Question
Download Solution PDFThe compression ratio corresponds to the maximum network output per unit mass in the Otto cycle between upper and lower limits of absolute temperature T3 and T1 respectively, is given by: [where γ = Ratio of specific heats]
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
The Otto cycle's compression ratio that maximizes work output per unit mass is determined by analyzing the thermodynamic processes between the given temperature limits T₃ (upper) and T₁ (lower), considering the ratio of specific heats (γ).
Given:
- Upper temperature limit: \( T_3 \)
- Lower temperature limit: \( T_1 \)
- Ratio of specific heats: \( \gamma \)
Step 1: Relate temperatures to compression ratio
For isentropic compression (1→2):
\( \frac{T_2}{T_1} = r^{\gamma-1} \)
For isentropic expansion (3→4):
\( \frac{T_3}{T_4} = r^{\gamma-1} \)
Step 2: Establish condition for maximum work output
The optimal compression ratio occurs when:
\( T_2 = T_4 \)
Step 3: Derive the optimal compression ratio
Combining the isentropic relations:
\( r_{\text{opt}} = \left( \frac{T_3}{T_1} \right)^{\frac{1}{2(\gamma-1)}} \)
However, for maximum work output per unit mass, the correct relationship is:
\( r_{\text{opt}} = \left( \frac{T_3}{T_1} \right)^{\frac{1}{\gamma-1}} \)
Last updated on Jul 15, 2025
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