Problem 7 - University Physics with Modern Physics

Question

The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8. (a) What is the ideal efficiency of the engine? Use \(\gamma\) = 1.40. (b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio?

Step-by-Step Solution

Answer

The increase in ideal efficiency is calculated from the difference between the two engine efficiencies using their specific compression ratios.

Step 1: Understand the formula for efficiency of an Otto-cycle engine

The efficiency of an ideal Otto cycle engine is given by the formula:\[ \eta = 1 - \left( \frac{1}{r^{\gamma-1}} \right) \]where \( \eta \) is the efficiency, \( r \) is the compression ratio, and \( \gamma \) is the heat capacity ratio (1.40 for this problem).

Step 2: Calculate the efficiency for SLK230

The compression ratio \( r \) for the Mercedes-Benz SLK230 is 8.8. Plug these values into the efficiency formula:\[ \eta_{SLK230} = 1 - \left( \frac{1}{8.8^{1.40-1}} \right) \]Simplify the exponent to 0.4:\[ \eta_{SLK230} = 1 - \left( \frac{1}{8.8^{0.4}} \right) \]Calculate \( 8.8^{0.4} \) and then complete the formula to find \( \eta_{SLK230} \).

Step 3: Calculate the efficiency for Viper GT2

The compression ratio \( r \) for the Dodge Viper GT2 is 9.6. Plug these values into the same formula:\[ \eta_{Viper} = 1 - \left( \frac{1}{9.6^{1.40-1}} \right) \]Again, simplify the exponent to 0.4:\[ \eta_{Viper} = 1 - \left( \frac{1}{9.6^{0.4}} \right) \]Calculate \( 9.6^{0.4} \) and then complete the formula to find \( \eta_{Viper} \).

Step 4: Calculate the increase in efficiency

Subtract the efficiency of the SLK230 from the efficiency of the Viper to find the increase in efficiency due to the higher compression ratio:\[ \Delta \eta = \eta_{Viper} - \eta_{SLK230} \]Compute this result once both efficiencies have been calculated.

Key Concepts

Compression Ratio

The compression ratio is a key parameter in understanding how efficient an Otto cycle engine can be. In simple terms, the compression ratio (\( r \) ) is the ratio of the volume of the engine's cylinder when the piston is at the bottom of its stroke (maximum volume) to the volume when it’s at the top of its stroke (minimum volume). This ratio is crucial because it influences the efficiency and power output of an engine.
The higher the compression ratio:

• The greater the volume change during the compression stroke, leading to more energy being converted from heat into mechanical work.
• The more the air-fuel mixture is compressed, which typically results in higher temperatures and pressures, thus increasing the thermal efficiency of the engine.

For example, in the exercise given, the Mercedes-Benz SLK230 has a compression ratio of 8.8, while the Dodge Viper GT2 has a ratio of 9.6. A higher compression ratio means the Viper should, theoretically, have a higher thermal efficiency.

Heat Capacity Ratio

The heat capacity ratio, also known as the adiabatic index or gamma (\( \gamma \)), is a measure of the specific heat of a gas at constant pressure divided by the specific heat at constant volume. For most gases, including the air-fuel mixture in car engines, this value is about 1.4. Understanding the heat capacity ratio is crucial because it affects how energy is transformed from heat to work in the Otto cycle.

• The value of \( \gamma \) determines how much work can be extracted from the engine's cycle, influencing efficiency.
• In the Otto cycle, higher values of \( \gamma \) suggest that the gas expands more vigorously, converting more heat into useful work.

In our specific problem, using a heat capacity ratio of 1.4 allows the calculation of efficiencies for the SLK230 and Viper GT2 engines via the given formula, impacting the comparison of their potential efficiencies.

Ideal Efficiency Calculation

Calculating the ideal efficiency of an Otto cycle engine involves using a specific formula to find out how effective an engine is in converting the fuel into work. The formula for ideal efficiency (\( \eta \)) is:\[\eta = 1 - \left( \frac{1}{r^{\gamma-1}} \right)\]Here, \( r \) is the compression ratio, and \( \gamma \) is the heat capacity ratio. This relationship shows that both a higher compression ratio and a higher heat capacity ratio are favorable for achieving better efficiency.

• For the Mercedes-Benz SLK230, with a compression ratio of 8.8 and \( \gamma = 1.4 \), we substitute these values into the formula to find its efficiency.
• Similarly, the Dodge Viper GT2, with its ratio of 9.6, can be analyzed for efficiency gains using the same approach.
• The difference in their efficiencies helps us understand the impact of small increments in compression ratio.

Ultimately, understanding how to calculate ideal efficiency enables engineers and enthusiasts alike to gauge engine performance and make informed improvements.