**4. General remarks, especially concerning entropy**

It is important to emphasize that the super-unity cyclic-heat-engine efficiencies *W<sup>D</sup>* <sup>1</sup>!*n,*max *=Q*<sup>1</sup> that can obtain with work output totally frictionally dissipated ð Þ if *n*≥3 are consistent with both the First and Second Laws of Thermodynamics. The two laws are *not* violated because, if the work output of a heat engine is frictionally dissipated as heat into a cooler reservoir, both laws allow this heat to be partially converted to work again if another, still cooler, reservoir is available.

In this Section 4 we do not restrict heat-engine efficiencies to the form given by Equation (5), nor necessarily assume efficiencies of the same form at each step *j* ! *j* þ 1 or *j* ! *j* þ *k* (1≤ *k*≤ *n* � *j*). The validity of this Section 4 requires only that the efficiency with all work sequestered, or at any one given step *j* ! *j* þ 1 whether work is sequestered or not, be within the Carnot limit, in accordance with the Second Law.

The extra work that is made available via frictional dissipation into cooler reservoirs is paid for by an extra increase in entropy. Consider the work available via heat-engine operation between reservoir *j* at temperature *Tj* and reservoir *j* þ 2 at temperature *Tj*þ<sup>2</sup> without versus with frictional dissipation into reservoir *j* þ 1 at temperature *Tj*þ<sup>1</sup> *Tj* >*Tj*þ<sup>1</sup> >*Tj*þ<sup>2</sup> . Without frictional dissipation a heat engine performs work

$$\mathbf{W}\_{\mathbf{j}\rightarrow\mathbf{j}+\mathbf{1}}=\mathbf{Q}\_{\mathbf{j}}\mathbf{e}\_{\mathbf{j}\rightarrow\mathbf{j}+\mathbf{1}}\tag{42}$$

*<sup>W</sup>D,*extra

<sup>1</sup>!*n,*max <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

*Thermodynamics and Energy Engineering*

<sup>1</sup>!*n,*max � *W*1!*<sup>n</sup>*

*T*1 � �*x=*ð Þ *<sup>n</sup>*�<sup>1</sup> " #

*T*1 � �*x=*ð Þ *<sup>n</sup>*�<sup>1</sup> " #

*Tn T*1 � �*<sup>x</sup>* *Tn T*1

� �*x=*ð Þ *<sup>n</sup>*�<sup>1</sup> " #

as *<sup>r</sup>* and setting *dWD,*extra

*d dr <sup>r</sup>*

) ð Þ *n* � 1 *r*

) *r* ¼ 1*:*

*<sup>n</sup>*�<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup>

*W<sup>D</sup>*

*Tn=T*1!<sup>0</sup>*,n* fixed

<sup>1</sup>!*n,*max <sup>&</sup>gt;0. Moreover, applying Eqs. (5), (36), and (38), note that

<sup>1</sup>!*n,*max ¼ ð Þ *n* � 1 *Q*<sup>1</sup> ¼ ð Þ *n* � 1 lim

Note that the values in Eqs. (36), (38), and (40) increase monotonically with increasing *n* and that the fulfillment of the inequality in Eq. (37) becomes monotonically easier with increasing *n*. Equation (40) is valid not only for Carnot efficiency (*x* ¼ 1) but even for Curzon-Ahlborn efficiency (*x* ¼ 1*=*2), indeed for any *x*

! 1 in the limit *Tn=T*<sup>1</sup> ! 0, albeit ever more slowly with decreasing *x*.

¼ lim

*Tn=T*1!<sup>0</sup>*,n* fixed

*Tn=T*1!<sup>0</sup>*,n* fixed

) *r*

� � � �*<sup>x</sup>* ( )

� �*x=*ð Þ *<sup>n</sup>*�<sup>1</sup>

� �*<sup>x</sup>* " #

� ð Þ *n* � 1

� *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *Tn*

� <sup>1</sup> � *Tn*

� 1 þ

<sup>1</sup>!*n,*max ≥0, with the equality obtaining if and only if

*Tn T*1

<sup>1</sup>!*n,*max � *<sup>W</sup>*<sup>1</sup>!*<sup>n</sup>* <sup>¼</sup> *<sup>W</sup>D,*extra

*<sup>n</sup>*�<sup>1</sup> � ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>r</sup>* � � <sup>¼</sup> <sup>0</sup>

*<sup>n</sup>*�<sup>2</sup> � ð Þ¼ *<sup>n</sup>* � <sup>1</sup> <sup>0</sup>

<sup>¼</sup> <sup>1</sup> ) *Tn*

*Tn=T*1!<sup>0</sup>*,n* fixed

*W*<sup>1</sup>!*<sup>n</sup>:*

*W*<sup>1</sup>!*<sup>n</sup>*

<sup>1</sup>!*n,*max � *W*<sup>1</sup>!*<sup>n</sup>* � �

> *T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup>

� �

*T*1 � � � �*<sup>x</sup>*

*T*1

*Tn T*1

<sup>1</sup>!*n,*max *=dr* ¼ 0 yields

≥0*:*

<sup>1</sup>!*n,*max ¼ 0. For,

*<sup>T</sup>*<sup>1</sup> ¼ 1. For all

*W*<sup>1</sup>!*<sup>n</sup>*

! 0 ⇔ 1 �

ln *<sup>T</sup>*<sup>1</sup> *Tn*

*:* (41)

(38)

(39)

(40)

<sup>¼</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *Tn*

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>1</sup> � *Tn*

¼ *Q*<sup>1</sup> *n* � 1 � ð Þ *n* � 1

¼ *Q*<sup>1</sup> *n* � 2 þ

<sup>1</sup>!*n,*max <sup>¼</sup> *<sup>W</sup>*<sup>1</sup>!*<sup>n</sup>* <sup>¼</sup> <sup>0</sup> ) *<sup>W</sup><sup>D</sup>*

*dWD,*extra <sup>1</sup>!*n,*max *dr* <sup>¼</sup> <sup>0</sup> )

<sup>1</sup>!*n,*max is minimized at 0 if *<sup>r</sup>* <sup>¼</sup> *Tn*

<sup>1</sup>!*n,*max <sup>¼</sup> lim

¼ð Þ *n* � 1 *Q*<sup>1</sup> � *Q*<sup>1</sup> ¼ ð Þ *n* � 2 *Q*<sup>1</sup> ¼ ð Þ *n* � 2 lim

finitely greater than 0 in the range 0<*x*≤ 1, because *Tn*

By contrast, even granting Carnot efficiency (*x* ¼ 1) [22]:

*Tn*

<sup>1</sup>!*n,*max <sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> ln *<sup>T</sup>*<sup>1</sup>

It is easily shown that *<sup>W</sup>D,*extra

*T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup>

*Tn*

*Tn*

*Tn T*1 � �*<sup>x</sup>=*ð Þ *<sup>n</sup>*�<sup>1</sup>

**138**

*<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>W</sup><sup>D</sup>*

denoting the ratio *Tn*

Thus *<sup>W</sup>D,*extra

) lim *Tn=T*1!<sup>0</sup>*,n* fixed

lim *<sup>n</sup>*!<sup>∞</sup>*, Tn=T*<sup>1</sup> fixed

*W<sup>D</sup>*

lim *Tn=T*1!<sup>0</sup>*,n* fixed *W<sup>D</sup>*

*WD,*extra

*<sup>T</sup>*<sup>1</sup> <sup>&</sup>lt; <sup>1</sup>*, WD,*extra

by employing the reservoir at temperature *Tj* as a hot reservoir and the reservoir at temperature *Tj*þ<sup>1</sup> as a cold reservoir. It rejects waste heat *Qj* � *Wj*!*j*þ<sup>1</sup> ¼ *Qj* 1 � ϵ*j*!*j*þ<sup>1</sup> to the reservoir at temperature *Tj*þ1. If a third reservoir at temperature *Tj*þ<sup>2</sup> and *Wj*!*j*þ<sup>1</sup> is sequestered, that is, not frictionally dissipated, a heat engine can then perform additional work:

$$\mathbf{W}\_{j+1 \to j+2} = \mathbf{Q}\_j (\mathbf{1} - \mathbf{e}\_{j \to j+1}) \mathbf{e}\_{j+1 \to j+2} \tag{43}$$

[In the last four steps of Eq. (50), we applied Eqs. (42), (45), (48), and (49).] Thus

In no case do we assume an efficiency with all work sequestered, or at any one

We note that, while frictional dissipation of work into intermediate reservoirs

extra), it seems to be of no help in reverse, that is, refrigerator or heat pump, operation. For, in refrigerator or heat pump operation, with an intermediate reservoir *j* þ 1 at temperature *Tj*þ<sup>1</sup>*, Qj*þ<sup>2</sup> þ *Wj*þ2!*j*þ<sup>1</sup> ¼ *Qj*þ<sup>1</sup>*, Qj*þ<sup>1</sup> þ *Wj*þ1!*<sup>j</sup>* ¼ *Qj*, hence *Qj*þ<sup>2</sup> þ *Wj*þ2!*j*þ<sup>1</sup> þ *Wj*þ1!*<sup>j</sup>* ¼ *Qj*þ<sup>2</sup> þ *Wj*þ2!*<sup>j</sup>* ¼ *Qj*. Without an intermediate

*Wj*þ2!*<sup>j</sup>* ¼ *Qj* is identical with or without an intermediate reservoir *j* þ 1 at temperature *Tj*þ1. With or without the intermediate reservoir *j* þ 1 at temperature *Tj*þ1, *all* of the energy must end up as *Qj*; thus, there is *none* left over to be frictionally dissipated. Hence the presence or absence of this intermediate reservoir makes no difference with respect to reverse, that is, refrigerator or heat pump, operation: See Ref. [1], Section 20-3; Ref. [2], Section 5.12 and Problem 5.22; Ref. [3], Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. [4], Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. [5], Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17; Ref. [6], Chapter XXI; Ref. [7], Sections 6.7, 6.8, 7.3, and 7.4); and Ref. [9], pp. 233–236 and Problems 1, 2, 4, 6, and 7 of Chapter 8. [Problem 2 of Chapter 8 in Ref. [9] considers absorption refrigeration, wherein the entire energy output is into an intermediate-temperature (most typically ambient-temperature) reservoir, and hence for which also there is *no* energy left over to be frictionally dissipated.]

We investigated the increased heat-engine efficiencies obtained via operation employing increasing numbers (≥ 3) of heat reservoirs and with work output totally frictionally dissipated into all reservoirs except the first, hottest, one at temperature *T*<sup>1</sup> and (possibly) also the last, coldest, one at temperature *Tn*. We emphasize again that our results are consistent with both the First and Second Laws of Thermodynamics. The two laws are *not* violated because, if the work output of a heat engine is frictionally dissipated as heat into a cooler reservoir, both laws allow this heat to be partially converted to work again if another, still cooler, reservoir is available.

We do, however, challenge an *over*statement of the Second Law that is sometimes made, namely, that energy can do work only once. Energy can indeed do work more than once, because the Second Law does not forbid recycling of energy, so long as total entropy does not decrease as a result. This criterion of non-decrease of total entropy *is* obeyed, as per Section 4. In no case do we assume an efficiency with all work sequestered, or at any one given step *j* ! *j* þ 1 whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except

While in this chapter we do not challenge the First or Second Laws of Thermodynamics, we should note that there have been many challenges to the Second Law,

that it is conserved—and we *never* violate conservation of energy).

extra in heat-engine operation (albeit at the expense of

extraϵ*j*þ1!*j*þ2*:* (51)

. The bottom line *Qj*þ<sup>2</sup> þ

extra <sup>¼</sup> *Tj*þ<sup>1</sup> *<sup>Δ</sup>S<sup>D</sup>*

given step *j* ! *j* þ *1* whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except

*W<sup>D</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

reservoir *j* þ 1 at temperature *Tj*þ<sup>1</sup>*, Qj*þ<sup>2</sup> þ *Wj*þ2!*<sup>j</sup>* ¼ *Qj*

can yield extra work *W<sup>D</sup>*

**5. Conclusion**

**141**

*ΔS<sup>D</sup>*

that it is conserved—and we *never* violate conservation of energy.)

by employing the reservoir at temperature *Tj*þ<sup>1</sup> as a hot reservoir and the reservoir at temperature *Tj*þ<sup>2</sup> as a cold reservoir. All told it can do work:

$$\begin{split} \mathcal{W}\_{j-j+2} &= \mathcal{W}\_{j-j+1} + \mathcal{W}\_{j+1-j+2} = \mathcal{Q}\_{j} \mathfrak{e}\_{j-j+1} + \mathcal{Q}\_{j} (\mathbf{1} - \mathfrak{e}\_{j-j+1}) \mathfrak{e}\_{j+1-j+2} \\ &= \mathcal{Q}\_{j} (\mathfrak{e}\_{j-j+1} + \mathfrak{e}\_{j+1-j+2} - \mathfrak{e}\_{j-j+1} \mathfrak{e}\_{j+1-j+2}) . \end{split} \tag{44}$$

With total frictional dissipation of *Wj*!*j*þ<sup>1</sup> into reservoir *j* þ 1 at temperature *Tj*þ1, we still have

$$\mathbf{W}\_{j \to j+1}^{D} = \mathbf{W}\_{j \to j+1} = \mathbf{Q}\_1 \mathbf{e}\_{j \to j+1}.\tag{45}$$

But now we let the work output *W<sup>D</sup> <sup>j</sup>*!*j*¼<sup>1</sup> ¼ *Q*1ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup> be totally frictionally dissipated into the reservoir at temperature *Tj*þ<sup>1</sup> (indicated via a superscript *D*). If there is a third reservoir at temperature *Tj*þ2, a heat engine can then perform additional work:

$$\mathbf{W}\_{j+1 \to j+2}^{D} = \mathbf{Q}\_1 \mathbf{e}\_{j+1 \to j+2}. \tag{46}$$

All told it can do work:

$$\begin{split} \mathbf{W}\_{j \rightarrow j+2}^{D} &= \mathbf{W}\_{j \rightarrow j+1}^{D} + \mathbf{W}\_{j+1 \rightarrow j+2}^{D} = \mathbf{Q}\_{j} \mathbf{e}\_{j \rightarrow j+1} + \mathbf{Q}\_{j} \mathbf{e}\_{j+1 \rightarrow j+2} \\ &= \mathbf{Q}\_{j} (\mathbf{e}\_{j \rightarrow j+1} + \mathbf{e}\_{j+1 \rightarrow j+2}). \end{split} \tag{47}$$

The extra work

$$\begin{aligned} \mathcal{W}\_{\text{extra}}^D &= \mathcal{W}\_{j+1 \to j+2}^D \\ &= Q\_j \mathfrak{e}\_{j \to j+1} \mathfrak{e}\_{j+1 \to j+2} \\ &= \mathcal{W}\_{j \to j+1} \mathfrak{e}\_{j+1 \to j+2} \\ &= \mathcal{W}\_{j \to j+1}^D \mathfrak{e}\_{j+1 \to j+2} \end{aligned} \tag{48}$$

is paid for by the extra increase in entropy owing to frictional dissipation into extra heat *Q<sup>D</sup>* extra of the work output as per Eqs. (42) and (45)

$$Q^{D}\_{\text{extra}} = W\_{j \to j+1} = W^{D}\_{j \to j+1} = Q\_{j} \epsilon\_{j \to j+1} \tag{49}$$

into reservoir *j* þ 1 at temperature *Tj*þ1. This extra increase in entropy is

$$
\Delta \mathbf{S}^{\rm D}\_{\rm extra} = \frac{\mathbf{Q}^{\rm D}\_{\rm extra}}{T\_{j+1}} = \frac{\mathbf{Q}\_{j} \mathbf{e}\_{j \to j+1}}{T\_{j+1}} = \frac{\mathbf{W}\_{j \to j+1}}{T\_{j+1}} = \frac{\mathbf{W}^{\rm D}\_{j \to j+1}}{T\_{j+1}} = \frac{\mathbf{W}^{\rm D}\_{\rm extra}}{\mathbf{e}\_{j+1 \to j+2} T\_{j+1}}.\tag{50}
$$

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs DOI: http://dx.doi.org/10.5772/intechopen.89047*

[In the last four steps of Eq. (50), we applied Eqs. (42), (45), (48), and (49).] Thus

$$\mathcal{W}^{D}\_{\text{extra}} = T\_{j+1} \Delta \mathbf{S}^{D}\_{\text{extra}} \epsilon\_{j+1 \to j+2}. \tag{51}$$

In no case do we assume an efficiency with all work sequestered, or at any one given step *j* ! *j* þ *1* whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except that it is conserved—and we *never* violate conservation of energy.)

We note that, while frictional dissipation of work into intermediate reservoirs can yield extra work *W<sup>D</sup>* extra in heat-engine operation (albeit at the expense of *ΔS<sup>D</sup>* extra), it seems to be of no help in reverse, that is, refrigerator or heat pump, operation. For, in refrigerator or heat pump operation, with an intermediate reservoir *j* þ 1 at temperature *Tj*þ<sup>1</sup>*, Qj*þ<sup>2</sup> þ *Wj*þ2!*j*þ<sup>1</sup> ¼ *Qj*þ<sup>1</sup>*, Qj*þ<sup>1</sup> þ *Wj*þ1!*<sup>j</sup>* ¼ *Qj*, hence *Qj*þ<sup>2</sup> þ *Wj*þ2!*j*þ<sup>1</sup> þ *Wj*þ1!*<sup>j</sup>* ¼ *Qj*þ<sup>2</sup> þ *Wj*þ2!*<sup>j</sup>* ¼ *Qj*. Without an intermediate reservoir *j* þ 1 at temperature *Tj*þ<sup>1</sup>*, Qj*þ<sup>2</sup> þ *Wj*þ2!*<sup>j</sup>* ¼ *Qj* . The bottom line *Qj*þ<sup>2</sup> þ *Wj*þ2!*<sup>j</sup>* ¼ *Qj* is identical with or without an intermediate reservoir *j* þ 1 at temperature *Tj*þ1. With or without the intermediate reservoir *j* þ 1 at temperature *Tj*þ1, *all* of the energy must end up as *Qj*; thus, there is *none* left over to be frictionally dissipated. Hence the presence or absence of this intermediate reservoir makes no difference with respect to reverse, that is, refrigerator or heat pump, operation: See Ref. [1], Section 20-3; Ref. [2], Section 5.12 and Problem 5.22; Ref. [3], Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. [4], Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. [5], Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17; Ref. [6], Chapter XXI; Ref. [7], Sections 6.7, 6.8, 7.3, and 7.4); and Ref. [9], pp. 233–236 and Problems 1, 2, 4, 6, and 7 of Chapter 8. [Problem 2 of Chapter 8 in Ref. [9] considers absorption refrigeration, wherein the entire energy output is into an intermediate-temperature (most typically ambient-temperature) reservoir, and hence for which also there is *no* energy left over to be frictionally dissipated.]

## **5. Conclusion**

by employing the reservoir at temperature *Tj* as a hot reservoir and the reservoir

 to the reservoir at temperature *Tj*þ1. If a third reservoir at temperature *Tj*þ<sup>2</sup> and *Wj*!*j*þ<sup>1</sup> is sequestered, that is, not frictionally dissipated, a heat engine

by employing the reservoir at temperature *Tj*þ<sup>1</sup> as a hot reservoir and the reser-

With total frictional dissipation of *Wj*!*j*þ<sup>1</sup> into reservoir *j* þ 1 at temperature

pated into the reservoir at temperature *Tj*þ<sup>1</sup> (indicated via a superscript *D*). If there is a third reservoir at temperature *Tj*þ2, a heat engine can then perform additional work:

*<sup>j</sup>*þ1!*j*þ<sup>2</sup> ¼ *Qj*

*j*þ1!*j*þ2 ¼ *Qj*ϵ*<sup>j</sup>*!*j*þ<sup>1</sup>ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup> ¼ *Wj*!*j*þ<sup>1</sup>ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>

*<sup>j</sup>*!*j*þ<sup>1</sup>ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>

*<sup>j</sup>*!*j*þ<sup>1</sup> ¼ *Qj*

<sup>¼</sup> *<sup>W</sup><sup>D</sup> j*!*j*þ1 *Tj*þ<sup>1</sup>

<sup>ϵ</sup>*j*þ1!*j*þ<sup>2</sup> (43)

<sup>ϵ</sup>*j*þ1!*j*þ<sup>2</sup>

*<sup>j</sup>*!*j*¼<sup>1</sup> ¼ *Q*1ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup> be totally frictionally dissi-

*<sup>j</sup>*!*j*þ<sup>1</sup> ¼ *Wj*!*j*þ<sup>1</sup> ¼ *Q*1ϵ*<sup>j</sup>*!*j*þ<sup>1</sup>*:* (45)

*<sup>j</sup>*þ1!*j*þ<sup>2</sup> ¼ *Q*1ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>*:* (46)

ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> þ *Qj*ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>

<sup>¼</sup> *<sup>W</sup><sup>D</sup>*

extra ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>*Tj*þ<sup>1</sup>

*:* (47)

(44)

(48)

ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> (49)

*:* (50)

at temperature *Tj*þ<sup>1</sup> as a cold reservoir. It rejects waste heat *Qj* � *Wj*!*j*þ<sup>1</sup> ¼

*Wj*þ1!*j*þ<sup>2</sup> ¼ *Qj* 1 � ϵ*j*!*j*þ<sup>1</sup>

voir at temperature *Tj*þ<sup>2</sup> as a cold reservoir. All told it can do work:

*Wj*!*j*þ<sup>2</sup> ¼ *Wj*!*j*þ<sup>1</sup> þ *Wj*þ1!*j*þ<sup>2</sup> ¼ *Qj*ϵ*j*!*j*þ<sup>1</sup> þ *Qj* 1 � ϵ*j*!*j*þ<sup>1</sup>

¼ *Qj* ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> þ ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup> � ϵ*<sup>j</sup>*!*j*þ<sup>1</sup>ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup> *:*

*W<sup>D</sup>*

*W<sup>D</sup>*

*<sup>j</sup>*!*j*þ<sup>1</sup> <sup>þ</sup> *<sup>W</sup><sup>D</sup>*

¼ *Qj* ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> þ ϵ*<sup>j</sup>*þ1!*j*þ<sup>2</sup>

extra <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

<sup>¼</sup> *<sup>W</sup><sup>D</sup>*

into reservoir *j* þ 1 at temperature *Tj*þ1. This extra increase in entropy is

<sup>¼</sup> *Wj*!*j*þ<sup>1</sup> *Tj*þ<sup>1</sup>

extra of the work output as per Eqs. (42) and (45)

extra <sup>¼</sup> *Wj*!*j*þ<sup>1</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

is paid for by the extra increase in entropy owing to frictional dissipation into

*W<sup>D</sup>*

But now we let the work output *W<sup>D</sup>*

All told it can do work:

The extra work

extra heat *Q<sup>D</sup>*

*ΔS<sup>D</sup>*

**140**

extra <sup>¼</sup> *<sup>Q</sup><sup>D</sup>*

extra *Tj*þ<sup>1</sup>

*W<sup>D</sup>*

*<sup>j</sup>*!*j*þ<sup>2</sup> <sup>¼</sup> *<sup>W</sup><sup>D</sup>*

*Q<sup>D</sup>*

<sup>¼</sup> *<sup>Q</sup> <sup>j</sup>*ϵ*<sup>j</sup>*!*j*þ<sup>1</sup> *Tj*þ<sup>1</sup>

*Qj* 1 � ϵ*j*!*j*þ<sup>1</sup>

*Tj*þ1, we still have

can then perform additional work:

*Thermodynamics and Energy Engineering*

We investigated the increased heat-engine efficiencies obtained via operation employing increasing numbers (≥ 3) of heat reservoirs and with work output totally frictionally dissipated into all reservoirs except the first, hottest, one at temperature *T*<sup>1</sup> and (possibly) also the last, coldest, one at temperature *Tn*. We emphasize again that our results are consistent with both the First and Second Laws of Thermodynamics. The two laws are *not* violated because, if the work output of a heat engine is frictionally dissipated as heat into a cooler reservoir, both laws allow this heat to be partially converted to work again if another, still cooler, reservoir is available.

We do, however, challenge an *over*statement of the Second Law that is sometimes made, namely, that energy can do work only once. Energy can indeed do work more than once, because the Second Law does not forbid recycling of energy, so long as total entropy does not decrease as a result. This criterion of non-decrease of total entropy *is* obeyed, as per Section 4. In no case do we assume an efficiency with all work sequestered, or at any one given step *j* ! *j* þ 1 whether work is sequestered or not, exceeding the Carnot efficiency, and hence we are within the restrictions of the Second Law. (The First Law, of course, puts no restrictions whatsoever on the recycling of energy, except that it is conserved—and we *never* violate conservation of energy).

While in this chapter we do not challenge the First or Second Laws of Thermodynamics, we should note that there have been many challenges to the Second Law, especially in recent years [41–46]. By contrast, the First Law has been questioned only in cosmological contexts [47–49] and with respect to fleeting violations thereof associated with the energy-time uncertainty principle [50, 51]. But there are contrasting viewpoints [50, 51] concerning the latter issue.

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