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"See appendix and references at end of this chapter.

"See appendix and references at end of this chapter.

2.1 The Carnot Cycle Even though he used the erroneous caloric theory, Car-not did list the correct steps in the cycle of the ideal heat engine. Therefore, the ideal heat engine is called the Carnot engine and the cycle is called the Carnot cycle for an engine operating between two heat reservoirs. These steps are enumerated below and are shown schematically in Fig. 5.7 and given in more detail for a special case in the appendix at the end of this chapter.

STEP 1. Isothermal heat intake (Isothermal means constant temperature.) The engine is at the temperature Tu of the hot reservoir, and slowly takes in heat, while its temperature is held constant. In order to do this, some other parameter of the engine system must change, and as a result the engine may do some useful work. (For example, in a steam engine the piston is forced out, increasing the volume of the cylinder.)

STEP 2. Adiabatic performance of work by the engine (Adiabatic means no heat can enter or leave.) The engine is completely thermally insulated from the heat reservoirs and the rest of the universe, and does some useful work. The performance of work means that energy (but not in the form of heat) leaves the engine, and therefore the internal energy content of the engine is less at the end of this step than at the beginning of this step. Because of the equation of state of the working substance of the engine, the temperature of the engine decreases. This step is terminated when the engine reaches the same temperature Tc as the low temperature or cold reservoir. (In Fig. 5.7, the piston is still moving outward.)

Energy flow

Engine operation

Engine operation

Work out

Work out

JHeat flowr

JHeat flowr

Figure 5.7. The Carnot cycle and associated energy flows. The left side of the figure shows the energy flow associated with diagrams on the right side of the figure. P, piston; WS, working substance; CR, connecting rod; FW, flywheel; Ins, insulation; Flm, flame; I, ice. (a) Isothermal heat intake from hot reservoir, piston moving outward, work done by engine, (b) Adiabatic work by engine, piston moving outward, (c) Isothermal heat discharge to cold reservoir, piston moving inward as work is done on the engine by the flywheel, (d) Adiabatic temperature increase, piston moving inward as more work is done on engine by the flywheel.

STEP 3. Isothermal heat discharge The engine is now at the temperature Tc of the cold reservoir, the thermal insulation is removed, and heat is slowly discharged to the low-temperature reservoir, while the temperature of the engine remains constant. Again, in order to do this, some other parameter of the engine must change, resulting in some work being done on the engine. It is at this step and the next step at which some of the work taken out of the engine must be returned to the engine. These two steps, which are necessary for an engine working in a cycle, can be regarded as the steps where nature reduces the efficiency from 100 percent to the thermodynamic values. (In Fig. 5.7, the piston is now

Heat source driven inward by the flywheel and connecting rod, decreasing the cylinder volume.)

STEP 4. Adiabatic temperature increase of engine The engine is again completely thermally insulated from the heat reservoirs and the rest of the universe. Work is done on the engine, usually by forcing a change in one of its parameters, resulting in an increase of temperature until it returns to the starting temperature, Tn, of the hot reservoir. The internal energy content of the engine is now the same as at the beginning of Step 1, and the engine is ready to repeat the cycle. (In Fig. 5.7, the piston continues its inward motion.)

It should be noted that there may be portions of an engine cycle in which heat is converted 100 percent to work. Step 1 of the Carnot cycle, when applied to the engine described in the appendix at the end of this chapter, is such a portion. In order for the engine to continue working, however, it must go through a complete cycle. In the process of going through the complete cycle, the second law of thermodynamics requires that a portion of the work be returned to the engine and transformed back into heat, some of which is then discharged to the low-temperature reservoir, and the rest used to return the engine to the starting point of the cycle. The important point is that this is true for all engines, regardless of how they are operated.

2.2 Efficiency of the Carnot Cycle Carnot had originally reasoned that the efficiency of the Carnot cycle should be proportional to the temperature difference between the hot and cold reservoirs, TH — Tc. Clausius corrected Carnot's analysis to show that, if the proper temperature scale is used (the so-called absolute temperature scale), the efficiency would be equal to the difference in temperature of the two reservoirs divided by the temperature of the hot reservoir; that is, the efficiency equals (TH - TC)/TH = 1 - TC/TH.

Using Clausius's formula for the Carnot or thermodynamic efficiency, it is possible to calculate the efficiency that could be expected if all friction, irreversibilities, and heat losses were eliminated from a modern electricity generating plant, and if it were operated in a Carnot cycle. The results of such calculations are also shown in Table 5.1. It can be seen that actual modern generating stations do get reasonably close to the Carnot efficiency for the temperatures at which they operate.

3. Possibilities for Improving Heat Engine Efficiencies

The only way to improve the theoretically possible efficiencies of heat engines is by either increasing Tn or decreasing Tc. On the planet Earth, Tc cannot be lower than 273 kelvins (32° Fahrenheit) because that is the freezing temperature of water, and present-day power plants use cooling water from a river or ocean for their low-temperature reservoirs. Theoretically, it would be possible to use some other coolant, say antifreeze (ethylene glycol) as used in automobile engines, which remains liquid at temperatures much lower than the freezing point of water. In order to do this, however, a refrigeration system must be used to lower the temperature of the antifreeze, so that it can be a very low-temperature reservoir. Unfortunately, energy is required to operate the refrigeration system. It can be shown from the second law of thermodynamics that the amount of energy required to operate the refrigeration system, which would have to be taken from the power plant, would more than offset the energy gain from the increased efficiency resulting from the lower value of Tc. In fact, the net effect of installing the refrigerator would be to decrease the overall efficiency below the Carnot efficiency.

Of course, one can imagine building an electricity generating station on an orbiting satellite in outer space, where Tc is only a few degrees absolute. It would then be necessary to consider how the electricity generated at the station would be transmitted to Earth.

Alternatively, it is necessary to consider the possibility of raising the temperature Th of the hot reservoir. This temperature cannot be higher than the melting temperature of the material out of which the heat engine is built. Actually Tu will have to be considerably less than the melting temperature, because most materials are rather weak near their melting points. If TH were about 1500 kelvins (melting point of steel is 1700 kelvins), and Tc were 300 kelvins, the Carnot efficiency is calculated to be 1 — 300/1500 = 0.8 = 80 percent. Because of the need for safety factors (to keep boilers from rupturing under high pressure associated with high temperatures), TH would be more reasonably set at 1000 kelvins, giving a Carnot efficiency of 70 percent. An actual engine might not achieve much more than % of the Carnot efficiency, yielding a maximum feasible efficiency in the range from 50 to 55 percent. Thus it can be expected that half or more of the fuel burned in a practical heat engine operating between two ' 'reasonable'' temperature reservoirs will be wasted so far as its convertibility into ' 'useful'' work is concerned. (See the data given in Table 5.1 for more specific examples.)

An idea has been proposed, which is in effect a three-reservoir heat engine to obtain more efficient transformation of the heat liberated at high temperatures. Some heat is added to the system at an intermediate temperature between Tn and Tc, for example, during step 4 outlined above for the Carnot cycle. This heat added at an intermediate temperature then helps to do the work on the engine to bring it back up to Tu. Step 4 would no longer be adiabatic. The result is that the heat entering the system at TH is then transformed more effectively into some other form of energy; however, the heat added at the intermediate temperature is transformed less effectively. Overall the efficiency is less than Carnot efficiency if all sources of heat are taken into account. In effect, Qc, the heat discharged to the low-temperature reservoir, comes mostly from the intermediate-temperature reservoir, rather than from the high-temperature reservoir. The advantage of the idea as proposed is that this intermediate-temperature reservoir would be a source of heat that requires that no additional fuel be burned; that is, the source would be either solar or geothermal heat, or some low-grade heat source that is already available for other reasons than energy conversion. (See the article by Powell et al. listed in the references at the end of this chapter.)

4. Types of Heat Engines

The ideas involved in heat engine theory are almost universal. There are many varieties of heat engines, and they can be classified according to several categories. For example, there are external combustion engines, such as reciprocating steam engines, steam turbines, and hot air engines; also internal combustion engines, such as the various types of gasoline and diesel engines, jet engines, and rocket engines. Any engine that involves the burning of some fuel or depends on temperature in order to operate, or in any way involves the liberation or extraction of heat and its conversion to some other form of energy, is a heat engine. Even the Earth's atmosphere is a giant engine that generates energy. (Wind energy is kinetic energy of systematic motion of air molecules, and thunderclouds contain electrical energy.) All heat engines are subject to the same rules concerning their maximum theoretical efficiency, which is completely governed by the theory developed by Carnot and corrected by Clausius. As already indicated, the ideal efficiency (the Carnot efficiency) for all these engines depends only on the temperatures of their two heat reservoirs and is independent of anything else or any details or features of engine construction.

A detailed discussion of a helium gas external combustion engine operating in a Carnot cycle is given in the appendix at the end of this chapter. It was the mathematical analysis of an engine of this type that permitted Clausius to derive the correct formula for engine efficiencies. Other than for theoretical purposes, an engine built specifically to conform as closely as possible to the Carnot cycle is not always practical. There is, in fact, a story about a ship's engine that was built to have an efficiency close to that of a Carnot engine. When it was installed in the ship for which it was designed, however, it was so heavy that the ship became overloaded and sank! (See Chapter XVII of Mott-Smith's book listed in the references.)

F. The Thermodynamic or Absolute Temperature Scale

In Section B2 above, thermometers and their scales were described as depending on temperature equilibrium and the equation of state of the thermometer system. It was also indicated that temperature scales could alternatively be defined in terms of the heat flow between two different temperatures. The Carnot cycle makes it possible to do just that. Instead of using the physical parameters of a thermometric substance such as mercury, it is possible to use the properties of heat and heat transformation.

Imagine two bodies at different temperatures. Imagine further a small Carnot engine that would use these two bodies as its heat reservoirs. According to the Carnot engine theory, the ratio of the heat flowing out of the engine to the heat flowing into the engine, QcIQn, should depend only on the temperatures of the two bodies. Lord Kelvin proposed that the temperature of the two bodies, Tc and rH, should be defined as proportional to their respective heat flows; that is, TCI Th = Qc/Qh. The temperature scale so defined is called a thermodynamic temperature scale. For example, suppose that a particular body is known to be at a temperature of 500 absolute, and that we wish to know the temperature of some cooler body. It might be imagined that a small Carnot engine is connected between the two bodies, and the heat flow in, QH, and the heat flow out, Qc, might be determined. If the ratio of Qc/Qn were V2, then according to Kelvin, the ratio of Tc/Th should be V2 also. Thus the temperature of the cooler body is V2 that of the hotter body or V2 of 500; that is, 250.

The thermodynamic temperature scale is defined earlier only in terms of ratios. To actually fix the scale it is necessary also to specify a size for the units of temperature (degrees). If there are to be 100 units between the freezing and boiling points of water (as in the Celsius temperature scale), then it turns out that on the thermodynamic scale the freezing temperature of water is 273 kelvins and the boiling temperature is 373 kelvins. This thermodynamic temperature scale is called the Kelvin scale.1 If there are to be 180° between the freezing and boiling points of water (as in the Fahrenheit scale), then it turns out that the freezing temperature of water is 491° and the boiling point is 671°. This thermodynamic scale is called the Rankine scale.

G. The Third Law of Thermodynamics

Because of the connection between absolute temperature and engine efficiencies, it is of interest to consider what would happen if the low-temperature reservoir were at absolute zero. As discussed above, the Carnot efficiency of a heat engine is equal to 1 — TC/TH. If Tc = 0, the Carnot efficiency would be 100 percent. Such reservoirs do not normally exist, however. As already pointed out, the energy expended in the attempt to make a cold reservoir at zero absolute temperature would be greater than the extra useful energy obtained by operating at zero absolute temperature. Nevertheless, it would be interesting to see whether the theory would be valid, even at that low temperature. Thus it is necessary to examine the possibilities of making such a reservoir.

It turns out that in order to make a low-temperature reservoir, the effectiveness of the refrigerator used to achieve the low temperature becomes less as the temperature goes down, and it takes longer and longer to reach lower temperatures as the temperature gets closer to absolute zero. In fact, this experience is summarized in the third law of thermodynamics: It is not possible to reach the absolute zero of temperature in a finite number of steps. This means that it is

!In common usage the units on a temperature scale are usually called degrees; by international agreement, however, the units on the Kelvin scale are called kelvins. The specification of the Kelvin scale is somewhat more complicated than described here, and involves a particular fixed point, the triple point of water. The Kelvin scale turns out to be identical with the absolute temperature scale, discussed in Optional Sections 5.1 and 5.2 above.

possible to get extremely close to absolute zero temperature, but it is always just beyond being attained. In other words, "absolute zero can be achieved, but it will take forever."2

The three laws of thermodynamics are sometimes summarized humorously in gambling terms. Comparing heat energy with the stakes in a gambling house, the laws then become:

1. "You can't win; you can only break even."

2. "You can only break even if you play long enough and are dealt the exactly correct set of cards."

3. "You should live so long as to get the right cards."

H. Energy Degradation and Unavailability

Even though energy is neither created nor destroyed, but only transformed, the concept of "lost" or "degraded" energy is useful. If a given heat engine is only 40 percent efficient, then the other 60 percent of the heat generated in operating the engine is not converted to ' 'useful'' forms, but is discharged to the low-temperature reservoir. Moreover, this heat cannot be recycled through the engine for another attempt to convert it into a useful form. Any heat that is discharged to a low-temperature reservoir is wasted, in that it is no longer available for transformation to other forms of energy (unless there is a still lower temperature reservoir accessible for use). The energy is said to be degraded. Indeed, a given amount of energy in the form of heat is degraded in comparison with the same amount of energy in some other form, because it is not completely convertible.

If energy in the form of heat flows out of a high-temperature reservoir directly to a lower temperature reservoir, it will become much more degraded than if it were to flow through a heat engine, where at least part of it is transformed to another form to be available for further transformations. As already mentioned, Clausius sought to describe the availability for transformation of the energy content of a system by coining the word entropy from Greek roots, meaning transformation content. In fact, he used the word to describe the unavailability of the system's energy for transformation. Thus a body or system or reservoir whose internal energy is less available for transformation has a greater entropy than a body of equal size whose internal energy is more available (i.e., available if a heat engine were set up between the body and another one at a lower temperature).

2Strictly speaking, it is preferable to say "zero Kelvin" rather than "zero absolute" and "thermodynamic temperature" rather than "absolute temperature," because the temperature scale is based on heat flow, which is a dynamic process. The use of words such as "absolute zero" can cause confusion, because in certain circumstances the question of "negative absolute temperature" arises. For further details, see Chapter 5 of the book by Zemansky, listed in the references at the end of this chapter.

There are two characteristics of the unavailability or degradation of energy: (1) that it must be in the form that is described as heat, and (2) that the lower the temperature of the body, the more unavailable its heat energy.

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