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# Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs

By Jack Denur

Submitted: June 28th 2019Reviewed: August 7th 2019Published: December 13th 2019

DOI: 10.5772/intechopen.89047

## 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 at the Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. , 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  to the Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. , Section 4-9) have been derived . But for definiteness and for simplicity, in this chapter, we employ the standard Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. , 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 . We note that heat-engine operation employing various numbers (≥ 3) of heat reservoirs  should not be confused with recycling heat engines’ frictionally dissipated work output 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 and which 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  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 much 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 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 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=Q11ϵ12to the reservoir at temperature T2. If there is a third reservoir at temperature T3 and W12is 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 the Curzon-Ahlborn efficiency [10, 11, 12] (see also Ref. , 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 the 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 the 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 ≤ j ≤ 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=Q11ϵ12to the reservoir at temperature T2. But now, in addition, we let the work output W12D=Q1ϵ12be 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), heat engines’ work outputs are typically totally frictionally dissipated on short time scales (see Ref. , Chapter VI (especially Sections 54, 60, and 61); and Ref. , 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.

With both the waste heat Q1W12D=Q11ϵ12has been rejected and the work W12D=Q1ϵ12has 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 ϵ23can 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. (W23Dmay 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 1in1and i<jn, and where x is a positive real number in the range 0<x1, applying Eqs. (5) and (10), we have:

W13D=W12D+W23=Q11T2T1x+1T3T2x=Q12T2T1xT3T2x.E11

We now maximize W13Dwith 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 T1and T3. Applying Eqs. (11) and (12), the maximum value W13,maxDof W13Dis

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<1/4for x=1and if T3/T1<1/16for 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/2as r and setting dW13,maxD,extra/dr=0yields

Thus W13,maxDis 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, W34Donly 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 1in1and i<jn, and where xis 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,optof Tjof reservoir j 1<j<n2jn1, which maximizes Wj1j+1D, is the geometric mean of Tj1and 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=8/27for x=1and if T4/T1<2/36=64/729for x=1/2. Also

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/3as r and setting dW14,maxD,extra/dr=0yields

Thus W14,maxD,extrais 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>1/4and 64/729>1/16).

Generalizing Eqs. (20)(30) for an n-reservoir system (n = any positive integer ≥ 3), we obtain:

Tj+1=TjTj+21/2,E31

where jis any positive integer in the range 1jn2and

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

where jis any positive integer in the range 1jn3. The respective temperatures T1and Tnof the extreme (hottest and coldest) reservoirs are assumed to be fixed. The temperatures T2through Tn1of 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 jis any positive integer in the range 1jn2we 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 jsuch that jis any positive integer in the range 1jn1. The first two lines of Eq. (35) pertain to any three adjacent heat reservoirs, and hence two appears in the exponents of the second line thereof; the third line of Eq. (35) pertains to all nheat reservoirs, and hence n1appears 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 ϵij=1Ti/Tjx, where iand jare positive integers in the respective ranges 1in1and i<jn, and where xis 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/n1as r and setting dW1n,maxD,extra/dr=0yields

Thus W1n,maxD,extrais 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=n1limTn/T10,nfixedW1nlimTn/T10,nfixedW1n,maxD,extra=limTn/T10,nfixedW1n,maxDW1n=n1Q1Q1=n2Q1=n2limTn/T10,nfixedW1n.E40

Note that the values in Eqs. (36), (38), and (40) increase monotonically with increasing nand 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 xfinitely greater than 0in the range 0<x1, because TnT1x/n101TnT1x/n11in the limit Tn/T10, albeit ever more slowly with decreasing x.

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

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

Note the linear divergence of W1n,maxDin the limit Tn/T10with nfixed as per Eq. (40) even not assuming Carnot efficiency, as contrasted with the paltry logarithmic divergence of W1n,maxDin the limit nwith Tn/T1fixed even granting Carnot efficiency as per the derivation  of Eq. (41) even assuming Carnot efficiency.

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/Tnbe 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,maxDin the limit nwith T1/Tnfixed 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 7times as much work output (to the nearest whole number) as its host star’s total energy output. Actually the limit nwith T1/Tnfixed 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 4concentric Dyson spheres [39, 40] can procure 4times 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 high—super-unity—heat-engine efficiencies W1n,maxD/Q1that can obtain with work output totally frictionally dissipated ifn3are 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+1or 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+1whether 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 jat temperature Tjand reservoir j+2at temperature Tj+2without versus with frictional dissipation into reservoir j+1at 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 Tjas a hot reservoir and the reservoir at temperature Tj+1as a cold reservoir. It rejects waste heat QjWjj+1=Qj1ϵjj+1to the reservoir at temperature Tj+1. If a third reservoir at temperature Tj+2and Wjj+1is 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+1as a hot reservoir and the reservoir at temperature Tj+2as 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+1into reservoir j+1at temperature Tj+1, we still have

Wjj=1D=Q1ϵjj+1.E45

But now we let the work output Wjj=1D=Q1ϵj+1j+2be 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 QextraDof the work output as per Eq. (42):

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

[In the last three steps of Eq. (50), we applied Eqs. (42), (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 WextraDin 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+1at 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+1at temperature Tj+1,Qj+2+Wj+2j=Qj. The bottom line Qj+2+Wj+2j=Qjis identical with or without an intermediate reservoir j+1at temperature Tj+1. With or without the intermediate reservoir j+1at 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. , Section 20-3; Ref. , Section 5.12 and Problem 5.22; Ref. , Sections 4.3, 4.4, and 4.7 (especially Section 4.7); Ref. , Sections 4-4, 4-5, and 4-6 (especially Section 4-6); Ref. , Sections 5-7-2, 6-2-2, 6-9-2, and 6-9-3, and Chapter 17; Ref. , Chapter XXI; Ref. , Sections 6.7, 6.8, 7.3, and 7.4); and Ref. , pp. 233--236 and Problems 1, 2, 4, 6, and 7 of Chapter 8. [Problem 2 of Chapter 8 in Ref.  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 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 + 1 whether work is sequestered or not, exceeding the Carnot efficiency, 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 its conservation).

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. 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.

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Jack Denur (December 13th 2019). Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs [Online First], IntechOpen, DOI: 10.5772/intechopen.89047. Available from: