## Steam Engine Cycles

As already pointed out, the usefulness of the helium gas heat engine is in the relative ease with which it can be mathematically analyzed. In actual applications, the steam engine is far more useful. Nevertheless the information derived from the helium gas engine is very important and universally applicable.

### 1. Carnot Cycle for a Steam Engine

Figure 5.13 shows a P-V diagram for a steam engine operating in a Carnot cycle. The points marked a, b, c, d correspond to the same points in Fig. 5.12 for the helium gas engine. It is quite clear that the cycle for water-steam shown in Fig. 5.13 is much different in appearance than that for helium in Fig. 5.12. Actually, the difference between the two cycles is even greater than it appears from a comparison of the figures, because it is necessary to draw the volume axis for Fig. 5.13 on a logarithmic scale (each unit along the scale does not just represent Logarithm of volume

Figure 5.13. P-V diagram for steam engine. Carnot cycle along solid path, abcda; Rankine cycle along dotted path abcd'a.

Logarithm of volume

Figure 5.13. P-V diagram for steam engine. Carnot cycle along solid path, abcda; Rankine cycle along dotted path abcd'a.

equal additional units of volume, but rather multiplicative factors) in order to keep the volume range within a reasonable size. Moreover, the Carnot cycle for water-steam can be reasonably drawn over only a restricted temperature range. Carnot himself felt that it would be essentially futile to build a steam engine operating in a Carnot cycle, with particular difficulty associated with achieving Step 4, the adiabatic compression step from d to a.

### 2. Rankine Cycle

Instead of evaluating actual steam engines against the Carnot cycle as a comparison standard, another ideal cycle, which is less efficient than the Carnot cycle, is used. This is shown also in Fig. 5.13 by the dotted extension to the solid cycle. Step 3, the isothermal discharge of heat to the low-temperature reservoir at temperature Tc, is prolonged to the point d' to its starting state a. The efficiency of the engine is less than Carnot efficiency because of the extra heat discharged between the points d' and d on the graph, and because of the extra heat added between the points d' and a.

### 3. Actual Steam Engine Cycles

An actual steam engine cycle is modified even further from the Rankine cycle. The adiabatic Step 2 from b to c in Fig. 5.13 would require a great change in the volume of the cylinder, making it excessively long. (Because volume is plotted on a logarithmic scale, each unit along the scale represents a factor of 10 change in volume.) Therefore, in order to keep the cylinder to a convenient size, Step 2 is terminated at the point c', and the pressure abruptly dropped by discharging heat at a temperature higher than Tc, thereby decreasing the efficiency further but not very much.

The P-V diagram for a typical steam engine cycle is shown in Fig. 5.14. In addition to the previously mentioned changes from Fig. 5.13, the "corners" of the cycle are "rounded" because of various heat leaks inherent in design. Even this cycle is an "idealized" cycle in that it does not take into account such factors Logarithm of volume

Figure 5.14. Idealized representation of actual steam-powered piston engine cycle.

Logarithm of volume

Figure 5.14. Idealized representation of actual steam-powered piston engine cycle.

as friction, turbulence in steam flow, substantial temperature differences between reservoirs and engine, and the normal rapid rate of engine operation.

### E. Other Engine Cycles

All heat engines operate according to a characteristic cycle. In the process of analyzing them, it is usually necessary to describe the engine in somewhat idealized terms.

For example, in the analysis of an internal combustion gasoline engine, the fact that the working substance (mixture of gasoline and air) is continuously being removed from the engine is ignored and the working substance is assumed to be conserved—only its temperature being changed. This is done to simplify the analysis. The cycle, of course, is neither Carnot nor Rankine. The following steps occur: (1) a heating step in which the pressure increases very rapidly while the volume remains constant; (2) an adiabatic expansion step in which work is done by the engine and the gas cools; (3) a further cooling step in which the pressure drops very quickly while the volume remains unchanged; and (4) an adiabatic compression step in which work is done on the engine, and the gas is returned to its original thermodynamic state. This particular cycle is called the Otto cycle, after one of the early German pioneers of gasoline engine development. Similarly, there is a Diesel cycle. There is also the Stirling cycle, which is used for external combustion engines with air as the working substance. The Stirling cycle is a very efficient cycle which, although proposed over a century ago, is currently the subject of much intensive research and development for possible automotive use. There are, of course, many other cycles that have been studied, but not all of them are based on gases in cylinders. Where other working substances are used, instead of graphing the cycle on a P-V diagram, other parameters are used to study the cycle. All such cycles can be compared on a similar basis by using temperature, T, and entropy, S, of the working substance as the parameters to be studied. Such a graph, called a T-S diagram, makes it possible to estimate engine efficiencies very simply.

Study Questions

1. What is meant by a parameter of a system?

2. One example of a system with its parameters would be an automobile tire inflated to a particular volume and pressure at a certain temperature. Can you give some other examples?

3. How is temperature distinguished from heat?

4. People sometimes talk about putting "cold" into a system. Can this be explained by the caloric theory or any other theory? Justify your answer.

5. What is meant by an equation of state? How is an equation of state used to determine the temperature of a system?

6. What is the first law of thermodynamics? How does it apply to systems that are not isolated?

7. Do all systems have the same equation of state?

8. What basic ideas are involved in the measurement of temperature?

9. In which direction is the natural flow of heat?

10. State several alternative forms of the second law of thermodynamics.

11. What is meant by perpetual motion of the second kind? Give some examples.

12. If you were told that some company had invented a new kind of electric power plant that was, in theory, three times more efficient than the best present-day power plants, would you invest money in that company?

13. How did Clausius modify Carnot's original analysis of the ideal heat engine?

14. Why is the Carnot engine important?

15. Define the terms isothermal and adiabatic.

16. How much heat can be transferred into a system during an adiabatic process?

17. Which is more efficient, a Carnot engine using steam or a Carnot engine using gasoline?

18. How much is the efficiency of an ideal heat engine?

19. How is temperature defined on the thermodynamic temperature scale?

20. It is sometimes said that absolute zero is the temperature at which everything is "frozen solid," that is, all motion ceases. As will be pointed out in Chapter 7, this statement is incorrect. What might be a better definition of absolute zero?

21. State the third law of thermodynamics.

22. Why is heat called a degraded form of energy?

23. Heat is allowed to flow from a high-temperature reservoir to a low-temperature reservoir. How can the resulting entropy increase be minimized?

24. What is the connection between entropy change and irreversibility?

25. When hot food items are placed into a working refrigerator, their entropy decreases. Is this a violation of the second law of thermodynamics? Justify your answer.

26. Why is entropy referred to as "time's arrow"?

27. A bullet has an adiabatic collision with a large block of wood and is stopped by the wood. What happens to the entropy of the bullet and why? What happens to the temperature of the bullet and why?

28. What is meant by "the energy distribution for the molecules of a system"?

29. Why should the equilibrium distribution be the most probable distribution?

30. What is the relationship between entropy, probability, and disorder?

31. Who is Maxwell's Demon and what does he do?

32. What is meant by the ' 'Heat Death'' of the universe and how is this related to the second law of thermodynamics?