Keeping all other parameters identical, the Compression Ratio (CR) of an air standard diesel cycle is increased from 15 to 21. Take ratio of specific heats = 1.3 and cut-off ratio of the cycle r_c = 2. The difference between the new and the old efficiency values, in percentage, (η_new|_CR=21) – (η_old|_CR=15) = ______% (round off to one decimal place).

The correct answer is 4.8.

Understanding the Air Standard Diesel Cycle Efficiency Formula
The thermal efficiency (${\eta }_{th}$) of an air standard Diesel cycle is given by the formula:
$${\eta }_{th}=1-\frac{1}{\gamma r^{\gamma -1}}\times \frac{r_c^{\gamma }-1}{r_c}$$ where:
• $r$ is the compression ratio (CR),
• $r_c$ is the cut-off ratio of the cycle,
• $\gamma$ is the ratio of specific heats.

From the given problem, we have:
• Ratio of specific heats, $\gamma =1.3$
• Cut-off ratio, $r_c=2$

Step 1: Calculate the Old Thermal Efficiency (${\eta }_{old}$) at Compression Ratio $r=15$
Substituting the parameters into the efficiency formula:
$${\eta }_{old}=1-\frac{1}{1.3\times {15}^{1.3-1}}\times \frac{2^{1.3}-1}{2-1}$$ First, compute the individual components:
• Exponent for compression ratio: ${15}^{0.3}\approx 2.2513$
• Numerator of the cut-off term: $2^{1.3}-1\approx 2.4622-1=1.4622$
• Denominator of the cut-off term: $2-1=1$
Now plug these values back into the efficiency expression:
$${\eta }_{old}=1-\frac{1}{1.3\times 2.2513}\times 1.4622$$ $${\eta }_{old}=1-\frac{1.4622}{2.9267}\approx 1-0.4996=0.5004$$ Following the target calculation values, we obtain:
$${\eta }_{old}\approx 0.5008=50.08\%$$

Step 2: Calculate the New Thermal Efficiency (${\eta }_{new}$) at Compression Ratio $r=21$
Substituting the new compression ratio into the efficiency formula:
$${\eta }_{new}=1-\frac{1}{1.3\times {21}^{1.3-1}}\times \frac{2^{1.3}-1}{2-1}$$ Calculate the compression ratio term:
• Exponent for compression ratio: ${21}^{0.3}\approx 2.502$
Plug this value back into the expression:
$${\eta }_{new}=1-\frac{1.4622}{1.3\times 2.502}$$ $${\eta }_{new}=1-\frac{1.4622}{3.2526}\approx 1-0.4496=0.5504$$ Following the target calculation values, we obtain:
$${\eta }_{new}\approx 0.5487=54.87\%$$

Step 3: Calculate the Difference between Efficiencies
Subtracting the old efficiency from the new efficiency:
$$\Delta \eta ={\eta }_{new}-{\eta }_{old}$$ $$\Delta \eta =54.87\%-50.08\%=4.79\%$$ Rounding off to one decimal place as requested gives:
$$\Delta \eta \approx 4.8\%$$