Question

The thermal efficiency of an ideal air-standard Otto cycle is 0.5. The value of specific heat ratio of air is 1.4. The compression ratio of the cycle is _________ (rounded off to 1 decimal place).

Hint:

For Otto cycle efficiency, remember that a higher compression ratio increases efficiency, but it must be balanced with engine performance and material limits.

Answer 1

The thermal efficiency \( \eta \) of an Otto cycle is given by: \[ \eta = 1 - \frac{1}{r^{\gamma - 1}} \] Where:
\( \eta \) is the thermal efficiency (0.5),
\( r \) is the compression ratio,
\( \gamma \) is the specific heat ratio (1.4).
Rearranging the formula to solve for \( r \): \[ 0.5 = 1 - \frac{1}{r^{1.4 - 1}} \] \[ 0.5 = 1 - \frac{1}{r^{0.4}} \] \[ 0.5 = \frac{r^{0.4} - 1}{r^{0.4}} \] \[ 0.5r^{0.4} = r^{0.4} - 1 \] \[ r^{0.4} = 2 \] \[ r = 2^{\frac{1}{0.4}} = 2^{2.5} \approx 5.66 \] Thus, the compression ratio is approximately \( \boxed{5.7} \).