Air-standard cycle relations: “The cut-off ratio is the reciprocal of the expansion ratio.” Assess this statement for an ideal Diesel cycle.

Introduction / Context:In the Diesel cycle, three ratios appear repeatedly: compression ratio r, cut-off ratio r_c, and expansion ratio r_e. Confusing their relationships can derail efficiency and work computations. This question examines the link among these quantities.

Given Data / Assumptions:

• Ideal air-standard Diesel cycle.
• State 2 (end of compression) at volume V_2; State 3 (end of heat addition) at volume V_3.
• State 1 at BDC (V_1 = V_s + V_c); State 4 at end of expansion (V_4).

Concept / Approach:By definition: r = V_1 / V_2, r_c = V_3 / V_2, and r_e = V_4 / V_3. Also, for the ideal cycle, V_4 = V_1 and V_2 = V_c. Hence r_e = (V_4 / V_3) = (V_1 / V_3) = (V_1 / V_2) / (V_3 / V_2) = r / r_c. Therefore, r_c is not the reciprocal of r_e unless r = 1, which never holds in practice. The statement is thus incorrect.

Step-by-Step Solution:Write definitions: r = V_1/V_2; r_c = V_3/V_2; r_e = V_4/V_3.Use volume identities: V_4 = V_1 (for a closed cycle), so r_e = (V_1/V_2) / (V_3/V_2).Simplify to r_e = r / r_c.Conclude: r_c = r / r_e → not simply 1/r_e.

Verification / Alternative check:Plug sample numbers: if r = 16 and r_c = 2, then r_e = 8. Clearly r_c (2) is not 1/r_e (0.125).

Why Other Options Are Wrong:“Correct”: contradicts r_e = r / r_c.

“Correct for Otto”: Otto cycle has no cut-off ratio at constant pressure; statement is inapplicable.

“Equals compression ratio”: not true; r is independent of r_c except via r_e relation.

Common Pitfalls:Assuming every pair of named ratios are reciprocals; forgetting that Diesel adds heat at constant pressure, introducing r_c explicitly.

Final Answer:Incorrect