Question

For the same compression ratio, the efficiency of diesel cycle approaches that of otto cycle as the Cut-off ratio approaches

• A. One
• B. Zero
• C. Infinity
• D. Ten

Hint:

At a constant compression ratio, the Otto cycle is always more efficient than the Diesel cycle.
As the cut-off ratio decreases toward 1, the Diesel cycle behaves more like an Otto cycle, and its efficiency increases.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The Otto and Diesel cycles are thermodynamic models used to evaluate spark-ignition and compression-ignition engines.
We can compare their thermal efficiencies by analyzing their respective equations under identical compression ratios.
Key Formula or Approach:
The thermal efficiency of the Diesel cycle (\(\eta_{\text{diesel}}\)) is given by: \[ \eta_{\text{diesel}} = 1 - \frac{1}{r^{\gamma-1}} \left[ \frac{\rho^\gamma - 1}{\gamma(\rho - 1)} \right] \] where:
- \(r\) is the compression ratio.
- \(\gamma\) is the specific heat ratio.
- \(\rho\) is the cut-off ratio.
The thermal efficiency of the Otto cycle (\(\eta_{\text{otto}}\)) is: \[ \eta_{\text{otto}} = 1 - \frac{1}{r^{\gamma-1}} \]

Step 2: Detailed Explanation:
The cut-off ratio (\(\rho\)) is the ratio of cylinder volumes after and before the combustion process.
Let us find the limit of the bracketed term in the Diesel efficiency equation as the cut-off ratio (\(\rho\)) approaches 1: \[ \lim_{\rho \to 1} \frac{\rho^\gamma - 1}{\gamma(\rho - 1)} \] Applying L'Hôpital's Rule to resolve the \(0/0\) indeterminate form: \[ \lim_{\rho \to 1} \frac{\frac{d}{d\rho}(\rho^\gamma - 1)}{\frac{d}{d\rho}(\gamma(\rho - 1))} = \lim_{\rho \to 1} \frac{\gamma \cdot \rho^{\gamma-1}}{\gamma} = \lim_{\rho \to 1} \rho^{\gamma-1} = 1^{\gamma-1} = 1 \] Substitute this limit back into the Diesel efficiency equation: \[ \eta_{\text{diesel}} \to 1 - \frac{1}{r^{\gamma-1}} \cdot (1) = \eta_{\text{otto}} \] This mathematical proof shows that as the cut-off ratio approaches 1, the combustion duration becomes zero, and the constant-pressure heat addition of the Diesel cycle becomes the constant-volume heat addition of the Otto cycle.

Step 3: Final Answer:
The efficiency of the Diesel cycle approaches that of the Otto cycle as the cut-off ratio approaches one.