**1. Introduction**

The efficiencies of heat-engine operation employing various numbers (≥ 2) of heat reservoirs are investigated. In Section 2, we discuss heat-engine operation with the work output of the heat engines sequestered. In Section 3, we discuss heat-engine operation with the work output of the heat engines being totally frictionally dissipated. We consider mainly heat engines whose efficiencies depend on ratios of a higher and lower temperature or on simple functions of such ratios. Examples include heat engines operating not only via the Carnot cycle [1–9] but also via the Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [2–9], and endoreversible heat engines operating at maximum power output assuming Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9). But we also provide brief comments concerning more general cases. Endoreversible heat-engine operation assumes irreversible heat flows directly proportional to temperature differences but otherwise reversible operation [10–12]. Although we do not employ them in this chapter, we note that generalizations of the Curzon-Ahlborn efficiency, and also various related efficiencies, have also been investigated [13–21]. In particular, we note that alternative results [21] to the Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9) have been derived [21]. But for definiteness and for simplicity, in this chapter,

we employ the standard Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9) for cyclic heat engines operating at maximum power output.

We show that, if a hot reservoir supplies a heat engine whose waste heat is discharged *and* whose work output is totally frictionally dissipated into a cooler reservoir, which in turn supplies heat-engine operation that discharges waste heat into a still cooler reservoir, the total work output can exceed the heat input from the initial hot reservoir. This extra work output increases with increasing numbers (≥ 3) of reservoirs. We also show that this obtains within the restrictions of the First and Second Laws of Thermodynamics.

*W*1!<sup>2</sup> þ *W*2!<sup>3</sup> ¼ *Q*1ϵ1!<sup>2</sup> þ *Q*1ð Þ 1 � ϵ1!<sup>2</sup> ϵ2!<sup>3</sup>

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

By contrast, if the heat engine operates in a single step at efficiency ϵ1!3, employing the reservoir at temperature *T*<sup>1</sup> as a hot reservoir and the reservoir at

Anticipating that we will eventually deal with *n* heat reservoirs, let us consider

*Tj* � �*<sup>x</sup>*

<sup>2</sup> � *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup>*

*T*2

� *<sup>T</sup>*<sup>3</sup> *T*2 � �*<sup>x</sup>*

¼ *W*<sup>1</sup>!3*:*

� �*<sup>x</sup>* h i

<sup>ϵ</sup>*i*!*<sup>j</sup>* <sup>¼</sup> <sup>1</sup> � *Ti*

where *i* and *j* are positive integers in the respective ranges 1 ≤ *i* ≤ *n* � 1 and *i* < *j* ≤ *n* and where *x* is a positive real number in the range 0 < *x* ≤ 1. Applying

Eqs. (3) and (5), *W*1!<sup>3</sup> = *W*1!<sup>2</sup> + *W*2!3, as we will now show. We have

8 >><

>>:

8 >><

>>:

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>Q</sup>*<sup>1</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

this additivity of *W* obtains for any number of steps, that is, we have

*<sup>W</sup>*<sup>1</sup>!*<sup>n</sup>* <sup>¼</sup> *<sup>W</sup>*<sup>1</sup>!<sup>2</sup> <sup>þ</sup> *<sup>W</sup>*<sup>2</sup>!<sup>3</sup> <sup>þ</sup> … <sup>þ</sup> *Wn*�1!*<sup>n</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

¼ *Q*<sup>1</sup>

<sup>1</sup> � *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup>* h i <sup>þ</sup> <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

� <sup>1</sup> � *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup>* h i <sup>1</sup> � *<sup>T</sup>*<sup>3</sup>

� <sup>1</sup> � *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup>*

> *T*1 � �*<sup>x</sup> T*<sup>3</sup>

*T*1 � � � �*<sup>x</sup>*

We note that *x* = 1 for the Carnot, Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [1–9] and *x* = 1/2 for endoreversible heat engines operating at Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9). For all of these cycles, the temperature in the numerator is that of the coldest available reservoir for a given cycle [1–12]. For the Carnot, Ericsson, and Stirling cycles, and for endoreversible heat engines operating at Curzon-Ahlborn efficiency, the temperature in the denominator is that of the hottest available reservoir for a given cycle [1–12]. For the air-standard Otto and air-standard Brayton cycles, the temperature in the denominator is that at the end of the adiabatic-compression process but before the addition of heat from the hottest available reservoir (substituting, in air-standard cycles, for combustion of fuel) [2–9] in a given cycle. The Second Law of Thermodynamics forbids *x* > 1 if the temperature in the numerator is that of the coldest available reservoir for a given cycle and the temperature in the denominator is that of the hottest available reservoir for a given cycle, because then the Carnot efficiency would be exceeded. Since for the aforementioned heat engines, and indeed for any heat engine for which Eq. (5) is applicable, *W*1!<sup>3</sup> = *W*1!<sup>2</sup> + *W*2!3,

� � � �*<sup>x</sup>*

temperature *T*<sup>3</sup> as a cold reservoir, it can do work

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

*W*<sup>1</sup>!<sup>2</sup> þ *W*<sup>2</sup>!<sup>3</sup> ¼ *Q*<sup>1</sup>

efficiencies of the form

**131**

<sup>¼</sup> *<sup>Q</sup>*1ð Þ <sup>ϵ</sup>1!<sup>2</sup> <sup>þ</sup> <sup>ϵ</sup>2!<sup>3</sup> � <sup>ϵ</sup>1!2ϵ2!<sup>3</sup> *:* (3)

*W*1!<sup>3</sup> ¼ *Q*1ϵ1!3*:* (4)

*T*2 � �*<sup>x</sup>* h i 9 >>=

>>;

*T*2 � �*<sup>x</sup>* h i

> � *<sup>T</sup>*<sup>3</sup> *T*2 � �*<sup>x</sup>*

> > <sup>þ</sup> *<sup>T</sup>*<sup>2</sup> *T*1 � �*<sup>x</sup> <sup>T</sup>*<sup>3</sup> *T*2

> > > *j*¼1

*Wj*!*j*þ1*:* (7)

*,* (5)

9 >>=

>>;

(6)

We fill in details and correct a few mistakes in an earlier, briefer, consideration of the efficiencies of heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22]. We note that heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22] should not be confused with recycling heat engines' frictionally dissipated work outputs into the hottest available reservoir [22–37], which is a *different* process that has been thoroughly investigated and discussed previously [22–37], and which we further investigate in another chapter [38] in this book.

We consider only cyclic heat engines. Noncyclic (necessarily one-time, singleuse) heat engines are not limited by the Carnot bound and can in principle operate at unit (100%) efficiency. A simple example is the one-time expansion of a gas pushing a piston. 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 exhaust gases resulting from combustion of the propellant). Even if the work output of a noncyclic engine could be frictionally dissipated and the resulting heat returned to the system, there would be, at best, restoration of temperature to its initial value but not restoration of the piston to its initial position. Hence the method investigated in this chapter is useless with respect to noncyclic heat engines.

General remarks, especially concerning entropy, are provided in Section 4. Concluding remarks are provided in Section 5.
