathrm{.}9],[2,15,4\mathrm{.}8],[3,22,8\mathrm{.}0]]$

These levels of removal are deemed unsatisfactory. The color entering the treatment plant typically averages about $10$ color units, and the design goal is to achieve a product water of $1$ color unit. What hydraulic residence time is needed to achieve the removal efficiency goal?