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Chapter 20: Problem 7

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鈥檚 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

For a refrigerator or air conditioner, the coefficient of performance \(K\) (often denoted as COP) is, as in Eq. (20.9), the ratio of cooling output \(Q_C\) 0 to the required electrical energy input \(W\) , both in joules. The coefficient of performance is also expressed as a ratio of powers, $$K = {(Q_C ) /t \over (W) /t}$$ where \(Q_C /t\) is the cooling power and \(W /t\) is the electrical power input to the device, both in watts. The energy efficiency ratio (\(EER\)) is the same quantity expressed in units of Btu for \(Q_C\) and \(W \cdot h\) for \(W\) . (a) Derive a general relationship that expresses \(EER\) in terms of \(K\). (b) For a home air conditioner, \(EER\) is generally determined for a 95\(^\circ\)F outside temperature and an 80\(^\circ\)F return air temperature. Calculate \(EER\) for a Carnot device that operates between 95\(^\circ\)F and 80\(^\circ\)F. (c) You have an air conditioner with an \(EER\) of 10.9. Your home on average requires a total cooling output of \(Q_C = 1.9 \times 10^{10} J\) per year. If electricity costs you 15.3 cents per \(kW \cdot h\), how much do you spend per year, on average, to operate your air conditioner? (Assume that the unit's \(EER\) accurately represents the operation of your air conditioner. A \(seasonal\) \(energy\) \(efficiency\) \(ratio\) (\(SEER\)) is often used. The \(SEER\) is calculated over a range of outside temperatures to get a more accurate seasonal average.) (d) You are considering replacing your air conditioner with a more efficient one with an \(EER\) of 14.6. Based on the \(EER\), how much would that save you on electricity costs in an average year?

CP A certain heat engine operating on a Carnot cycle absorbs 410 J of heat per cycle at its hot reservoir at 135\(^\circ\)C and has a thermal efficiency of 22.0%. (a) How much work does this engine do per cycle? (b) How much heat does the engine waste each cycle? (c) What is the temperature of the cold reservoir? (d) By how much does the engine change the entropy of the world each cycle? (e) What mass of water could this engine pump per cycle from a well 35.0 m deep?

A typical coal-fired power plant generates 1000 MW of usable power at an overall thermal efficiency of 40%. (a) What is the rate of heat input to the plant? (b) The plant burns anthracite coal, which has a heat of combustion of 2.65 \(\times\) 10\(^7\) J/kg. How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river is at 18.0\(^\circ\)C before it reaches the power plant and 18.5\(^\circ\)C after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

An object of mass \(m_1\), specific heat \(c_1\), and temperature \(T_1\) is placed in contact with a second object of mass \(m_2\), specific heat \(c_2\), and temperature \(T_2\) > \(T_1\). As a result, the temperature of the first object increases to \(T\) and the temperature of the second object decreases to \(T'\). (a) Show that the entropy increase of the system is $$\Delta S = m_1c_1 ln {T \over T_1} + m_2c_2 ln {T' \over T_2}$$ and show that energy conservation requires that $$m_1c_1 (T - T_1) = m_2c_2 (T_2 - T')$$ (b) Show that the entropy change \(\Delta S\), considered as a function of \(T\), is a \(maximum\) if \(T = T'\), which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of randomness.

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.)
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