Basic Mechanical Engineering (3110006)

Question-3b-OR

BE | Semester-1 Summer-2020 | 11-06-2020

Q3) (b)

Figure shows the Otto cycle plotted on p – V diagram. The Otto cycle has following main processes:

Adiabatic compression process (1 – 2)

At state point 1 cylinder is full of gas with volume $V_{1}$ , pressure $p_{1}$ and temperaure $T_{1}$ .
Piston moves from BDC to TDC and an ideal gas is compressed adiabatially to state point 2 through compression ratio, $r = \frac{V_{1}}{V_{2}}$ .
For adiabatic compression process (1 – 2), $p_{1} V_{1}^{γ} = p_{2} V_{2}^{γ}$
and $\frac{T_{2}}{T_{1}} = {(\frac{V_{1}}{V_{2}})}^{γ - 1} = r^{γ - 1}$

$T_{1} = \frac{T_{2}}{r^{γ - 1}}$

Constant volume heat addition (2 – 3)

Heat is added at constant volume from an external heat source, both pressure and temperature are increases.
$\frac{p_{2}}{T_{2}} = \frac{p_{3}}{T_{3}}$
Heat supplied to the gas during constant volume process (2 – 3),
$Q_{s} = Q_{2 - 3} = {mC}_{v} (T_{3} - T_{2})$

Adiabatic expansion process (3 – 4)

The increased high pressure exerts a greater amount of force on the piston and pushes it towards the BDC.
Expansion of working fluid takes place isentropically and work done by the gas.
The volume ratio $r = \frac{V_{4}}{V_{3}}$ is called expansion ratio.
For adiabatic expansion process (3 – 4), $p_{3} V_{3}^{γ} = p_{4} V_{4}^{γ}$
and $\frac{T_{3}}{T_{4}} = {(\frac{V_{4}}{V_{3}})}^{γ - 1} = {(\frac{V_{1}}{V_{2}})}^{γ - 1} = r^{γ - 1}$

$T_{4} = \frac{T_{3}}{r^{γ - 1}}$

Constant volume heat addition (4 – 1)

Heat is rejected to the external sink at constant volume. This process is so controlled that ultimately the working fluid comes to its initial state 1 and the cycle is repeated
$\frac{p_{4}}{T_{4}} = \frac{p_{1}}{T_{1}}$
Heat rejected from the gas during constant volume process (4 – 1),
$Q_{r} = Q_{4 - 1} = {mC}_{v} (T_{4} - T_{1})$

The expression of efficiency of Otto cycle:

η = \frac{Net work done}{Heat Supplied} = \frac{W_{net}}{Q_{s}} = \frac{Q_{s} - Q_{r}}{Q_{s}}

η = \frac{{mC}_{v} (T_{3} - T_{2}) - {mC}_{v} (T_{4} - T_{1})}{{mC}_{v} (T_{3} - T_{2})}

η = 1 - \frac{(T_{4} - T_{1})}{(T_{3} - T_{2})}

Substitute the value of

T_{1}

and

T_{4}

η = 1 - \frac{(\frac{T_{4}}{r^{γ - 1}} - \frac{T_{2}}{r^{γ - 1}})}{(T_{3} - T_{2})}

η = 1 - \frac{1}{r^{γ - 1}}

Above equation proved that the efficiency of Otto cycle is a function of compression ratio.