Derive the efficiency of air standard Otto cycle.

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written 3.4 years ago by

DevarenjiniP • 3.2k

• modified 3.4 years ago

Otto cycle

1-2 Isentropic Compression

2-3 Constant Volume Heating

3-4 Isentropic Expansion

4-1 Constant Volume Heating

$η_.otto ={\frac{Heat supplied-Heat rejected}{Heat supplied}} \\$

$= \cfrac {m Cv (T3-T2) - m Cv (T4-T1)} {m Cv (T3-T2)}$ $$=\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 $\require{enclose} {\enclose{circle}{1}}\\$

$η_.otto ={\frac{1-T3.\cfrac{1}{r}^{\gamma -1}-T2.\cfrac{1}{r}^{\gamma -1}}{T3-T2}}\\$

$={\frac{1-\cfrac{1}{r}^{\gamma -1}{(T3-T2)}}{T3-T2}}=1-\cfrac{1}{r}^{\gamma -1}\\$

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