**3. Carnot and Curzon-Ahlborn efficiencies without high-temperature recharge (HTR)**

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

$$
\epsilon\_{\text{Carnot,std}} = \frac{W}{Q\_H} = \frac{T\_H - T\_C}{T\_H} = \mathbf{1} - \frac{T\_C}{T\_H} \equiv \mathbf{1} - R\_T \tag{1}
$$

and

*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 tem-

Henceforth if HTR is employed we construe the terms "the highest practicable

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

**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

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

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

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

done against a conservative force. This misconception is erroneous.

temperature for HTR" and "the hot reservoir" to be synonymous.

perature for HTR.

entirely different issue.

remarks are provided in Section 7.

*Thermodynamics and Energy Engineering*

has been done is an entirely different issue.

energy while it operates at constant speed.

**112**

$$\epsilon\_{\rm CA,std} = \frac{\mathcal{W}}{Q\_H} = \frac{T\_H^{1/2} - T\_C^{1/2}}{T\_H^{1/2}} = \mathbf{1} - \left(\frac{T\_C}{T\_H}\right)^{1/2} \equiv \mathbf{1} - \mathcal{R}\_T^{1/2}.\tag{2}$$

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

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

$$R\_{e1} = \frac{\epsilon\_{\text{Carnot,std}}}{\epsilon\_{\text{CA,std}}} = \frac{\mathbf{1} - R\_T}{\mathbf{1} - R\_T^{1/2}}.\tag{3}$$

*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. This yields

$$\lim\_{R\_{\Gamma}\to 1} R\_{\epsilon1} = \lim\_{\delta \to 0} R\_{\epsilon1} = \lim\_{\delta \to 0} \frac{1 - (1 - \delta)}{1 - (1 - \delta)^{1/2}} = \frac{\delta}{1 - \left(1 - \frac{1}{2}\delta\right)} = \frac{\delta}{\frac{1}{2}\delta} = 2. \tag{4}$$

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

hot reservoir, the work output *W*, the waste heat *QC* exhausted to its cold reservoir, the efficiency ϵCarnot,std, the entropy change Δ*SH*,Carnot,std of its hot reservoir, the entropy change Δ*SC*,Carnot,std of its cold reservoir, and the total entropy change Δ*S*total,std ¼ Δ*SH*,Carnot,std þ Δ*SC*,Carnot,std are related in accordance with

$$\begin{aligned} \mathbf{W} &= \mathbf{Q}\_H - \mathbf{Q}\_C = \mathbf{Q}\_H \mathbf{e}\_{\text{Carnot,std}} = \mathbf{Q}\_H (\mathbf{1} - \mathbf{R}\_T) \\ &\Rightarrow \mathbf{Q}\_C = \mathbf{R}\_T \mathbf{Q}\_H. \end{aligned} \tag{5}$$

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.] Apart from such rare exceptional cases, in the operation of *any* cyclic

> ¼ � *QH TH*

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

Note that for any *RT* >0, the inequality Δ*S*total,ultimate >0 is stronger than the inequality Δ*S*total,CA,std > 0. In particular, note that as *RT* ! 1, Δ*S*total,ultimate ! 0 more slowly than Δ*S*total,CA,std ! 0, while as *RT* ! 0, Δ*S*total,ultimate ! ∞ faster than

**4. Carnot and Curzon-Ahlborn efficiencies with high-temperature**

Consider first a reversible heat engine operating at Carnot efficiency. If the engine's work output *W* ¼ *QH*ð Þ 1 � *RT* is frictionally dissipated into its hot reservoir (at temperature *TH*), then the *net* heat input *QH*,net required from its hot reservoir is reduced from *QH* to *QH* � *W* ¼ *QH* � *QH*ð Þ¼ 1 � *RT QH*� *QHRT* ¼ *QC*. Hence, with the help of Eqs. (1) and (5), the efficiency ϵCarnot,HTR of a Carnot

If, as is almost always the case, a cyclic heat engine's work output *W* is totally frictionally dissipated as heat immediately or on short time scales [16, 17], the engine's efficiency can be increased if this dissipation is not at the temperature of its cold reservoir but instead at the highest practicable temperature. 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, for cyclic heat engines *powered by a cold reservoir*, employing ambient as the *hot* reservoir [39–57], frictional dissipation at a higher temperature than ambient probably *would* be practicable. For *these* cyclic heat engines, employment of HTR could boost *TH*, the temperature of the hot reservoir, from ambient to the the highest practicable temperature for HTR. In this Section 4, we take *TH*, the temperature of the hot reservoir, to be the highest practicable temperature for frictional dissipation

1 *RT* � 1 

<sup>¼</sup> *QH TH*

*TH*

<sup>þ</sup> *QC* <sup>þ</sup> *<sup>W</sup> TC*

> <sup>¼</sup> *QH TH*

¼ � *QH TH*

*R*�<sup>1</sup> *<sup>T</sup>* � <sup>1</sup>

<sup>þ</sup> *QH TC*

(9)

heat engine operating at *any* efficiency with*out* HTR—whether reversible at Carnot efficiency, endoreversible at Curzon-Ahlborn efficiency, or otherwise—not only is the waste heat *QC* immediately exhausted into the cold reservoir, but also the heat engine's work output *W* is frictionally dissipated into the cold reservoir immediately or on short time scales [16, 17]. Thus, apart from such rare exceptional cases, the *ultimate* total entropy increase Δ*S*total,ultimate resulting from the operation of *any* heat engine with*out* HTR at *any* efficiency occurs as soon as the work is done

or shortly thereafter and is [1–7, 16, 17]:

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

*TH*

¼ � *QH TH*

of a cyclic heat engine's work output into heat.

engine operating with HTR is

**115**

<sup>þ</sup> *QC TC* þ *W TC*

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

<sup>þ</sup> *QH RTTH*

<sup>&</sup>gt; <sup>Δ</sup>*S*total,CA,std <sup>¼</sup> *QH*

> Δ*S*total,Carnot,std ¼ 0*:*

<sup>Δ</sup>*S*total,ultimate ¼ � *QH*

Δ*S*total,CA,std ! ∞.

**recharge (HTR)**

and

$$
\Delta \mathbf{S}\_{\text{total,Carnot,std}} = \Delta \mathbf{S}\_{H, \text{Carnot,std}} + \Delta \mathbf{S}\_{\text{C,Carnot,std}} = -\frac{\mathbf{Q}\_H}{T\_H} + \frac{\mathbf{Q}\_C}{T\_C} = -\frac{\mathbf{Q}\_H}{T\_H} + \frac{\mathbf{R}\_T \mathbf{Q}\_H}{R\_T T\_H} = \mathbf{0}.\tag{6}
$$

We note that, in most derivations (in textbooks or elsewhere) of ϵCarnot,std, Δ*S*total,Carnot,std ¼ 0 is invoked as the initial Second Law assumption or postulate and is employed along with the initial First Law postulate *W* ¼ *QH* � *QC* [1–7].

Similarly, by the First and Second Laws of Thermodynamics, for a standard endoreversible heat engine operating (without HTR) at Curzon-Ahlborn efficiency, the heat input *QH* from its hot reservoir, the work output *W*, the waste heat *QC* exhausted to its cold reservoir, the efficiency ϵCA,std, the entropy change Δ*SH*,CA,std of its hot reservoir, the entropy change Δ*SC*,CA,std of its cold reservoir, and the total entropy change Δ*S*total,CA,std ¼ Δ*SH*,CA,std þ Δ*SC*,CA,std are related in accordance with

$$\begin{aligned} \mathbf{W} &= \mathbf{Q}\_H - \mathbf{Q}\_C = \mathbf{Q}\_H \mathbf{e}\_{\mathbf{CA}, \text{std}} = \mathbf{Q}\_H \left( \mathbf{1} - \mathbf{R}\_T^{1/2} \right) \\\\ \Rightarrow \mathbf{Q}\_C &= \mathbf{R}\_T^{1/2} \mathbf{Q}\_H. \end{aligned} \tag{7}$$

and

$$
\Delta \mathbf{S}\_{\text{total,CA,std}} = \Delta \mathbf{S}\_{H,\text{CA,std}} + \Delta \mathbf{S}\_{\text{C,CA,std}} = -\frac{\mathbf{Q}\_H}{T\_H} + \frac{\mathbf{Q}\_C}{T\_C} = -\frac{\mathbf{Q}\_H}{T\_H} + \frac{\mathbf{R}\_T^{1/2} \mathbf{Q}\_H}{R\_T T\_H}
$$

$$
= \frac{\mathbf{Q}\_H}{T\_H} \left(\frac{\mathbf{1}}{\mathbf{R}\_T^{1/2}} - \mathbf{1}\right) = \frac{\mathbf{Q}\_H}{T\_H} \left(\mathbf{R}\_T^{-1/2} - \mathbf{1}\right) \tag{8}
$$

$$
> \Delta \mathbf{S}\_{\text{total,Carm,std}} = \mathbf{0}.
$$

Note that for any *RT* in general and as *RT* ! 0 in particular, Δ*S*total,Carnot,std ¼ 0; by contrast, for any *RT* in general Δ*S*total,CA,std >0 (Δ*S*total,CA,std ! 0 only in the limit *RT* ! 1), and as *RT* ! 0, Δ*S*total,CA,std ! ∞.

As we have already noted, heat engines' work outputs are, in almost all cases, totally frictionally dissipated as heat immediately or on short time scales [16, 17]. For example, an automobile 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

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

hot reservoir, the work output *W*, the waste heat *QC* exhausted to its cold reservoir, the efficiency ϵCarnot,std, the entropy change Δ*SH*,Carnot,std of its hot reservoir, the entropy change Δ*SC*,Carnot,std of its cold reservoir, and the total entropy change

*W* ¼ *QH* � *QC* ¼ *QH*ϵCarnot,std ¼ *QH*ð Þ 1 � *RT*

We note that, in most derivations (in textbooks or elsewhere) of ϵCarnot,std, Δ*S*total,Carnot,std ¼ 0 is invoked as the initial Second Law assumption or postulate and

efficiency, the heat input *QH* from its hot reservoir, the work output *W*, the waste heat *QC* exhausted to its cold reservoir, the efficiency ϵCA,std, the entropy change Δ*SH*,CA,std of its hot reservoir, the entropy change Δ*SC*,CA,std of its cold reservoir, and the total entropy change Δ*S*total,CA,std ¼ Δ*SH*,CA,std þ Δ*SC*,CA,std are related in

*<sup>W</sup>* <sup>¼</sup> *QH* � *QC* <sup>¼</sup> *QH*ϵCA,std <sup>¼</sup> *QH* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup>

*TH*

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

Note that for any *RT* in general and as *RT* ! 0 in particular, Δ*S*total,Carnot,std ¼ 0; by contrast, for any *RT* in general Δ*S*total,CA,std >0 (Δ*S*total,CA,std ! 0 only in the limit

As we have already noted, heat engines' work outputs are, in almost all cases, totally frictionally dissipated as heat immediately or on short time scales [16, 17]. For example, an automobile 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

<sup>þ</sup> *QC TC*

is employed along with the initial First Law postulate *W* ¼ *QH* � *QC* [1–7]. Similarly, by the First and Second Laws of Thermodynamics, for a standard

endoreversible heat engine operating (without HTR) at Curzon-Ahlborn

*<sup>T</sup> QH:*

<sup>¼</sup> *QH TH*

) *QC* <sup>¼</sup> *RTQH:* (5)

<sup>þ</sup> *QC TC* ¼ � *QH TH* þ *RTQH RTTH*

*T* � �

> ¼ � *QH TH* þ *R*<sup>1</sup>*=*<sup>2</sup> *<sup>T</sup> QH RTTH*

¼ 0*:* (6)

(7)

(8)

*TH*

Δ*S*total,std ¼ Δ*SH*,Carnot,std þ Δ*SC*,Carnot,std are related in accordance with

<sup>Δ</sup>*S*total,Carnot,std <sup>¼</sup> <sup>Δ</sup>*SH*,Carnot,std <sup>þ</sup> <sup>Δ</sup>*SC*,Carnot,std ¼ � *QH*

*Thermodynamics and Energy Engineering*

) *QC* <sup>¼</sup> *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup>

<sup>Δ</sup>*S*total,CA,std <sup>¼</sup> <sup>Δ</sup>*SH*,CA,std <sup>þ</sup> <sup>Δ</sup>*SC*,CA,std ¼ � *QH*

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

> Δ*S*total,Carnot,std ¼ 0*:*

� 1 !

<sup>¼</sup> *QH TH*

*RT* ! 1), and as *RT* ! 0, Δ*S*total,CA,std ! ∞.

and

accordance with

and

**114**

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.] Apart from such rare exceptional cases, in the operation of *any* cyclic heat engine operating at *any* efficiency with*out* HTR—whether reversible at Carnot efficiency, endoreversible at Curzon-Ahlborn efficiency, or otherwise—not only is the waste heat *QC* immediately exhausted into the cold reservoir, but also the heat engine's work output *W* is frictionally dissipated into the cold reservoir immediately or on short time scales [16, 17]. Thus, apart from such rare exceptional cases, the *ultimate* total entropy increase Δ*S*total,ultimate resulting from the operation of *any* heat engine with*out* HTR at *any* efficiency occurs as soon as the work is done or shortly thereafter and is [1–7, 16, 17]:

$$\begin{split} \Delta \mathbf{S}\_{\text{total,ultinate}} &= -\frac{\mathbf{Q}\_{H}}{T\_{H}} + \frac{\mathbf{Q}\_{C}}{T\_{C}} + \frac{\mathbf{W}}{T\_{C}} = -\frac{\mathbf{Q}\_{H}}{T\_{H}} + \frac{\mathbf{Q}\_{C} + \mathbf{W}}{T\_{C}} = -\frac{\mathbf{Q}\_{H}}{T\_{H}} + \frac{\mathbf{Q}\_{H}}{T\_{C}} \\ &= -\frac{\mathbf{Q}\_{H}}{T\_{H}} + \frac{\mathbf{Q}\_{H}}{R\_{T}T\_{H}} = \frac{\mathbf{Q}\_{H}}{T\_{H}} \left(\frac{\mathbf{1}}{R\_{T}} - \mathbf{1}\right) = \frac{\mathbf{Q}\_{H}}{T\_{H}} \left(R\_{T}^{-1} - \mathbf{1}\right) \\ &> \Delta \mathbf{S}\_{\text{total,CA,std}} = \frac{\mathbf{Q}\_{H}}{T\_{H}} \left(R\_{T}^{-1/2} - \mathbf{1}\right) \\ &> \Delta \mathbf{S}\_{\text{total,Carnot,std}} = \mathbf{0}. \end{split} \tag{9}$$

Note that for any *RT* >0, the inequality Δ*S*total,ultimate >0 is stronger than the inequality Δ*S*total,CA,std > 0. In particular, note that as *RT* ! 1, Δ*S*total,ultimate ! 0 more slowly than Δ*S*total,CA,std ! 0, while as *RT* ! 0, Δ*S*total,ultimate ! ∞ faster than Δ*S*total,CA,std ! ∞.
