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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?

Short Answer

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

Step by step solution

01

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).
02

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 \( \eta_{SLK230} \).
03

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 \( \eta_{Viper} \).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A 15.0-kg block of ice at 0.0\(^\circ\)C melts to liquid water at 0.0\(^\circ\)C inside a large room at 20.0\(^\circ\)C. Treat the ice and the room as an isolated system, and assume that the room is large enough for its temperature change to be ignored. (a) Is the melting of the ice reversible or irreversible? Explain, using simple physical reasoning without resorting to any equations. (b) Calculate the net entropy change of the system during this process. Explain whether or not this result is consistent with your answer to part (a).

A person with skin of surface area 1.85 m\(^2\) and temperature 30.0\(^\circ\)C is resting in an insulated room where the ambient air temperature is 20.0\(^\circ\)C. In this state, a person gets rid of excess heat by radiation. By how much does the person change the entropy of the air in this room each second? (Recall that the room radiates back into the person and that the emissivity of the skin is 1.00.)

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work \(W\) is performed by the engine?

A 4.50-kg block of ice at 0.00\(^\circ\)C falls into the ocean and melts. The average temperature of the ocean is 3.50\(^\circ\)C, including all the deep water. By how much does the change of this ice to water at 3.50\(^\circ\)C alter the entropy of the world? Does the entropy increase or decrease? (\(Hint\): Do you think that the ocean temperature will change appreciably as the ice melts?)

(a) Calculate the theoretical efficiency for an Otto-cycle engine with \(\gamma\) = 1.40 and \(r\) = 9.50. (b) If this engine takes in 10,000 J of heat from burning its fuel, how much heat does it discard to the outside air?

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