A plug flow reactor packed with cell flocs is used for the production of products. The diameter of the reactor is $8m$ and the substrate is converted into the product only. The size of the cell floc is small enough to neglect the boundary layer effect. Inlet flow does not contain the product and operation is being operated in a steady state condition. Answer the following questions using the data given below.

Data:

$F=300\frac{L}{h}r,S_0=100\frac{g}{L},q_p=0\mathrm{.}3g\frac{P}{g}x-hr,$

$Y_{\frac{P}{S}}=0\mathrm{.}4mg\frac{P}{m}gS,x=50\frac{g}{L}$

Derive the expression for the substrate concentration profile in the axial direction.

Derive the expression for the product concentration profile in the axial direction.

Determine the length of the reactor $(m)$ for $90\%$ conversion of the substrate.