A gasoline engine, utilizing cold air, recorded a work of 431 BTU/lb at a maximum temperature of 3,273 K and 1112 deg F temperature at the end of constant volume heat rejection. What is the compression ratio? 

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.

A gasoline engine, utilizing cold air, recorded a work of 431 BTU/lb at a maximum temperature of 3,273 K and 1112 deg F temperature at the end of constant volume heat rejection. What is the compression ratio?

Solve the problem of determining the compression ratio of a gasoline engine with 431 BTU/lb work output, a maximum temperature of 3273 K, and a final temperature of 1112°F after heat rejection. Learn about the air-standard Otto cycle and key thermodynamic formulas in this detailed solution.

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