Examine the process of sweetening coke$-$oven gas $($COG$)$ as shown in the given diagram. The

primary goal of this process is to eliminate acidic contaminants, especially hydrogensulfide, from

the COG$-$ a combination of $H2,CH4,CO,N2,NH3,CO2,$ and $H2S.$ Hydrogen sulfideis particularly

problematic due to its corrosive properties and its role in generating SO$2$ emissionswhen COG is

combusted. Given that ammonia is present in COG and selectively absorbs H$2$S$,$ aqueous

ammonia is proposed as an effective solvent $($identified as process lean$-$stream, S$1).$ It is

advantageous for the ammonia extracted from the sour gas to offset a significant amount of

ammonia lost in the system, thereby minimizing the requirement for additional ammonia.

Additionally, an external source of MSA $($chilled methanol, S$2)$ can be used alongside the

aqueous ammonia solution if necessary. The COG purification process entails scrubbing the sour

COG, R$1,$ using adequate amounts of aqueous ammonia and$/$or chilled methanol to capture the

necessary hydrogen sulfide. Acid gases are then removed from these solvents, and the refreshed

MSAs are circulated back into the system. The acid gases extracted are processed in a Claus unit

to recover elemental sulfur from the hydrogen sulfide. To comply with air pollution control

standards, the residual gases exiting the Claus unit, R$2,$ must be processed to further reduce any

remaining hydrogen sulfide. The data for these streams is summarized in below Table.

Using the pinch diagram with ${}_1={}_2=0\mathrm{.}0001,$ determine the minimum cost of MSAs

needed for the desulfurization of R$1$ and R$2.$ Also, identify the position of the pinch point by using

cascade analysis method.