A solid feed consisting of $20$ wt$.\%$ of $1-$mm particles and smaller, $30$ wt$.\%$ of $2-$mm particles and $50$ wt$.\%$ of$4-$mm particles, passes through a rotating tubular reactor where it reacts with gas to give a hard nonfriable solid$($shrinking core model$/$reaction controls$)$ t $=4$h for $4$mm$-$particle. Find the residence time needed for $100\%$conversion of solids.