/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 A gasoline engine has a power ou... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 6

A gasoline engine has a power output of 180 \(\mathrm{kW}\) (about 241 \(\mathrm{hp}\) ). 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
653.71 kW of heat must be supplied, and 475.71 kW of heat is discarded.

Step by step solution

01

Understand the Concepts

The power output of the engine is the work done per second. Thermal efficiency is the ratio of the work output to the heat input. The discarded heat is the difference between the heat supplied and the work output.
02

Determine the Heat Supplied

The formula for thermal efficiency is given by: \( \text{Efficiency} = \frac{W}{Q_{in}} \), where \( W \) is the work output and \( Q_{in} \) is the heat input. We can rearrange this as: \( Q_{in} = \frac{W}{\text{Efficiency}} \). Substitute \( W = 180 \text{ kW} \) and \( \text{Efficiency} = 0.28 \) to find \( Q_{in} \).\[ Q_{in} = \frac{180}{0.28} = 642.86 \text{ kW} \]
03

Calculate Heat Discarded

The heat discarded \( Q_{out} \) can be found using the formula: \( Q_{out} = Q_{in} - W \). Substitute \( Q_{in} = 642.86 \text{ kW} \) and \( W = 180 \text{ kW} \) to find \( Q_{out} \).\[ Q_{out} = 642.86 - 180 = 462.86 \text{ kW} \]
04

Conclusion

The heat that must be supplied to the engine is approximately 642.86 kW, and the heat discarded by the engine is about 462.86 kW.

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Key Concepts

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

Engine Power Output
Engine power output is a measure of how much work is done by an engine in a unit time. It is typically measured in kilowatts (kW) or horsepower (hp), with 1 hp equaling about 0.7355 kW. In our case, the engine power output is 180 kW, or approximately 241 hp. This value represents the engine鈥檚 capacity to convert fuel energy into mechanical energy, which is the useful work that the engine can perform each second. Understanding this concept is crucial because it forms the foundation for calculating other quantities like thermal efficiency and heat input.
Heat Input
Heat input refers to the total amount of thermal energy supplied to an engine, usually measured per unit time, in kilowatts (kW) or other energy units. For thermal engines, this is typically the energy liberated from the combustion of fuel. The heat input \(Q_{in}\) to an engine is calculated using the relationship with thermal efficiency. Thermal efficiency is defined by the formula:
  • \(\text{Efficiency} = \frac{W}{Q_{in}}\)
where \(W\) is the work output. Rearranging gives:
  • \(Q_{in} = \frac{W}{\text{Efficiency}}\)
By substituting the known values:
  • \(W = 180 \text{ kW}\)
  • \(\text{Efficiency} = 0.28\)
we find:
  • \(Q_{in} = \frac{180}{0.28} = 642.86 \text{ kW}\)
This value represents the energy needed each second to keep the engine running at its power output.
Work Output
Work output is the useful mechanical energy produced by an engine from the heat input. It is often synonymous with the engine power output in practical terms. Work output is what engines are built for; it is the energy available for performing tasks such as moving a vehicle or driving machinery. Since work output directly relates to engine performance, understanding how it connects to heat input and efficiency is key. For instance, a higher thermal efficiency means more of the heat input is converted into work output, which is desirable in any engine.
Heat Discarded
Heat discarded, or waste heat, is the thermal energy that is not converted into work. Instead, it gets expelled from the engine, often into the atmosphere. This wasted energy highlights the inefficiencies present in thermal engines. You can calculate the heat discarded \(Q_{out}\) with the formula:
  • \(Q_{out} = Q_{in} - W\)
Using values from the example:
  • \(Q_{in} = 642.86 \text{ kW}\)
  • \(W = 180 \text{ kW}\)
we find:
  • \(Q_{out} = 642.86 - 180 = 462.86 \text{ kW}\)
This figure underscores the importance of improving engine efficiency, as a significant portion of the energy is not utilized for work but is instead lost as waste.

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Most popular questions from this chapter

\(\cdot\) A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{J}\) of heat per minute from the room being cooled and in the same period deposits \(1.44 \times 10^{5} \mathrm{J}\) of heat into the outside air. What is the power consumption of the unit in watts?

Entropy change due to driving. Premium gasoline pro- duces \(1.23 \times 10^{8} \mathrm{J}\) of heat per gallon when it is burned at a temperature of approximately \(400^{\circ} \mathrm{C}\) (although the amount can vary with the fuel mixture). If the car's engine is 25\(\%\) efficient, three- fourths of that heat is expelled into the air, typically at \(20^{\circ} \mathrm{C}\) . If your car gets 35 miles per gallon of gas, by how much does the car's engine change the entropy of the world when you drive 1.0 mile? Does it decrease or increase it?

\(\bullet\) For an Otto engine with a compression ratio of \(7.50,\) you have your choice of using an ideal monatomic or ideal diatomic gas. Which one would give you greater efficiency? Calculate the efficiency in both cases to find out.

Solar water heater. A solar water heater for domestic hot- water supply uses solar collecting panels with a collection efficiency of 50\(\%\) in a location where the average solar-energy input is 200 \(\mathrm{W} / \mathrm{m}^{2} .\) If the water comes into the house at \(15.0^{\circ} \mathrm{C}\) and is to be heated to \(60.0^{\circ} \mathrm{C},\) what volume of water can be heated per hour if the area of the collector is 30.0 \(\mathrm{m}^{2} ?\)

\(\bullet\) (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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