Product $P$ is produced from reactant $R$ according to the reaction

$2RP+W$

Unfortunately, both the reactant and the product decompose to form byproduct B according to the reactions

$RB+P$

$P2B+W$

Only $50\%$ conversion of $R$ is achieved in the reactor when a fresh feed containing $1$ mole inert I to $11$ moles $R$ is used. The unreacted $R$ and inerts are separated from the reactor products and recycled. Some of the unreacted $R$ and inerts are purged to limit the inerts level in the combined reactor feed to $12$mol$\%.$ If the reactor outlet analyzes $21$ mole $\%P$ and $7$mole$\%B,$ do the following:

a$.$ Perform degrees of freedom analysis on all the units and the overall process.

b$.$ Calculate all the flow rates in the process for a fresh feed rate of $100mo\frac{l}{h}r.$

c$.$ Calculate the overall conversion of $R.$