Question

A diesel engine has following specifications:
i) Compression ratio (r) = 14:1
ii) Fuel cut off at 0.08 of the stroke
iii) Relative efficiency = 52%
What is the cut off ratio for the engine

• A. 1.02
• B. 2.04
• C. 3.06
• D. 4.08

Hint:

Always remember the direct formula:
\[ \rho = 1 + \text{cut-off fraction} \times (r - 1) \] This allows you to bypass volume derivations and calculate the answer in under a minute.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The cut-off ratio ($\rho$) of a Diesel cycle is the ratio of the cylinder volume at the end of the combustion process (fuel injection cut-off) to the clearance volume.
It is related to the compression ratio and the cut-off point, which is expressed as a fraction of the engine's stroke.
Key Formula or Approach:
The volumes are related by:
\[ r = \frac{V_1}{V_c} = \frac{V_c + V_s}{V_c} = 1 + \frac{V_s}{V_c} \implies \frac{V_s}{V_c} = r - 1 \] The volume at cut-off ($V_3$) is:
\[ V_3 = V_c + x \cdot V_s \] The cut-off ratio ($\rho$) is:
\[ \rho = \frac{V_3}{V_c} = 1 + x \cdot \left( \frac{V_s}{V_c} \right) = 1 + x \cdot (r - 1) \] where:
$r$ = compression ratio = 14
$x$ = cut-off fraction of the stroke = 0.08
$V_s$ = swept (stroke) volume
$V_c$ = clearance volume

Step 2: Detailed Explanation:
Substitute the given values into the cut-off ratio formula:
\[ \rho = 1 + 0.08 \times (14 - 1) \] \[ \rho = 1 + 0.08 \times 13 \] \[ \rho = 1 + 1.04 = 2.04 \] The relative efficiency of $52\%$ is extra information and is not needed to calculate the geometric cut-off ratio.

Step 3: Final Answer:
The cut-off ratio ($\rho$) for the engine is $2.04$.