Isooctane $(2,2,4-$trimethylpentane$)$ is one of several octane isomers whose chemical formula is$($CH:$)3$CCCH$($CH$3)2.$ Isooctane is the standard $100$ point on the octane rating scale for gasoline fuel efficiency. Therefore, it is frequently added to gasoline in relatively large proportions to increase the knock resistance of fuel.One of the main processes of isooctane production is the dimerization and hydrogenation of isobutene. The balanced chemical reaction is:CH$,2$ CH$,-$ C$=$CH$,+$ H$2-$CH$,$CH$,$CH$,$ CCH CH CH$,$CH$,$An undesirable reaction of hydrogenation of isobutene to isobutane takes place as follows:CH$,-$ C$=$ CH$,+$ H$2$:CH$,$CH$,-$ CH $-$ CHThe flowsheet of the process is shown in figure $1.$ Two fresh feed streams are fed to the process, the first is isobutane that contains a small percentage of inerts. The second feed stream is pure hydrogen. The two fresh feed streams are mixed in the first mixer $($M$-1),$ and the product from the first mixer is mixed with the recycle stream in the second mixer producing the reactor feed. The flow rate of the reactor feed stream is $6000$ mol$/$hr$,$ the mole ratio of isobutene to hydrogen is$1\mathrm{.}71$:$1,$ and the compositions of isobutene and isobutane in the same stream are $56\mathrm{.}31\%$ and $1\mathrm{.}88\%$ respectively. The conversion of isobutene in the reactor is found to be $80\%$ and the yield of isooctane is $76\%.$The reactor effluent is fed to the first distillation column which separates all the hydrogen, $65\%$ of isobutane, $60\%$ of isobutene and $50\%$ of the inerts in the top product. The bottom product from the first distillation column is fed to the second distillation column which separates all the isobutaneand isobutene in the top product and all the isooctane in the bottom product. The composition of the final product isooctane is $97\mathrm{.}6$ mole $\%.$The top product from the first distillation column is fed to a splitter to purge $30\%$ of the stream to reduce the inerts concentration in the reactor feed. The recycle stream is mixed with the mixed fresh feed in the second mixer.Note that some streams do not contain the components name written on them, you can find themout yourself. The written components on a stream are ALL the components present in that streams.Do not add components to those already written on a stream

Calculate the composition of feed isobutene stream

$.$ Calculate the overall conversion of isobutane