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Chapter 20: Problem 4

A gasoline engine has a power output of 180 \(kW\)(about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

Short Answer

Expert verified
(a) 642.86 kW; (b) 462.86 kW.

Step by step solution

01

Understanding Power and Efficiency

The power output of the engine is the useful work done per second. The thermal efficiency formula is given by \(\eta = \frac{W}{Q_{in}}\), where \(\eta\) is the efficiency, \(W\) is the work done or power output, and \(Q_{in}\) is the heat energy supplied per second. The engine's power output is 180 kW, and its thermal efficiency is 28%.
02

Calculate Heat Supplied Per Second (Part a)

Rearrange the formula for efficiency to find the heat input: \(Q_{in} = \frac{W}{\eta}\). Substituting the values, \(Q_{in} = \frac{180 \, \text{kW}}{0.28} = 642.86 \, \text{kW}\). Therefore, approximately 642.86 kW of heat must be supplied per second.
03

Calculate Heat Discarded Per Second (Part b)

The heat discarded \(Q_{out}\) can be calculated using the relation \(Q_{out} = Q_{in} - W\). From part a, \(Q_{in} = 642.86 \, \text{kW}\) and \(W = 180 \, \text{kW}\). Therefore, \(Q_{out} = 642.86 \, \text{kW} - 180 \, \text{kW} = 462.86 \, \text{kW}\). So, approximately 462.86 kW of heat is discarded per second.

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Key Concepts

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

Heat Engine
A heat engine is a fascinating piece of technology used to convert thermal energy into mechanical work. Essentially, it captures heat from a high-temperature source, performs work, and then releases some of the heat at a lower temperature. The power process is much like what happens in a gasoline engine.
In this scenario, the gasoline engine receives thermal energy from burning fuel. It uses part of this energy to produce useful mechanical work, such as moving your car. However, not all energy is transformed into work; some energy is inevitably lost as waste heat.
Key features of a heat engine include:
  • Source of thermal energy: Typically fuel combustion in gasoline engines.
  • Conversion to work: Mechanical power that moves a vehicle.
  • Heat rejection: The unwanted heat leaving the engine, often resulting in energy loss.
Understanding how a heat engine works helps in appreciating the importance of efficiency, as engineers constantly seek to minimize waste and maximize work output.
Thermal Efficiency
Thermal efficiency is a crucial measure of a heat engine's performance. It describes how effectively the engine converts heat into useful work. In general, efficiency is about maximizing output for a given input.To calculate thermal efficiency (\(\eta \)), we use the formula:\[\eta = \frac{W}{Q_{in}}\]where:
  • \(W\) is the work output (power output in kW).
  • \(Q_{in}\) is the heat supplied to the engine (in kW).
For example, if an engine has a thermal efficiency of 28%, it means 28% of the heat energy provided is converted into work, while the rest is lost.
Enhancing thermal efficiency helps engines to perform better, reduce fuel consumption, and lower emissions. Understanding and optimizing efficiency parameters allows engineers to contribute to more environmentally friendly and cost-effective technologies.
Heat Transfer
Heat transfer is a key aspect and inevitable component of heat engines. It involves the movement of heat from one place to another, which can be from a hot source to engine components or from the engine to a cooler sink.
In a gasoline engine, besides converting heat into work, a significant portion of the heat is lost as waste. This process is referred to as heat rejection. Being able to manage this heat transfer efficiently ensures that the engine remains functional without overheating.
Heat transfer occurs in several ways:
  • Conduction: Heat moves through solid parts of the engine.
  • Convection: Heat transfers to fluids like air or coolant surrounding the engine.
  • Radiation: Emission of heat in the form of radiation, especially critical for cooling.
All these heat transfer methods are crucial for the engine's cooling system, which keeps the engine at a safe operating temperature.

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Most popular questions from this chapter

A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 \(L\), and the temperature is 300 \(K\). Assume that the oxygen may be treated as an ideal gas. The oxygen is carried through the following processes: (i) Heated at constant pressure from the initial state (state 1) to state 2, which has \(T\) = 450 K. (ii) Cooled at constant volume to 250 \(K\) (state 3). (iii) Compressed at constant temperature to a volume of 4.00 \(L\) (state 4). (iv) Heated at constant volume to 300 \(K\), which takes the system back to state 1. (a) Show these four processes in a \(pV\)-diagram, giving the numerical values of \(p\) and \(V\) in each of the four states. (b) Calculate \(Q\) and \(W\) for each of the four processes. (c) Calculate the net work done by the oxygen in the complete cycle. (d) What is the efficiency of this device as a heat engine? How does this compare to the efficiency of a Carnot cycle engine operating between the same minimum and maximum temperatures of 250 \(K\) and 450 \(K\)?

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?

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 refrigerator has a coefficient of performance of 2.10. In each cycle it absorbs 3.10 \(\times\) 10\(^4\) J of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

A Carnot engine is operated between two heat reservoirs at temperatures of 520 \(K\) and 300 \(K\). (a) If the engine receives 6.45 \(kJ\) of heat energy from the reservoir at 520 \(K\) in each cycle, how many joules per cycle does it discard to the reservoir at 300 \(K\)? (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?
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