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Chapter 20: Problem 64

Consider a Diesel cycle that starts (at point \(a\) in Fig. 20.7\()\) with air at temperature \(T_{a}\) The air may be treated as an ideal gas. (a) If the temperature at point \(c\) is \(T_{c}\) , derive an expression for the efficiency of the cycle in terms of the compression ratio \(r .\) (b) What is the efficiency if \(T_{a}=300 \mathrm{K}, T_{c}=950 \mathrm{K}, \gamma=1.40\) and \(r=21.0 ?\)

Short Answer

Expert verified
The efficiency of the Diesel cycle is approximately 85.9%.

Step by step solution

01

Define the Diesel Cycle Process

The Diesel cycle consists of four processes: isentropic compression (from a to b), constant volume heat addition (from b to c), isentropic expansion (from c to d), and constant pressure heat rejection (from d to a). We need to find the efficiency for this cycle.
02

Identify Important Relations

The efficiency \( \eta \) of a Diesel cycle is given by the formula: \[\eta = 1 - \frac{1}{r^{\gamma - 1}} \left(\frac{T_c}{T_b} - 1 \right) \] where \( r \)\ is the compression ratio, \( T_c \)\ is the maximum temperature, and \( \gamma \) is the specific heat ratio (Cp/Cv).
03

Calculate Efficiency for the Given Values

Substitute the given values: \( T_a = 300 \text{ K}, \, T_c = 950 \text{ K}, \, \gamma = 1.40, \, r = 21.0 \) into the efficiency formula. First, find \( T_b \) by noting that \( T_b = T_a \, r^{\gamma-1} \). \[ T_b = 300 \, \times \, 21^{1.40 - 1} \approx 660.5 \text{ K} \]
04

Compute the Efficiency

Now use the efficiency formula: \[ \eta = 1 - \frac{1}{21^{0.40}} \left(\frac{950}{660.5} - 1 \right) \] Calculate: \[ \eta \approx 1 - \frac{1}{3.1} \left(1.438 - 1 \right) \] Further simplify: \[ \eta \approx 1 - \frac{0.438}{3.1} \approx 1 - 0.141 \approx 0.859 \] Thus, the efficiency is approximately 85.9%.

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Key Concepts

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

Ideal Gas Processes
The Diesel cycle relies on processes often described by the ideal gas law, which is a fundamental principle in thermodynamics. An ideal gas is defined by its state variables: pressure, volume, and temperature, which are related through the equation \( PV = nRT \). This means that for a given mass of an ideal gas, if you know two of these variables, you can calculate the third. In the context of the Diesel cycle:
  • Compression and expansion are isentropic, meaning they occur without any heat transfer. The ideal gas assumption simplifies these calculations.
  • The cycle involves constant volume and constant pressure processes, which are straightforward to analyze using the ideal gas law.
Understanding ideal gas processes is crucial because it allows engineers to predict how a gas will behave when subject to changes in temperature, volume, or pressure. The idealization helps create simpler models that are quite accurate for gases like air at room temperature.
Compression Ratio
Compression ratio, denoted by \( r \), is a key parameter in evaluating the efficiency of internal combustion engines, including Diesel engines. It is defined as the ratio of the total volume within a cylinder when the piston is at the bottom of its stroke to the volume within the cylinder when the piston is at the top of its stroke:\[ r = \frac{V_{max}}{V_{min}}.\]
  • A higher compression ratio implies that the air-fuel mixture is compressed to a smaller volume before ignition, which generally increases the efficiency of the cycle.
  • High compression ratios lead to better fuel efficiency, translating to more power output for the same amount of fuel.
  • In Diesel engines, the compression ratio can be higher than in gasoline engines because the air is compressed to high temperatures needed for auto-ignition of the fuel.
The efficiency of the Diesel cycle itself can be significantly increased by optimizing the compression ratio, making it an essential factor in engine design and thermodynamic analyses.
Specific Heat Ratio
The specific heat ratio, represented by \( \gamma \), is the ratio of the specific heat at constant pressure \( C_p \) to the specific heat at constant volume \( C_v \):\[ \gamma = \frac{C_p}{C_v}.\]
  • This ratio is crucial in evaluating adiabatic processes, such as those found in the Diesel cycle, where no heat is transferred to the surroundings.
  • A higher \( \gamma \) value leads to more efficient compression and expansion processes, which is typical for gases like dry air used in Diesel engines.
  • For an ideal gas undergoing adiabatic compression, the temperature rise and the work done are directly related to \( \gamma \).
In engineering applications, knowing \( \gamma \) helps predict how a gas will behave during compression and expansion, which are critical to optimizing the performance and efficiency of engines.

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Most popular questions from this chapter

An experimental power plant at the Natural Energy Laboratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water temperatures are \(27^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C},\) respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 \(\mathrm{kW}\) of power, at what rate must he extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of \(10^{\circ} \mathrm{C}\) . What must be the flow rate of cold water through the system? Give your answer in \(\mathrm{kg} / \mathrm{h}\) and in \(\mathrm{L} / \mathrm{h}\) .

A Carnot engine whose high-temperature reservoir is at 620 \(\mathrm{K}\) takes in 550 \(\mathrm{J}\) of heat at this temperature in each cycle and gives up 335 \(\mathrm{J}\) to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir? (c) What is the thermal efficiency of the cycle?

Heat Pump. A heat pump is a heat engine run in reverse. In winter it pumps heat from the cold air outside into the warmer air inside the building, maintaining the building at a comfortable temperature. In summer it pumps heat from the cooler air inside the building to the warmer air outside, acting as an air conditioner. (a) If the outside temperature in winter is \(-5.0^{\circ} \mathrm{C}\) and the inside temperature is \(17.0^{\circ} \mathrm{C}\) , how many joules of heat will the heat pump deliver to the inside for each joule of electrical energy used to run the unit, assuming an ideal Carnot cycle? (b) Suppose you have the option of using electrical resistance heating rather than a heat pump. How much electrical energy would you need in order to deliver the same amount of heat to the inside of the house as in part (a)? Consider a Carnot heat pump delivering heat to the inside of a house to maintain it at \(68^{\circ} \mathrm{F}\) . Show that the heat pump delivers less heat for each joule of electrical energy used to operate the unit as the outside temperature decreases. Notice that this behavior is opposite to the dependence of the efficiency of a Carnot heat engine on the difference in the reservoir temperatures. Explain why this is so.

A \(10.0\) -L gas tank containing 3.20 moles of ideal He gas at \(20.0^{\circ} \mathrm{C}\) is placed inside a completely evacuated, insulated bell jar of volume 35.0 L. A small hole in the tank allows the He to leak out into the jar until the gas reaches a final equilibrium state with no more leakage. (a) What is the change in entropy of this system due to the leaking of the gas? (b) Is the process reversible or irreversible? How do you know?

A room air conditioner has a coefficient of performance of 2.9 on a hot day and uses 850 \(\mathrm{W}\) of electrical power. (a) How many ioules of heat does the air conditioner remove from the room in one minute? (b) How many joules of heat does the air conditioner deliver to the hot outside air in one minute? (c) Explain why your answers to parts (a) and (b) are not the same.
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