Otto Cycle Efficiency Basics

This question involves calculating the thermal efficiency of an Otto cycle. The provided data includes the compression ratio ($r = 9$), initial pressure ($P_1 = 100$ kPa), initial temperature ($T_1 = 10\text{°C}$), and the highest temperature reached during heat addition ($T_3 = 2500$ K).

For an ideal Otto cycle, the thermal efficiency ($\eta_{th}$) is determined by the compression ratio ($r$) and the ratio of specific heats ($\gamma$). The standard formula is:

$$ \eta_{th} = 1 - \frac{1}{r^{(\gamma-1)}} $$

For calculations involving air as the working fluid (air-standard analysis), the value of $\gamma$ is typically assumed to be $1.4$.

Calculating Thermal Efficiency for Otto Cycle (r=9)

Identify Necessary Parameters:

• Compression Ratio, $r = 9$
• Specific Heat Ratio, $\gamma = 1.4$

It's important to note that the initial conditions ($P_1$, $T_1$) and the maximum temperature ($T_3$) are not required to determine the ideal thermal efficiency using the standard formula based on the compression ratio. These values would be relevant for other performance calculations like work output or heat rejected.

Step 1: Input values into the efficiency formula.

Substitute the known values of $r$ and $\gamma$ into the formula:

$$ \eta_{th} = 1 - \frac{1}{9^{(1.4-1)}} $$

Step 2: Simplify the exponent term.

Calculate the exponent value:

$$ \gamma - 1 = 1.4 - 1 = 0.4 $$

The formula simplifies to:

$$ \eta_{th} = 1 - \frac{1}{9^{0.4}} $$

Step 3: Evaluate the term $9^{0.4}$.

Using a calculator or logarithm rules, we find:

$$ 9^{0.4} \approx 2.40817 $$

Step 4: Calculate the thermal efficiency.

Substitute the calculated value back into the equation:

$$ \eta_{th} = 1 - \frac{1}{2.40817} $$

$$ \eta_{th} \approx 1 - 0.41525 $$

$$ \eta_{th} \approx 0.58475 $$

Step 5: Express the efficiency in percentage.

To express the efficiency as a percentage, multiply by 100:

$$ \eta_{th} \approx 0.58475 \times 100\% = 58.475\% $$

Final Otto Cycle Efficiency Result

The calculation shows that the thermal efficiency of the Otto cycle with a compression ratio of 9 is approximately 58.475%. This value corresponds closely to the option 58.47%.