Currently, the significant concern within the energy industry is the carbon emissions stemming from natural gas power plants, highlighting the urgency for process optimization. Power plants generate high$-$pressure and high$-$temperature steam $($T$=600$ and P$=57$ bar$)$ in a boiler, utilizing the heat derived from burning natural gas$($CH$4)*$ in a furnace. The high temperature steam from the boiler is introduced to a turbine to harness power $($total plant power capacity $=25\mathrm{.}4$ MW$).$ The saturated vapour leaving the turbine at $9\mathrm{.}3$ bar is directed into a condenser to decrease the final temperature and pressure of the stream to $60$ and $8$ bar respectively. The condensed water is pressurized to the final value of $61$ bar with temperature of $73$ using pumps. The high pressure water from the pumping site is introduced to an economizer $($heat exchanger$)$ to preheat the fluid reaching the temperature of $160$ considering $2$ bar pressure head loss, and finally, the hot water is recycled back to the boiler to produce steam.

$*$The heat requirement for the boiler would be provided by the furnace burring natural gas $($CH$4)$ as shown in equation $(1).$ It should be noted that $35\%$ of the heat reaction would be wasted in the furnace due to inefficiency of heat isolation. Moreover, the flue gas $($mixture of water and CO$2)$ from the furnace with $930$ temperature would be used as source of heat in the economizer to preheat the water stream before entering the boiler.

Finally, the cooling water required for the condenser in the plant is provided from a lake nearby with $23$ temperature which would be heated up to $77$ in the heat exchanger and recycled back to the lake.

CH$4+2$ O$2->$ CO$2+2$ H$2$O $+890$ kj$,$

heat of combustion $=890$ kj$/1$mol CH$4$

$,$Flue gas mixture: CO$2$ & H$2$O

Boiler efficiency in heat transfer $65\%$

$($i$)$ Draw a process flowchart of power plant to produce $25\mathrm{.}4$ MW power from natural gas, labelling all the knowns and unknowns and state all your assumptions.

$($ii$)$ Write and solve the relevant energy or mass balance equations to answer the following, stating all your assumptions:

$$ How much natural gas in terms of kg$/$s is required to produce the power?

$$ What is the mass flow rate of water being recycled in the power plant $($kg$/$s$)?$

$$ What would be the power requirement for pumping site, economizer preheating system, steam boiler, and condenser in terms of MW$?$

$$ How much cooling water $($kg$/$s$)$ is required in the heat exchanger to condense the output steam from the turbine?

$$ What would be the final temperature of the flue gas leaving the economizer heat exchanger?

Please provide more caculations on unknowns and knowns.