15.8

Heat Engines

A heat engine is any device that uses heat to perform work. It has three essential features:

1. | Heat is supplied to the engine at a relatively high input temperature from a place called the hot reservoir.
2. | Part of the input heat is used to perform work by the working substance of the engine, which is the material within the engine that actually does the work (e.g., the gasoline-air mixture in an automobile engine).
3. | The remainder of the input heat is rejected to a place called the cold reservoir, which has a temperature lower than the input temperature.

Figure 15-12 illustrates these features. The symbol QH refers to the input heat, and the subscript H indicates the hot reservoir. Similarly, the symbol QC stands for the rejected heat, and the subscript C denotes the cold reservoir. The symbol W refers to the work done. The vertical bars enclosing each of these three symbols are included to emphasize that we are concerned here with the absolute values, or magnitudes, of the symbols. Thus, |QH| indicates the magnitude of the input heat, |QC| denotes the magnitude of the rejected heat, and |W| stands for the magnitude of the work done. Problem solving insight. Since |QH|, |QC|, and |W| refer to magnitudes only, they never have negative values assigned to them when they appear in equations.

Figure 15-12 This schematic representation of a heat engine shows the input heat () that originates from the hot reservoir, the work () that the engine does, and the heat () mat the engine rejects to the cold reservoir.

The physics of a heat engine.

To be highly efficient, a heat engine must produce a relatively large amount of work from as little input heat as possible. Therefore, the efficiency e of a heat engine is defined as the ratio of the magnitude of the work |W| done by the engine to the magnitude of the input heat |QH|:

| (15.11)

If the input heat were converted entirely into work, the engine would have an efficiency of 1.00, since |W| = |QH|; such an engine would be 100% efficient. Efficiencies are often quoted as percentages obtained by multiplying the ratio |W|/|QH| by a factor of 100. Thus, an efficiency of 68% would mean that a value of 0.68 is used for the efficiency in Equation 15.11.
An engine, like any device, must obey the principle of conservation of energy. Some of the engine’s input heat |QH| is converted into work |W|, and the remainder |QC| is rejected to the cold reservoir. If there are no other losses in the engine, the principle of energy conservation requires that

| (15.12)

Solving this equation for |W| and substituting the result into Equation 15.11 leads to the following alternative expression for the efficiency e of a heat engine:

| (15.13)

Example 6 illustrates how the concepts of efficiency and energy conservation are applied to a heat engine.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 6 | | | An Automobile Engine

An automobile engine has an efficiency of 22.0% and produces 2510 J of work. How much heat is rejected by the engine? Reasoning Energy conservation indicates that the amount of heat rejected to the cold reservoir is the part of the input heat that is not converted into work. The work is given, and the input heat can be obtained since the efficiency of the engine is also given.
Knowns and Unknowns The following data are available:

Description | Symbol | Value
Efficiency of engine | e | 22.0% (0.220)
Magnitude of work | |W| | 2510 J
Unknown Variable
Magnitude of rejected heat | |QC| | ?

Modeling the Problem

| Step 1 The Conservation of Energy
| According to the energy-conservation principle, the magnitudes of the input heat |QH|, the work done |W|, and the rejected heat |QC| are related according to (Equation 15.12). Solving for |QC| gives Equation 15.1 at the right. In this result, |W| is known, but |QH| is not, although it will be evaluated in Step 2.

| (1)

| Step 2 Engine Efficiency
| Equation 15.11 gives the engine efficiency as . Solving for |QH|, we find that

which can be substituted into Equation 15.1 as shown in the right column.

Solution Combining the results of each step algebraically, we find that

The magnitude of the rejected heat, then, is

Related Homework: Problems 41, 43, 76

Problem solving insight When efficiency is stated as a percentage (e.g., 22.0%), it must be converted to a decimal fraction (e.g., 0.220) before being used in an equation.

In Example 6, less than one-quarter of the input heat is converted into work because the efficiency of the automobile engine is only 22.0%. If the engine were 100% efficient, all the input heat would be converted into work. Unfortunately, nature does not permit 100%-efficient heat engines to exist, as the next section discusses.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.