Open access peer-reviewed chapter

# Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs

Written By

Jack Denur

Submitted: 28 June 2019 Reviewed: 07 August 2019 Published: 13 December 2019

DOI: 10.5772/intechopen.89047

From the Edited Volume

## Thermodynamics and Energy Engineering

Edited by Petrică Vizureanu

Chapter metrics overview

View Full Metrics

## Abstract

The efficiencies of heat-engine operation employing various numbers (≥ 2) of heat reservoirs are investigated. Operation with the work output of the heat engines sequestered, as well as with it being totally frictionally dissipated, is discussed. We consider mainly heat engines whose efficiencies depend on ratios of a higher and lower temperature or on simple functions of such ratios but also provide brief comments concerning more general cases. 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.

### Keywords

• First Law of Thermodynamics
• Second Law of Thermodynamics
• heat engines
• work
• heat
• entropy
• multiple heat reservoirs

## 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, 2, 3, 4, 5, 6, 7, 8, 9] but also via the Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [2, 3, 4, 5, 6, 7, 8, 9], and endoreversible heat engines operating at maximum power output assuming Curzon-Ahlborn efficiency [10, 11, 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, 11, 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, 14, 15, 16, 17, 18, 19, 20, 21]. In particular, we note that alternative results [21] to the Curzon-Ahlborn efficiency [10, 11, 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, 11, 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.

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, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], which is a different process that has been thoroughly investigated and discussed previously [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], and which we further investigate in another chapter [38] in this book.

We consider only cyclic heat engines. Noncyclic (necessarily one-time, single-use) 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.

## 2. Multiple-reservoir heat-engine efficiencies with work output sequestered

We designate the temperatures of the heat reservoirs via subscripts, with T1 being the temperature of the initial, hottest, reservoir, T2 the temperature of the second-hottest reservoir, T3 the temperature of the third-hottest reservoir, etc., and Tn the temperature of the nth, coldest, reservoir.

Let a heat engine operate between two reservoirs, extracting heat Q1 from a hot reservoir at temperature T1 and rejecting waste heat to a cold reservoir at temperature T2. If its efficiency is ϵ12, its work output is

W12=Q1ϵ12.E1

It rejects waste heat Q1W12=Q11ϵ12 to the reservoir at temperature T2. If there is a third reservoir at temperature T3 and W12 is sequestered, that is, not frictionally dissipated, and if the efficiency of heat-engine operation between the second and third reservoirs is ϵ23, a heat engine can then perform additional work

W23=Q11ϵ12ϵ23E2

by employing the reservoir at temperature T2 as a hot reservoir and the reservoir at temperature T3 as a cold reservoir. All told it can do work:

W12+W23=Q1ϵ12+Q11ϵ12ϵ23=Q1ϵ12+ϵ23ϵ12ϵ23.E3

By contrast, if the heat engine operates in a single step at efficiency ϵ13, employing the reservoir at temperature T1 as a hot reservoir and the reservoir at temperature T3 as a cold reservoir, it can do work

W13=Q1ϵ13.E4

Anticipating that we will eventually deal with n heat reservoirs, let us consider efficiencies of the form

ϵij=1TiTjx,E5

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), W1→3 = W1→2 + W2→3, as we will now show. We have

W12+W23=Q11T2T1x+1T3T2x1T2T1x1T3T2x=Q12T2T1xT3T2x1T2T1xT3T2x+T2T1xT3T2x=Q11T2T1xT3T2x=Q11T3T1x=W13.E6

We note that x = 1 for the Carnot, Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [1, 2, 3, 4, 5, 6, 7, 8, 9] and x = 1/2 for endoreversible heat engines operating at Curzon-Ahlborn efficiency [10, 11, 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 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, 3, 4, 5, 6, 7, 8, 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, W1→3 = W1→2 + W2→3, this additivity of W obtains for any number of steps, that is, we have

W1n=W12+W23++Wn1n=j=1n1Wjj+1.E7

For more complex efficiencies than those of Eq. (5), for example, those of the Diesel and dual cycles, which are functions of more than two temperatures, and also for some more complex efficiencies that are functions of two temperatures, the equality of Eq. (7) may not always obtain [3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 19]. But whether or not the equality of Eq. (7) obtains, the Second Law of Thermodynamics requires that, whichever reservoirs are employed, the efficiency with all work outputs sequestered, whether Wjj+1/Qj (1 ≤ j ≤ n − 1), Wjj+k/Qj (1 ≤  n − 1 and 1  k  n − j), or W1→n/Q1, cannot exceed the Carnot limit.

## 3. Multiple-reservoir heat-engine efficiencies with work output totally frictionally dissipated

Let a heat engine operate between two reservoirs, extracting heat Q1 from a hot reservoir at temperature T1 and rejecting waste heat to a cold reservoir at temperature T2. If its efficiency is ϵ12, its work output is

W12=Q1ϵ12.E8

It rejects waste heat Q1W12=Q11ϵ12 to a reservoir at temperature T2. But now, in addition, we let the work output W12D=Q1ϵ12 be totally frictionally dissipated and rejected into the reservoir at temperature T2 (indicated via a superscript D). This is in fact by far the most common mode of heat-engine operation. With rare exceptions (e.g., a heat engine’s work output being sequestered for a long time interval as gravitational potential energy in the construction of a building, or essentially permanently in the launching of a spacecraft), heat engines’ work outputs are typically totally frictionally dissipated immediately or on short time scales (see Ref. [6], Chapter VI (especially Sections 54, 60, and 61); and Ref. [7], Sections 6.9–6.14 and 16.8). Indeed, this is true of almost all engines, heat engines or otherwise. The work outputs of all engines of vehicles (automobiles, trains, ships, submarines, aircraft, etc.) operating at constant speed, and of all factory and appliance engines operating at constant speed, are immediately and continually frictionally dissipated. The work output temporarily sequestered as kinetic energy when a vehicle accelerates, or when a factory or appliance engine is turned on, is frictionally dissipated a short time later when the vehicle decelerates, or when the factory or appliance engine is turned off.

If both the waste heat Q1W12D=Q11ϵ12 has been rejected and the work output W12D=Q1ϵ12 has been totally frictionally dissipated into the reservoir at temperature T2, and there is a third reservoir at temperature T3, a heat engine operating at efficiency ϵ23 can then perform additional work

W23=Q1ϵ23E9

by employing the reservoir at temperature T2 as a hot reservoir and the reservoir at temperature T3 as a cold reservoir. (W23D may or may not be frictionally dissipated, so it only optionally carries the superscript D.) All told the total work output is

W13D=W12D+W23D=Q1ϵ12+Q1ϵ23=Q1ϵ12+ϵ23.E10

If ϵij=1Ti/Tjx, where i and j are positive integers in the respective ranges 1in1 and i<jn, and where x is a positive real number in the range 0<x1, applying Eqs. (5) and (10), we have:

W13D=W12D+W23D=Q11T2T1x+1T3T2x=Q12T2T1xT3T2x.E11

We now maximize W13D with respect to T2:

dW13DdT2=0ddT22T2T1xT3T2x=0ddT2T2T1+T3T2=01T1T3T22=0T2,opt=T1T31/2.E12

Thus, the optimum value T2,opt of T2, which maximizes W13D, is the geometric mean of T1 and T3. Applying Eqs. (11) and (12), the maximum value W13,maxD of W13D is

W13,maxD=Q12T1T31/2T1xT3T1T31/2x=Q12T3T1x/2T3T1x/2=Q122T3T1x/2=2Q11T3T1x/2.E13

Note that

W13,maxD>Q1ifT3T1x/2<12T3T1<122/x.E14

This obtains if T3/T1<1/4 for x=1 and if T3/T1<1/16 for x=1/2. Also, comparing the last line of Eq. (6) with Eq. (13), we find for the maximum extra work W13,maxD,extra:

W13,maxD,extra=W13,maxDW13=2Q11T3T1x/2Q11T3T1x=Q121T3T1x/21T3T1x=Q122T3T1x/21+T3T1x=Q11+T3T1x2T3T1x/20.E15

It is easily shown that W13,maxD,extra0, with the equality obtaining if and only if T3T1=1W13,maxD=W13=0W13,maxDW13=W13,maxD,extra=0. For, denoting the ratio T3T1x/2 as r and setting dW13,maxD,extra/dr=0 yields

Thus W13,maxD is minimized at 0 if r=T3T1x/2=1T3T1=1. For all T3T1<1,W13,extraD>0. Moreover, applying Eqs. (5), (13), and (15), note that

limT3/T10W13,maxD=2Q1=2limT3/T10W13limT3/T10W13,maxD,extra=2Q1Q1=Q1=limT3/T10W13.E17

Now consider heat-engine operation employing four heat reservoirs, with all work totally frictionally dissipated (except possibly at the last step; thus, W34D only optionally carries the superscript D). Thus we have

W14D=W12D+W23D+W34D=Q1ϵ12+Q1ϵ23+Q1ϵ34=Q1ϵ12+ϵ23+ϵ34.E18

If ϵij=1Ti/Tjx, where i and j are positive integers in the respective ranges 1in1 and i<jn, and where x is a positive real number in the range 0<x1, applying Eqs. (5) and (18), we have:

W14D=W12D+W23D+W34D=Q11T2T1x+1T3T2x+1T4T3x=Q13T2T1xT3T2xT4T3x.E19

We wish to maximize W14D. Based on Eq. (12) and the associated discussions, the optimum value Tj,opt of Tj of reservoir j 1<j<n2jn1, which maximizes Wj1j+1D, is the geometric mean of Tj1 and Tj+1. Thus we have

T2,opt=T1T3,opt1/2E20

and

T3,opt=T2,optT41/2.E21

Applying Eqs. (20) and (21), we obtain

T2,optT1=T1T3,opt1/2T1=T3,optT11/2E22

and

T4T3,opt=T4T2,optT41/2=T4T2,opt1/2.E23

Applying Eqs. (20)(23), we obtain

T3,optT2,opt=T3,optT1T3,opt1/2=T3,optT11/2=T2,optT41/2T2,opt=T4T2,opt1/2T3,optT11/2=T4T2,opt1/2T2,optT1=T3,optT2,opt=T4T3,opt.E24

Applying Eqs. (22)(24), we obtain

T4T1=T2T1T3T2T4T3in general=T2,optT1T3,optT2,optT4T3,optin particular=T2,optT13=T3,optT2,opt3=T4T3,opt3T2,optT1=T3,optT2,opt=T4T3,opt=T4T11/3.E25

Applying Eqs. (19) and (25), we obtain

W14,maxD=Q133T4T1x/3=3Q11T4T1x/3.E26

We now slightly modify Eqs. (14)(17) to apply for our four-reservoir system. We obtain

W14,maxD>Q1ifT4T1x/3<23T4T1<233/x.E27

This obtains if T4/T1<2/33=8/27 for x=1 and if T4/T1<2/36=64/729 for x=1/2. Also, applying Eqs. (5) and (26),

W14,maxD,extra=W14,maxDW14=3Q11T4T1x/3Q11T4T1x=Q131T4T1x/31T4T1x=Q133T4T1x/31+T4T1x=Q12+T4T1x3T4T1x/30.E28

It is easily shown that W14,maxD,extra0, with the equality obtaining if and only if T4T1=1W14,maxD=W14=0W14,maxDW14=W14,maxD,extra=0. For, denoting the ratio T4T1x/3 as r and setting dW14,maxD,extra/dr=0 yields

Thus W14,maxD,extra is minimized at 0 if r=T4T1x/3=1T4T1=1. For all T4T1<1,W14,maxD,extra>0. Moreover, applying Eqs. (5), (26), and (28), note that

limT4/T10W14,maxD=3Q1=3limT4/T10W14limT4/T10W14,maxD,extra=3Q1Q1=2Q1=2limT4/T10W14.E30

Comparing Eqs. (13)(17) with Eqs. (26)(30), note the larger values in Eqs. (26), (28), and (30) than in Eqs. (13), (15), and (17), respectively, and the easier fulfillment of the inequality in Eq. (27) than in Eq. (14) (concerning the latter point: 8/27>1/4 and 64/729>1/16).

Generalizing Eqs. (20)(30) for an n-reservoir system (n = any positive integer ≥ 4), we obtain:

Tj+1=TjTj+21/2,E31

where j is any positive integer in the range 1jn2 and

Tj+2=Tj+1Tj+31/2,E32

where j is any positive integer in the range 1jn3. The respective temperatures T1 and Tn of the extreme (hottest and coldest) reservoirs are assumed to be fixed. The temperatures T2 through Tn1 of all intermediate reservoirs are all assumed to be optimized in accordance with Eqs. (31) and (32). With that understood, for brevity and to avoid using different subscripts for the extreme and intermediate reservoirs, the subscript “opt” is omitted in Eqs. (31)(35). Applying Eqs. (31) and (32), we obtain:

Tj+1Tj=TjTj+21/2Tj=Tj+2Tj1/2E33

and

Tj+2Tj+1=Tj+2TjTj+21/2=Tj+2Tj1/2.E34

Applying Eqs. (33) and (34), and recognizing that Eqs. (33) and (34) obtain for all values of j such that j is any positive integer in the range 1jn2, we obtain:

Tj+2Tj+1=Tj+1TjTj+2Tj=Tj+1TjTj+2Tj+1=Tj+1Tj2Tj+1Tj=Tj+2Tj1/2TnT1=Tj+1Tjn1Tj+1Tj=TnT11/n1.E35

The first two lines of Eq. (35) obtain for all values of j such that j is any positive integer in the range 1jn2, and the third line of Eq. (35) obtain for all values of j such that j is any positive integer in the range 1jn1. The first two lines of Eq. (35) pertain to any three adjacent heat reservoirs, and hence 2 appears in the exponents of the second line thereof; the third line of Eq. (35) pertains to all n heat reservoirs, and hence n1 appears in the exponents thereof. The second and third lines of Eq. (35) mutually justify each other: the third line of Eq. (35)must obtain because the second line thereof obtains for all values of j; and, conversely, given that the third line of Eq. (35) obtains, the second line thereof must obtain for all values of j.

If, as per Eq. (5), ϵij=1Ti/Tjx, where i and j are positive integers in the respective ranges 1in1 and i<jn, and where x is a positive real number in the range 0<x1, then, applying Eqs. (5) and (31)(35), we now generalize Eqs. (13)(17) and (26)(30), as well as the associated discussions, to apply for our n-reservoir system. We obtain:

W1n,maxD=n1Q11TnT1x/n1,E36
W1n,maxD>Q1ifTnT1x/n1<n2n1TnT1<n2n1n1/x,E37

and

W1n,maxD,extra=W1n,maxDW1n=n1Q11TnT1x/n1Q11TnT1x=Q1n11TnT1x/n11TnT1x=Q1n1n1TnT1x/n11+TnT1x=Q1n2+TnT1xn1TnT1x/n10.E38

It is easily shown that W1n,maxD,extra0, with the equality obtaining if and only if TnT1=1W1n,maxD=W1n=0W1n,maxDW1n=W1n,maxD,extra=0. For, denoting the ratio TnT1x/n1 as r and setting dW1n,maxD,extra/dr=0 yields

Thus W1n,maxD,extra is minimized at 0 if r=TnT1x/n1=1TnT1=1. For all TnT1<1,W1n,maxD,extra>0. Moreover, applying Eqs. (5), (36), and (38), note that

limTn/T10,nfixedW1n,maxD=n1Q1=n1 limTn/T10,nfixedW1nlimTn/T10,nfixed W1n,maxD,extra=limTn/T10,nfixed W1n,maxDW1n=n1Q1Q1=n2Q1=n2limTn/T10,nfixedW1n.E40

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=1) but even for Curzon-Ahlborn efficiency (x=1/2), indeed for any x finitely greater than 0 in the range 0<x1, because TnT1x/n101TnT1x/n11 in the limit Tn/T10, albeit ever more slowly with decreasing x.

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

limn,Tn/T1fixedW1n,maxD=Q1lnT1Tn=limTn/T10,nfixedW1nlnT1Tn.E41

Note the linear divergence of W1n,maxD in the limit Tn/T10 with n fixed as per Eq. (40) even not assuming Carnot efficiency, as contrasted with the paltry logarithmic divergence of W1n,maxD in the limit n with Tn/T1 fixed even granting Carnot efficiency as per the derivation [22] of Eq. (41).

But we note that the temperature of the cosmic background radiation is only 2.7K, while the most refractory materials remain solid at temperatures slightly exceeding 2700K. This provides a temperature ratio of T1/Tn103Tn/T1103. Could even larger values of T1/Tn 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. Yet even with the paltry logarithmic divergence of W1n,maxD in the limit n with T1/Tn fixed as per Eq. (41) and even with a temperature ratio of T1/Tn103Tn/T1103, assuming Carnot efficiency by Eq. (41)W1n,maxD/Q1ln1037. Hence by Eq. (41) an advanced civilization employing 7 concentric Dyson spheres [39, 40] can procure 7 times as much work output (to the nearest whole number) as its host star’s total energy output. Actually the limit n with T1/Tn fixed is not sufficiently closely approached to apply Eq. (41): we should instead apply Eq. (36). Applying Eq. (36) and assuming Carnot efficiency with T1/Tn103Tn/T1103, W1n,maxD/Q14. Hence by Eq. (36) an advanced civilization employing 4 concentric Dyson spheres [39, 40] can procure 4 times as much work output (to the nearest whole number) as its host star’s total energy output.

## 4. General remarks, especially concerning entropy

It is important to emphasize that the super-unity cyclic-heat-engine efficiencies W1n,maxD/Q1 that can obtain with work output totally frictionally dissipated ifn3 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 jj+1 or jj+k (1knj). The validity of this Section 4 requires only that the efficiency with all work sequestered, or at any one given step jj+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+2 at temperature Tj+2 without versus with frictional dissipation into reservoir j+1 at temperature Tj+1Tj>Tj+1>Tj+2. Without frictional dissipation a heat engine performs work

Wjj+1=Qjϵjj+1E42

by employing the reservoir at temperature Tj as a hot reservoir and the reservoir at temperature Tj+1 as a cold reservoir. It rejects waste heat QjWjj+1=Qj1ϵjj+1 to the reservoir at temperature Tj+1. If a third reservoir at temperature Tj+2 and Wjj+1 is sequestered, that is, not frictionally dissipated, a heat engine can then perform additional work:

Wj+1j+2=Qj1ϵjj+1ϵj+1j+2E43

by employing the reservoir at temperature Tj+1 as a hot reservoir and the reservoir at temperature Tj+2 as a cold reservoir. All told it can do work:

Wjj+2=Wjj+1+Wj+1j+2=Qjϵjj+1+Qj1ϵjj+1ϵj+1j+2=Qjϵjj+1+ϵj+1j+2ϵjj+1ϵj+1j+2.E44

With total frictional dissipation of Wjj+1 into reservoir j+1 at temperature Tj+1, we still have

Wjj+1D=Wjj+1=Q1ϵjj+1.E45

But now we let the work output Wjj=1D=Q1ϵj+1j+2 be totally frictionally dissipated into the reservoir at temperature Tj+1 (indicated via a superscript D). If there is a third reservoir at temperature Tj+2, a heat engine can then perform additional work:

Wj+1j+2D=Q1ϵj+1j+2.E46

All told it can do work:

Wjj+2D=Wjj+1D+Wj+1j+2D=Qjϵjj+1+Qjϵj+1j+2=Qjϵjj+1+ϵj+1j+2.E47

The extra work

is paid for by the extra increase in entropy owing to frictional dissipation into extra heat QextraD of the work output as per Eqs. (42) and (45)

into reservoir j+1 at temperature Tj+1. This extra increase in entropy is

[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 given step j → j + 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 WextraD in heat-engine operation (albeit at the expense of ΔSextraD), 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+1 at temperature Tj+1,Qj+2+Wj+2j+1=Qj+1,Qj+1+Wj+1j=Qj, hence Qj+2+Wj+2j+1+Wj+1j=Qj+2+Wj+2j=Qj. Without an intermediate reservoir j+1 at temperature Tj+1,Qj+2+Wj+2j=Qj. The bottom line Qj+2+Wj+2j=Qj is identical with or without an intermediate reservoir j+1 at temperature Tj+1. With or without the intermediate reservoir j+1 at temperature Tj+1, 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

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 T1 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 overstatement 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 + 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, 42, 43, 44, 45, 46]. By contrast, the First Law has been questioned only in cosmological contexts [47, 48, 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.

## Acknowledgments

I am very grateful to Dr. Donald H. Kobe, Dr. Paolo Grigolini, Dr. Daniel P. Sheehan, Dr. Bruce N. Miller, and Dr. Marlan O. Scully and for many very helpful and thoughtful insights, as well as for very perceptive and valuable discussions and communications, which greatly helped my understanding of thermodynamics and statistical mechanics. Also, I am indebted to them, as well as to Dr. Bright Lowry, Dr. John Banewicz, Dr. Bruno J. Zwolinski, Dr. Roland E. Allen, Dr. Abraham Clearfield, Dr. Russell Larsen, Dr. James H. Cooke, Dr. Wolfgang Rindler, Dr. Richard McFee, Dr. Nolan Massey, and Dr. Stan Czamanski for lectures, discussions, and/or communications from which I learned very much concerning thermodynamics and statistical mechanics. I thank Dr. Stan Czamanski and Dr. S. Mort Zimmerman for the very interesting general scientific discussions over many years. I also thank Dan Zimmerman, Dr. Kurt W. Hess, and Robert H. Shelton for the very interesting general scientific discussions at times. Additionally, I thank Robert H. Shelton for very helpful advice concerning diction.

## Conflict of interest

The author declares no conflict of interest.

## References

1. 1. Walker J, Halliday D, Resnick R. Fundamentals of Physics. 11th extended ed. Hoboken, NJ: John Wiley & Sons; 2018, Chapters 18 and 20 (especially Sections 20-2 and 20-3)
2. 2. Reif F. Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hill; 1965. (reissued: Long Grove, IL: Waveland Press; 2009), Sections 5-11 ad 5-12, and Problems 5.22 through 5.26
3. 3. Callen HC. Thermodynamics: New York: John Wiley & Sons; 1960, Chapter 4
4. 4. Callen HC. Thermodynamics and an Introduction to Thermostatistics. 2nd ed. New York: John Wiley & Sons; 1985, Chapter 4
5. 5. Wark K, Richards DE. Thermodynamics. 6th ed. Boston, MA: WCB/McGraw-Hill; 1999, Chapters 6, 8, 9, and 15--17
6. 6. Faries VM. Applied Thermodynamics. Revised Ed. New York, NY: MacMillan; 1949, Chapters V–VIII, XVII, and XIX
7. 7. Zemansky, MW, Dittman RH. Adapted by Chattopadhyay AK. Heat and Thermodynamics, 8th ed. Chennai, India: McGraw Hill Education (India); 2011. (Seventeenth reprint 2018), Chapters 6 and 7
8. 8. Baierlein R. Thermal Physics. Cambridge, UK: Cambridge University Press; 1999, Chapters 2 and 3
9. 9. Kittel C, Kroemer H. Thermal Physics. 2nd ed. San Francisco, CA: W. H. Freeman and Company; 1980, Chapter 8
10. 10. Curzon FL, Ahlborn B. Efficiency of a Carnot engine at maximum power output. American Journal of Physics. 1975;43:22-24. DOI: 10.1119/1.10023
11. 11. Vaudrey A, Lanzetta F, Feidt M. Reitlinger and the origins of the efficiency at maximum power output for heat engines. Journal of Non-Equilibrium Thermodynamics. 2014;39:199-203. DOI: 10.1515/jnet-2014-0018
12. 12. Endoreversible thermodynamics [Online]. Available from: https://www.wikipedia.org/ [Accessed: 16 March 2020]
13. 13. De Vos A. Efficiency of some heat engines at maximum-power conditions. American Journal of Physics. 1985;53:570-573. DOI: 10.1119/1.14240
14. 14. Gordon JM. Maximum power-point characteristics of heat engines as a general thermodynamic problem. American Journal of Physics. 1989;57:1136-1142. DOI: 10.1119/1.16130
15. 15. Gordon JM. Observations on efficiency of heat engines operating at maximum power. American Journal of Physics. 1990;58:370-375. DOI: 10.1119/1.16175
16. 16. Schmiedl T, Seifert U. Efficiency at maximum power: An analytically solvable model for stochastic heat engines. Europhysics Letters. 2008;81:20003. DOI: 10.1209/0295-5075/81/20003
17. 17. Tu ZC. Efficiency at maximum power of Feynman’s ratchet as an engine. Journal of Physics A. 2008;41:312003. DOI: 10.1088/1751-8113/41/31/312003
18. 18. Leff HS. Thermal efficiency at maximum work output: New results for old heat engines. American Journal of Physics. 1987;55:602-610. DOI: 10.1119/1.15071
19. 19. Ouerdane H, Apertet Y, Goupil C, Lecoeur P. Continuity and boundary conditions in thermodynamics: From Carnot’s efficiency to efficiencies at maximum power. European Physical Journal - Special Topics. 2015;55:839-862. DOI: 10.1140/epjst/e2015-02431-x
20. 20. Parrando JMR, Ouerdane H, et al. Debate. Continuity and boundary conditions in thermodynamics: From Carnot’s efficiency to efficiencies at maximum power. European Physical Journal - Special Topics. 2015;224:862-864
21. 21. Apertet Y, Ouerdane H, Goupil C, Lecoeur Ph. True nature of the Curzon-Ahlborn efficiency. Physical Review E. 2017;96:022119. DOI: 10.1103/PhysRevE.96.022119
22. 22. Denur J. The apparent “super-Carnot” efficiency of hurricanes: Nature’s steam engine versus the steam locomotive. American Journal of Physics. 2011;79:631-643. DOI: 10.1119/13534841 (especially Section VI)
23. 23. Emanuel K. Divine Wind. Oxford, UK: Oxford University Press; 2005 (especially Chapter 10)
24. 24. Emanuel K. Hurricanes: Tempests in a greenhouse. Physics Today. 2006;59(8):74-75. DOI: 10.1063/1.2349743
25. 25. Emanuel K. Tropical cyclones. Annual Review of Earth and Planetary Sciences. 2003;31:75-104. DOI: 10.1146/annurev.earth.31.100901.141259
26. 26. Emanuel K. Thermodynamic control of hurricane intensity. Nature. 1999;401:665-669. DOI: 10.1038./44326
27. 27. Bister M, Emanuel KA. Dissipative heating and hurricane intensity. Meteorology and Atmospheric Physics. 1998;65:223-230
28. 28. Zhang DL, Altshuler E. The effects of dissipative heating on hurricane intensity. Monthly Weather Review. 1999;127:3032-3038
29. 29. Emanuel K. Response of tropical cyclone activity to climate change: Theoretical basis. In: Murmane RJ, Liu K-B, editors. Hurricanes and Typhoons: Past, Present, and Future. New York: Columbia University Press; 2004, pp. 395–407
30. 30. Emanuel KA, Speer K, Rotunno R, Srivastava R, Molina M. Hypercanes: A possible link in global extinction scenarios. Journal of Geophysical Research-Atmospheres. 1995;100:13755-13765. DOI: 10.1029/95JD01368
31. 31. Emanuel K, Callagham J, Otto PA. A hypothesis for redevelopment of warm-core cyclones over northern Australia. Monthly Weather Review. 2008;136:3863-3872. DOI: 10.1175/2008MWR2409.1
32. 32. Kieu C. Revisiting dissipative heating in tropical cyclone maximum potential intensity. Quarterly Journal of the Royal Meteorological Society. 2015;141:2497-2504. DOI: 10.1002/qj.2534
33. 33. Apertet Y, Ouerdane H, Goupil C, Lecoeur P. Efficiency at maximum power of thermally coupled heat engines. Physical Review E. 2012;85:041144. DOI: 10.1103/PhysRevE.85041144
34. 34. Makarieva AM, Gorshkov VC, Li B-L, Nobre AD. A critique of some modern applications of the Carnot heat engine concept: The dissipative engine cannot exist. Proceedings of the Royal Society A. 2010;466:1893-1902. DOI: 10.1098/rspa.2009.0581
35. 35. Bejan A. Thermodynamics of heating. Proceedings of the Royal Society A. 2019;475:20180820. DOI: 10.1098/rspa.2018.0820
36. 36. Bister M, Renno N, Pauluis O, Emanuel K. Comment on Makarieva et al. ‘A critique of some modern applications of the Carnot heat engine concept: The dissipative engine cannot exist’. Proceedings of the Royal Society A. 2011;467:1-6. DOI: 10.1098/rspa.2010.0087
37. 37. Ozawa H, Shimokawa S. Thermodynamics of a tropical cyclone: Generation and dissipation of mechanical energy in a self-driven convection system. Tellus A. 2015;67:24216. DOI: 10.3402/tellusa.v67.24216. 15 pages
38. 38. Denur J. Improving heat-engine performance via high-temperature recharge. In: Vizureanu P, Academic editor. Applied Thermodynamics and Energy Engineering. London, UK: IntechOpen; 2019
39. 39. Dyson Sphere. Available from: https://www.wikipedia.org/ [Accessed: 16 March 2020]
40. 40. Dyson Spheres in Popular Culture. Available from: https://www.wikipedia.org/ [Accessed: 16 March 2020]
41. 41. Sheehan DP, editor. Quantum limits to the second law. In: AIP Conference Proceedings Volume 643; Melville, NY: American Institute of Physics; 2002
42. 42. Nikulov AV, Sheehan DP, editors. Special issue: Quantum limits to the second law of thermodynamics. Entropy 2004;6(1)
43. 43. Čápek V, Sheehan DP. Challenges to the Second Law of Thermodynamics: Theory and Experiment. Dordrecht, The Netherlands: Springer; 2005
44. 44. Sheehan DP, editor. Special issue: The second law of thermodynamics: Foundations and status. Foundations of Physics. 2007;37(12)
45. 45. Sheehan DP, editor. Second law of thermodynamics: Status and challenges. In AIP Conference Proceedings Volume 1411; Melville, NY: American Institute of Physics; 2011
46. 46. Sheehan DP, editor. Special issue: Limits to the second law of thermodynamics: Experiment and theory. Entropy. 2017;19
47. 47. Harrison ER. Mining energy in an expanding universe. The Astrophysical Journal. 1995;446:63-66
48. 48. Sheehan DP, Kriss VG. Energy Emission by Quantum Systems in an Expanding FRW Metric [Online]. Available from: arXiv:astroph/0411299v1 [Accessed: 16 March 2020]
49. 49. Parry R. Extracting Energy from the Expanding Universe: Can we Avoid the Heat Death? Honours Physics 2015. Sydney, Australia: Sydney Institute for Astronomy, School of Physics, The University of Sydney; 2015
50. 50. Griffiths DJ, Schroeter DF. Introduction to Quantum Mechanics. 3rd ed. Cambridge, UK: Cambridge University Press; 2018, Section 3.5.3
51. 51. Hagmann MJ. Distribution of times for barrier traversal caused by energy fluctuations. Journal of Applied Physics. 1993;74:7302-7305

Written By

Jack Denur

Submitted: 28 June 2019 Reviewed: 07 August 2019 Published: 13 December 2019