An ideal gas with heat capacity ratio of 2 is used in an ideal Otto-cycle which operates between minimum and maximum temperatures of 200 K and 1800 K. What is the compression ratio of the cycle for maximum work output?

Concept:

An ideal Otto-cycle is shown on the T-S diagram 

T1 = Temperature at the compressor inlet (minimum temperature)

T2 =  Temperature at the compressor outlet

T3 = Temperature after the heat addition (maximum temperature)

T4 = Temperature after the expansion

1-2 is the isentropic process 

\(\begin{array}{l} \therefore {T_1}V_1^{\gamma - 1} = {T_2}V_2^{\gamma - 1}\\ \left( {\frac{{{T_1}}}{{{T_2}}}} \right) = {\left( {\frac{{{V_2}}}{{{V_1}}}} \right)^{\gamma - 1}}\\ \frac{{{V_1}}}{{{V_2}}} = {\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^{\frac{1}{{\gamma - 1}}}} \end{array}\)

\( \frac{{{V_1}}}{{{V_2}}} \) is the compression ratio

For maximum work output \({T_2} = {T_4} = \sqrt {{T_1}\;{T_3}} \)

\(r = {\left( {\frac{{{T_3}}}{{{T_1}}}} \right)^{\frac{1}{{2\left( {\gamma - 1} \right)}}}} \)

\(r = {\left( {\frac{{{T_{max}}}}{{{T_{min}}}}} \right)^{\frac{1}{{2\left( {\gamma - 1} \right)}}}} \)

Calculation:

Given:

Heat capacity ratio, γ = C_p/C_v = 2

T_max = 1800 K, T_min = 200 K

For maximum work output for Otto cycle:

\(r = {\left( {\frac{{{T_{max}}}}{{{T_{min}}}}} \right)^{\frac{1}{{2\left( {\gamma - 1} \right)}}}} = {\left( {\frac{{1800}}{{200}}} \right)^{\frac{1}{{2\left( {2 - 1} \right)}}}} = {9^{\frac{1}{2}}} = 3\)