(B) As the cut-off ratio $r_c$ increases at a fixed compression ratio $r$, the thermal efficiency decreases
(C) When $r_c \to 1$, the air-standard Diesel cycle efficiency approaches the Otto cycle efficiency for the same compression ratio
(D) Heat addition takes place at constant volume while heat rejection takes place at constant pressure
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
1. The Diesel cycle thermal efficiency formula is $\eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}} [\frac{r_c^\gamma - 1}{\gamma(r_c - 1)}]$. (Statement A True)
2. Since $\frac{r_c^\gamma - 1}{\gamma(r_c - 1)} > 1$ for all $r_c > 1$ and increases monotonically with $r_c$, increasing $r_c$ increases the bracketed factor, thereby reducing $\eta_{th}$. (Statement B True)
3. By L'Hôpital's rule, $\lim_{r_c \to 1} \frac{r_c^\gamma - 1}{\gamma(r_c - 1)} = 1$, which reduces to $1 - \frac{1}{r^{\gamma-1}}$, the Otto cycle efficiency. (Statement C True)
4. In the Diesel cycle, heat addition occurs at constant pressure ($P = C$), and heat rejection occurs at constant volume ($v = C$). (Statement D False)