I. Analysis on Combustion of Methane $(100$ Points$)$

Recall the analysis that was done on the combustion of $C{H}_4$ for the cogeneration cycle $($Bejan

book$).$ The reaction resulted in a fuel$-$to$-$air ratio of $0\mathrm{.}0321,$ where the air inlet temperature

to the combustion chamber was $850K,$ and the combustion was to proceed at a temperature

of $1520K.$ The amount of heat loss from the combustion chamber was approximated as being

$2\%$ of the methane lower heating value $($LHV$).$ For that case, the reaction would have been:

$C{H}_4+6\mathrm{.}4143O_2+24\mathrm{.}1371N_2+0\mathrm{.}0093C{O}_2+5\mathrm{.}9190H_{2}O(g)$

$4\mathrm{.}4143O_2+24\mathrm{.}1371N_2+1\mathrm{.}0093C{O}_2+2\mathrm{.}5919H_{2}O(g)$

Recall for the stoichiometric reaction of $C{H}_4,O_2$ ath $=2.$ Recall excess $O_2$ is:

$\%$ Excess $O_2=\frac{(\text{m}\text{o}\text{l}\text{e}\text{s}{O}_2)-(\text{m}\text{o}\text{l}\text{e}\text{s}{O}_2{)}_{\text{t}\text{h}\text{e}\text{o}\text{r}\text{.}}}{(\text{m}\text{o}\text{l}\text{e}\text{s}{O}_2{)}_{\text{t}\text{h}\text{a}\text{n}\text{r}}}100$

Re$-$do the combustion analysis for the following cases $($for all cases, assume the inlet

temperature $T_3$ to remain at $850K)$:

For the above $0\mathrm{.}0321$ fuel$-$to$-$air ratio, determine what the combustion temperature $T_4$

would have bee if the combustion chamber was to be perfectly insulated.

If the reaction proceeded with only $20\%$ excess $O_2($i$.$e$.20\%$ excess air$)$ how high

would the combustion temperature get? Assume the combustion chamber to be

insulated in this case.

If the reaction proceeded with only $5\%$ excess $O_2,$ how high would the combustion

temperature get? Again, assume the combustion chamber to be insulated.

What happens to the combustion temperature if the reaction is stiochrometric?