$8\mathrm{.}30$ A synthesis gas is upgraded to a higher methane content in the recycle system shown in Figure P$8\mathrm{.}30.$ The feed gas contains a small amount of methane and analyzes $22\%CO,13\%CO,$ and $65\%H_2(mol\%)$ on a methane$-$free basis. The product stream $($stream $6)$ analyzes $($all mol $\%)$ CO $5\%,H_{2}9\%,C{H}_{4}50\%,C{O}_{2}27\%,$ and $H_{2}O9\%.$ Both feed and product are gases at $200F.$ In the reactor the two reactions

$CO+3H_{2}C{H}_4+H_{2}O$

$CO+H_{2}OC{O}_2+H_2$

take place. The reactor effluent is cooled to $500F$ in exchanger $1$ and then is further cooled in exchanger $2.$ Part of the effluent is split off as product. The remainder is sent to a separator in which the stream is cooled to $90F$

Problems

$535$

Figure P$8\mathrm{.}30$

and $95\%$ of the $H_{2}O$ is separated as liquid. The remaining gas stream is reheated in exchanger $2,$ mixed with the fresh feed, and returned to the reactor. Assume that the mixer, the splitter, exchanger $2,$ and the reactor operate adiabatically. Use average $C_p$ values as given on the flowsheet.

$($a$)$ Construct a degree$-$of$-$freedom table and show that the process is correctly specified.

$($b$)$ Determine a calculation order for the problem.

$($c$)$ Calculate the reactor outlet temperature.