A food air drying was dried in a tray dryer with a surface area of $0\mathrm{.}4m^2.$ The sample weight in the completely dry state is $2,345kg$ of dry solids. The wet sample weight at equilibrium was measured as $2,620kg($water $+$ dry solid$).$ In the drying test, the following data were obtained.

$\begin{array}{cc}\text{T}\text{i}\text{m}\text{e}(Hours)& \text{m}\text{a}\text{s}\text{s}(kilogram)\\ 0& 3\mathrm{.}644\\ 0\mathrm{.}30& 3\mathrm{.}500\\ 0\mathrm{.}50& 3\mathrm{.}382\\ 0\mathrm{.}83& 3\mathrm{.}190\\ 1\mathrm{.}13& 3\mathrm{.}015\\ 2\mathrm{.}20& 2\mathrm{.}905\\ 3\mathrm{.}50& 2\mathrm{.}770\\ 5\mathrm{.}50& 2\mathrm{.}700\\ 8\mathrm{.}0& 2\mathrm{.}650\\ 12\mathrm{.}0& 2\mathrm{.}620\end{array}$

a$)$ Calculate the free moisture content $x(kg$ water $/$kg dry solid$)$ for each data and plot $x$ versus $t.($Hint:To calculate the free moisture content, please consider the equilibrium moisture content$)$

b$)$ Measure the slopes, calculate the drying rate $($ kg of water$/m^2.h)$ and plot the graph $R$ versus $x.$

c$)$ Using this drying rate curve, we can calculate the total time required to dry the sample from $x=0\mathrm{.}3$ to $x=0\mathrm{.}04$ Calculate.

Use graphical integration for a period of decreasing speed.