Styrene $($C$8$H$8)$ is produced from dehydrogenation reaction of ethylbenzene $($C$8$H$10)$ with the present of catalyst. A general procedure of producing it is as follows:

A fresh feed stream consists of C$8$H$10$ and steam $($H$2$O$)$ at a mole ratio of $1$:$1$ is mixed with a recycle stream and enters a reactor $($R$).$ The steam acts as an inert and cleaning compound for the catalyst surface. The production process is designed for $70-80\%$ overall conversion and single$-$pass conversion of $30-40\%.$ The reactor outlet stream enters a first separator $($S$1)$ which produces $2$ streams. The first stream consists of H$2$ and H$2$O is discarded for further process, while the second stream consists of C$8$H$8$ and C$8$H$10$ enters a second separator $($S$2).$ The S$2$ produces $2$ streams, the first stream consists of C$8$H$8$ and C$8$H$10$ is collected as product while the second stream consists of $90$ mol$\%$ C$8$H$10$ and the balance C$8$H$8$ is recycled and mixed with the fresh feed stream in a mixer $($M$)$

$-$ write a material balance proposal for the plant. The material balance proposal should consider the following steps:

a$)$ A process$($ flow diagram$)$ of the process and the chemical reaction involved.

b$)$ Evaluate material balance of the whole process by stating your own basis of calculation. This include calculating all $($molar flow rates of the stream $)$and $($compositions for each stream$).$ Validate all answer.