**1. Introduction, overview, and general considerations**

Perfect (reversible) cyclic heat engines operate at Carnot efficiency [1–7]. Perfect (reversible) nonheat engines and noncyclic (necessarily one-time, singleuse) heat engines operate at unit (100%) efficiency. A simple example of a noncyclic heat engine is the one-time expansion of a gas pushing a piston. (If the expansion is isothermal, the heat is supplied from the internal energy of a reservoir; if is adiabatic, the heat is supplied from the internal energy of the gas itself. A polytropic expansion is intermediate between these two extremes.) Other examples include rockets: the piston (payload) is launched into space by a one-time power stroke (but typically most of the work output accelerates the exhaust gases, not the payload) and firearms: the piston (bullet) is accelerated by a one-time power stroke and then discarded (but some, typically less than with rockets, of the work output

accelerates the gases resulting from combustion of the propellant). (Some rocket engines, e.g., those employed in the Space Shuttle and by SpaceX, can be refurbished and reused, but both the first use and each subsequent refurbishment and reuse constitute a necessarily one-time, single-use of a *non*cyclic rocket heat engine.)

(in Ref. [6], see pp. 11–12, 60–65, and 263–265, especially pp. 263–265). The entropy increase resulting from frictional dissipation of work *W* at temperature *T*, namely Δ*S* ¼ *W=T*, decreases monotonically with increasing *T* but is positive for any finite

We should emphasize that the entropy increase Δ*S* ¼ *W=T* associated with work *W* being frictionally dissipated at temperature *T* is *always* Δ*S* ¼ *W=T* and hence *always* decreases monotonically with increasing *T*. Whether the coefficient of friction is small or large *makes no difference* in Δ*S* ¼ *W=T*. Of course, all other things being equal, if the coefficient of friction is small, frictional dissipation of *W* will take longer than if the coefficient of friction is large. But the entropy increase Δ*S* ¼ *W=T* associated with work *W* being frictionally dissipated at temperature *T* is the same whether the coefficient of friction is small or large. Even with coefficients of friction in the low range, *W* will typically be frictionally dissipated immediately or

Of course, efficiency is highest if work *W*, whether supplied via a heat engine or otherwise, is not frictionally dissipated at all. This would obtain, for example, in perfect (reversible) regenerative braking of an electrically-powered motor vehicle, with the motor operating backward as a generator during braking. It also would obtain, for example, if a noncyclic rocket heat engine's work output is perfectly (reversibly) sequestered as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload) or if a cyclic heat engine's work output is perfectly (reversibly) sequestered as gravitational potential energy in the construction of a building. But of course in practice (as opposed to in principle) *total* avoidance of frictional dissipation of work [and also of additional losses, e.g., due to irreversible heat flows engendered by finite temperature differences (no insulation is perfect)]

Although we do not consider them in this chapter, we should note that: (a) There

A misconception pertaining to the efficiencies of engines (heat engines or oth-

In Section 3, we review the work outputs, efficiencies, and entropy productions of Carnot (reversible) and Curzon-Ahlborn (endoreversible) heat engines, first without frictional dissipation of a heat engine's work output and then with frictional dissipation thereof into its cold reservoir. In Section 3, we do not consider frictional dissipation of a heat engine's work output at the highest practicable temperature,

In Section 4, we discuss the work outputs, efficiencies, and entropy productions of Carnot and Curzon-Ahlborn heat engines operating with frictional dissipation of a heat engine's work output at the highest practicable temperature—which we dub as high-temperature recharge (HTR)—and the improvements thereof over those

are generalizations of the Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) at maximum power output both for macroscopic heat engines [18–20] and for microscopic heat engines [21, 22], with irreversible heat flows *not* necessarily directly proportional to temperature differences. (b) There are analyses of maximum heat-engine work output per cycle (as opposed to maximum power output) [23]. Comprehensive discussions concerning the Curzon-Ahlborn efficiency and generalizations thereof are provided in Refs. [24–26]. Some, but not all, such generalized efficiencies [18–26] do not differ greatly from the Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9). In particular, we note that alternative results [26] to the Curzon-Ahlborn efficiency have been derived [26]. But for definiteness and for simplicity, in this chapter, we employ the standard Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) for cyclic heat engines

*T*—and the Second Law requires only that Δ*S*≥0 [1–7].

*Improving Heat-Engine Performance via High-Temperature Recharge*

*DOI: http://dx.doi.org/10.5772/intechopen.89913*

on short time scales.

is not possible.

**109**

operating at maximum power output.

erwise) is discussed and corrected in Section 2.

which we dub as high-temperature recharge (HTR).

But a usually necessary [1–7], although not always sufficient, requirement to achieve reversibility is that a heat engine, whether cyclic or noncyclic—indeed any engine, heat engine or otherwise—must operate infinitely slowly, i.e., quasistatically [1–7]. And infinitely slow operation, which implies infinitesimally small power output, is obviously impractical [1–7]. Indeed, some types of friction, such as sliding and rolling friction, do *not* vanish as the speed of operation becomes infinitely slow [8–11] [see also Ref. [1], Sections 5-2, 6-1, 6-2, 8-1, 11-1, and 11-2; Ref. [2], Section 4.2 (especially the 3rd and 4th paragraphs) and Figure 4.3; Ref. [3], Problem 4.2-1; and Ref. [4], Section 3.6]. In such cases, reversibility does *not* obtain *even with* infinitely slow, i.e., quasi-static, operation [8–11]. By contrast, for example, in cosmology, reversibility may in some cases obtain even with processes occurring at finite rates [12], but of course this is not relevant with respect to practical heat engines.

Most real heat engines operate, if not at maximum power output, then at least closer to maximum power output than to maximum efficiency. Assuming endoreversibility (irreversible heat flows directly proportional to finite temperature differences but otherwise reversible operation), at maximum power output cyclic heat engines operate at Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9). The work outputs of heat engines—indeed of *all* engines, heat engines or otherwise—are in almost cases totally frictionally dissipated as heat immediately or on short time scales [16, 17]. For example, an automobile's cyclic heat engine's work output in initially accelerating the automobile is typically frictionally dissipated only a short time later the next time the automobile decelerates; its work output while the automobile travels at constant speed is immediately and continually frictionally dissipated. [Rare exceptions include, for example, a noncyclic rocket heat engine's work output being sequestered essentially permanently as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload) and a cyclic heat engine's work output being sequestered for a long time interval as gravitational potential energy in the construction of a building.]

We note that the *work* output of *any* engine (heat engine or otherwise) can be dissipated *only via friction*. Additional losses can, and almost always if not always, also occur, for example, irreversible heat losses engendered by finite temperature differences (no insulation is perfect). [The Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) takes into account losses due to irreversible heat flows directly proportional to finite temperature differences but assumes otherwise reversible operation.] But such heat losses are *not* work. An engine's *work* output per se can be dissipated *only via friction*. This is true because work is a *force* exerted through a distance: thus work can be dissipated *only* by an opposing *force* that is *non*conservative. And *non*conservative *force* is *friction*. [It might be contended that, ultimately, friction is the electromagnetic force, which is conservative. But for all typical macroscopic motions, for which the kinetic energy in any given degree of freedom greatly exceeds *kBT* (*kB* is Boltzmann's constant,*T* is the temperature), friction is *effectively non*conservative.]

But if a heat engine's work output must be frictionally dissipated, it is best to dissipate it not at the temperature of its cold reservoir but instead at the highest practicable temperature. This is consistent with the Second Law of Thermodynamics, which allows frictional dissipation of work into heat at *any temperature* [1–7]

accelerates the gases resulting from combustion of the propellant). (Some rocket

refurbished and reused, but both the first use and each subsequent refurbishment and reuse constitute a necessarily one-time, single-use of a *non*cyclic rocket heat

But a usually necessary [1–7], although not always sufficient, requirement to achieve reversibility is that a heat engine, whether cyclic or noncyclic—indeed any engine, heat engine or otherwise—must operate infinitely slowly, i.e., quasistatically [1–7]. And infinitely slow operation, which implies infinitesimally small power output, is obviously impractical [1–7]. Indeed, some types of friction, such as sliding and rolling friction, do *not* vanish as the speed of operation becomes infinitely slow [8–11] [see also Ref. [1], Sections 5-2, 6-1, 6-2, 8-1, 11-1, and 11-2; Ref. [2], Section 4.2 (especially the 3rd and 4th paragraphs) and Figure 4.3; Ref. [3], Problem 4.2-1; and Ref. [4], Section 3.6]. In such cases, reversibility does *not* obtain *even with* infinitely slow, i.e., quasi-static, operation [8–11]. By contrast, for example, in cosmology, reversibility may in some cases obtain even with processes occurring at finite rates [12], but of course this is not relevant with respect to

Most real heat engines operate, if not at maximum power output, then at least

endoreversibility (irreversible heat flows directly proportional to finite temperature differences but otherwise reversible operation), at maximum power output cyclic heat engines operate at Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9). The work outputs of heat engines—indeed of *all* engines, heat engines or otherwise—are in almost cases totally frictionally dissipated as heat immediately or on short time scales [16, 17]. For example, an automobile's cyclic heat engine's work output in initially accelerating the automobile is typically frictionally dissipated only a short time later the next time the automobile decelerates; its work output while the automobile travels at constant speed is immediately and continually frictionally dissipated. [Rare exceptions include, for example, a noncyclic rocket heat engine's work output being sequestered essentially permanently as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload) and a cyclic heat engine's work output being sequestered for a long time interval as

We note that the *work* output of *any* engine (heat engine or otherwise) can be dissipated *only via friction*. Additional losses can, and almost always if not always, also occur, for example, irreversible heat losses engendered by finite temperature differences (no insulation is perfect). [The Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) takes into account losses due to irreversible heat flows directly proportional to finite temperature differences but assumes otherwise reversible operation.] But such heat losses are *not* work. An engine's *work* output per se can be dissipated *only via friction*. This is true because work is a *force* exerted through a distance: thus work can be dissipated *only* by an opposing *force* that is *non*conservative. And *non*conservative *force* is *friction*. [It might be contended that, ultimately, friction is the electromagnetic force, which is conservative. But for all typical macroscopic motions, for which the kinetic energy in any given degree of freedom greatly exceeds *kBT* (*kB* is Boltzmann's constant,*T* is the temperature),

But if a heat engine's work output must be frictionally dissipated, it is best to dissipate it not at the temperature of its cold reservoir but instead at the highest practicable temperature. This is consistent with the Second Law of Thermodynamics, which allows frictional dissipation of work into heat at *any temperature* [1–7]

closer to maximum power output than to maximum efficiency. Assuming

gravitational potential energy in the construction of a building.]

friction is *effectively non*conservative.]

**108**

engines, e.g., those employed in the Space Shuttle and by SpaceX, can be

engine.)

*Thermodynamics and Energy Engineering*

practical heat engines.

(in Ref. [6], see pp. 11–12, 60–65, and 263–265, especially pp. 263–265). The entropy increase resulting from frictional dissipation of work *W* at temperature *T*, namely Δ*S* ¼ *W=T*, decreases monotonically with increasing *T* but is positive for any finite *T*—and the Second Law requires only that Δ*S*≥0 [1–7].

We should emphasize that the entropy increase Δ*S* ¼ *W=T* associated with work *W* being frictionally dissipated at temperature *T* is *always* Δ*S* ¼ *W=T* and hence *always* decreases monotonically with increasing *T*. Whether the coefficient of friction is small or large *makes no difference* in Δ*S* ¼ *W=T*. Of course, all other things being equal, if the coefficient of friction is small, frictional dissipation of *W* will take longer than if the coefficient of friction is large. But the entropy increase Δ*S* ¼ *W=T* associated with work *W* being frictionally dissipated at temperature *T* is the same whether the coefficient of friction is small or large. Even with coefficients of friction in the low range, *W* will typically be frictionally dissipated immediately or on short time scales.

Of course, efficiency is highest if work *W*, whether supplied via a heat engine or otherwise, is not frictionally dissipated at all. This would obtain, for example, in perfect (reversible) regenerative braking of an electrically-powered motor vehicle, with the motor operating backward as a generator during braking. It also would obtain, for example, if a noncyclic rocket heat engine's work output is perfectly (reversibly) sequestered as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload) or if a cyclic heat engine's work output is perfectly (reversibly) sequestered as gravitational potential energy in the construction of a building. But of course in practice (as opposed to in principle) *total* avoidance of frictional dissipation of work [and also of additional losses, e.g., due to irreversible heat flows engendered by finite temperature differences (no insulation is perfect)] is not possible.

Although we do not consider them in this chapter, we should note that: (a) There are generalizations of the Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) at maximum power output both for macroscopic heat engines [18–20] and for microscopic heat engines [21, 22], with irreversible heat flows *not* necessarily directly proportional to temperature differences. (b) There are analyses of maximum heat-engine work output per cycle (as opposed to maximum power output) [23]. Comprehensive discussions concerning the Curzon-Ahlborn efficiency and generalizations thereof are provided in Refs. [24–26]. Some, but not all, such generalized efficiencies [18–26] do not differ greatly from the Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9). In particular, we note that alternative results [26] to the Curzon-Ahlborn efficiency have been derived [26]. But for definiteness and for simplicity, in this chapter, we employ the standard Curzon-Ahlborn efficiency [13–15] (see also Ref. [3], Section 4-9) for cyclic heat engines operating at maximum power output.

A misconception pertaining to the efficiencies of engines (heat engines or otherwise) is discussed and corrected in Section 2.

In Section 3, we review the work outputs, efficiencies, and entropy productions of Carnot (reversible) and Curzon-Ahlborn (endoreversible) heat engines, first without frictional dissipation of a heat engine's work output and then with frictional dissipation thereof into its cold reservoir. In Section 3, we do not consider frictional dissipation of a heat engine's work output at the highest practicable temperature, which we dub as high-temperature recharge (HTR).

In Section 4, we discuss the work outputs, efficiencies, and entropy productions of Carnot and Curzon-Ahlborn heat engines operating with frictional dissipation of a heat engine's work output at the highest practicable temperature—which we dub as high-temperature recharge (HTR)—and the improvements thereof over those

obtainable (as per Section 3) without HTR. Cases wherein HTR is practicable include, but are not necessarily limited to, (a) hurricanes, which via HTR are rendered more powerful than they would otherwise be [27–37], (b) thermoelectric generators [38], and (c) heat engines *powered by a cold reservoir*, employing ambient as the *hot* reservoir, for example, heat engines powered by the evaporation of water [39–51] or by liquid nitrogen [52], ocean-thermal-energy-conversion (OTEC) heat engines [53–56], and heat engines powered by the cold of outer space [57].

done via forward operation of cyclic heat engines). {See Ref. [1], Section 20-3; Ref. [2], Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. [3], Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. [5], Section 5.12 and Problem 5.22; Ref. [7], pp. 233–236 and Problems 1, 2, 4, 6, and 7 of Chapter 8; Ref. [16], Chapter XXI; Ref. [17], Sections 6.7, 6.8, 7.3, and 7.4; and Ref. [54], Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17. [Problem 2 of Chapter 8 in Ref. [7] considers absorption refrigeration, wherein the entire energy output is into an *intermediate*-temperature (most typically ambient-temperature) reservoir, and hence for which HTR is *even more strongly never practicable*.]} They also are not practicable for cyclic heat engines in cases wherein a cyclic heat engine's work output is not frictionally dissipated immediately or on short time scales [16, 17], for example, as gravitational potential energy sequestered for a long time interval in the construction of a building. For a building once erected typically remains standing for a century or longer. Even if, when it is finally torn down, its gravitational potential energy were to be totally frictionally dissipated into a hot reservoir, it is simply impracticable to wait that long. Thus HTR is not practicable in all cases. But in the many cases wherein cyclic heat engines' work outputs are frictionally dissipated immediately or on short time scales [16, 17], practicability obtains: improved conversion—and reconversion—of frictionally dissipated heat into work, and hence improved cyclic heat-engine performance, can then obtain. Since HTR is *never* practicable for noncyclic (necessarily one-time, single-use) heat engines such as rockets or firearms, or for reverse operation of cyclic heat engines as refrigerators or heat pumps, henceforth we will (except where otherwise

*Improving Heat-Engine Performance via High-Temperature Recharge*

*DOI: http://dx.doi.org/10.5772/intechopen.89913*

mentioned) focus exclusively on forward operation of cyclic heat engines.

*immediately* and hence *both* time intervals are *zero*.)

**111**

Note that the primarily relevant time scale pertaining to "in the many cases wherein cyclic heat engines' work outputs are frictionally dissipated immediately or on short time scales [16, 17]" is (i) the time interval between a cyclic heat engine's work output and frictional dissipation of this work output [16, 17], *not* (ii) the time interval required for frictional dissipation per se. The time interval (ii) is *zero* in all cases wherein work is done against the nonconservative force of friction and hence frictionally dissipated *immediately*. Indeed, in *all* cases of steady-state engine operation against friction (e.g., an automobile traveling at constant speed) *both* time intervals are *zero*. This is by far the most common mode of engine operation. Even work output sequestered when an engine is started or when an automobile accelerates is typically frictionally dissipated only a short time later, when the engine is turned off or when the automobile decelerates. Work output sequestration for a century or longer can obtain (as gravitational potential energy) in the construction of buildings and essentially permanently in the launchings of spacecraft—but these are rare exceptions. Thus we focus on time interval (i): a necessary (but not sufficient) condition for HTR to be practicable is that the time interval (i) be zero or at most short. (This condition is automatically met in steady-state engine operation, wherein work is frictionally dissipated

But for cyclic heat engines whose work outputs typically are frictionally dissipated immediately or on short time scales [16, 17], HTR often *is* practicable. For cyclic heat engines employing ambient as the *cold* reservoir, the existent hot reservoir is likely already at the practicable upper temperature limit. Hence for these cyclic heat engines, HTR at the temperature of the hot reservoir could increase efficiency, but HTR at a still higher temperature probably would not be practicable. By contrast, consider cyclic heat engines *powered by a cold reservoir*, employing ambient as the *hot* reservoir, for example, cyclic heat engines powered by the evaporation of water [39–51] or by liquid nitrogen [52], ocean thermal-energy-conversion (OTEC) heat engines [53–56], and heat engines powered by the cold of outer space [57]. For *these* cyclic heat engines, HTR at a higher temperature than ambient probably

Concerning (a) in the immediately preceding paragraph, on the one hand, the importance of HTR (dubbed as "dissipative heating") has been confirmed in a study of Hurricane Andrew (1992) [36], and, as one might expect, "dissipative heating appears to be a more important process in intense hurricanes, such as Andrew, than weak ones" [36]. But, on the other hand, more recently it has been contended [37] that, while HTR exists in hurricanes, it is of lesser importance than previously supposed [27–36]. [There are occasional speculations concerning extracting useful energy from hurricanes (with or without help from HTR) and also freshwater. But, of course, except for (strongly built!) windmills and ocean-wavepowered generators for extracting energy and reservoirs for extracting freshwater, this is beyond currently available (and perhaps even currently foreseeable) technology. To the extent that HTR increases wind speeds in hurricanes, it increases the power flux density available to (strongly built!) windmills and ocean-wavepowered generators: wind power flux density is proportional to the cube of the wind speed (and directly proportional to the air density). But, at least for the time being, the main (or perhaps even only) employment of HTR in hurricanes is by the hurricanes themselves, to increase their wind speeds, whether as previously supposed [27–36] or to a lesser degree [37].

To the best knowledge of the author, the concept of HTR was first partially and qualitatively broached by Spanner (see Ref. [6], pp. 11–12, 60–65, and 263–265, especially pp. 263–265) and, later, was first fully and quantitatively expounded and developed by Emanuel [27–34] in the course of his research concerning hurricane science. It was subsequently employed by Apertet et al. [38] for increasing efficiencies of thermoelectric generators. In these works [6, 27–34, 38] and in related works [35–37, 58–62], the concept is not dubbed HTR, but of course it is the concept itself, and not the dubbing it with a name, that is important. To the best knowledge of the author, the concept has not been dubbed HTR (dubbed, if at all, as "dissipative heating") in the previous literature. Heat engines employing it have previously been dubbed "dissipative engines" (see, e.g., Refs. [58–60]).

The increases in efficiency attainable via HTR are not practicable if frictional dissipation of work into other than the cold reservoir is not practicable. Thus they are *never* practicable for noncyclic (necessarily one-time, single-use) heat engines: however the work output of a noncyclic (necessarily one-time, single-use) heat engine might be frictionally dissipated, the heat thereby generated can*not* restore the engine to its initial state. Moreover in many cases the work outputs of noncyclic (necessarily one-time, single-use) heat engines are not frictionally dissipated *at all*, at least not during practicable time scales, for example, a noncyclic rocket heat engine's work output is sequestered essentially permanently as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload). They also are *never* practicable for reverse operation of cyclic heat engines as refrigerators or heat pumps, because for both refrigerators and heat pumps, the *total* energy output (the work *W*, plus the heat *QC* extracted from a cold reservoir at the expense of *W* as required by the Second Law of Thermodynamics) *always* is deposited as heat *Q <sup>H</sup>* into the hot reservoir (*Q <sup>H</sup>* = *QC* þ *W*): thus there is *never* any *additional* energy to be deposited into the hot reservoir (as there is from frictional dissipation of work

### *Improving Heat-Engine Performance via High-Temperature Recharge DOI: http://dx.doi.org/10.5772/intechopen.89913*

obtainable (as per Section 3) without HTR. Cases wherein HTR is practicable include, but are not necessarily limited to, (a) hurricanes, which via HTR are rendered more powerful than they would otherwise be [27–37], (b) thermoelectric generators [38], and (c) heat engines *powered by a cold reservoir*, employing ambient as the *hot* reservoir, for example, heat engines powered by the evaporation of water [39–51] or by liquid nitrogen [52], ocean-thermal-energy-conversion (OTEC) heat

engines [53–56], and heat engines powered by the cold of outer space [57]. Concerning (a) in the immediately preceding paragraph, on the one hand, the importance of HTR (dubbed as "dissipative heating") has been confirmed in a study of Hurricane Andrew (1992) [36], and, as one might expect, "dissipative heating appears to be a more important process in intense hurricanes, such as Andrew, than weak ones" [36]. But, on the other hand, more recently it has been contended [37] that, while HTR exists in hurricanes, it is of lesser importance than previously supposed [27–36]. [There are occasional speculations concerning

extracting useful energy from hurricanes (with or without help from HTR) and also freshwater. But, of course, except for (strongly built!) windmills and ocean-wavepowered generators for extracting energy and reservoirs for extracting freshwater, this is beyond currently available (and perhaps even currently foreseeable) technology. To the extent that HTR increases wind speeds in hurricanes, it increases the

To the best knowledge of the author, the concept of HTR was first partially and qualitatively broached by Spanner (see Ref. [6], pp. 11–12, 60–65, and 263–265, especially pp. 263–265) and, later, was first fully and quantitatively expounded and developed by Emanuel [27–34] in the course of his research concerning hurricane science. It was subsequently employed by Apertet et al. [38] for increasing efficiencies of thermoelectric generators. In these works [6, 27–34, 38] and in related works [35–37, 58–62], the concept is not dubbed HTR, but of course it is the concept itself, and not the dubbing it with a name, that is important. To the best knowledge of the author, the concept has not been dubbed HTR (dubbed, if at all, as "dissipative heating") in the previous literature. Heat engines employing it have previously been

The increases in efficiency attainable via HTR are not practicable if frictional dissipation of work into other than the cold reservoir is not practicable. Thus they are *never* practicable for noncyclic (necessarily one-time, single-use) heat engines: however the work output of a noncyclic (necessarily one-time, single-use) heat engine might be frictionally dissipated, the heat thereby generated can*not* restore the engine to its initial state. Moreover in many cases the work outputs of noncyclic (necessarily one-time, single-use) heat engines are not frictionally dissipated *at all*, at least not during practicable time scales, for example, a noncyclic rocket heat engine's work output is sequestered essentially permanently as kinetic and gravitational potential energy in the launching of a spacecraft (but typically most of the kinetic energy accelerates the exhaust gases, not the payload). They also are *never* practicable for reverse operation of cyclic heat engines as refrigerators or heat pumps, because for both refrigerators and heat pumps, the *total* energy output (the work *W*, plus the heat *QC* extracted from a cold reservoir at the expense of *W* as required by the Second Law of Thermodynamics) *always* is deposited as heat *Q <sup>H</sup>* into the hot reservoir (*Q <sup>H</sup>* = *QC* þ *W*): thus there is *never* any *additional* energy to be deposited into the hot reservoir (as there is from frictional dissipation of work

power flux density available to (strongly built!) windmills and ocean-wavepowered generators: wind power flux density is proportional to the cube of the wind speed (and directly proportional to the air density). But, at least for the time being, the main (or perhaps even only) employment of HTR in hurricanes is by the hurricanes themselves, to increase their wind speeds, whether as previously

supposed [27–36] or to a lesser degree [37].

*Thermodynamics and Energy Engineering*

dubbed "dissipative engines" (see, e.g., Refs. [58–60]).

**110**

done via forward operation of cyclic heat engines). {See Ref. [1], Section 20-3; Ref. [2], Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. [3], Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. [5], Section 5.12 and Problem 5.22; Ref. [7], pp. 233–236 and Problems 1, 2, 4, 6, and 7 of Chapter 8; Ref. [16], Chapter XXI; Ref. [17], Sections 6.7, 6.8, 7.3, and 7.4; and Ref. [54], Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17. [Problem 2 of Chapter 8 in Ref. [7] considers absorption refrigeration, wherein the entire energy output is into an *intermediate*-temperature (most typically ambient-temperature) reservoir, and hence for which HTR is *even more strongly never practicable*.]} They also are not practicable for cyclic heat engines in cases wherein a cyclic heat engine's work output is not frictionally dissipated immediately or on short time scales [16, 17], for example, as gravitational potential energy sequestered for a long time interval in the construction of a building. For a building once erected typically remains standing for a century or longer. Even if, when it is finally torn down, its gravitational potential energy were to be totally frictionally dissipated into a hot reservoir, it is simply impracticable to wait that long. Thus HTR is not practicable in all cases. But in the many cases wherein cyclic heat engines' work outputs are frictionally dissipated immediately or on short time scales [16, 17], practicability obtains: improved conversion—and reconversion—of frictionally dissipated heat into work, and hence improved cyclic heat-engine performance, can then obtain.

Since HTR is *never* practicable for noncyclic (necessarily one-time, single-use) heat engines such as rockets or firearms, or for reverse operation of cyclic heat engines as refrigerators or heat pumps, henceforth we will (except where otherwise mentioned) focus exclusively on forward operation of cyclic heat engines.

Note that the primarily relevant time scale pertaining to "in the many cases wherein cyclic heat engines' work outputs are frictionally dissipated immediately or on short time scales [16, 17]" is (i) the time interval between a cyclic heat engine's work output and frictional dissipation of this work output [16, 17], *not* (ii) the time interval required for frictional dissipation per se. The time interval (ii) is *zero* in all cases wherein work is done against the nonconservative force of friction and hence frictionally dissipated *immediately*. Indeed, in *all* cases of steady-state engine operation against friction (e.g., an automobile traveling at constant speed) *both* time intervals are *zero*. This is by far the most common mode of engine operation. Even work output sequestered when an engine is started or when an automobile accelerates is typically frictionally dissipated only a short time later, when the engine is turned off or when the automobile decelerates. Work output sequestration for a century or longer can obtain (as gravitational potential energy) in the construction of buildings and essentially permanently in the launchings of spacecraft—but these are rare exceptions. Thus we focus on time interval (i): a necessary (but not sufficient) condition for HTR to be practicable is that the time interval (i) be zero or at most short. (This condition is automatically met in steady-state engine operation, wherein work is frictionally dissipated *immediately* and hence *both* time intervals are *zero*.)

But for cyclic heat engines whose work outputs typically are frictionally dissipated immediately or on short time scales [16, 17], HTR often *is* practicable. For cyclic heat engines employing ambient as the *cold* reservoir, the existent hot reservoir is likely already at the practicable upper temperature limit. Hence for these cyclic heat engines, HTR at the temperature of the hot reservoir could increase efficiency, but HTR at a still higher temperature probably would not be practicable. By contrast, consider cyclic heat engines *powered by a cold reservoir*, employing ambient as the *hot* reservoir, for example, cyclic heat engines powered by the evaporation of water [39–51] or by liquid nitrogen [52], ocean thermal-energy-conversion (OTEC) heat engines [53–56], and heat engines powered by the cold of outer space [57]. For *these* cyclic heat engines, HTR at a higher temperature than ambient probably

*would* be practicable. For *these* cyclic heat engines, employment of HTR could boost the temperature of the hot reservoir from ambient to the highest practicable temperature for HTR.

In general, when an engine (heat engine or otherwise) is turned on, part of its work output is sequestered as its own kinetic energy and the kinetic energy of any equipment that it might be operating. But this kinetic energy is frictionally dissipated as heat when the engine is turned off, so the net effect of the on/off process is frictional dissipation of this temporarily sequestered kinetic energy, the same as the instantaneous and continual frictional dissipation of the engine's work output while it operates at constant speed between the time it is turned on and the time it is

**3. Carnot and Curzon-Ahlborn efficiencies without high-temperature**

The standard (without high-temperature recharge or HTR) Carnot efficiency ϵCarnot,std and Curzon-Ahlborn efficiency ϵCA,std corresponding to heat-engine operation between a hot reservoir at temperature *TH* and a cold reservoir at temperature

> <sup>¼</sup> *TH* � *TC TH*

<sup>¼</sup> <sup>1</sup> � *TC TH*

<sup>¼</sup> <sup>1</sup> � *TC TH* <sup>1</sup>*=*<sup>2</sup>

> <sup>¼</sup> <sup>1</sup> � *RT* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup> *T*

> > <sup>1</sup> � <sup>1</sup> � <sup>1</sup>

<sup>2</sup> *<sup>δ</sup>* <sup>¼</sup> *<sup>δ</sup>*

1

<sup>2</sup> *<sup>δ</sup>* <sup>¼</sup> <sup>2</sup>*:* (4)

*R*ϵ<sup>1</sup> increases monotonically with increasing *R*T. In the limit *RT* ! 0, it is obvious that *R*ϵ<sup>1</sup> ! 1. In the limit *RT* ! 1, *R*ϵ<sup>1</sup> ! 2. The latter limit is most easily demonstrated by setting *RT* ¼ 1 � *δ*, letting *δ* ! 0, and applying the binomial theorem.

<sup>1</sup> � ð Þ <sup>1</sup> � *<sup>δ</sup>* <sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> *<sup>δ</sup>*

By the First and Second Laws of Thermodynamics, for a standard reversible heat engine operating (without HTR) at Carnot efficiency, the heat input *QH* from its

We define the *temperature* ratio between a heat engine's cold and hot reservoirs as *RT* � *TC=TH*. Obviously in all cases 0 ≤*RT* ≤1. The case *RT* ¼ 1 is of no interest because it corresponds to zero efficiency of any heat engine. The case *RT* ¼ 0 is unattainable because there is no cold reservoir at absolute zero (0 K) [63], and even if there was [63], it would no longer be at 0 K the instant after a heat engine began operating and exhausting its waste heat into it [see also Ref. [2], Chapter 10 (especially Section 10.4); Ref. [3], Chapter 11 (especially Section 11-3); and Ref. [4], Chapter 14 (especially Sections 14.3–14.5 and 14.7)]. Hence we confine our atten-

� 1 � *RT* (1)

*:* (3)

*<sup>T</sup> :* (2)

� <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup>

*TC* are given by the respective well-known formulas:

*Improving Heat-Engine Performance via High-Temperature Recharge*

*DOI: http://dx.doi.org/10.5772/intechopen.89913*

<sup>ϵ</sup>Carnot,std <sup>¼</sup> *<sup>W</sup>*

*QH*

<sup>ϵ</sup>CA,std <sup>¼</sup> *<sup>W</sup>*

tion to the range 0 <*RT* < 1.

*QH*

<sup>¼</sup> *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup>

The *efficiency* ratio *R*ϵ<sup>1</sup> between these efficiencies is

*R*ϵ<sup>1</sup> ¼ lim *δ*!0

*<sup>R</sup>*ϵ<sup>1</sup> <sup>¼</sup> <sup>ϵ</sup>Carnot,std ϵCA,std

1 � ð Þ 1 � *δ*

*<sup>H</sup>* � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> *C T*<sup>1</sup>*=*<sup>2</sup> *H*

turned off.

and

This yields

**113**

lim *RT*!1

*R*ϵ<sup>1</sup> ¼ lim *δ*!0

**recharge (HTR)**

Henceforth if HTR is employed we construe the terms "the highest practicable temperature for HTR" and "the hot reservoir" to be synonymous.

Recapitulation and generalization are provided in Section 5. A reply to criticisms [58, 59] of HTR is provided in Section 6 and in references cited therein. Concluding remarks are provided in Section 7.

### **2. Correcting a misconception pertaining to the efficiencies of engines**

The efficiency of any engine in general is its work output (force-times-distance output) divided by its energy input, and the efficiency of a heat engine in particular it is its work output (force-times-distance output) divided by its heat-energy input. What happens to an engine's work output after the work has been done is an entirely different issue.

Work can be done either against a conservative force, in which case it is sequestered, or against the nonconservative force of friction, in which case it is dissipated as heat. To re-emphasize, in either case—whether the opposing force is conservative or nonconservative—the efficiency of any engine in general is its work output (force-times-distance output) divided by its energy input, and the efficiency of a heat engine in particular it is its work output (force-times-distance output) divided by its heat-energy input. What happens to an engine's work output after the work has been done is an entirely different issue.

In this Section 2, we wish to correct a misconception that is sometimes made, according to which an engine's efficiency can exceed zero only if its work output is done against a conservative force. This misconception is erroneous.

In the vast majority of cases, for almost all engines on Earth, work is done against the nonconservative force of friction, and hence instantaneously and continually dissipated as heat. The engines work at steady state, and while working, their internal energy and the internal energy of any equipment they might be operating do not change. Consider, for example, the engine of any automobile, train, ship, submarine, or aircraft traveling at constant speed, any factory or workshop engine such as a power saw operating at constant speed, or any domestic appliance engine such as that of a dishwasher, refrigerator, etc., operating at constant speed. According to the erroneous misconception that an engine's efficiency is zero if its work output is done against the nonconservative force of friction, the efficiency of all of these engines—indeed of almost all engines on Earth—would falsely be evaluated at zero. If their efficiencies were truly zero, they could do zero work against *any* opposing force, conservative or nonconservative, i.e., they could not operate *at all*. A specific example is the following: If the efficiency of an engine (heat engine or otherwise) attempting to maintain an automobile at constant speed was zero, the engine could do zero work against friction, and the automobile's speed would also be zero.

Only in rare cases, such as the construction of buildings and the launchings of spacecraft, is the work done even against conservative forces (e.g., gravity, inertia, etc.) sequestered for any significant lengths of time. Even in most cases wherein work is done against a conservative force, it is frictionally dissipated a short time later. For example, the work done in accelerating an automobile against its own inertia is typically frictionally dissipated as heat a short time later the next time the automobile decelerates. The net effect of the acceleration/deceleration process is frictional dissipation of the automobile's temporarily sequestered kinetic energy, the same as the instantaneous and continual frictional dissipation of its kinetic energy while it operates at constant speed.

*Improving Heat-Engine Performance via High-Temperature Recharge DOI: http://dx.doi.org/10.5772/intechopen.89913*

In general, when an engine (heat engine or otherwise) is turned on, part of its work output is sequestered as its own kinetic energy and the kinetic energy of any equipment that it might be operating. But this kinetic energy is frictionally dissipated as heat when the engine is turned off, so the net effect of the on/off process is frictional dissipation of this temporarily sequestered kinetic energy, the same as the instantaneous and continual frictional dissipation of the engine's work output while it operates at constant speed between the time it is turned on and the time it is turned off.
