**4. Carnot and Curzon-Ahlborn efficiencies with high-temperature recharge (HTR)**

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

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 engine operating with HTR is

$$\begin{aligned} \text{e.}\_{\text{Carnot,HTR}} = \frac{W}{Q\_{H,\text{net}}} = \frac{W}{Q\_H - W} = \frac{W}{Q\_C} = \frac{Q\_H - Q\_C}{Q\_C} = \frac{Q\_H}{Q\_C} - \mathbf{1} = \frac{T\_H}{T\_C} - \mathbf{1} = R\_T^{-1} - \mathbf{1} \\ \Rightarrow R\_{c2} = \frac{\mathbf{e}\_{\text{Carnot,HTR}}}{\mathbf{e}\_{\text{Carnot,std}}} = \frac{R\_T^{-1} - \mathbf{1}}{\mathbf{1} - R\_T} = R\_T^{-1} > \mathbf{1}. \end{aligned} \tag{10}$$

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

*RT* < <sup>1</sup>

entropy is positive:

by the entropy ratio

**117**

*QH*,net

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

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

) *<sup>R</sup>*ϵ<sup>3</sup> <sup>¼</sup> <sup>ϵ</sup>CA,HTR

ϵCA,std

<sup>Δ</sup>*S*total,CA,HTR ¼ � *QH*

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

<sup>¼</sup> *<sup>W</sup>*

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

*QC*

unity in the limit *RT* ! 1 to <sup>∞</sup> in the limit *RT* ! 0. If *RT* <sup>&</sup>lt; <sup>1</sup>

*QH* � *<sup>W</sup>* <sup>¼</sup> *<sup>W</sup>*

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

not violated because no energy is created (or destroyed): ϵCA,HTR >1 if

*TH* þ *W TH*

� *QH* � *<sup>W</sup> TH*

> 1 *RT* � 1

1 *TC* � 1 *TH* 

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

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

> Δ*S*total,Carnot,std ¼ 0*:*

<sup>¼</sup> *QC TC*

¼ *QC*

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

<sup>¼</sup> *QH TH*

*RS*<sup>2</sup> <sup>¼</sup> <sup>Δ</sup>*S*total,CA,HTR Δ*S*total,ultimate

Eqs. (11), (14), and (15), it is larger by the entropy ratio

*RS*<sup>3</sup> <sup>¼</sup> <sup>Δ</sup>*S*total,CA,HTR Δ*S*total,Carnot,HTR

*QC*

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

<sup>¼</sup> *<sup>W</sup> QH* *QH QC* ¼

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

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

The efficiency ϵCA,HTR increases monotonically from zero in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. And the efficiency ratio *R*ϵ<sup>3</sup> increases monotonically from

Yet the First and Second Laws of Thermodynamics are *not* violated. The First Law is

<sup>4</sup> ⇔*R*ϵ<sup>3</sup> >2 obtains via recycling and reusing the same energy, not via the creation of new energy. The Second Law is not violated because the change in total

> <sup>þ</sup> *QC TC*

> > <sup>¼</sup> *QC TC*

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

*TH*

In Eq. (14), we applied Eqs. (7) and (8). Yet, also applying Eq. (9), Δ*S*total,CA,HTR, while not merely positive but greater than Δ*S*total,CA,std, is smaller than Δ*S*total,ultimate

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

But, as one would expect, Δ*S*total,CA,HTR is larger than Δ*S*total,Carnot,HTR. Applying

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

There is one more efficiency ratio that is of interest. Applying Eqs. (10) and (13):

*R*�<sup>1</sup>

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

1 � *RT*

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

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

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

*<sup>T</sup>* <sup>¼</sup> *<sup>R</sup>*�<sup>1</sup>

*<sup>T</sup> :* (15)

*<sup>S</sup>*<sup>2</sup> *:* (16)

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

� *QC TH*

> 1 *RTTH*

� 1 *TH*

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

*QH*

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

*QH QC*

<sup>4</sup> ⇔*R*ϵ<sup>3</sup> >2, ϵCA,HTR >1.

(13)

(14)

Note that not only is *R*ϵ<sup>2</sup> >1 in all cases, but also that in some cases also ϵCarnot,HTR >1. The efficiency ϵCarnot,HTR increases monotonically from zero in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. And the efficiency ratio *R*ϵ<sup>2</sup> increases monotonically from unity in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. If *RT* < <sup>1</sup> <sup>2</sup> ⇔*R*ϵ<sup>2</sup> >2, ϵCarnot,HTR >1. Yet the First and Second Laws of Thermodynamics are *not* violated. The First Law is not violated because no energy is created (or destroyed): ϵCarnot,HTR >1 if *RT* < <sup>1</sup> <sup>2</sup> ⇔*R*ϵ<sup>2</sup> >2 obtains via recycling and reusing the same energy, not via the creation of new energy. The Second Law is not violated because the change in total entropy is positive:

$$\begin{aligned} \Delta \mathbf{S}\_{\text{total,Carnot,HTR}} &= -\frac{\mathbf{Q}\_{H}}{T\_{H}} + \frac{\mathbf{W}}{T\_{H}} + \frac{\mathbf{Q}\_{C}}{T\_{C}} \\ &= \frac{\mathbf{Q}\_{C}}{T\_{C}} - \frac{\mathbf{Q}\_{H} - \mathbf{W}}{T\_{H}} = \frac{\mathbf{Q}\_{C}}{T\_{C}} - \frac{\mathbf{Q}\_{C}}{T\_{H}} \\ &= \mathbf{Q}\_{C} \left( \frac{1}{T\_{C}} - \frac{1}{T\_{H}} \right) = \mathbf{R}\_{T} \mathbf{Q}\_{H} \left( \frac{1}{\mathbf{R}\_{T} T\_{H}} - \frac{1}{T\_{H}} \right) \\ &= \frac{\mathbf{R}\_{T} \mathbf{Q}\_{H}}{T\_{H}} \left( \frac{1}{\mathbf{R}\_{T}} - 1 \right) = \frac{\mathbf{R}\_{T} \mathbf{Q}\_{H}}{T\_{H}} \left( \frac{1 - \mathbf{R}\_{T}}{\mathbf{R}\_{T}} \right) \\ &= \frac{\mathbf{Q}\_{H}}{T\_{H}} (1 - \mathbf{R}\_{T}) \\ &> \Delta \mathbf{S}\_{\text{total,Carnot,std}} = \mathbf{0}. \end{aligned} \tag{11}$$

In Eq. (11), we applied Eqs. (5) and (6). Yet, also applying Eq. (9), Δ*S*total,Carnot,HTR, while positive and hence greater than Δ*S*total,Carnot,std ¼ 0, is smaller than Δ*S*total,ultimate by the *entropy* ratio

$$R\_{S1} = \frac{\Delta S\_{\text{total,Carnot,HTR}}}{\Delta S\_{\text{total,ultimate}}} = \frac{\mathbf{1} - R\_T}{R\_T^{-1} - \mathbf{1}} = R\_T. \tag{12}$$

The entropy ratio *RS*<sup>1</sup> decreases monotonically from unity in the limit *RT* ! 1 to zero in the limit *RT* ! 0.

It may be of interest to note that ϵCarnot,HTR of Eq. (10) is the multiplicative inverse of the Carnot coefficient of performance of a refrigerator, whereas, by contrast, ϵCarnot,std of Eq. (1) is the multiplicative inverse of the Carnot coefficient of performance of a heat pump. These relations may be of interest. But, as we have noted in Section 1, HTR is *never* practicable for reverse operation of cyclic heat engines as refrigerators or heat pumps.

Consider next an endoreversible heat engine operating at Curzon-Ahlborn efficiency. If the engine's work output *<sup>W</sup>* <sup>¼</sup> *QH* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup> *T* 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* � *<sup>W</sup>* <sup>¼</sup> *QH* � *QH* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup> *T* <sup>¼</sup>

*QHR*<sup>1</sup>*=*<sup>2</sup> *<sup>T</sup>* ¼ *QC*. Hence, with the help of Eqs. (2) and (7), the efficiency ϵCA,HTR of a Curzon-Ahlborn engine operating with HTR is

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

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

*RT* < <sup>1</sup>

*QH*,net

*Thermodynamics and Energy Engineering*

ϵCarnot,std

) *<sup>R</sup>*ϵ<sup>2</sup> <sup>¼</sup> <sup>ϵ</sup>Carnot,HTR

destroyed): ϵCarnot,HTR >1 if *RT* < <sup>1</sup>

than Δ*S*total,ultimate by the *entropy* ratio

zero in the limit *RT* ! 0.

refrigerators or heat pumps.

*QHR*<sup>1</sup>*=*<sup>2</sup>

**116**

because the change in total entropy is positive:

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

<sup>¼</sup> *<sup>W</sup>*

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

*QH* � *<sup>W</sup>* <sup>¼</sup> *<sup>W</sup>*

*QC*

<sup>¼</sup> *<sup>R</sup>*�<sup>1</sup> *<sup>T</sup>* >1*:*

Note that not only is *R*ϵ<sup>2</sup> >1 in all cases, but also that in some cases also ϵCarnot,HTR >1. The efficiency ϵCarnot,HTR increases monotonically from zero in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. And the efficiency ratio *R*ϵ<sup>2</sup> increases monotonically from unity in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. If

are *not* violated. The First Law is not violated because no energy is created (or

*TH* þ *W TH*

1 *TC* � 1 *TH* 

<sup>¼</sup> *QC TC*

¼ *QC*

<sup>¼</sup> *RTQH TH*

<sup>¼</sup> *QH TH*

*RS*<sup>1</sup> <sup>¼</sup> <sup>Δ</sup>*S*total,Carnot,HTR Δ*S*total,ultimate

efficiency. If the engine's work output *<sup>W</sup>* <sup>¼</sup> *QH* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup>

Curzon-Ahlborn engine operating with HTR is

same energy, not via the creation of new energy. The Second Law is not violated

� *QH* � *<sup>W</sup> TH*

> 1 *RT* � 1

ð Þ 1 � *RT*

> Δ*S*total,Carnot,std ¼ 0*:*

Δ*S*total,Carnot,HTR, while positive and hence greater than Δ*S*total,Carnot,std ¼ 0, is smaller

The entropy ratio *RS*<sup>1</sup> decreases monotonically from unity in the limit *RT* ! 1 to

It may be of interest to note that ϵCarnot,HTR of Eq. (10) is the multiplicative inverse

of the Carnot coefficient of performance of a refrigerator, whereas, by contrast, ϵCarnot,std of Eq. (1) is the multiplicative inverse of the Carnot coefficient of performance of a heat pump. These relations may be of interest. But, as we have noted in Section 1, HTR is *never* practicable for reverse operation of cyclic heat engines as

Consider next an endoreversible heat engine operating at Curzon-Ahlborn

into its hot reservoir (at temperature *TH*), then the *net* heat input *QH*,net required

*<sup>T</sup>* ¼ *QC*. Hence, with the help of Eqs. (2) and (7), the efficiency ϵCA,HTR of a

from its hot reservoir is reduced from *QH* to *QH* � *<sup>W</sup>* <sup>¼</sup> *QH* � *QH* <sup>1</sup> � *<sup>R</sup>*<sup>1</sup>*=*<sup>2</sup>

In Eq. (11), we applied Eqs. (5) and (6). Yet, also applying Eq. (9),

<sup>2</sup> ⇔*R*ϵ<sup>2</sup> >2, ϵCarnot,HTR >1. Yet the First and Second Laws of Thermodynamics

<sup>þ</sup> *QC TC*

> <sup>¼</sup> *QC TC*

¼ *RTQH*

<sup>¼</sup> *RTQH TH*

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

> *T*

� *QC TH*

> 1 *RTTH*

� 1 *TH*

¼ *RT:* (12)

is frictionally dissipated

*T* 

¼

1 � *RT RT* 

<sup>¼</sup> *QH* � *QC QC*

<sup>¼</sup> *QH QC*

<sup>2</sup> ⇔*R*ϵ<sup>2</sup> >2 obtains via recycling and reusing the

� <sup>1</sup> <sup>¼</sup> *TH TC*

� <sup>1</sup> <sup>¼</sup> *<sup>R</sup>*�<sup>1</sup>

*<sup>T</sup>* � 1

(10)

(11)

$$\epsilon\_{\text{CA,HTR}} = \frac{W}{Q\_{H,\text{net}}} = \frac{W}{Q\_H - W} = \frac{W}{Q\_C} = \frac{W}{Q\_H} \frac{Q\_H}{Q\_C} = \frac{Q\_H \left(1 - R\_T^{1/2}\right)}{Q\_H} \frac{Q\_H}{Q\_C}$$

$$= \left(1 - R\_T^{1/2}\right) \frac{Q\_H}{Q\_C} = \left(1 - R\_T^{1/2}\right) \frac{Q\_H}{R\_T^{1/2} Q\_H} = \frac{1 - R\_T^{1/2}}{R\_T^{1/2}}\tag{13}$$

$$= R\_T^{-1/2} - 1$$

$$\Rightarrow R\_{\text{c3}} = \frac{\epsilon\_{\text{CA,HTR}}}{\epsilon\_{\text{CA,std}}} = \frac{R\_T^{-1/2} - 1}{1 - R\_T^{1/2}} = R\_T^{-1/2}.$$

The efficiency ϵCA,HTR increases monotonically from zero in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. And the efficiency ratio *R*ϵ<sup>3</sup> increases monotonically from unity in the limit *RT* ! 1 to <sup>∞</sup> in the limit *RT* ! 0. If *RT* <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> ⇔*R*ϵ<sup>3</sup> >2, ϵCA,HTR >1. Yet the First and Second Laws of Thermodynamics are *not* violated. The First Law is not violated because no energy is created (or destroyed): ϵCA,HTR >1 if *RT* < <sup>1</sup> <sup>4</sup> ⇔*R*ϵ<sup>3</sup> >2 obtains via recycling and reusing the same energy, not via the creation of new energy. The Second Law is not violated because the change in total entropy is positive:

$$\begin{split} \Delta S\_{\text{total,CA,HTR}} &= -\frac{Q\_H}{T\_H} + \frac{W}{T\_H} + \frac{Q\_C}{T\_C} \\ &= \frac{Q\_C}{T\_C} - \frac{Q\_H - W}{T\_H} = \frac{Q\_C}{T\_C} - \frac{Q\_C}{T\_H} \\ &= Q\_C \left(\frac{1}{T\_C} - \frac{1}{T\_H}\right) = R\_T^{1/2} Q\_H \left(\frac{1}{R\_T T\_H} - \frac{1}{T\_H}\right) \\ &= \frac{R\_T^{1/2} Q\_H}{T\_H} \left(\frac{1}{R\_T} - 1\right) \\ &= \frac{Q\_H}{T\_H} \left(R\_T^{-1/2} - R\_T^{1/2}\right) \\ &> \Delta S\_{\text{total,CA,std}} = \frac{Q\_H}{T\_H} \left(R\_T^{-1/2} - 1\right) \\ &> \Delta S\_{\text{total,CA,std}} = 0. \end{split} \tag{14}$$

In Eq. (14), we applied Eqs. (7) and (8). Yet, also applying Eq. (9), Δ*S*total,CA,HTR, while not merely positive but greater than Δ*S*total,CA,std, is smaller than Δ*S*total,ultimate by the entropy ratio

$$R\_{\rm S2} = \frac{\Delta S\_{\rm total, CA, HTR}}{\Delta S\_{\rm total,ultimate}} = \frac{R\_T^{-1/2} - R\_T^{1/2}}{R\_T^{-1} - \mathbf{1}} = R\_T^{1/2}.\tag{15}$$

But, as one would expect, Δ*S*total,CA,HTR is larger than Δ*S*total,Carnot,HTR. Applying Eqs. (11), (14), and (15), it is larger by the entropy ratio

$$R\_{\rm S3} = \frac{\Delta S\_{\rm total, CA,HTR}}{\Delta S\_{\rm total,Carnot,HTR}} = \frac{R\_T^{-1/2} - R\_T^{1/2}}{1 - R\_T} = R\_T^{-1/2} = R\_{\rm S2}^{-1}.\tag{16}$$

There is one more efficiency ratio that is of interest. Applying Eqs. (10) and (13):

*Thermodynamics and Energy Engineering*

$$R\_{e4} = \frac{\epsilon\_{\text{Carnot,HTR}}}{\epsilon\_{\text{CA,HTR}}} = \frac{R\_T^{-1} - \mathbf{1}}{R\_T^{-1/2} - \mathbf{1}}.\tag{17}$$

diminution of <sup>Δ</sup>*<sup>S</sup>* <sup>¼</sup> *<sup>W</sup>*

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

*TC*

<sup>¼</sup> *<sup>W</sup> TH*

<sup>Δ</sup>*<sup>X</sup>* <sup>¼</sup> *TC* � <sup>ð</sup>diminution of <sup>Δ</sup>*S*Þ ¼ *TCW* <sup>1</sup>

� *<sup>W</sup> TH*

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

*R*�<sup>1</sup>

And the corresponding saving of exergy or free energy is

in the limit *RT* ! 0, (diminution of Δ*S*) ! ∞ and Δ*X* ! *W*.

<sup>¼</sup> *<sup>W</sup>* <sup>1</sup> *TC*

� 1 *TH* 

*TC* � 1 *TH* 

Note that in the limit *RT* ! 1, both (diminution of Δ*S*) ! 0 and Δ*X* ! 0, while

Consider work *W* from *any* source, heat engine or otherwise, frictionally dissipated into heat at temperature *TH* [6]. The temperature *TH* could be generated via the friction itself [6]. Thus frictional dissipation of work *W* from *any* source, heat engine or otherwise, could, in at least in principle, generate an arbitrarily high temperature *TH* [6]. In the limit *TH* ! ∞ (with *TC* fixed) or equivalently in the limit *RT* ! 0, not only the standard (non-HTR) Carnot efficiency ϵCarnot,std but even the standard (non-HTR) Curzon-Ahlborn efficiency ϵCA,std approaches unity, and entropy production even given the standard (non-HTR) Curzon-Ahlborn efficiency ϵCA,std approaches zero. And both HTR efficiencies, ϵCarnot,HTR and ϵCA,HTR, approach infinity (ϵCarnot,HTR approaching infinity at higher order), with entropy production

corresponding to both approaching zero (that corresponding to ϵCarnot,HTR approaching zero at higher order). Thus in the limit *RT* ! 0, work frictionally dissipated into heat can completely be reconverted back into work by a heat engine. When this work is, in turn, frictionally dissipated, the process can be repeated over and over again—indefinitely in the limit *RT* ! 0. We emphasize again: no energy is created (or destroyed)—energy is merely recycled—hence the First Law of Thermodynamics is not violated [6]. No decrease in entropy occurs—Δ*S* ¼ *W=TH* >0 for any finite *TH*—hence the Second Law of Thermodynamics is not violated [6]: as per Eqs. (20) and (21), Δ*S* is diminished but still remains positive. To re-emphasize, it is the *diminution* of Δ*S* ¼ *W=T* at higher *T*: *W=TH* <*W=TC* (notwithstanding that Δ*S* still remains positive: *W=TH* >0) that yields increased efficiency via HTR, within the restriction of frictional dissipation of *W* being unavoidable.

Consider the following thought experiment. If an automobile travels at constant speed, the work output of its engine is immediately and continually frictionally dissipated, but the work was done and the efficiency was *W=QH*, not zero. In the operation of automobiles at constant speed, *W* is immediately and continually frictionally dissipated to ambient (the cold reservoir), and hence the consequent entropy increase is Δ*S* ¼ *W=TC*. (Of course, there are in practice other entropy increases accompanying the operation of automobiles, e.g., owing to irreversible heat flows engendered by finite temperature differences.) But if *W* could instead be frictionally dissipated into the cylinders of an automobile's engine during power strokes (of course this is impracticable), the entropy increase would be diminished from Δ*S* ¼ *W=TC* to Δ*S* ¼ *W=TH*, and the required heat input would be reduced from *Q <sup>H</sup>* to *QH* � *W* = *QC*. Hence via HTR the efficiency would be increased from *W=QH* to *W=*ð Þ *QH* � *W* ¼ *W=QC*, and thus also the required fuel consumption would be decreased by the ratio *QC=QH*. Even though this is obviously impracticable, given that Δ*S* ¼ *W=TH* >0

the Second Law allows it, so we can at least do it as a thought experiment.

**119**

If a heat engine's work output is frictionally dissipated into its hot reservoir, the net heat input required from the hot reservoir is reduced from *QH* to *QH* � *W*,

<sup>¼</sup> *<sup>W</sup>* <sup>1</sup>

*<sup>T</sup>* � <sup>1</sup> *:* (20)

*RTTH*

<sup>¼</sup> *<sup>W</sup>* <sup>1</sup> � *TC*

� 1 *TH*

*TH* 

¼ *W*ð Þ 1 � *RT :*

(21)

The efficiency ratio *R*ϵ<sup>4</sup> increases monotonically from 2 in the limit *RT* ! 1 to ∞ in the limit *RT* ! 0. The latter limit is obvious. The former limit is most easily demonstrated by setting *RT* ¼ 1 � *δ*, letting *δ* ! 0, and applying the binomial theorem. This yields

$$\lim\_{\delta \to -1} R\_{e4} = \lim\_{\delta \to 0} R\_{e4} = \lim\_{\delta \to 0} \frac{\left(\mathbf{1} - \boldsymbol{\delta}\right)^{-1} - \mathbf{1}}{\left(\mathbf{1} - \boldsymbol{\delta}\right)^{-1/2} - \mathbf{1}} = \frac{\left(\mathbf{1} + \boldsymbol{\delta}\right) - \mathbf{1}}{\left(\mathbf{1} + \frac{1}{2}\boldsymbol{\delta}\right) - \mathbf{1}} = \frac{\boldsymbol{\delta}}{\frac{1}{2}\boldsymbol{\delta}} = \mathbf{2}.\tag{18}$$

Note that: (a) Applying Eqs. (3), (4), (17), and (18), in the limit *RT* ! 1, *R*ϵ<sup>1</sup> ! 2 from below; by contrast *R*ϵ<sup>4</sup> ! 2 from above. (b) Applying Eqs. (3), (4), (17), and (18), in the limit *RT* ! 0, *R*ϵ<sup>1</sup> ! 1 but *R*ϵ<sup>4</sup> ! ∞. (c) Applying Eqs. (10) and (13), in the limit *RT* ! 0, <sup>ϵ</sup>Carnot,HTR ! <sup>∞</sup> as *<sup>R</sup>*�<sup>1</sup> *<sup>T</sup>* but <sup>ϵ</sup>CA,HTR ! <sup>∞</sup> only as *<sup>R</sup>*�1*=*<sup>2</sup> *<sup>T</sup>* ; thus, while both approach ∞, ϵCarnot,HTR does so at higher order—hence as *RT* ! 0, *R*ϵ<sup>4</sup> ! ∞.

As an aside, it may be of interest to note, applying Eqs. (3) and (17), that

$$\frac{R\_{c4}}{R\_{c1}} = \frac{\frac{R\_T^{-1} - 1}{R\_T^{-1/2} - 1}}{\frac{1 - R\_T}{1 - R\_T}} = \frac{R\_T^{-1} - 1}{1 - R\_T} \times \frac{1 - R\_T^{1/2}}{R\_T^{-1/2} - 1} \tag{19}$$

$$= R\_T^{-1} \times R\_T^{1/2} = R\_T^{-1/2}.$$

### **5. Recapitulation and generalization**

Efficiency is of course 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 (necessarily one-time, single-use) 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.

The Second Law of Thermodynamics allows frictional dissipation of work into heat at *any temperature* [1–7] (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]. The diminution of Δ*S* ¼ *W=T* at higher *T* is, ultimately, what yields increased efficiency via HTR, within the restriction of frictional dissipation of *W* being unavoidable. The diminution of Δ*S* ¼ *W=T* via frictional dissipation of *W* at *TH* as opposed to at *TC* ¼ *RTTH* is

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

*<sup>R</sup>*ϵ<sup>4</sup> <sup>¼</sup> <sup>ϵ</sup>Carnot,HTR ϵCA,HTR

theorem. This yields

lim *RT*!1

*R*ϵ<sup>4</sup> ¼ lim *δ*!0

*Thermodynamics and Energy Engineering*

the limit *RT* ! 0, <sup>ϵ</sup>Carnot,HTR ! <sup>∞</sup> as *<sup>R</sup>*�<sup>1</sup>

*R*ϵ<sup>4</sup> ¼ lim *δ*!0

> *R*ϵ<sup>4</sup> *R*ϵ<sup>1</sup> ¼

**5. Recapitulation and generalization**

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

<sup>¼</sup> *<sup>R</sup>*�<sup>1</sup>

differences (no insulation is perfect)] is not possible.

*TH* as opposed to at *TC* ¼ *RTTH* is

**118**

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

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

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

*<sup>T</sup>* but <sup>ϵ</sup>CA,HTR ! <sup>∞</sup> only as *<sup>R</sup>*�1*=*<sup>2</sup>

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

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

�

¼ *δ* 1

The efficiency ratio *R*ϵ<sup>4</sup> increases monotonically from 2 in the limit *RT* ! 1 to ∞

Note that: (a) Applying Eqs. (3), (4), (17), and (18), in the limit *RT* ! 1, *R*ϵ<sup>1</sup> ! 2 from below; by contrast *R*ϵ<sup>4</sup> ! 2 from above. (b) Applying Eqs. (3), (4), (17), and (18), in the limit *RT* ! 0, *R*ϵ<sup>1</sup> ! 1 but *R*ϵ<sup>4</sup> ! ∞. (c) Applying Eqs. (10) and (13), in

both approach ∞, ϵCarnot,HTR does so at higher order—hence as *RT* ! 0, *R*ϵ<sup>4</sup> ! ∞. As an aside, it may be of interest to note, applying Eqs. (3) and (17), that

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

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

Efficiency is of course 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 (necessarily one-time, single-use) 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

The Second Law of Thermodynamics allows frictional dissipation of work into

heat at *any temperature* [1–7] (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]. The diminution of Δ*S* ¼ *W=T* at higher *T* is, ultimately, what yields increased efficiency via HTR, within the restriction of frictional dissipation of *W* being unavoidable. The diminution of Δ*S* ¼ *W=T* via frictional dissipation of *W* at

in the limit *RT* ! 0. The latter limit is obvious. The former limit is most easily demonstrated by setting *RT* ¼ 1 � *δ*, letting *δ* ! 0, and applying the binomial

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

*:* (17)

<sup>2</sup> *<sup>δ</sup>* <sup>¼</sup> <sup>2</sup>*:* (18)

*<sup>T</sup>* ; thus, while

(19)

$$\begin{aligned} \text{dimension of } \Delta \mathbf{S} &= \frac{W}{T\_C} - \frac{W}{T\_H} = \mathcal{W} \left( \frac{1}{T\_C} - \frac{1}{T\_H} \right) = \mathcal{W} \left( \frac{1}{R\_T T\_H} - \frac{1}{T\_H} \right) \\ &= \frac{W}{T\_H} (R\_T^{-1} - 1). \end{aligned} \tag{20}$$

And the corresponding saving of exergy or free energy is

$$
\Delta X = T\_C \times \text{(dimension of } \Delta \text{S)} = T\_C W \left(\frac{1}{T\_C} - \frac{1}{T\_H}\right) = W \left(1 - \frac{T\_C}{T\_H}\right) = W(1 - R\_T). \tag{21}
$$

Note that in the limit *RT* ! 1, both (diminution of Δ*S*) ! 0 and Δ*X* ! 0, while in the limit *RT* ! 0, (diminution of Δ*S*) ! ∞ and Δ*X* ! *W*.

Consider work *W* from *any* source, heat engine or otherwise, frictionally dissipated into heat at temperature *TH* [6]. The temperature *TH* could be generated via the friction itself [6]. Thus frictional dissipation of work *W* from *any* source, heat engine or otherwise, could, in at least in principle, generate an arbitrarily high temperature *TH* [6]. In the limit *TH* ! ∞ (with *TC* fixed) or equivalently in the limit *RT* ! 0, not only the standard (non-HTR) Carnot efficiency ϵCarnot,std but even the standard (non-HTR) Curzon-Ahlborn efficiency ϵCA,std approaches unity, and entropy production even given the standard (non-HTR) Curzon-Ahlborn efficiency ϵCA,std approaches zero. And both HTR efficiencies, ϵCarnot,HTR and ϵCA,HTR, approach infinity (ϵCarnot,HTR approaching infinity at higher order), with entropy production corresponding to both approaching zero (that corresponding to ϵCarnot,HTR approaching zero at higher order). Thus in the limit *RT* ! 0, work frictionally dissipated into heat can completely be reconverted back into work by a heat engine. When this work is, in turn, frictionally dissipated, the process can be repeated over and over again—indefinitely in the limit *RT* ! 0. We emphasize again: no energy is created (or destroyed)—energy is merely recycled—hence the First Law of Thermodynamics is not violated [6]. No decrease in entropy occurs—Δ*S* ¼ *W=TH* >0 for any finite *TH*—hence the Second Law of Thermodynamics is not violated [6]: as per Eqs. (20) and (21), Δ*S* is diminished but still remains positive. To re-emphasize, it is the *diminution* of Δ*S* ¼ *W=T* at higher *T*: *W=TH* <*W=TC* (notwithstanding that Δ*S* still remains positive: *W=TH* >0) that yields increased efficiency via HTR, within the restriction of frictional dissipation of *W* being unavoidable.

Consider the following thought experiment. If an automobile travels at constant speed, the work output of its engine is immediately and continually frictionally dissipated, but the work was done and the efficiency was *W=QH*, not zero. In the operation of automobiles at constant speed, *W* is immediately and continually frictionally dissipated to ambient (the cold reservoir), and hence the consequent entropy increase is Δ*S* ¼ *W=TC*. (Of course, there are in practice other entropy increases accompanying the operation of automobiles, e.g., owing to irreversible heat flows engendered by finite temperature differences.) But if *W* could instead be frictionally dissipated into the cylinders of an automobile's engine during power strokes (of course this is impracticable), the entropy increase would be diminished from Δ*S* ¼ *W=TC* to Δ*S* ¼ *W=TH*, and the required heat input would be reduced from *Q <sup>H</sup>* to *QH* � *W* = *QC*. Hence via HTR the efficiency would be increased from *W=QH* to *W=*ð Þ *QH* � *W* ¼ *W=QC*, and thus also the required fuel consumption would be decreased by the ratio *QC=QH*. Even though this is obviously impracticable, given that Δ*S* ¼ *W=TH* >0 the Second Law allows it, so we can at least do it as a thought experiment.

If a heat engine's work output is frictionally dissipated into its hot reservoir, the net heat input required from the hot reservoir is reduced from *QH* to *QH* � *W*, and hence via HTR the engine's efficiency is increased from *W=QH* to *W=*ð Þ¼ *QH* � *W W=QC*, which can indeed exceed the Carnot limit—even though the efficiency *W=QH* of the *initial* production of work must be within the Carnot limit. If the temperature *TC* of the cold reservoir is only a small fraction of the the temperature *TH* of the hot reservoir, *W=QH* can be almost as large as unity or equivalently *W* can be almost as large as *QH*, and hence *W=*ð Þ¼ *QH* � *W W=QC* can *greatly* exceed the Carnot limit.

Perhaps the simplest and most straightforward reply to these criticisms [58, 59] is that provided by Spanner (see Ref. [6], pp. 11–12, 60–65, and 263–265, especially pp. 263–265): Friction resulting from dissipation of work can in principle generate arbitrarily high temperature *TH* without violating the Second Law of Thermodynamics: The entropy increase resulting from frictional dissipation of work *W* at temperature *TH*, namely, Δ*S* ¼ *W=TH*, decreases monotonically with increasing *TH* but is positive for any finite *TH*—and the Second Law requires only that Δ*S*≥ 0 [6]. A heat engine operating between this high temperature *TH* and a low (cold-reservoir) temperature *TC* arbitrarily close to absolute zero (0 K) can in principle recover essentially all of the frictional dissipation as work [6]—and the recycling of energy from work to heat via frictional dissipation and then back to work via the heat engine can in principle then be repeated essentially indefinitely [6]. No energy is created (or destroyed)—energy is merely recycled—hence the First Law of Thermodynamics is not violated [6]. No decrease in entropy occurs—Δ*S* ¼ *W=TH* >0 for any finite *TH*—

As has been previously emphasized [35], it is only recycling of a heat engine's *waste heat QC* into its hot reservoir at *TH* instead of rejection thereof into its cold reservoir at *TC*—*not* recycling of heat generated by frictional dissipation of its *work output W* back into its hot reservoir at *TH*—that would violate the Second Law of Thermodynamics. Recharging *W* to the hot reservoir does *not* violate the Second Law, because the entropy change Δ*S* ¼ *W=TH* is positive—albeit less strongly pos-

There is one caveat: the entropy increase Δ*S* ¼ *W=TH* > 0 owing to frictional dissipation of *W* at *TH* could in principle be employed to pay for pumping a heat engine's waste heat *QC* from *TC* to *TH*, but no capability to do work would be gained by this procedure. For, even if this procedure could be executed perfectly (reversibly), e.g., via a perfect (reversible) heat pump, we would have [applying Eqs. (1)

> 1 *TC* � 1 *TH*

¼ *QH*ð Þ¼ 1 � *RT QH*ϵCarnot,std*:*

What Eq. (22) brings to light is that the operation of the heat pump, even if perfect (reversible), results merely in the recovery of *W*. But *W* is recoverable more simply by avoiding this unnecessary procedure, as per Section 5 and the first three

We provided introductory remarks, an overview, and general considerations in Section 1. A misconception pertaining to the efficiencies of engines (heat engines or

¼ *QH*

*TC TH* <sup>¼</sup> *<sup>W</sup> TH*

*TH TC* � 1 

� *QC*

*TH* � *TC TCTH*

¼ 0

(22)

itive than Δ*S* ¼ *W=TC* that obtains if *W* is frictionally dissipated into the cold reservoir. Only recharging *QC* to the hot reservoir would violate the Second Law, because the entropy change Δ*S* ¼ *QC=TH* � *QC=TC* would be negative. And recycling of a heat engine's *waste heat QC* into its hot reservoir at *TH* instead of its rejection into its cold reservoir at *TC* has *never* been claimed

hence the Second Law of Thermodynamics is not violated [6].

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

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

[27–38, 60, 62].

and (5)]

<sup>Δ</sup>*S*total <sup>¼</sup> *<sup>W</sup>*

) *W* ¼ *QC*

**7. Conclusion**

**121**

*TH*

� *QC TC*

<sup>¼</sup> *QH* <sup>1</sup> � *TC*

paragraphs of this Section 6.

*TH* � *TC TC*

*TH* 

<sup>þ</sup> *QC TH*

<sup>¼</sup> *<sup>W</sup> TH*

¼ *QC*

� *QC*

*TH TC* � 1 

We note that the temperature of the cosmic background radiation is only 2*:*7 K, while the most refractory materials remain solid at temperatures slightly exceeding 2700 K. This provides a temperature ratio of *RT* � *TC=TH* <sup>≈</sup> <sup>10</sup>�3. Could even smaller values of *RT* � *TC=TH* be possible, at least in principle? Perhaps, maybe, if frictional dissipation of work into heat might somehow be possible into a gaseous hot reservoir at temperatures exceeding the melting point or even the critical temperature (the maximum boiling point at any pressure) of even the most refractory material.

While in this chapter we do not challenge the Second Law, we do challenge an *over*statement of the Second Law that is sometimes made: that energy can do work only once. This *over*statement is *false*. Energy can indeed do work more than once, in principle up to an infinite number of times, and even in practice many more times than merely once, before its ability to do work is totally dissipated. Consider these three examples: (i) Energy can do work in an infinite number of times in perfect (reversible) regenerative braking of an electrically-powered motor vehicle, with the motor operating backward as a generator during braking. Even with realworld less-than-perfect (less than completely reversible) regenerative braking, energy can do work many more times than merely once before its ability to do work is totally dissipated. (ii) Energy can do work in an infinite number of times in perfect (reversible) HTR (in the limit *RT* ! 0). Even with real-world less-thanperfect (less than completely reversible) HTR (finite but small *RT* >0), energy can do work many more times than merely once before its ability to do work is totally dissipated. (iii) Energy can do work in an infinite number of times in perfect (reversible) thermal recharge of intermediate heat reservoirs—not to be confused with HTR discussed in this present chapter—see Section VI of Ref. [35] and the improved treatment in another chapter [61] in this book. Even with real-world lessthan-perfect (less than completely reversible) thermal recharge of intermediate heat reservoirs, energy can do work more times than merely once before its ability to do work is totally dissipated.
