To determine the compression ratio ($r$) for the gasoline engine, we need to use the air-standard Otto cycle model, which is commonly used to represent gasoline engines. The compression ratio is given by the formula:

$r = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$

Where:

• $T_2$ is the temperature at the end of the heat rejection process (which is related to the lower temperature of the cycle),
• $T_1$ is the temperature at the beginning of compression,
• $\gamma$ is the ratio of specific heats (for air, $\gamma \approx 1.4$).

Given:

1. Work per unit mass ($W = 431 \, \text{BTU/lb}$).
2. Maximum temperature in the cycle ($T_3 = 3273 \, \text{K}$).
3. Temperature at the end of constant volume heat rejection ($T_2 = 1112 \, \text{°F}$).

Step-by-step approach:

1. Convert the given temperatures to consistent units (Kelvin):

   • $T_2 = 1112 \, \text{°F}$ → Convert to Kelvin: $T_2 = \left(1112 - 32\right) \times \frac{5}{9} + 273.15 = 866.67 + 273.15 = 1139.82 \, \text{K}$

   • $T_3 = 3273 \, \text{K}$.

2. Assumptions: Since we have the maximum temperature $T_3$ and temperature after heat rejection $T_2$, we can apply the following Otto cycle relationships.

   The work done in the cycle $W$ is related to the temperature difference: $W = c_v (T_3 - T_2)$ where $c_v$ is the specific heat at constant volume (for air, $c_v \approx 0.171 \, \text{BTU/lb-°R}$).

   We can solve this equation for $T_3$ and then proceed to find the compression ratio.

Let me calculate that for you.The compression ratio ($r$) of the gasoline engine is approximately 171.58.

Would you like further details or clarifications? Here are some related questions that can expand your understanding:

1. How does the compression ratio affect engine efficiency?
2. What is the significance of the air-standard Otto cycle in engine design?
3. How does the specific heat ratio ($\gamma$) influence the engine's performance?
4. What assumptions are made in the air-standard Otto cycle model?
5. How can we convert between different temperature scales in thermodynamic problems?

Tip: A higher compression ratio typically leads to better engine efficiency, but there are practical limits due to fuel combustion constraints.