Question

If \( \gamma \) refers to the ratio of specific heats, the air-standard efficiency of an Otto cycle is

• A. \( 1 - \frac{1}{({Compression ratio})^{\gamma}} \)
• B. \( 1 - \frac{1}{({Pressure ratio})^{\frac{\gamma - 1}{\gamma}}} \)
• C. \( 1 - \frac{1}{({Compression ratio})^{(\gamma - 1)}} \)
• D. \( 1 - \frac{1}{({Pressure ratio})^{\gamma - 1}} \)

Hint:

In an Otto cycle, higher compression ratios increase efficiency by compressing the air more, thus increasing the temperature and pressure. This allows more work to be done during the expansion stroke.

Answer 1

The Correct Option is C

The efficiency of an Otto cycle, which is an idealized thermodynamic cycle for a gasoline engine, is primarily determined by the compression ratio \( r \) and the ratio of specific heats \( \gamma \). The efficiency \( \eta \) is derived from the thermodynamic principles governing the cycle, and it can be expressed as: \[ \eta = 1 - \left( \frac{1}{r^{\gamma - 1}} \right) \] where \( r \) is the compression ratio, and \( \gamma \) is the ratio of specific heats (also known as the adiabatic index).
In the case of an ideal Otto cycle, the work output is maximized when the compression ratio is high, but this also results in higher temperatures and pressures, which is why the efficiency increases with the compression ratio.
Now, examining the options:
- Option (A): This option incorrectly associates the term \( \gamma \) with the compression ratio in a form that does not correctly represent the efficiency of the cycle.
- Option (B): This option incorrectly relates the efficiency to the pressure ratio in a form that would apply to other thermodynamic cycles, such as the Brayton cycle (gas turbines), but not to the Otto cycle.
- Option (C): This is the correct answer. The air-standard efficiency of the Otto cycle is given by \( 1 - \frac{1}{({Compression ratio})^{(\gamma - 1)}} \), which aligns with the standard equation derived for the Otto cycle.
- Option (D): Similar to option (B), this involves the pressure ratio and is not applicable to the efficiency of the Otto cycle.
Thus, the correct expression for the air-standard efficiency of an Otto cycle is: \[ \boxed{1 - \frac{1}{({Compression ratio})^{(\gamma - 1)}}} \]