A rotary batch dryer is initially charged with $500$ kg granular food material containing $16\%$ moisture by mass. At this time, $100$ kg$/$ min of dry air with absolute humidity of $0\mathrm{.}01$ kg moisture$/$ kg dry air enters the dryer and leaves with the same flow rate. The rate of drying $($kg moisture$/$ hour$)$ is known to be proportional $($ X$-$XE$),$ where X and XE are the moisture content and the equilibrium moisture content of the food material $($ both in kg moisture $/$ kg dry solid$)$ respectivel. The equilibrium moisture content of food material is given that XE$=0\mathrm{.}04$ kg H$2$O$/$ kg dry solid. The food material was absorve to contain $8\%$ moisture after $2$ hours of drying. Determine

a$)$ The moisture content x as function of time.

b$)$ The time, in hours, needed to dry the material to $6\%$moisture by mass.

c$)$ The humidity $($ kg moisture$/$ kg dry air$)$ of the exit air as a function of time.