OTTO CYCLE PROCESSES

1-2 Isentropic Compression

2-3 Constant Volume Heating

3-4 Isentropic Expansion

4-1 Constant Volume Heating

OTTO CYCLE EFFICIENCY ($η_.otto$)

$$η_.otto ={\frac{Heat supplied-Heat rejected}{Heat supplied}} \\$$ $$ = \cfrac {m Cv (T3-T2) - m Cv (T4-T1)} {m Cv (T3-T2)}$$ $$=\cfrac{T3-T2-(T4-T1)}{T3-T2} \$$

$$=1-\frac{(T4-T1)}{T3-T2}\rightarrow \require{enclose} {\enclose{circle}{1}}\\ $$

Now,we have, Compression ratio, $$ r =\cfrac{V1}{V2}$$

Expansion ratio, $$ r1 =\cfrac{V4}{V3}$$

∴ V1=V4 and V2=V3

Compression ratio=Expansion ratio

For adiabatic compression process ,1-2 $$ \cfrac{T2}{T1}=\cfrac{V1}{V2}^{\gamma -1}$$ $$= r^{\gamma -1}$$

For adiabatic expansion process ,3-4 $$ \cfrac{T4}{T3}=\cfrac{V3}{V4}^{\gamma -1}$$ $$= \cfrac{V2}{V1}^{\gamma -1}$$ $$=\cfrac{1}{r}^{\gamma -1}$$

Substitute the above values in $ \require{enclose} {\enclose{circle}{1}}\\ $ $$η_.otto ={\frac{1-T3.\cfrac{1}{r}^{\gamma -1}-T2.\cfrac{1}{r}^{\gamma -1}}{T3-T2}}\\ $$ $$ ={\frac{1-\cfrac{1}{r}^{\gamma -1}{(T3-T2)}}{T3-T2}}=1-\cfrac{1}{r}^{\gamma -1}\\ $$